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This article is part of the series Jean Mawhin’s Achievements in Nonlinear Analysis.

Open Access Research

Noncommutativity of mappings in hybrid fixed point results

Hemant Kumar Pathak1 and Rosana Rodríguez-López2*

Author Affiliations

1 School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur, C.G., 492010, India

2 Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, Santiago de Compostela, 15782, Spain

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Boundary Value Problems 2013, 2013:145  doi:10.1186/1687-2770-2013-145

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/145


Received:10 December 2012
Accepted:25 May 2013
Published:11 June 2013

© 2013 Pathak and Rodríguez-López; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this note, some coincidence and common fixed points of nonlinear hybrid mappings have been obtained under certain noncommutativity conditions of mappings. Our results improve several known results in the field of hybrid fixed point theory.

MSC: 54H25, 47H10, 54C60.

Keywords:
coincidence point; fixed point; occasionally coincidentally idempotent; multi-valued mappings

Introduction

As a generalization of the Banach fixed point theorem, Nadler’s contraction principle has lead to an excellent fixed point result in the area of nonlinear analysis. Some other works focused on fixed point results for multi-valued mappings are, for instance, [1-5]. Coincidence and common fixed points of nonlinear hybrid contractions (i.e., contractions involving single-valued and multi-valued mappings) have been recently studied by many authors. To mention some of the achievements, we cite, for example, [6-12].

The concept of commutativity of single-valued mappings [13] was extended in [14] to the setting of a single-valued mapping and a multi-valued mapping on a metric space. This concept of commutativity has been further generalized by different authors, viz weakly commuting [15], compatible [16], weakly compatible [8]. It is interesting to note that in all the results obtained so far concerning common fixed points of hybrid mappings the (single-valued and multi-valued) mappings under consideration satisfy either the commutativity condition or one of its generalizations (see, for instance, [6-10]). In this note, we show the existence of fixed points of hybrid contractions which do not satisfy any of the commutativity conditions or its above-mentioned generalizations. Our result extends and improves several well-known results in the field of hybrid fixed point theory. Some other recent related references are [17,18], where common fixed point theorems for hybrid mappings on a symmetric space are proved under the assumptions of weak compatibility and occasional weak compatibility. Some analogous results for the case of contractivity conditions of integral type are presented in [19-21] and generalized contractive hybrid pairs are considered in [22]. Finally, in [23], fixed point results are proved in topological vector space valued cone metric spaces (with nonnormal cones).

Preliminaries

For a metric space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M1">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M3">View MathML</a> denote respectively the hyper-space of non-empty closed bounded and non-empty closed subsets of X, where H is the Hausdorff metric induced by d. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M5">View MathML</a>, we shall use the following notations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M6">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M7">View MathML</a>

We recall some definitions.

Definition 1 Mappings f and T are said to be commuting at a point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M8">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M9">View MathML</a>. The mappings f and T are said to be commuting on X if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M10">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M11">View MathML</a>.

Definition 2 Mappings f and T are said to be weakly commuting at a point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M8">View MathML</a> if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M13">View MathML</a>

The mappings f and T are said to be weakly commuting on X if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M14">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M11">View MathML</a>.

Definition 3 The mappings f and T are said to be compatible if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M16">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M18">View MathML</a>, whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M19">View MathML</a> is a sequence in X such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M21">View MathML</a>, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M22">View MathML</a>.

Definition 4 The mappings f and T are said to be f-weak compatible if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M16">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M24">View MathML</a> and the following limits exist and satisfy the inequalities:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M25">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M26">View MathML</a>,

whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M27">View MathML</a> is a sequence in X such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M28">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M29">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M30">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M31">View MathML</a> denote the set of all coincidence points of the mappings f and T, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M32">View MathML</a>.

Definition 5 The mappings f and T are said to be coincidentally commuting if they commute at their coincidence points.

Definition 6 Mappings f and T are said to be coincidentally idempotent if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M33">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M34">View MathML</a>, that is, if f is idempotent at the coincidence points of f and T.

