SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Multiple positive solutions for first-order impulsive singular integro-differential equations on the half line in a Banach space

Yanlai Chen1* and Baoxia Qin2

Author Affiliations

1 School of Economics, Shandong University, Jinan, Shandong, 250100, People’s Republic of China

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

For all author emails, please log on.

Boundary Value Problems 2013, 2013:69  doi:10.1186/1687-2770-2013-69

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/69


Received:22 December 2012
Accepted:18 March 2013
Published:2 April 2013

© 2013 Chen and Qin; licensee Springer

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

Abstract

In this paper, the author discusses the multiple positive solutions for an infinite three-point boundary value problem of first-order impulsive superlinear singular integro-differential equations on the half line in a Banach space by means of the fixed-point theorem of cone expansion and compression with norm type.

MSC: 45J05, 34G20, 47H10.

Keywords:
impulsive singular integro-differential equation in a Banach space; infinite three-point boundary value problem; fixed-point theorem of cone expansion and compression with norm type

1 Introduction

In recent years, multiple solutions of boundary value problems for impulsive differential equations in scalar spaces had been extensively studied (see, for example, [1-3]). In recent papers [4] and [5], Professor D. Guo discussed two infinite boundary value problems for nth-order impulsive nonlinear singular integro-differential equations of mixed type on the half line in a Banach space. By constructing a bounded closed convex set, apart from the singularities, and using the Schauder fixed-point theorem, he obtained the existence of positive solutions for the infinite boundary value problems. But such equations are sublinear, and there are no results on existence of two positive solutions. Now, in this paper, we shall discuss the existence of two positive solutions for first-order superlinear singular equations by means of a different method, i.e., by using the fixed-point theorem of cone expansion and compression with norm type (see [6,7]), and the key point is to introduce a new cone Q.

Let E be a real Banach space and P be a cone in E, which defines a partial ordering in E by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M1">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M2">View MathML</a>. P is said to be normal if there exists a positive constant N such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M3">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M4">View MathML</a>, where θ denotes the zero element of E, and the smallest N is called the normal constant of P. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M1">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M6">View MathML</a>, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M7">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M8">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M9">View MathML</a>. For details on cone theory, see [7].

Consider the infinite three-point boundary value problem for a first-order impulsive nonlinear singular integro-differential equation of mixed type on the half line in E:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M10">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M13">View MathML</a> , <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M17">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M18">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M21">View MathML</a> (for some m), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M22">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M23">View MathML</a>

(2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M27">View MathML</a> denotes the set of all nonnegative numbers. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M28">View MathML</a> denotes the jump of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M29">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M30">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M31">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M32">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M33">View MathML</a> represent the right and left limits of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M29">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M30">View MathML</a>, respectively. In the following, we always assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M36">View MathML</a>

(3)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M37">View MathML</a>

(4)

(where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M38">View MathML</a> means <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M40">View MathML</a>), i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M41">View MathML</a> is singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M42">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M43">View MathML</a>. We also assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M44">View MathML</a>

(5)

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M45">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M18">View MathML</a>) are singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M43">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M48">View MathML</a> = {<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M49">View MathML</a> is a map from J into E such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M29">View MathML</a> is continuous at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M51">View MathML</a>, left continuous at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M30">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M32">View MathML</a> exists, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M18">View MathML</a>} and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M55">View MathML</a>. It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M56">View MathML</a> is a Banach space with norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M57">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M58">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M59">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M60">View MathML</a> and Q are two cones in space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M56">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M62">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M63">View MathML</a> is called a positive solution of the infinite three-point boundary value problem (1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M64">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M65">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M29">View MathML</a> satisfies (1). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M67">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M68">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M69">View MathML</a>.

2 Several lemmas

Let us list some conditions.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M70">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M72">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M73">View MathML</a>

In this case, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M74">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M75">View MathML</a>) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M76">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M77">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M78">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M79">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M80">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M81">View MathML</a>) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M82">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M18">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M84">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M85">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M86">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M87">View MathML</a>) For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M88">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M89">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M90">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M91">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M18">View MathML</a>) are relatively compact in E, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M93">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M94">View MathML</a>.

