This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

Theory of fractional hybrid differential equations with linear perturbations of second type

Hongling Lu, Shurong Sun*, Dianwu Yang and Houshan Teng

Author Affiliations

School of Mathematical Sciences, University of Jinan, Jinan, Shandong, 250022, P.R. China

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Boundary Value Problems 2013, 2013:23  doi:10.1186/1687-2770-2013-23


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/23


Received:30 November 2012
Accepted:22 January 2013
Published:11 February 2013

© 2013 Lu et al.; 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 paper, we develop the theory of fractional hybrid differential equations with linear perturbations of second type involving Riemann-Liouville differential operators of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M1">View MathML</a>. An existence theorem for fractional hybrid differential equations is proved under the φ-Lipschitz condition. Some fundamental fractional differential inequalities which are utilized to prove the existence of extremal solutions are also established. Necessary tools are considered and the comparison principle which will be useful for further study of qualitative behavior of solutions is proved.

MSC: 34A40, 34A12, 34A99.

Keywords:
fractional differential inequalities; existence theorem; comparison principle

1 Introduction

Fractional differential equations have been of great interest recently. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications; see [1-14]. Although the tools of fractional calculus have been available and applicable to various fields of study, there are few papers on the investigation of the theory of fractional differential equations; see [15-19]. The differential equations involving Riemann-Liouville differential operators of fractional order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M1">View MathML</a> are very important in modeling several physical phenomena [20-22] and therefore seem to deserve an independent study of their theory parallel to the well-known theory of ordinary differential equations.

In recent years, quadratic perturbations of nonlinear differential equations have attracted much attention. The importance of the investigations of hybrid differential equations lies in the fact that they include several dynamic systems as special cases. This class of hybrid differential equations includes the perturbations of original differential equations in different ways. There have been many works on the theory of hybrid differential equations, and we refer the readers to the articles [23-29]. Dhage and Lakshmikantham [24] discussed the following first-order hybrid differential equation with linear perturbations of first type:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M3">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M5">View MathML</a>. Dhage and Jadhav [25] discussed the following first-order hybrid differential equation with linear perturbations of second type:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M6">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M5">View MathML</a>. They established the existence and uniqueness results and some fundamental differential inequalities for hybrid differential equations initiating the study of theory of such systems and proved utilizing the theory of inequalities, its existence of extremal solutions and a comparison result.

From the above works, we develop the theory of fractional hybrid differential equations involving Riemann-Liouville differential operators of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M1">View MathML</a>. In this paper, we initiate the basic theory of fractional hybrid differential equations of mixed perturbations of second type involving three nonlinearities and prove the basic result such as the strict and nonstrict fractional differential inequalities, an existence theorem and maximal and minimal solutions etc. We claim that the results of this paper are a basic and important contribution to the theory of nonlinear fractional differential equations.

2 Fractional hybrid differential equation

Let ℝ be a real line and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M10">View MathML</a> be a bounded interval in ℝ for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M11">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M12">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M13">View MathML</a> denote the class of continuous functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M14">View MathML</a>.

Definition 2.1[19]

The Riemann-Liouville fractional derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M15">View MathML</a> of a continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M16">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M17">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M19">View MathML</a> denotes the integer part of number α, provided that the right-hand side is pointwise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M20">View MathML</a>.

Definition 2.2[19]

The Riemann-Liouville fractional integral of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M15">View MathML</a> of a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M22">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M23">View MathML</a>

provided that the right-hand side is pointwise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M20">View MathML</a>.

We consider fractional hybrid differential equations (in short FHDE) involving Riemann-Liouville differential operators of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M1">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M26">View MathML</a>

(2.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M27">View MathML</a>.

By a solution of FHDE (2.1), we mean a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M28">View MathML</a> such that

(i) the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M29">View MathML</a> is continuous for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M30">View MathML</a>, and

(ii) x satisfies the equations in (2.1).

The theory of strict and nonstrict differential inequalities related to ODEs and hybrid differential equations is available in the literature (see [24,25,28,29]). It is known that differential inequalities are useful for proving the existence of extremal solutions of ODEs and hybrid differential equations defined on J.

3 Existence result

In this section, we prove the existence results for FHDE (2.1) on the closed and bounded interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M31">View MathML</a> under mixed Lipschitz and compactness conditions on the nonlinearities involved in it.

We place FHDE (2.1) in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M32">View MathML</a> of continuous real-valued functions defined on J. Define a supremum norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M33">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M32">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M35">View MathML</a>. Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M36">View MathML</a> is a Banach algebra with respect to the above norm.