Definition 7 Mappings f and T are said to be occasionally coincidentally idempotent (or, in brief, oci) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M33">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M36">View MathML</a>.

It should be remarked that coincidentally idempotent pairs of mappings are occasionally coincidentally idempotent, but the converse is not necessarily true as shown in Example 18 of this note.

Main results

We recall the following lemma.

Lemma 8[8]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M37">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M38">View MathML</a>bef-weak compatible. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M39">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M40">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M41">View MathML</a>for allx, yinY, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M43">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M44">View MathML</a>.

We remark that the above-mentioned lemma has been used in [8,9] and [10] to prove the existence of fixed points of hybrid mappings. However, we have noticed some typos in its original statement which have been rectified in the above statement without altering the proof.

Next, we prove a fixed point result for hybrid mappings under a general integral-type contractivity condition. In contrast to [20], we avoid the complete character of the base space X, and we introduce hybrid mappings. With respect to the study in [21], we consider here occasionally coincidentally idempotent mappings.

Theorem 9LetYbe an arbitrary non-empty set, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M1">View MathML</a>be a metric space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M46">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M47">View MathML</a>be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M48">View MathML</a>

(1)

that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M49">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M50">View MathML</a>

(2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M51">View MathML</a>

(3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M52">View MathML</a>is a Lebesgue measurable mapping which is nonnegative, summable on each compact interval and such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M53">View MathML</a>

(4)

which trivially implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M54">View MathML</a>

(5)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M55">View MathML</a>

(6)

Suppose also that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M56">View MathML</a>

(7)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M57">View MathML</a>is such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M58">View MathML</a>

(8)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M59">View MathML</a>

(9)

ThenTandfhave a coincidence point. Further, iffandTare occasionally coincidentally idempotent, thenfandThave a common fixed point.

Proof In view of (1) and Nadler’s remark in [24], given the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M60">View MathML</a>, we can construct two sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M61">View MathML</a> in Y and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M62">View MathML</a> in X such that, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M63">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M64">View MathML</a>

Indeed, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M65">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M66">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M67">View MathML</a>. Besides, given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M68">View MathML</a>, by Nadler’s remark in [24] and using that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M69">View MathML</a>, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M70">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M71">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M72">View MathML</a> for a certain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M73">View MathML</a>. The continuation of this process allows to construct the two above-mentioned sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M61">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M62">View MathML</a> inductively.

We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M76">View MathML</a> is a Cauchy sequence. Using the inequality in (2) and also property (7), which is trivially valid for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M77">View MathML</a>, it follows, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M78">View MathML</a>, that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M79">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M80">View MathML</a>

Suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M81">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M82">View MathML</a>, hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M83">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M84">View MathML</a>

so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M85">View MathML</a>

where we have also used (6) (a consequence of (4)), (7), (8) and (9). The previous inequalities imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M86">View MathML</a>

which is a contradiction. In consequence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M87">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M88">View MathML</a>, by hypothesis, and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M89">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M90">View MathML</a>. This is clear from the following inequality, valid for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M91">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M92">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M93">View MathML</a>

which tends to zero as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M94">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M90">View MathML</a> is complete, then the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M89">View MathML</a> has a limit in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M90">View MathML</a>, say u. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M98">View MathML</a> and prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M99">View MathML</a>.

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M100">View MathML</a>, then, by (2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M101">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M102">View MathML</a>

Here, we have used that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M103">View MathML</a>, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M105">View MathML</a>, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M107">View MathML</a> due to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M108">View MathML</a> and Tw closed, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M109">View MathML</a>

Hence, for n large enough, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M110">View MathML</a>

Making n tend to +∞ in the previous inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M111">View MathML</a>

and, therefore, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M112">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M107">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M114">View MathML</a>, which is a contradiction. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M115">View MathML</a>, that is, w is a coincidence point for T and f.