Remark Obviously, condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M87">View MathML</a>) is satisfied automatically when E is finite dimensional.

Remark It is clear: If condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M70">View MathML</a>) is satisfied, then the operators T and S defined by (2) are bounded linear operators from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M56">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M56">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M100">View MathML</a>; moreover, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M101">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M102">View MathML</a>.

We shall reduce the infinite three-point boundary value problem (1) to an impulsive integral equation. To this end, we consider the operator A defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M103">View MathML</a>

(6)

In what follows, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M105">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M106">View MathML</a>).

Lemma 1Let conePbe normal and conditions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M70">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M87">View MathML</a>) be satisfied. Then operatorAdefined by (6) is a continuous operator from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M109">View MathML</a>intoQ; moreover, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M111">View MathML</a>is relatively compact.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M112">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M113">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M114">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M115">View MathML</a>

so,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M116">View MathML</a>

(7)

where N denotes the normal constant of cone P, and consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M117">View MathML</a>

(8)

By condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M75">View MathML</a>) and (8), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M119">View MathML</a>

(9)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M120">View MathML</a>

which implies the convergence of the infinite integral

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M121">View MathML</a>

(10)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M122">View MathML</a>

(11)

On the other hand, by condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M81">View MathML</a>) and (8), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M124">View MathML</a>

(12)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M125">View MathML</a>

which implies the convergence of the infinite series

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M126">View MathML</a>

(13)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M127">View MathML</a>

(14)

It follows from (6), (11), and (14) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M128">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M129">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M130">View MathML</a>

(15)

Moreover, by (6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M131">View MathML</a>

(16)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M132">View MathML</a>

(17)

It is clear,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M133">View MathML</a>

(18)

so, (17) and (18) imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M134">View MathML</a>

(19)

It follows from (16) and (19) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M135">View MathML</a>

(20)

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M136">View MathML</a>, i.e., A maps <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M109">View MathML</a> into Q.

Now, we are going to show that A is continuous. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M138">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M139">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M140">View MathML</a>). Write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M141">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M142">View MathML</a>) and we may assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M143">View MathML</a>

So, by (7),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M144">View MathML</a>

(21)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M145">View MathML</a>

(22)

Similar to (15), it is easy to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M146">View MathML</a>

(23)

It is clear that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M147">View MathML</a>

(24)

and, similar to (9) and observing (21) and (22), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M148">View MathML</a>

(25)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M149">View MathML</a>

It follows from (24), (25), and the dominated convergence theorem that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M150">View MathML</a>

(26)

On the other hand, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M151">View MathML</a>, we can choose a positive integer j such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M152">View MathML</a>

(27)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M153">View MathML</a>

And then, choose an positive integer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M154">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M155">View MathML</a>

(28)

From (27), (28), and observing condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M81">View MathML</a>) and (21), (22), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M157">View MathML</a>

hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M158">View MathML</a>

(29)

It follows from (23), (26), and (29) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M159">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M140">View MathML</a>, and the continuity of A is proved.

Finally, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M111">View MathML</a> is relatively compact, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M69">View MathML</a> are arbitrarily given. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M163">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M164">View MathML</a>). Then, by (7),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M165">View MathML</a>

(30)

Similar to (9), (12), (15), and observing (30), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M166">View MathML</a>

(31)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M167">View MathML</a>

(32)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M168">View MathML</a>

(33)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M169">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M170">View MathML</a>

Consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M171">View MathML</a> for any fixed i. By (6) and (31), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M172">View MathML</a>

(34)

which implies that the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M173">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M164">View MathML</a>) defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M175">View MathML</a>

(35)

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M176">View MathML</a> denotes the right limit of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M177">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M178">View MathML</a>) are equicontinuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M179">View MathML</a>. On the other hand, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M151">View MathML</a>, choose a sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M181">View MathML</a> and a sufficiently large positive integer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M182">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M183">View MathML</a>