We prove the existence of a solution for FHDE (2.1) by a fixed point theorem in the Banach algebra due to Dhage [30].

Definition 3.1 Let X be a Banach space. A mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M37">View MathML</a> is called φ-Lipschitzian if there exists a continuous and nondecreasing function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M38">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M39">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M40">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M41">View MathML</a>.

Further, if φ satisfies the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M43">View MathML</a>, then T is called a nonlinear contraction with a control function φ.

Lemma 3.1[30]

LetSbe a nonempty, closed convex and bounded subset of the Banach algebraXand let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M44">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M45">View MathML</a>be two operators such that

(a) Ais nonlinear contraction,

(b) Bis completely continuous,

(c) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M46">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M47">View MathML</a>.

Then the operator equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M48">View MathML</a>has a solution inS.

We consider the following hypotheses in what follows.

(A0) The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M49">View MathML</a> is increasing in ℝ for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a>.

(A1) There exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M51">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M52">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M54">View MathML</a>.

(A3) There exists a continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M55">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M56">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M30">View MathML</a>.

Lemma 3.2[19]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M1">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M59">View MathML</a>.

(H1) The equality<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M60">View MathML</a>holds.

(H2) The equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M61">View MathML</a>

holds almost everywhere onJ.

The following lemma is useful in what follows.

Lemma 3.3Assume that hypothesis (A0) holds. Then, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M55">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M1">View MathML</a>, the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M64">View MathML</a>is a solution of the FHDE

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M65">View MathML</a>

(3.1)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M66">View MathML</a>

(3.2)

if and only ifxsatisfies the hybrid integral equation (HIE)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M67">View MathML</a>

(3.3)

Proof Let x be a solution of the Cauchy problem (3.1) and (3.2). Since the Riemann-Liouville fractional integral <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M68">View MathML</a> is a monotone operator, thus we apply the fractional integral <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M68">View MathML</a> on both sides of (3.1). By Lemma 3.2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M70">View MathML</a>

then by (3.2), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M71">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M72">View MathML</a>

Thus, (3.3) holds.

Conversely, assume that x satisfies HIE (3.3). Then applying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M73">View MathML</a> on both sides of (3.3), (3.1) is satisfied. Again, substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M74">View MathML</a> in (3.3) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M75">View MathML</a>

The map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M49">View MathML</a> is increasing in ℝ for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a>, the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M78">View MathML</a> is injective in ℝ, hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M79">View MathML</a>. The proof is completed. □

Now, we are in a position to prove the following existence theorem for FHDE (2.1).

Theorem 3.1Assume that hypotheses (A0)-(A2) hold. Then FHDE (2.1) has a solution defined onJ.

Proof Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M80">View MathML</a> and define a subset S of X defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M81">View MathML</a>

(3.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M82">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M83">View MathML</a>.

Clearly, S is a closed, convex and bounded subset of the Banach algebra X. Now, using the hypotheses (A0)-(A2), it can be shown by an application of Lemma 3.3, FHDE (2.1) is equivalent to the nonlinear HIE

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M84">View MathML</a>

(3.5)

Define two operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M44">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M45">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M87">View MathML</a>

(3.6)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M88">View MathML</a>

(3.7)

Then HIE (3.5) is transformed into the operator equation as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M89">View MathML</a>

(3.8)

We will show that the operators A and B satisfy all the conditions of Lemma 3.1.

First, we show that A is a Lipschitz operator on X with the Lipschitz constant L. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M90">View MathML</a>. Then by hypothesis (A1),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M91">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a>. Taking supremum over t, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M93">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M90">View MathML</a>. This shows that A is a nonlinear contraction on X with a control function φ defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M95">View MathML</a>.

Next, we show that B is a compact and continuous operator on S into X. First, we show that B is continuous on S. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M96">View MathML</a> be a sequence in S converging to a point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M97">View MathML</a>. Then, by the Lebesgue dominated convergence theorem,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M98">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a>. This shows that B is a continuous operator on S.

Now, we show that B is a compact operator on S. It is enough to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M100">View MathML</a> is a uniformly bounded and equicontinuous set in X. On the one hand, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M97">View MathML</a> be arbitrary. Then by hypothesis (A2),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M102">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a>. Taking supremum over t,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M104">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M97">View MathML</a>. This shows that B is uniformly bounded on S.