Although this fact is not relevant to the proof, we note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M116">View MathML</a> since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M117">View MathML</a>

Indeed,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M118">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M119">View MathML</a>

therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M120">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M121">View MathML</a> and, by the properties of ψ, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M122">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M123">View MathML</a>. From the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M62">View MathML</a>, we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M125">View MathML</a> for every n and, therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M126">View MathML</a>, so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M127">View MathML</a> is asymptotically T-regular with respect to f. However, this property can be deduced directly from the fact that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M128">View MathML</a>

Now, if f and T are occasionally coincidentally idempotent, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M129">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M130">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M131">View MathML</a>

(10)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M132">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M133">View MathML</a>, then from inequality (10) and using (5) (which is guaranteed by (4)), we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M134">View MathML</a>

which is a contradiction. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M135">View MathML</a>. Thus we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M136">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M137">View MathML</a>, i.e., fw is a common fixed point of f and T. □

Let Φ denote the family of maps ϕ from the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M138">View MathML</a> of nonnegative real numbers to itself such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M139">View MathML</a>

(11)

Corollary 10LetYbe an arbitrary non-empty set, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M1">View MathML</a>be a metric space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M46">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M47">View MathML</a>be such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M143">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M144">View MathML</a>

(12)

for allx, yinY, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M145">View MathML</a> (satisfying (11) for a certain<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M146">View MathML</a>),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M147">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M52">View MathML</a>is a Lebesgue measurable mapping which is nonnegative, summable on each compact interval and such that (4) holds. Suppose also that (7), (8) and (9) hold for a certain<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M57">View MathML</a>andqdetermined by (11). ThenTandfhave a coincidence point. Further, iffandTare occasionally coincidentally idempotent, thenfandThave a common fixed point.

Proof It is a consequence of Theorem 9 since (11) and (12) imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M150">View MathML</a>

for all x, y in Y and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M151">View MathML</a>. □

Remark 11 The condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M152">View MathML</a>

(13)

implies the validity of hypothesis (7) in Theorem 9 for the particular case of γ the identity mapping. Moreover, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M153">View MathML</a>, hypotheses (8) and (9) are trivially satisfied for this choice of γ. Indeed, using that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M153">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M155">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M156">View MathML</a>

Remark 12 Assuming (8), condition (9) is trivially valid if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M157">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M158">View MathML</a> or, equivalently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M159">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M160">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M161">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M162">View MathML</a>. Note that this last condition is trivially valid for γ the identity mapping. Moreover, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M163">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M162">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M165">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M162">View MathML</a> and, therefore, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M167">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M168">View MathML</a>, obtaining (8) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M169">View MathML</a>.

Remark 13 According to Remark 12, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M151">View MathML</a> fixed and ψ satisfying (4), an admissible function γ can be obtained by taking

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M171">View MathML</a>

provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M169">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M163">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M162">View MathML</a>.

Example 14 Taking ψ as the constant function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M175">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M176">View MathML</a>, in the statement of Theorem 9, condition (7) is reduced to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M177">View MathML</a>

so that we must choose γ as a nonnegative function satisfying that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M178">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M179">View MathML</a> (obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M169">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M151">View MathML</a>) in order to guarantee conditions (7), (8) and (9).

Example 15 A simple calculation provides that, for the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M182">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M176">View MathML</a>, condition (7) is written as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M184">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M179">View MathML</a> and, therefore, in this case condition (8) is never fulfilled. If we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M186">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M176">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M188">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M189">View MathML</a> fixed, then (7) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M190">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M162">View MathML</a>.

Example 16 Now, we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M192">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M176">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M188">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M195">View MathML</a> are fixed. Note that the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M196">View MathML</a> has already been studied in Example 14. In this case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M195">View MathML</a>, condition (7) is reduced to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M198">View MathML</a>

which is equivalent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M199">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M179">View MathML</a>. Note that this inequality implies, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M153">View MathML</a>, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M169">View MathML</a>. If we add the hypothesis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M203">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M162">View MathML</a>, then we guarantee the validity of conditions (8) and (9) due to Remark 12. Hence, we can take any nonnegative function γ satisfying that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M205">View MathML</a>

Of course, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M178">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M207">View MathML</a> are valid choices.

Example 17 Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M208">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M176">View MathML</a>. Condition (7) is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M210">View MathML</a>

that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M211">View MathML</a>

Now, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M162">View MathML</a> fixed, we calculate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M213">View MathML</a>, which is obviously positive, and we check that its value is equal to z.

It is easy to prove that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M162">View MathML</a> fixed, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M215">View MathML</a> is decreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M216">View MathML</a>. Indeed, the sign of its derivative coincides with the sign of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M217">View MathML</a> and also with the sign of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M218">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M219">View MathML</a>. Now, the function τ is strictly negative on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M216">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M221">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M222">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M223">View MathML</a>.

Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M224">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M162">View MathML</a>; in consequence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M226">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M162">View MathML</a>. Therefore, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M228">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M229">View MathML</a>, then (7) follows. Note also that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M151">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M169">View MathML</a>. Finally, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M151">View MathML</a>, if we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M57">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M234">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M162">View MathML</a>, we deduce the validity of (7), (8) and (9).

The following example shows that Theorem 9 is a proper generalization of the fixed point results in [7-10].

Example 18 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M236">View MathML</a> be endowed with the Euclidean metric, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M238">View MathML</a> be defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M239">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M240">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M241">View MathML</a> be defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M242">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M243">View MathML</a>. Then mappings f and T are not commuting and also do not satisfy any of its generalizations, viz weakly commuting, compatibility, weak compatibility. Also the mappings f and T are not coincidentally commuting. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M244">View MathML</a>, but <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M245">View MathML</a> and so f and T are not coincidentally idempotent, but <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M246">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M247">View MathML</a> thus f and T are occasionally coincidentally idempotent. For all x and y in X, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M248">View MathML</a>

Note that these inequalities are valid if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M249">View MathML</a>

which is satisfied taking, for instance, the constant function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M250">View MathML</a>. On the other hand, γ is chosen as the identity map and it satisfies (8) and (9).

Note that 0 is a common fixed point of f and T. We remark that the results of [7-9] and [10] cannot be applied to these mappings f and T.

Theorem 19In Theorem 9, we can assume, instead of condition (2), one of the inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M251">View MathML</a>

(14)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M252">View MathML</a>

(15)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M253">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M254">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M167">View MathML</a>.

Similarly, in Corollary 10, we can consider one of the contractivity conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M256">View MathML</a>

(16)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M257">View MathML</a>

(17)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M253">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M254">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M260">View MathML</a> (satisfying (11) for a certain<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M167">View MathML</a>) and the conclusion follows.

Proof It follows from the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M262">View MathML</a>

and the nonnegative character of a, b and ψ. Indeed, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M263">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M264">View MathML</a>

hence, for instance,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M265">View MathML</a>

Note that, in cases (16) and (17), it is not necessary to assume the nondecreasing character of the function ϕ since, using that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M145">View MathML</a>, we deduce (14) and (15), respectively. □

Of course, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M267">View MathML</a> is admissible in the results of this paper.

Note that, taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M268">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M269">View MathML</a> in the inequalities of Theorem 19, we obtain the corresponding contractivity conditions of Theorem 9 and Corollary 10. On the other hand, taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M270">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M271">View MathML</a> in Theorem 19, we have the following results, which are also corollaries of Theorem 9.

Corollary 20Let Y be an arbitrary non-empty set, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M272">View MathML</a>be a metric space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M38">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M274">View MathML</a>be such that conditions (1), (3) hold and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M275">View MathML</a>

(18)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M276">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M277">View MathML</a>is a Lebesgue measurable mapping which is nonnegative, summable on each compact interval and such that (4) holds. Assume also that (7), (8) and (9) are fulfilled for a certain<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M57">View MathML</a>. ThenfandThave a coincidence point. Further, if fandTare occasionally coincidentally idempotent, thenfandThave a common fixed point.

Corollary 21Let Y be an arbitrary non-empty set, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M272">View MathML</a>be a metric space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M38">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M281">View MathML</a>be such that conditions (1), (3) hold and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M282">View MathML</a>

(19)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M145">View MathML</a> (satisfying (11) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M284">View MathML</a>) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M277">View MathML</a>is a Lebesgue measurable mapping which is nonnegative, summable on each compact interval and such that (4) holds. Assume also that (7), (8) and (9) are fulfilled for a certain<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M57">View MathML</a>. ThenfandThave a coincidence point. Further, if fandTare occasionally coincidentally idempotent, thenfandThave a common fixed point.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M287">View MathML</a> be a function having the following property (see, for instance, [6,25]):

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M288">View MathML</a>) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M289">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M290">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M291">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M292">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M293">View MathML</a>.

This property obviously holds if η is continuous since η attains its maximum (less than 1) on each compact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M294">View MathML</a>.

Definition 22 A sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M127">View MathML</a> is said to be asymptotically T-regular with respect to f if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M296">View MathML</a>.

The following theorem is related to the main results of Hu [[25], Theorem 2], Jungck [14], Kaneko [26], Nadler [[24], Theorem 5] and Beg and Azam [[6], Theorem 5.4 and Corollary 5.5].

Theorem 23LetYbe an arbitrary non-empty set, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M272">View MathML</a>be a metric space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M298">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M299">View MathML</a>be such that condition (1) holds and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M300">View MathML</a>

(20)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M301">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M287">View MathML</a>satisfies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M288">View MathML</a>) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M304">View MathML</a>is nonincreasing.

Suppose also thatTxis a compact set for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M305">View MathML</a>.

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M90">View MathML</a>is complete, then

(i) there exists an asymptoticallyT-regular sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M307">View MathML</a>with respect tofinY,

(ii) fandThave a coincidence point.

Further, if fandTare occasionally coincidentally idempotent, thenfandThave a common fixed point.

Proof For some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M308">View MathML</a> in Y, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M309">View MathML</a> and choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M310">View MathML</a> in Y such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M311">View MathML</a>. Then, by (20), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M312">View MathML</a>

Using (1), we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M73">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M314">View MathML</a> and satisfying that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M315">View MathML</a>

hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M316">View MathML</a>

Note that, in the previous inequalities, we have used that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M317">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M318">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M319">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M19">View MathML</a> is asymptotically T-regular with respect to f.

By induction, we construct a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M307">View MathML</a> in Y and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M322">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M90">View MathML</a> such that, for every n,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M324">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M325">View MathML</a>.

Also, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M326">View MathML</a>

It follows that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M327">View MathML</a> is decreasing and converges to its greatest lower bound, say t. Clearly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M328">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M176">View MathML</a>, then by the property (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M288">View MathML</a>) of η, there will exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M290">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M291">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M333">View MathML</a>

For this <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M334">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M335">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M336">View MathML</a>, whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M337">View MathML</a>. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M338">View MathML</a>, whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M337">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M340">View MathML</a>. Then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M341">View MathML</a> , we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M342">View MathML</a>

which contradicts the assumption that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M343">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M344">View MathML</a>; i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M345">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M123">View MathML</a>. Hence the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M347">View MathML</a> is asymptotically T-regular with respect to f.

We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M348">View MathML</a> is a Cauchy sequence. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M349">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M350">View MathML</a>, then, by the nonincreasing character of ψ, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M351">View MathML</a>

Now, we recall that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M352">View MathML</a>

for every n, which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M353">View MathML</a>

Following this procedure, we prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M354">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M355">View MathML</a>

We check that the right-hand side in the last inequality tends to 0 as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M356">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M357">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M123">View MathML</a>, it suffices to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M359">View MathML</a> is bounded (uniformly on n, m). Indeed, we check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M360">View MathML</a> is bounded for any sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M361">View MathML</a> with nonnegative terms and tending to 0 as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M362">View MathML</a>, using the property (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M363">View MathML</a>) of the function η. Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M364">View MathML</a>, by (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M363">View MathML</a>), there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M366">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M367">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M368">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M369">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M370">View MathML</a>, given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M371">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M372">View MathML</a> such that, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M373">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M374">View MathML</a>. This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M375">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M373">View MathML</a>.