(36)

We have, by (35), (6), (31), (32), and (36),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M184">View MathML</a>

(37)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M185">View MathML</a>

(38)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M186">View MathML</a>

(39)

It follows from (37), (38), (39), and ([8], Theorem 1.2.3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M187">View MathML</a>

(40)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M188">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M189">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M190">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M191">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M192">View MathML</a> denotes the Kuratowski measure of noncompactness of bounded set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M193">View MathML</a> (see [[8], Section 1.2]). Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M194">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M195">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M196">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M197">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M198">View MathML</a>, we see that, by condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M87">View MathML</a>),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M200">View MathML</a>

(41)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M201">View MathML</a>

(42)

It follows from (40) to (42) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M202">View MathML</a>

which implies by virtue of the arbitrariness of ϵ that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M203">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M204">View MathML</a>.

By the Ascoli-Arzela theorem (see [[8], Theorem 1.2.5]), we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M205">View MathML</a> is relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M206">View MathML</a>, hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M173">View MathML</a> has a subsequence which is convergent uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M208">View MathML</a>, so, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M209">View MathML</a> has a subsequence which is convergent uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M210">View MathML</a>. Since i may be any positive integer, so, by diagonal method, we can choose a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M211">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M209">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M211">View MathML</a> is convergent uniformly on each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M214">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M18">View MathML</a>). Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M216">View MathML</a>

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M217">View MathML</a>. By (33), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M218">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M219">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M220">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M151">View MathML</a> be arbitrarily given and choose a sufficiently large positive number τ such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M222">View MathML</a>

(43)

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M223">View MathML</a>, we have, by (6),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M224">View MathML</a>

which implies by virtue of (31), (32), and (43) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M225">View MathML</a>

(44)

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M226">View MathML</a> in (44), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M227">View MathML</a>

(45)

On the other hand, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M211">View MathML</a> converges uniformly to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M229">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M230">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M226">View MathML</a>, there exists a positive integer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M232">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M233">View MathML</a>

(46)

It follows from (44) to (46) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M234">View MathML</a>

(47)

By (46) and (47), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M235">View MathML</a>

hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M236">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M226">View MathML</a>, and the relative compactness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M111">View MathML</a> is proved. □

Lemma 2Let conePbe normal and conditions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M70">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M87">View MathML</a>) be satisfied. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M241">View MathML</a>is a positive solution of the infinite three-point boundary value problem (1) if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M112">View MathML</a>is a solution of the following impulsive integral equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M243">View MathML</a>

(48)

i.e., uis a fixed point of operatorAdefined by (6) in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M109">View MathML</a>.

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M245">View MathML</a>, it is easy to get the following formula:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M246">View MathML</a>

(49)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M241">View MathML</a> be a positive solution of the infinite three-point boundary value problem (1). By (1) and (49), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M248">View MathML</a>

(50)

We have shown in the proof of Lemma 1 that the infinite integral (10) and the infinite series (13) are convergent, so, by taking limits as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M249">View MathML</a> in both sides of (50), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M250">View MathML</a>

(51)

On the other hand, by (1) and (50), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M251">View MathML</a>

(52)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M252">View MathML</a>

(53)

It follows from (51) to (53) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M253">View MathML</a>

and, substituting it into (50), we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M29">View MathML</a> satisfies equation (48), i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M255">View MathML</a>.

Conversely, assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M112">View MathML</a> is a solution of Equation (48). We have, by (48),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M257">View MathML</a>

(54)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M258">View MathML</a>

(55)

Moreover, by taking limits as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M249">View MathML</a> in (48), we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M260">View MathML</a> exists and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M261">View MathML</a>

(56)

It follows from (54) to (56) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M262">View MathML</a>

On the other hand, direct differentiation of (48) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M263">View MathML</a>

and, it is clear, by (48),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M264">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M265">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M29">View MathML</a> satisfies (1). Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M112">View MathML</a>, so (7) holds and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M268">View MathML</a>, hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M64">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M65">View MathML</a>. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M29">View MathML</a> is a positive solution of the infinite three-point boundary value problem (1). □

Lemma 3 (The fixed-point theorem of cone expansion and compression with norm type; see [[6], Theorem 3] or [[7], Theorem 2.3.4])

LetPbe a cone in real Banach spaceEand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M272">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M273">View MathML</a>be two bounded open sets inEsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M274">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M275">View MathML</a>, and operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M276">View MathML</a>be completely continuous, whereθdenotes the zero element ofEand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M277">View MathML</a>denotes the closure of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M278">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M279">View MathML</a>). Suppose that one of the following two conditions is satisfied:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M280">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M281">View MathML</a>denotes the boundary of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M278">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M279">View MathML</a>).

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M284">View MathML</a>

ThenAhas at least one fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M285">View MathML</a>.

Remark 1 Lemma 3 is different from the Krasnoselskii fixed-point theorem of cone expansion and compression (see [[9], Theorem 44.1]). In Krasnoselskii’s theorem, the condition corresponding to (a) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M286">View MathML</a>

It is clear, conditions (a) and (a′) are independent each other. On the other hand, in Krasnoselskii’s theorem, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M272">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M273">View MathML</a> are balls with center θ.

3 Main theorems

Let us list more conditions.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M289">View MathML</a>) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M290">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M291">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M292">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M293">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M294">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M295">View MathML</a>

Remark 2 Condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M289">View MathML</a>) means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M41">View MathML</a> is superlinear with respect to u.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M298">View MathML</a>) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M299">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M300">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M301">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M302">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M303">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M304">View MathML</a>

Theorem 1Let conePbe normal and conditions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M70">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M298">View MathML</a>) be satisfied. Assume that there exists a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M307">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M308">View MathML</a>

(57)

whereNdenotes the normal constant ofP, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M309">View MathML</a>

(58)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M310">View MathML</a>

(59)

(for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M311">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M312">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M313">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M314">View MathML</a>; see conditions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M75">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M81">View MathML</a>)). Then the infinite three-point boundary value problem (1) has at least two positive solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M317">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M318">View MathML</a>.

Proof By Lemma 1 and Lemma 2, operator A defined by (6) is continuous from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M109">View MathML</a> into Q and we need to prove that A has two fixed points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M320">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M321">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M109">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M323">View MathML</a>.

By condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M289">View MathML</a>), there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M325">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M326">View MathML</a>

(60)

so,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M327">View MathML</a>

(61)

Choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M328">View MathML</a>

(62)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M329">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M330">View MathML</a>, we have by (7) and (62),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M331">View MathML</a>

(63)

so, (6), (63), (61), and (7) imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M332">View MathML</a>

(64)

and consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M333">View MathML</a>

(65)

By condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M298">View MathML</a>), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M335">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M336">View MathML</a>

(66)

so,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M337">View MathML</a>

(67)

Choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M338">View MathML</a>

(68)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M329">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M340">View MathML</a>, we have by (68) and (7),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M341">View MathML</a>

(69)

so, we get by (6), (69), and (67),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M342">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M343">View MathML</a>

and consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M344">View MathML</a>

(70)

On the other hand, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M329">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M346">View MathML</a>, by condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M75">View MathML</a>), condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M81">View MathML</a>), (58), and (59), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M349">View MathML</a>

(71)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M350">View MathML</a>

(72)

It is clear, by (17),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M351">View MathML</a>

(73)

It follows from (71) to (73) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M352">View MathML</a>

(74)

Thus, (74) and (57) imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M353">View MathML</a>

(75)

From (62) and (68), we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M354">View MathML</a>, and by Lemma 1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M355">View MathML</a> is completely continuous, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M356">View MathML</a>, hence, (65), (70), (75), and Lemma 3 imply that A has two fixed points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M357">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M358">View MathML</a>. The proof is complete. □

Theorem 2Let conePbe normal and conditions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M70">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M87">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M298">View MathML</a>) be satisfied. Assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M362">View MathML</a>

(76)

uniformly for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M363">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M364">View MathML</a>

(77)

(for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M311">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M312">View MathML</a>, see conditions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M75">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M81">View MathML</a>)). Then the infinite three-point boundary value problem (1) has at least one positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M369">View MathML</a>.

Proof As in the proof of Theorem 1, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M370">View MathML</a> such that (70) holds (in this case, we put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M371">View MathML</a> in (66) and (68)). On the other hand, by (76) and (77), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M372">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M373">View MathML</a>

(78)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M374">View MathML</a>

(79)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M375">View MathML</a>

(80)

Choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M376">View MathML</a>

(81)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M329">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M378">View MathML</a>, we have by (7) and (81),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M379">View MathML</a>

so, (78) and (79) imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M380">View MathML</a>

(82)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M381">View MathML</a>

(83)

It follows from (73), conditions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M75">View MathML</a>), condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M81">View MathML</a>), (82), (83), and (80) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M384">View MathML</a>

and consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M385">View MathML</a>

(84)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M386">View MathML</a> by virtue of (81), we conclude from (70), (84), and Lemma 3 that A has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M387">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M388">View MathML</a>. The theorem is proved. □

Example 1

Consider the infinite system of scalar first-order impulsive singular integro-differential equations of mixed type on the half line:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M389">View MathML</a>

(85)

Conclusion Infinite system (85) has at least two positive solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M390">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M164">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M392">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M164">View MathML</a>) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M394">View MathML</a>

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M395">View MathML</a> with norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M396">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M397">View MathML</a>. Then P is a normal cone in E with normal constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M398">View MathML</a>, and infinite system (85) can be regarded as an infinite three-point boundary value problem of form (1). In this situation, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M399">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M400">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M401">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M402">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M18">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M404">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M405">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M406">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M407">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M408">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M409">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M410">View MathML</a>, in which

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M411">View MathML</a>

(86)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M412">View MathML</a>

(87)

It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M17">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M18">View MathML</a>) and condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M70">View MathML</a>) is satisfied and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M417">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M418">View MathML</a>. We have, by (86),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M419">View MathML</a>

(88)

so, observing the inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M420">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M421">View MathML</a>

which implies that condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M75">View MathML</a>) is satisfied for

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M423">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M424">View MathML</a>

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M425">View MathML</a>

By (87), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M426">View MathML</a>

(89)

so,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M427">View MathML</a>

which implies that condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M81">View MathML</a>) is satisfied for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M429">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M430">View MathML</a>

On the other hand, (86) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M431">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M432">View MathML</a>

(90)

so, we see that condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M289">View MathML</a>) is satisfied for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M434">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M435">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M436">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M437">View MathML</a> and condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M298">View MathML</a>) is satisfied for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M439">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M440">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M441">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M442">View MathML</a>. In addition, from (90), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M443">View MathML</a>

which implies that (3) and (4) hold, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M41">View MathML</a> is singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M42">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M43">View MathML</a>. Moreover, from (87), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M447">View MathML</a>

and so,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M448">View MathML</a>

which implies that (5) holds, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M45">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M18">View MathML</a>) are singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M43">View MathML</a>. Now, we check that condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M87">View MathML</a>) is satisfied. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M88">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M89">View MathML</a> be fixed, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M455">View MathML</a> be any sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M456">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M457">View MathML</a>. Then, we have, by (86) and (88),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M458">View MathML</a>

(91)

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M459">View MathML</a> is bounded, and, by diagonal method, we can choose a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M460">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M461">View MathML</a>

(92)

which implies by virtue of (91) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M462">View MathML</a>

(93)

Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M463">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M151">View MathML</a> be given. Choose a positive integer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M154">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M466">View MathML</a>

(94)

By (92), we see that there exists a positive integer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M232">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M468">View MathML</a>

(95)

It follows from (91) to (95) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M469">View MathML</a>

hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M470">View MathML</a> in E as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M471">View MathML</a>. Thus, we have proved that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M456">View MathML</a> is relatively compact in E. Similarly, by using (89), we can prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M473">View MathML</a> is relatively compact in E. Hence, condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M87">View MathML</a>) is satisfied. Finally, we check that inequality (57) is satisfied for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M371">View MathML</a>. In this case,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M476">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M477">View MathML</a>

so,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M478">View MathML</a>

i.e., inequality (57) is satisfied for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M371">View MathML</a>. Hence, our conclusion follows from Theorem 1. □

Example 2

Consider the infinite system of scalar first order impulsive singular integro-differential equations of mixed type on the half line:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M480">View MathML</a>

(96)

Conclusion Infinite system (96) has at least one positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M390">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M164">View MathML</a>) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M483">View MathML</a>

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M395">View MathML</a> with norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M396">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M486">View MathML</a>. Then P is a normal cone in E with normal constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M398">View MathML</a>, and infinite system (96) can be regarded as an infinite three-point boundary value problem of form (1) in E. In this situation, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M399">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M489">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M401">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M491">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M18">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M493">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M494">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M495">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M496">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M497">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M409">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M499">View MathML</a>, in which

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M500">View MathML</a>

(97)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M501">View MathML</a>

(98)

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M17">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M18">View MathML</a>) and condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M70">View MathML</a>) is satisfied and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M506">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M418">View MathML</a>. We have, by (97) and (98),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M508">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M509">View MathML</a>

so, observing

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M510">View MathML</a>

we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M511">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M512">View MathML</a>

which imply that conditions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M75">View MathML</a>) is satisfied for

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M514">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M515">View MathML</a>

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M516">View MathML</a>

and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M81">View MathML</a>) is satisfied for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M518">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M519">View MathML</a>) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M520">View MathML</a>

By (97), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M521">View MathML</a>

(99)

so, condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M298">View MathML</a>) is satisfied for

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M523">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M524">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M525">View MathML</a>. Moreover, (99) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M526">View MathML</a>

so, (3) and (4) are satisfied, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M41">View MathML</a> is singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M42">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M43">View MathML</a>. Similarly, (98) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M530">View MathML</a>

so, (5) is satisfied, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M45">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M18">View MathML</a>) are singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M43">View MathML</a>. Similar to the discussion in Example 1, we can prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M456">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M473">View MathML</a> (for fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M88">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M89">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M18">View MathML</a>) are relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M539">View MathML</a>, so, condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M87">View MathML</a>) is satisfied. On the other hand, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/69/mathml/M541">View MathML</a>

so, (76) is satisfied. Moreover, it is clear that (77) is satisfied. Hence, our conclusion follows from Theorem 2. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors typed, read, and approved the final manuscript.

Acknowledgements

The author would like to thank Professor D. Guo for his valuable suggestions.

References

  1. Xian, X, O’Regan, D, Agarwal, RP: Multiplicity results via topological degree for impulsive boundary value problems under a non-well ordered upper and lower solution condition. Bound. Value Probl.. 2008, Article ID 197205 (2008)

  2. Agarwal, RP, Franco, D, O’Regan, D: Singular boundary value problems for first and second order impulsive differential equations. Aequ. Math.. 69, 83–96 (2005). Publisher Full Text OpenURL

  3. 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

  4. Guo, D: Existence of positive solutions for nth-order nonlinear impulsive singular integro-differential equations in Banach spaces. Nonlinear Anal.. 68, 2727–2740 (2008). Publisher Full Text OpenURL

  5. 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

  6. Guo, D: Some fixed point theorems of expansion and compression type with applications. In: Lakshmikantham V (ed.) Nonlinear Analysis and Applications, pp. 213–221. Dekker, New York (1987)

  7. Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones, Academic Press, Boston (1988)

  8. Guo, D, Lakshmikantham, V, Liu, XZ: Nonlinear Integral Equations in Abstract Spaces, Kluwer Academic, Dordrecht (1996)

  9. Krasnoselskii, MA, Zabreiko, PP: Geometrical Methods of Nonlinear Analysis, Springer, New York (1984)