On the other hand, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M106">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M107">View MathML</a>. Then, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M97">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M109">View MathML</a>

Hence, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M110">View MathML</a>, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M111">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M112">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M106">View MathML</a> and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M97">View MathML</a>. This shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M100">View MathML</a> is an equicontinuous set in X. Now, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M100">View MathML</a> is a uniformly bounded and equicontinuous set in X, so it is compact by the Arzela-Ascoli theorem. As a result, B is a complete continuous operator on S.

Next, we show that hypothesis (c) of Lemma 3.1 is satisfied. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M97">View MathML</a>. Then, by assumption (A1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M118">View MathML</a>

Taking supremum over t,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M119">View MathML</a>

Thus, all the conditions of Lemma 3.1 are satisfied and hence the operator equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M48">View MathML</a> has a solution in S. As a result, FHDE (2.1) has a solution defined on J. This completes the proof. □

4 Fractional hybrid differential inequalities

We discuss a fundamental result relative to strict inequalities for FHDE (2.1).

Lemma 4.1[17]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M121">View MathML</a>be locally Hölder continuous such that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M122">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M123">View MathML</a>

(4.1)

Then it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M124">View MathML</a>

(4.2)

Theorem 4.1Assume that hypothesis (A0) holds. Suppose that there exist functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M125">View MathML</a>that are locally Hölder continuous such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M126">View MathML</a>

(4.3)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M127">View MathML</a>

(4.4)

one of the inequalities being strict. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M128">View MathML</a>

(4.5)

implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M129">View MathML</a>

(4.6)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a>.

Proof Suppose that inequality (4.4) is strict. Assume that the claim is false. Then there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M131">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M132">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M133">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M134">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M135">View MathML</a>.

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M136">View MathML</a>

Then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M137">View MathML</a> and by virtue of hypothesis (A0), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M138">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M135">View MathML</a>. Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M140">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M141">View MathML</a>, we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M142">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M141">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M144">View MathML</a>. Then by Lemma 4.1, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M145">View MathML</a>. By (4.3) and (4.4), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M146">View MathML</a>

This is a contradiction to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M133">View MathML</a>. Hence, the conclusion (4.6) is valid and the proof is complete. □

The next result is concerned with nonstrict fractional differential inequalities which require a kind of one-sided φ-Lipshitz condition.

Theorem 4.2Assume that the conditions of Theorem 4.1 hold. Suppose that there exists a real number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M148">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M149">View MathML</a>

(4.7)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M150">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M151">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M152">View MathML</a>implies, provided<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M153">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M154">View MathML</a>

(4.8)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a>.

Proof

We set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M156">View MathML</a>

for small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M110">View MathML</a>, so that we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M158">View MathML</a>

(4.9)

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M159">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M160">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a>. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M162">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M153">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M165">View MathML</a>

Also, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M166">View MathML</a>. Hence, by an application of Theorem 4.1 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M167">View MathML</a> yields that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M168">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a>. By the arbitrariness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M110">View MathML</a>, taking the limits as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M171">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M172">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a>. This completes the proof. □

Remark 4.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M174">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M175">View MathML</a>. We can easily verify that f and g satisfy the condition (4.7).

5 Existence of maximal and minimal solutions

In this section, we prove the existence of maximal and minimal solutions for FHDE (2.1) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M176">View MathML</a>. We need the following definition in what follows.

Definition 5.1 A solution r of FHDE (2.1) is said to be maximal if for any other solution x to FHDE (2.1), one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M177">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a>. Similarly, a solution ρ of FHDE (2.1) is said to be minimal if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M179">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a>, where x is any solution of FHDE (2.1) on J.

We discuss the case of a maximal solution only, as the case of a minimal solution is similar and can be obtained with the same arguments with appropriate modifications. Given an arbitrary small real number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M110">View MathML</a>, consider the following initial value problem of FHDE of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M1">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M183">View MathML</a>

(5.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M184">View MathML</a>.

An existence theorem for FHDE (5.1) can be stated as follows.

Theorem 5.1Assume that hypotheses (A0)-(A2) hold. Then, for every small number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M110">View MathML</a>, FHDE (5.1) has a solution defined onJ.

Proof The proof is similar to Theorem 3.1 and we omit the details. □

Our main existence theorem for a maximal solution for FHDE (2.1) is as follows.

Theorem 5.2Assume that hypotheses (A0)-(A2) hold. Then FHDE (2.1) has a maximal solution defined onJ.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M186">View MathML</a> be a decreasing sequence of positive real numbers such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M187">View MathML</a>. By Theorem 5.1, then there exists a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M188">View MathML</a> of the FHDE defined on J

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M189">View MathML</a>

(5.2)

Then, for any solution u of FHDE (2.1), any solution of auxiliary problem (5.2) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M190">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M191">View MathML</a>. By Theorem 4.2, we infer that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M192">View MathML</a>

(5.3)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M194">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M195">View MathML</a>, then by Theorem 4.2, we infer that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M196">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M188">View MathML</a> is a decreasing sequence of positive real numbers, the limit

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M198">View MathML</a>

(5.4)

exists. We show that the convergence in (5.4) is uniform on J. To finish, it is enough to prove that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M188">View MathML</a> is equicontinuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M200">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M201">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M107">View MathML</a> be arbitrary. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M203">View MathML</a>

Since f is continuous on a compact set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M204">View MathML</a>, it is uniformly continuous there. Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M205">View MathML</a>

uniformly for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M206">View MathML</a>.

Therefore, from the above inequality, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M207">View MathML</a>

uniformly for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M206">View MathML</a>. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M209">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a>.

Next, we show that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M211">View MathML</a> is a solution of FHDE (2.1) defined on J. Now, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M188">View MathML</a> is a solution of FHDE (5.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M213">View MathML</a>

(5.5)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a>. Taking the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M215">View MathML</a> in above Eq. (5.5) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M216">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a>. Thus, the function r is a solution of FHDE (2.1) on J. Finally, from inequality (5.3), it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M218">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a>. Hence, FHDE (2.1) has a maximal solution on J. This completes the proof. □

6 Comparison theorems

The main problem of differential inequalities is to estimate a bound for the solution set for the differential inequality related to FHDE (2.1). In this section, we prove that the maximal and minimal solutions serve as bounds for the solutions of the related differential inequality to FHDE (2.1) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M220">View MathML</a>.

Theorem 6.1Assume that hypotheses (A0)-(A2) hold. Suppose that there exists a real number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M148">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M222">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M150">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M151">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M153">View MathML</a>. Furthermore, if there exists a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M226">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M227">View MathML</a>

(6.1)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M228">View MathML</a>

(6.2)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a>, whereris a maximal solution of FHDE (2.1) onJ.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M110">View MathML</a> be arbitrary small. By Theorem 5.2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M231">View MathML</a> is a maximal solution of FHDE (5.1) and the limit

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M232">View MathML</a>

(6.3)

is uniform on J and the function r is a maximal solution of FHDE (2.1) on J. Hence, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M233">View MathML</a>

From the above inequality, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M234">View MathML</a>

(6.4)

Now, we apply Theorem 4.2 to inequalities (6.1) and (6.4) and conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M235">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a>. This further, in view of limit (6.3), implies that inequality (6.2) holds on J. This completes the proof. □

Theorem 6.2Assume that hypotheses (A0)-(A2) hold. Suppose that there exists a real number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M148">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M238">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M150">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M151">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M241">View MathML</a>. Furthermore, if there exists a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M226">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M243">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M244">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a>, whereρis a minimal solution of FHDE (2.1) onJ.

Note that Theorem 6.1 is useful to prove the boundedness and uniqueness of the solutions for FHDE (2.1) on J. A result in this direction is as follows.

Theorem 6.3Assume that hypotheses (A0)-(A2) hold. Suppose that there exists a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M246">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M247">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M150">View MathML</a>. If an identically zero function is the only solution of the differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M249">View MathML</a>

(6.5)

then FHDE (2.1) has a unique solution onJ.

Proof By Theorem 3.1, FHDE (2.1) has a solution defined on J. Suppose that there are two solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M250">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M251">View MathML</a> of FHDE (2.1) existing on J. Define a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M252">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M253">View MathML</a>

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M254">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M256">View MathML</a>

for almost everywhere <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M258">View MathML</a>.

Now, we apply Theorem 6.1 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M174">View MathML</a> to get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M260">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a>. This gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M262">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a>. Then we can get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M264">View MathML</a> in view of hypothesis (A0). This completes the proof. □

7 Existence of extremal solutions in a vector segment

Sometimes it is desirable to have knowledge of the existence of extremal positive solutions for FHDE (2.1) on J. In this section, we prove the existence of maximal and minimal positive solutions for FHDE (2.1) between the given upper and lower solutions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M31">View MathML</a>. We use a hybrid fixed point theorem of Dhage [26] in ordered Banach spaces for establishing our results. We need the following preliminaries in what follows.

A nonempty closed set K in a Banach algebra X is called a cone with vertex 0 if

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M266">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M267">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M268">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M269">View MathML</a>,

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M270">View MathML</a>, where 0 is the zero element of X,

(iv) A cone K is called positive if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M271">View MathML</a>, where ∘ is a multiplication composition in X.

We introduce an order relation ‘≤’ in X as follows. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M90">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M273">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M274">View MathML</a>. A cone K is called normal if the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M33">View MathML</a> is semi-monotone increasing on K, that is, there is a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M276">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M277">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M278">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M273">View MathML</a>. It is known that if the cone K is normal in X, then every order-bounded set in X is norm-bounded. The details of cones and their properties appear in Heikkilä and Lakshmikantham [31].

Lemma 7.1[26]

LetKbe a positive cone in a real Banach algebraXand let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M280">View MathML</a>be such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M281">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M282">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M283">View MathML</a>.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M284">View MathML</a>, the order interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M285">View MathML</a> is a set in X given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M286">View MathML</a>

Definition 7.1 A mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M287">View MathML</a> is said to be nondecreasing or monotone increasing if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M273">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M289">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M290">View MathML</a>.

We use the following fixed point theorems of Dhage [27] for proving the existence of extremal solutions for IVP (2.1) under certain monotonicity conditions.

Lemma 7.2[27]

LetKbe a cone in a Banach algebraXand let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M284">View MathML</a>be such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M292">View MathML</a>. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M293">View MathML</a>are two nondecreasing operators such that

(a) Ais a nonlinear contraction,

(b) Bis completely continuous,

(c) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M294">View MathML</a>for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M295">View MathML</a>.

Further, if the coneKis positive and normal, then the operator equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M48">View MathML</a>has a least and a greatest positive solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M285">View MathML</a>.

We equip the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M32">View MathML</a> with the order relation ≤ with the help of a cone K defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M299">View MathML</a>

(7.1)

It is well known that the cone K is positive and normal in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M300">View MathML</a>. We need the following definitions in what follows.

Definition 7.2 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M301">View MathML</a> is called a lower solution of FHDE (2.1) defined on J if it satisfies (4.3). Similarly, a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M302">View MathML</a> is called an upper solution of FHDE (2.1) defined on J if it satisfies (4.4). A solution to FHDE (2.1) is a lower as well as an upper solution for FHDE (2.1) defined on J and vice versa.

We consider the following set of assumptions:

(B0) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M303">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M304">View MathML</a>.

(B1) FHDE (2.1) has a lower solution a and an upper solution b defined on J with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M292">View MathML</a>.

(B2) The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M306">View MathML</a> is increasing in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M307">View MathML</a> almost everywhere for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a>.

(B3) The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M309">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M310">View MathML</a> are nondecreasing in x almost everywhere for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a>.

(B4) There exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M312">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M313">View MathML</a>.

We remark that hypothesis (B4) holds in particular if f is continuous and g is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M314">View MathML</a>-Carathéodory on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M315">View MathML</a>.

Theorem 7.1Suppose that assumptions (A1) and (B0)-(B4) hold. Then FHDE (2.1) has a minimal and a maximal positive solution defined onJ.

Proof Now, FHDE (2.1) is equivalent to integral equation (3.5) defined on J. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M316">View MathML</a>. Define two operators A and B on X by (3.6) and (3.7) respectively. Then the integral equation (3.5) is transformed into an operator equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M317">View MathML</a> in the Banach algebra X. Notice that hypothesis (B0) implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M318">View MathML</a>. Since the cone K in X is normal, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M285">View MathML</a> is a norm bounded set in X. Now it is shown, as in the proof of Theorem 3.1, that A is a Lipschitzian with the Lipschitz constant L and B is a completely continuous operator on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M285">View MathML</a>. Again, hypothesis (B3) implies that A and B are nondecreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M285">View MathML</a>. To see this, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M290">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M273">View MathML</a>. Then, by hypothesis (B3),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M324">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a>. Similarly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M326">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a>. So, A and B are nondecreasing operators on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M285">View MathML</a>. Lemma 7.1 and hypothesis (B3) together imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M329">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M295">View MathML</a>. As a result, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M332">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M50">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M295">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M294">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M295">View MathML</a>.

Now, we apply Lemma 7.2 to the operator equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M337">View MathML</a> to yield that FHDE (2.1) has a minimal and a maximal positive solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/23/mathml/M285">View MathML</a> defined on J. This completes the proof. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

This research is supported by the Natural Science Foundation of China (11071143, 60904024, 61174217), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119) and supported by Shandong Provincial Natural Science Foundation (ZR2012AM009, ZR2010AL002, ZR2011AL007), also supported by Natural Science Foundation of Educational Department of Shandong Province (J11LA01).

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