In consequence, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M377">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M378">View MathML</a>

and this expression is bounded independently of m, n.

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M379">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M90">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M90">View MathML</a> is complete, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M379">View MathML</a> converges to some p in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M90">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M384">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M385">View MathML</a>. Next, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M386">View MathML</a>

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M387">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M388">View MathML</a>. Thus we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M389">View MathML</a>. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M390">View MathML</a>.

Now, if f and T are occasionally coincidentally idempotent, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M391">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M392">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M393">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M135">View MathML</a>. It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M395">View MathML</a>. Hence, fw is a common fixed point of T and f. □

Now we state some fixed point theorems for Kannan-type multi-valued mappings which extend and generalize the corresponding results of Shiau et al.[10] and Beg and Azam [6,27]. A proper blend of the proof of Theorem 9 and those of [[10], Th. 6, Th. 7, Th. 8 respectively] and [[9], Theorems 3.1, 3.2, 3.3] will complete the proof.

Theorem 24LetYbe an arbitrary non-empty set, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M396">View MathML</a>be a metric space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M298">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M398">View MathML</a>be such that (1) holds and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M399">View MathML</a>

(21)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M400">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M401">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M402">View MathML</a>) are bounded on bounded sets, ris some fixed positive real number and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M277">View MathML</a>is a Lebesgue measurable mapping which is summable on each compact interval and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M404">View MathML</a>for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M405">View MathML</a>. Suppose that there exists an asymptoticallyT-regular sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M19">View MathML</a>with respect tofinY. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M407">View MathML</a>is complete or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M408">View MathML</a>

(22)

thenfandThave a coincidence point. Further, iffandTare occasionally coincidentally idempotent, thenfandThave a common fixed point.

Proof By hypotheses,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M409">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M19">View MathML</a> is asymptotically T-regular with respect to f in Y, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M411">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M412">View MathML</a> are bounded sequences and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M413">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M414">View MathML</a>, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M415">View MathML</a>. This provides the property <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M416">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M417">View MathML</a>, so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M418">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M2">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M420">View MathML</a> is complete, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M421">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M422">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M123">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M424">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M425">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M426">View MathML</a>

where the number of terms containing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M427">View MathML</a> is a finite number depending on r, and therefore fixed. Calculating the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M123">View MathML</a> and taking into account that the length of the intervals in the last integral tends to zero, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M429">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M430">View MathML</a>

and, by the properties of ψ, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M431">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M432">View MathML</a> and u is a coincidence point.

Now, suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M90">View MathML</a> is complete. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M434">View MathML</a> is closed and bounded for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M435">View MathML</a>. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M436">View MathML</a> fixed. By the results in [24], we can affirm that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M437">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M438">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M439">View MathML</a>.

Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M440">View MathML</a>, we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M437">View MathML</a> and, for this <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M442">View MathML</a> fixed, we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M438">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M444">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M445">View MathML</a>

By the hypothesis on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M19">View MathML</a> and the Cauchy character of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M418">View MathML</a>, we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M448">View MathML</a> is a Cauchy sequence. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M90">View MathML</a> is complete, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M450">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M451">View MathML</a>. By hypotheses, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M452">View MathML</a> for every n, hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M453">View MathML</a>

and taking the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M454">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M455">View MathML</a>

In this case,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M456">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M457">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M432">View MathML</a>. Now, if f and T are coincidentally idempotent, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M459">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M460">View MathML</a>. Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M461">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M462">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M463">View MathML</a>

Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M464">View MathML</a>

obtaining <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M465">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M466">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M467">View MathML</a>, we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M468">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M469">View MathML</a>. In consequence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M470">View MathML</a> and fw is a common fixed point of T and f. □

Remark 25 In the statement of Theorem 24, condition (22) can be replaced by the more general one

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M471">View MathML</a>

To complete the proof with this more general hypothesis, take into account that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M472">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M473">View MathML</a> is a closed set in X and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M474">View MathML</a>. Using that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M90">View MathML</a> is complete, we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M476">View MathML</a> is complete. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M418">View MathML</a> is a sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M478">View MathML</a> and it is a Cauchy sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M476">View MathML</a>. Therefore, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M480">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M481">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M123">View MathML</a>. Note also that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M483">View MathML</a> is a closed set in the complete space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M90">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M483">View MathML</a> is complete and, therefore, a closed set, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M486">View MathML</a>. Once we have proved that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M422">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M123">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M489">View MathML</a>, the proof follows analogously.

Theorem 26In addition to the hypotheses of Theorem 24, suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M434">View MathML</a>is compact for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M491">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M492">View MathML</a>is a cluster point of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M89">View MathML</a>, thenzis a coincidence point offandT.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M494">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M495">View MathML</a>, this is possible since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M434">View MathML</a> is compact. It is obvious that a cluster point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M89">View MathML</a> is a cluster point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M62">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M492">View MathML</a> be a cluster point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M89">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M62">View MathML</a>, then we check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M502">View MathML</a>, where u is obtained in the proof of Theorem 24. Note that, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M503">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M504">View MathML</a>

hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M505">View MathML</a>

In consequence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M506">View MathML</a>

Using that there exists a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M507">View MathML</a> converging to fz, the properties of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M19">View MathML</a> and the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M509">View MathML</a>

then, taking the limit when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M510">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M511">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M502">View MathML</a>. To prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M390">View MathML</a>, using that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M514">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M515">View MathML</a>

This implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M516">View MathML</a>

and, by the properties of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M517">View MathML</a> and ψ, we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M518">View MathML</a>, which proves that z is a coincidence point of f and T. □

The following result extends [[10], Theorem 3.3].

Theorem 27LetYbe an arbitrary non-empty set, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M396">View MathML</a>be a metric space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M298">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M398">View MathML</a>be such that (1) and (21) hold, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M522">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M402">View MathML</a>) are bounded on bounded sets and such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M524">View MathML</a>

ris some fixed positive real number and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M525">View MathML</a>is a Lebesgue measurable mapping which is summable on each compact interval, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M526">View MathML</a>for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M527">View MathML</a>. Suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M528">View MathML</a>

(23)

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M407">View MathML</a>is complete or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M90">View MathML</a>is complete, thenfandThave a coincidence point. Further, iffandTare occasionally coincidentally idempotent, thenfandThave a common fixed point.

Proof Using Theorem 24, it suffices to prove that there exists an asymptotically T-regular sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M19">View MathML</a> with respect to f in Y. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M532">View MathML</a> and take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M19">View MathML</a> in Y such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M534">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M63">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M536">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M537">View MathML</a>

or also, using the hypothesis on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M538','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M538">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M517">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M540">View MathML</a>

The properties of ψ imply that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M541">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M435">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M543">View MathML</a> is nonincreasing and bounded below. Therefore it is convergent to the infimum, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M544','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M544">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M19">View MathML</a> is asymptotically T-regular with respect to f in Y. □

Remark 28 Note that condition (23) in Theorem 27 cannot be replaced by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M546','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M546">View MathML</a>

since the infimum taking the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M547','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M547">View MathML</a> could be positive (we calculate the infimum in a smaller set).

Remark 29 In Theorem 27, condition (23) can be replaced by the following:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M548">View MathML</a>

(24)

Indeed, since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M549','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M549">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M550','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M550">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M551">View MathML</a>.

Remark 30 In Theorem 27, if we are able to obtain a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M19">View MathML</a> with an infinite number of terms which are different, then we can relax condition (23) to the following:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M553','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/145/mathml/M553">View MathML</a>

(25)

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed to each part of this work equally and read and approved the final version of the manuscript.

Acknowledgements

Dedicated to Professor Jean Mawhin, on the occasion of his seventieth birthday.

We thank the editor, the anonymous referees and also Professor Stojan Radenović for their helpful comments and suggestions. This research was partially supported by the University Grants Commission, New Delhi, India; Ministerio de Economía y Competitividad, project MTM2010-15314, and co-financed by EC fund FEDER.

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