SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Variational approach to a class of impulsive differential equations

Dajun Guo

Author Affiliations

Department of Mathematics, Shandong University, Jinan, Shandong, 250100, People’s Republic of China

Boundary Value Problems 2014, 2014:37  doi:10.1186/1687-2770-2014-37


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/37


Received:16 December 2013
Accepted:14 January 2014
Published:7 February 2014

© 2014 Guo; 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 article, the author discusses the existence of solutions for a class of impulsive differential equations by means of a variational approach different from earlier approaches.

MSC: 34B37, 45G10, 47H30, 47J30.

Keywords:
impulsive differential equation; integral equation; variational method; critical point theory

1 Introduction

The theory of impulsive differential equations has been emerging as an important area of investigation in recent years [1-3]. There is a vast literature on the existence of solutions by using topological methods, including fixed point theorems, Leray-Schauder degree theory, and fixed point index theory [4-15]. But it is quite difficult to apply the variational approach to an impulsive differential equation; therefore, there was no result in this area for a long time. Only in the recent five years, there appeared a few articles which dealt with some impulsive differential equations by using variational methods [16-20]. Motivated by [17], in this article we shall use a different variational approach to discuss the existence of solutions for a class of impulsive differential equations and we only deal with classical solutions.

Consider the boundary value problem (BVP) for the second-order nonlinear impulsive differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M1">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M6">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M7">View MathML</a>) are any real numbers, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M8">View MathML</a> is a real function defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M9">View MathML</a>, where R denotes the set of all real numbers, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M8">View MathML</a> is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M11">View MathML</a>, left continuous at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M12">View MathML</a>, i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M13">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M14">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M15">View MathML</a>), and the right limit at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M12">View MathML</a> exists, i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M17">View MathML</a>

(denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M18">View MathML</a>) exists for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M14">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M7">View MathML</a>). <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M21">View MathML</a> denotes the jump of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M22">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M12">View MathML</a>, i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M24">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M25">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M26">View MathML</a> represent the right and left limits of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M22">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M12">View MathML</a>, respectively. Similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M29">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M30">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M31">View MathML</a> represent the right and left limits of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M32">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M12">View MathML</a>, respectively. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M34">View MathML</a> = {<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M35">View MathML</a> is a real function on J such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M22">View MathML</a> is continuous at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M37">View MathML</a>, left continuous at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M12">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M25">View MathML</a> exists, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M15">View MathML</a>} and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M41">View MathML</a> = {<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M42">View MathML</a> is continuous at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M37">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M31">View MathML</a> exist, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M15">View MathML</a>}. A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M47">View MathML</a> is called a solution of BVP (1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M22">View MathML</a> satisfies (1).

Let us list some conditions.

(H1) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M49">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M50">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M51">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M52">View MathML</a>

(H2) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M53">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M54">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M55">View MathML</a>

Lemma 1<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M47">View MathML</a>is a solution of BVP (1) if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M57">View MathML</a>is a solution of the integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M58">View MathML</a>

(2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M59">View MathML</a>

(3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M60">View MathML</a>

(4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M61">View MathML</a>

(5)

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M47">View MathML</a>, we have the formula (see [21], Lemma 1(b))

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M63">View MathML</a>

(6)

So, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M47">View MathML</a> is a solution of BVP (1), then, by (1) and (6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M65">View MathML</a>

(7)

It is clear, by (5), that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M66">View MathML</a>

(8)

so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M67">View MathML</a>

(9)

Substituting (9) into (7), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M68">View MathML</a>

(10)

By virtue of (5), we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M57">View MathML</a> (in fact, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M70">View MathML</a>) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M71">View MathML</a>

so, letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M72">View MathML</a> in (10), we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M73">View MathML</a>

(11)

Substituting (11) into (10), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M74">View MathML</a>

so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M75">View MathML</a> is a solution of the integral equation (2).

Conversely, suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M57">View MathML</a> is a solution of (2), i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M77">View MathML</a>

(12)

By (4), it is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M78">View MathML</a> is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M79">View MathML</a>, so differentiation of (12) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M80">View MathML</a>

(13)

Differentiating again, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M81">View MathML</a>

(14)

From (13) we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M82">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M83">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M7">View MathML</a>) exist and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M85">View MathML</a>

(15)

It follows from (4), (5), (12), (14), and (15) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M47">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M22">View MathML</a> satisfies (1). □

Lemma 2Let condition (H1) be satisfied. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M88">View MathML</a>is a solution of the integral equation (2), then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M57">View MathML</a>.

Proof It is clear, for function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M90">View MathML</a> defined by (5),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M91">View MathML</a>

(16)

By (4), (5), (16), and condition (H1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M92">View MathML</a>

so,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M93">View MathML</a>

(17)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M94">View MathML</a>

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M95">View MathML</a> satisfies the Caratheodory condition, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M95">View MathML</a> is measurable with respect to t on J for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M97">View MathML</a> and is continuous with respect to v on R for almost <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M98">View MathML</a> (in fact, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M95">View MathML</a> is discontinuous only at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M12">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M7">View MathML</a>)), so (17) implies [22,23] that the operator g defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M102">View MathML</a>

(18)

is bounded and continuous from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M103">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M104">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M105">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M106">View MathML</a>).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M88">View MathML</a> be a solution of the integral equation (2). Then by the Hölder inequality,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M108">View MathML</a>

which implies by virtue of the uniform continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M109">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M110">View MathML</a> that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M57">View MathML</a>. □

2 Variational approach

Theorem 1If conditions (H1) and (H2) are satisfied, then BVP (1) has at least one solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M47">View MathML</a>.

Proof By Lemma 1 and Lemma 2, we need only to show that the integral equation (2) has a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M88">View MathML</a>. The integral equation (2) can be written in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M114">View MathML</a>

(19)

where G is the linear integral operator defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M115">View MathML</a>

(20)

and the nonlinear operator g is defined by (18), which is bounded and continuous from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M103">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M104">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M118">View MathML</a>). It is well known that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M109">View MathML</a> is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M120">View MathML</a> positive-definite kernel with eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M121">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M122">View MathML</a>) and, by the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M109">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M124">View MathML</a>

(21)

so [22,23] the linear operator G defined by (20) is completely continuous from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M125">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M125">View MathML</a> and also from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M104">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M103">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M129">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M130">View MathML</a> (the positive square-root operator of G) is completely continuous from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M125">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M103">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M133">View MathML</a> denotes the adjoint operator of H, which is completely continuous from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M104">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M125">View MathML</a>. We now show that (19) has a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M88">View MathML</a> is equivalent to the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M137">View MathML</a>

(22)

has a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M138">View MathML</a>. In fact, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M88">View MathML</a> is a solution of (19), i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M140">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M141">View MathML</a>, so, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M142">View MathML</a> and u is a solution of (22). Conversely, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M138">View MathML</a> is a solution of (22), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M144">View MathML</a>, so, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M145">View MathML</a> and v is a solution of (19). Consequently, we need only to show that (22) has a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M138">View MathML</a>. It is well known [22,23] that the functional Φ defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M147">View MathML</a>

(23)

is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M148">View MathML</a> functional on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M125">View MathML</a> and its Fréchet derivative is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M150">View MathML</a>

(24)

Hence we need only to show that there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M138">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M152">View MathML</a> (θ denotes the zero element of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M125">View MathML</a>), i.e.u is a critical point of functional Φ.

By (4), (5), (16), and condition (H1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M154">View MathML</a>

(25)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M155">View MathML</a>

(26)

So, (25), (26), and condition (H2) imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M156">View MathML</a>

(27)

It is well known [24],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M157">View MathML</a>

(28)

where G is defined by (20) and is regarded as a positive-definite operator from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M125">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M125">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M160">View MathML</a> denotes the largest eigenvalue of G. It follows from (23), (27), and (28) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M161">View MathML</a>

(29)

which implies by virtue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M53">View MathML</a> (see condition (H2)) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M163">View MathML</a>

(30)

So, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M164">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M165">View MathML</a>

(31)

It is well known [22,23] that the ball <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M166">View MathML</a> is weakly closed and weakly compact and the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M167">View MathML</a> is weakly lower semicontinuous, so, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M168">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M169">View MathML</a>

(32)

It follows from (31) and (32) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M170">View MathML</a>

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M171">View MathML</a> and the theorem is proved. □

Example 1 Consider the BVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M172">View MathML</a>

(33)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M6">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M7">View MathML</a>) are any real numbers.

Conclusion BVP (33) has at least one solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M47">View MathML</a>.

Proof Evidently, (33) is a BVP of the form (1) with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M180">View MathML</a>

(34)

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M181">View MathML</a>. By (34), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M182">View MathML</a>

(35)

Moreover, it is well known that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M183">View MathML</a>

(36)

So, (35) and (36) imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M184">View MathML</a>

and consequently, condition (H1) is satisfied for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M185">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M186">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M187">View MathML</a>. On the other hand, choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M188">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M189">View MathML</a>

(37)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M190">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M191">View MathML</a>, so,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M192">View MathML</a>

(38)

By (35), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M193">View MathML</a>

(39)

It follows from (38) and (39) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M194">View MathML</a>

(40)

Since, by virtue of (37),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M195">View MathML</a>

we see that (40) implies that condition (H2) is satisfied for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M196">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M197">View MathML</a>. Hence, our conclusion follows from Theorem 1. □

By using the Mountain Pass Lemma and the Minimax Principle established by Ambrosetti and Rabinowitz [25,26], we have obtained in [23] the existence of a nontrivial solution and the existence of infinitely many nontrivial solutions for a class of nonlinear integral equations. Since (2) is a special case of such nonlinear integral equations, we get the following result for (2).

Lemma 3 (Special case of Theorem 1 and Theorem 2 in [23])

Suppose the following.

(a) There exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M49">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M51">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M201">View MathML</a>

(b) There exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M202">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M203">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M204">View MathML</a>

(c) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M205">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M206">View MathML</a>uniformly for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M98">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M208">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M209">View MathML</a>uniformly for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M98">View MathML</a>.

Then the integral equation (2) has at least one nontrivial solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M103">View MathML</a>. If, in addition,

(d) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M212">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M213">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M97">View MathML</a>.

Then the integral equation (2) has infinite many nontrivial solutions in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M103">View MathML</a>.

Let us list more conditions for the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M8">View MathML</a>.

(H3) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M202">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M203">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M219">View MathML</a>

(H4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M220">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M221">View MathML</a> uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M98">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M223">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M224">View MathML</a> uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M98">View MathML</a>.

(H5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M226">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M213">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M14">View MathML</a>.

Theorem 2Suppose that conditions (H1), (H3), and (H4) are satisfied. Then BVP (1) has at least one solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M47">View MathML</a>. If, in addition, condition (H5) is satisfied, then BVP (1) has infinitely many solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M230">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M122">View MathML</a>).

Proof In the proof of Lemma 2, we see that condition (H1) implies condition (a) of Lemma 3 (see (17)). On the other hand, it is clear that conditions (H3), (H4), (H5) are the same as conditions (b), (c), (d) in Lemma 3, respectively. Hence the conclusion of Theorem 2 follows from Lemma 3, Lemma 2, and Lemma 1. □

Example 2 Consider the BVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M232">View MathML</a>

(41)

Conclusion BVP (41) has infinite many solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M230">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M122">View MathML</a>).

Proof Obviously, (41) is a BVP of form (1). In this situation, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M237">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M238">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M239">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M240">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M241">View MathML</a>

(42)

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M8">View MathML</a> is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M11">View MathML</a>, left continuous at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M244">View MathML</a>, and the right limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M245">View MathML</a> exists. By (42), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M246">View MathML</a>

so, condition (H1) is satisfied for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M247">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M248">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M249">View MathML</a>. By (5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M250">View MathML</a>

(43)

so, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M251">View MathML</a> and (42) and (43) imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M252">View MathML</a>

(44)

and, consequently, (H3) is satisfied for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M253">View MathML</a> and any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/37/mathml/M203">View MathML</a>. On the other hand, from (44) we see that conditions (H4) and (H5) are all satisfied. Hence, our conclusion follows from Theorem 2. □

Competing interests

The author declares that they have no competing interests.

Acknowledgements

Research was supported by the National Nature Science Foundation of China (No. 10671167).

References

  1. Lakshmikantham, V, Bainov, DD, Simeonov, PS: Theory of Impulsive Differential Equations, World Scientific, Singapore (1989)

  2. Samoilenko, AM, Perestyuk, NA: Impulsive Differential Equations, World Scientific, Singapore (1995)

  3. Benchohra, M, Henderson, J, Ntouyas, SK: Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, New York (2006)

  4. Agarwal, RP, O’Regan, D: Multiple nonnegative solutions for second order impulsive differential equations. Appl. Math. Comput.. 114, 51–59 (2000). Publisher Full Text OpenURL

  5. Yan, B: Boundary value problems on the half line with impulses and infinite delay. J. Math. Anal. Appl.. 259, 94–114 (2001). Publisher Full Text OpenURL

  6. Agarwal, RP, O’Regan, D: A multiplicity result for second order impulsive differential equations via the Leggett Williams fixed point theorem. Appl. Math. Comput.. 161, 433–439 (2005). Publisher Full Text OpenURL

  7. Kaufmann, ER, Kosmatov, N, Raffoul, YN: A second-order boundary value problem with impulsive effects on an unbounded domain. Nonlinear Anal.. 69, 2924–2929 (2008). Publisher Full Text OpenURL

  8. Guo, D: Positive solutions of an infinite boundary value problem for nth-order nonlinear impulsive singular integro-differential equations in Banach spaces. Nonlinear Anal.. 70, 2078–2090 (2009). Publisher Full Text OpenURL

  9. Guo, D: Multiple positive solutions for first order impulsive superlinear integro-differential equations on the half line. Acta Math. Sci. Ser. B. 31(3), 1167–1178 (2011). Publisher Full Text OpenURL

  10. Guo, D, Liu, X: Multiple positive solutions of boundary value problems for impulsive differential equations. Nonlinear Anal.. 25, 327–337 (1995). Publisher Full Text OpenURL

  11. Guo, D: Multiple positive solutions for first order nonlinear impulsive integro-differential equations in a Banach space. Appl. Math. Comput.. 143, 233–249 (2003). Publisher Full Text OpenURL

  12. Guo, D: Multiple positive solutions of a boundary value problem for nth-order impulsive integro-differential equations in Banach spaces. Nonlinear Anal.. 63, 618–641 (2005). Publisher Full Text OpenURL

  13. Xu, X, Wang, B, O’Regan, D: Multiple solutions for sub-linear impulsive three-point boundary value problems. Appl. Anal.. 87, 1053–1066 (2008). Publisher Full Text OpenURL

  14. Jankowski, J: Existence of positive solutions to second order four-point impulsive differential problems with deviating arguments. Comput. Math. Appl.. 58, 805–817 (2009). Publisher Full Text OpenURL

  15. Liu, Y, O’Regan, D: Multiplicity results using bifurcation techniques for a class of boundary value problems of impulsive differential equations. Commun. Nonlinear Sci. Numer. Simul.. 16, 1769–1775 (2011)

  16. Tian, Y, Ge, W: Applications of variational methods to boundary value problem for impulsive differential equations. Proc. Edinb. Math. Soc.. 51, 509–527 (2008)

  17. Nieto, JJ, O’Regan, D: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl.. 10, 680–690 (2009). Publisher Full Text OpenURL

  18. Zhang, Z, Yuan, R: An application of variational methods to Dirichlet boundary value problem with impulses. Nonlinear Anal., Real World Appl.. 11, 155–162 (2010). Publisher Full Text OpenURL

  19. Chen, H, Sun, J: An application of variational method to second-order impulsive differential equations on the half line. Appl. Math. Comput.. 217, 1863–1869 (2010). Publisher Full Text OpenURL

  20. Bai, L, Dai, B: Existence and multiplicity of solutions for an impulsive boundary value problem with a parameter via critical point theory. Math. Comput. Model.. 53, 1844–1855 (2011). Publisher Full Text OpenURL

  21. Guo, D: A class of second-order impulsive integro-differential equations on unbounded domain in a Banach space. Appl. Math. Comput.. 125, 59–77 (2002). Publisher Full Text OpenURL

  22. Krasnoselskii, MA: Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon, Oxford (1964)

  23. Guo, D: The number of nontrivial solutions to Hammerstein nonlinear integral equations. Chin. Ann. Math., Ser. B. 7(2), 191–204 (1986)

  24. Zaanen, AC: Linear Analysis, Interscience, New York (1958)

  25. Ambrosetti, A, Rabinowitz, PH: Dual variational method in critical point theory and applications. J. Funct. Anal.. 14, 349–381 (1973). Publisher Full Text OpenURL

  26. Rabinowitz, PH: Variational methods for nonlinear eigenvalue problems. Varenna, Italy. (1974)