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Solutions for nth-order boundary value problems of impulsive singular nonlinear integro-differential equations in Banach spaces

Yanlai Chen1*, Tingqiu Cao1 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, 250013, People’s Republic of China

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Boundary Value Problems 2014, 2014:128  doi:10.1186/1687-2770-2014-128

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/128


Received:2 January 2014
Accepted:6 May 2014
Published:20 May 2014

© 2014 Chen et al.; licensee Springer.

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

Abstract

Not considering the Green’s function, the present study starts to construct a cone formed by a nonlinear term in Banach spaces, and through the cone creates a convex closed set. We obtain the existence of solutions for the boundary values problems of nth-order impulsive singular nonlinear integro-differential equations in Banach spaces by applying the Mönch fixed point theorem. An example is given to illustrate the main results.

MSC: 45J05, 34G20, 47H10.

Keywords:
impulsive singular integro-differential equation; Banach spaces; boundary value problem; Mönch fixed point theorem; measure of noncompactness

1 Introduction and preliminaries

By using the Schauder fixed point theorem, Guo [1] obtained the existence of solutions of initial value problems for nth-order nonlinear impulsive integro-differential equations of mixed type on an infinite interval with infinite number of impulsive times in a Banach space. In [2], by using the fixed point theorem in a cone, Chen and Qin investigated the existence of multiple solutions for a class of boundary value problems of singular nonlinear integro-differential equations of mixed type in Banach spaces. For singular differential equations in Banach spaces please see [3-9]. Generally based on Green’s function to construct a cone, but using the cone to study different nonlinear terms, we encountered difficulties, especially in infinite dimensional Banach spaces. In this paper, informed by the characteristics of the nonlinear term we construct a new cone, and through this cone create a convex closed set. On the new convex closed set, we apply the Mönch fixed point theorem to investigate the existence of solutions for the boundary value problems of nth-order impulsive singular nonlinear integro-differential equations in Banach spaces. Finally, an example of scalar second-order impulsive integro-differential equations for an infinite system is offered. Because of difficulties of compactness arising from impulsiveness and the use of nth-order integro-differential equations, a space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M1">View MathML</a> is introduced. Let E be a real Banach space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M2">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M3">View MathML</a> := {<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M4">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M5">View MathML</a> continuous at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M6">View MathML</a>, left continuous at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M7">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M8">View MathML</a> exists, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M9">View MathML</a>}. