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Periodic boundary value problems for second-order impulsive integro-differential equations with integral jump conditions

Chatthai Thaiprayoon12*, Decha Samana12 and Jessada Tariboon23

Author Affiliations

1 Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok, 10520, Thailand

2 Centre of Excellence in Mathematics, CHE, Sri Ayutthaya Road, Bangkok, 10400, Thailand

3 Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, 10800, Thailand

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Boundary Value Problems 2012, 2012:122  doi:10.1186/1687-2770-2012-122

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/122


Received:24 June 2012
Accepted:11 October 2012
Published:24 October 2012

© 2012 Thaiprayoon 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

This paper is concerned with the existence of extremal solutions of periodic boundary value problems for second-order impulsive integro-differential equations with integral jump conditions. We introduce a new definition of lower and upper solutions with integral jump conditions and prove some new maximum principles. The method of lower and upper solutions and the monotone iterative technique are used.

MSC: 34B37, 34K10, 34K45.

Keywords:
impulsive integro-differential equation; lower and upper solutions; periodic boundary value problem; monotone iterative technique

1 Introduction

Differential equations which have impulse effects describe many evolution processes that abruptly change their state at a certain moment. In recent years, impulsive differential equations have become more important tools in some mathematical models of real processes and phenomena studied in physics, biotechnology, chemical technology, population dynamics and economics; see [1-5]. Many papers have been published about existence analysis of periodic boundary value problems of first and second order for impulsive ordinary or functional or integro-differential equations. We refer the readers to the papers [6-29]. More recent works on existence results of impulsive problems with integral boundary conditions can be found in [30-35] and the reference therein. This literature has lead to significant development of a general theory for impulsive differential equations.

The monotone iterative technique coupled with the method of upper and lower solutions has been used to study the existence of extremal solutions of periodic boundary value problems for second-order impulsive equations; see, for example, [36-41]. This method has been also used to study abstract nonlinear problems; see [42]. However, in most of these papers concerned with applications of the monotone iterative technique to second-order periodic boundary value problems with impulses, the authors assume that the jump conditions at impulse point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M1">View MathML</a> of solution values and the derivative of solution values depend on the left-hand limits of solutions or the slope of solutions themselves, such as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M5">View MathML</a>.

In this paper, we consider the periodic boundary value problem for second-order impulsive integro-differential equation (PBVP) with integral jump conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M6">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M8">View MathML</a> is continuous everywhere except at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M11">View MathML</a> exist, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M19">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M20">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M26">View MathML</a>.

In [43,44], the authors discussed some kinds of first-order impulsive problems with the integral jump condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M27">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M30">View MathML</a>. We note that the jump condition (1.2) depends on functionals of path history before impulse points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M1">View MathML</a> and after the past impulse points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M32">View MathML</a>. The aim of our research is to deal with the integral jump conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M33">View MathML</a>

(1.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M19">View MathML</a>. The integral jump condition (1.3) means that a sudden change of solution values and the derivative of solution values at impulse point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M1">View MathML</a> depend on the area under the curves of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M38">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M39">View MathML</a> between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M40">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M41">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M42">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M43">View MathML</a>, respectively. It should be noticed that the impulsive effects of PBVP (1.1) have memory of the past states.

This paper is organized as follows. Firstly, we introduce a new concept of lower and upper solutions. After that, we establish some new comparison principles and discuss the existence and uniqueness of the solutions for second-order impulsive integro-differential equations with integral jump conditions. By using the method of upper and lower solutions and the monotone iterative technique, we obtain the existence of an extreme solution of PBVP (1.1). Finally, we give an example to illustrate the obtained results.

2 Preliminaries

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M45">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M46">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M47">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M48">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M49">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M50">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M51">View MathML</a> are Banach spaces with the norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M52">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M53">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M54">View MathML</a>. A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M55">View MathML</a> is called a solution of PBVP (1.1) if it satisfies (1.1).

Definition 2.1 We say that the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M56">View MathML</a> are lower and upper solutions of PBVP (1.1), respectively, if there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M18">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M64">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M65">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M66">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M67">View MathML</a>

Now we are in the position to establish some new comparison principles which play an important role in the monotone iterative technique.

Lemma 2.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M55">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M69">View MathML</a>

(2.1)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M61">View MathML</a>are constants and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M77">View MathML</a>, and they satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M78">View MathML</a>

(2.2)

Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80">View MathML</a>.

Proof Suppose, to the contrary, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M81">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80">View MathML</a>. We divide the proof into two cases:

Case (i). There exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M83">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M84">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M85">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80">View MathML</a>.

From (2.1), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M87">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M88">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M89">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M38">View MathML</a> is nondecreasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80">View MathML</a> and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M92">View MathML</a>. However, by (2.1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M93">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M94">View MathML</a>, which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M95">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M97">View MathML</a>, a contradiction.

Case (ii). There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M98">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M100">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M101">View MathML</a>, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M102">View MathML</a>, for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M103">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M104">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M105">View MathML</a>. Without loss of generality, we only consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M104">View MathML</a>. For the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M105">View MathML</a> the proof is similar. It follows that

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M109">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M112">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M39">View MathML</a> is strictly increasing on J, which contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M114">View MathML</a>. Then there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M115">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M116">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M117">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M118">View MathML</a>. By mean value theorem, we have

Summing up the above inequalities, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M120">View MathML</a>

(2.3)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M122">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M123">View MathML</a> by using the method to get (2.3), then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M124">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M125">View MathML</a>, then the above method together with (2.1), (2.3) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M126">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M127">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M128">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M129">View MathML</a>. We first assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M130">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M131">View MathML</a>. By the mean value theorem, we have

Summing up, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M133">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M134">View MathML</a>

which contradicts (2.2).

For the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M135">View MathML</a>, the proof is similar, and thus we omit it. This completes the proof. □

Lemma 2.2Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M55">View MathML</a>satisfies

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M61">View MathML</a>are constants and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M77">View MathML</a>, and they satisfy (2.2). Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M79">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M148">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80">View MathML</a>, and define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M150">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M151">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M152">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M153">View MathML</a>. If we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M154">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M155">View MathML</a> and the proof is complete. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M156">View MathML</a>, then we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M157">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M158">View MathML</a>. Indeed, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M30">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M160">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M161">View MathML</a>

Meanwhile, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M162">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M153">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M164">View MathML</a>

Then by Lemma 2.1, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M165">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80">View MathML</a>. □

Consider the linear PBVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M169">View MathML</a>

(2.4)

where constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M175">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M176">View MathML</a> are constants and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M77">View MathML</a>.

Lemma 2.3<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M181">View MathML</a>is a solution of (2.4) if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M182">View MathML</a>is a solution of the following impulsive integral equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M183">View MathML</a>

(2.5)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M184">View MathML</a>

This proof is similar to the proof of Lemma 2.1 in [36], and we omit it.

Lemma 2.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M61">View MathML</a>are constants and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M77">View MathML</a>. If

(2.6)

(2.7)

then (2.4) has a unique solutionxinE.

Proof For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M195">View MathML</a>, we define an operator F by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M196">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M197">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M198">View MathML</a> are given by Lemma 2.3. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M199">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M200">View MathML</a>

By computing directly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M201">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M202">View MathML</a>

On the other hand, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M203">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M204">View MathML</a>

Similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M205">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M206">View MathML</a>

By the Banach fixed-point theorem, F has a unique fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M207">View MathML</a>, and by Lemma 2.3, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M208">View MathML</a> is also the unique solution of (2.4). This completes the proof. □

3 Main results

In this section, we establish existence criteria for solutions of PBVP (1.1) by the method of lower and upper solutions and the monotone iterative technique. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M56">View MathML</a>, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M210">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M211">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80">View MathML</a>. In such a case, we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M213">View MathML</a>.

Theorem 3.1Suppose that the following conditions hold:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M214">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M215">View MathML</a>are lower and upper solutions for PBVP (1.1), respectively, such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M216">View MathML</a>.

(H2) The functionfsatisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M217">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M219">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M220">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M221">View MathML</a>.

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M226">View MathML</a>are constants, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M112">View MathML</a>, and they satisfy (2.2), (2.6) and (2.7).

(H4) The functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M230">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M231">View MathML</a>satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M232">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M233">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M236">View MathML</a>.

Then there exist monotone sequences<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M237">View MathML</a>which converge inEto the extreme solutions of PBVP (1.1) in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M238">View MathML</a>, respectively.

Proof For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M239">View MathML</a>, we consider linear PBVP (2.4) with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M240">View MathML</a>

By Lemma 2.4, PBVP (2.4) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M55">View MathML</a>. We define an operator A from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M242">View MathML</a> to E by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M243">View MathML</a>. We complete the proof in four steps.

Step 1. We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M244">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M245">View MathML</a>. We only prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M244">View MathML</a> since the second inequality can be proved in a similar manner.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M247">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M248">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M249">View MathML</a> satisfies

We finish Step 1 in two cases.

Case 1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M251">View MathML</a>, which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M252">View MathML</a>

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M214">View MathML</a> is a lower solution of PBVP (1.1), then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M88">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M255">View MathML</a>

and

Then by Lemma 2.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M257">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M258">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M244">View MathML</a>.

Case 2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M260">View MathML</a>, which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M261">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M262">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M263">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M264">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M265">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M266">View MathML</a>

Then by Lemma 2.2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M257">View MathML</a>, which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M258">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M244">View MathML</a>.

Step 2. We prove that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M270">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M271">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M272">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M273">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M274">View MathML</a>, then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M88">View MathML</a>, and by (H2), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M276">View MathML</a>

From (H3), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M277">View MathML</a>

Applying Lemma 2.1, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M257">View MathML</a>, which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M271">View MathML</a>.

Step 3. We show that PBVP (1.1) has solutions.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M280">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M281">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M282">View MathML</a> . Following the first two steps, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M283">View MathML</a>

Obviously, each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M284">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M285">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M286">View MathML</a>) satisfies

and

Thus, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M289">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M208">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M291">View MathML</a>

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M289">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M208">View MathML</a> satisfy PBVP (1.1).

Step 4. We show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M289">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M208">View MathML</a> are extreme solutions of PBVP (1.1).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M39">View MathML</a> be any solution of PBVP (1.1), which satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M297">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80">View MathML</a>. Suppose that there exists a positive integer n such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M300">View MathML</a>. Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M301">View MathML</a>, then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M88">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M303">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M304">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M305">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M306">View MathML</a>

Still by Lemma 2.1, we have for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M308">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M309">View MathML</a>. Similarly, we can prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M310">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M312">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80">View MathML</a>, which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M314">View MathML</a>. The proof is complete. □

4 An example

In this section, in order to illustrate our results, we consider an example.

Example 4.1 Consider the following PBVP:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M315">View MathML</a>

(4.1)

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M316">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M317">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M318">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M319">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M320">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M321">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M322">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M323">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M324">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M325">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M326">View MathML</a> are lower and upper solutions for (4.1), respectively, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M210">View MathML</a>.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M328">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M329">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M330">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M331">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M332">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M80">View MathML</a>. It is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M334">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M335">View MathML</a>

whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M336">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M337">View MathML</a>.

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M338">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M339">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M340">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M341">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M342">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M343">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M344">View MathML</a>

Therefore, (4.1) satisfies all the conditions of Theorem 3.1. So, PBVP (4.1) has minimal and maximal solutions in the segment <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M242">View MathML</a>.

Substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M214">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M215">View MathML</a> into monotone iterative scheme, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M348">View MathML</a>

(4.2)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M349">View MathML</a>

(4.3)

After using the variational iteration method [45] for (4.2), (4.3), the approximate solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M249">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M351">View MathML</a> can be illustrated as Figure 1 and Figure 2, respectively.

thumbnailFigure 1. Time history of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M352">View MathML</a>.

thumbnailFigure 2. Time history of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/122/mathml/M353">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally in this article. They read and approved the final manuscript.

Acknowledgements

This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.

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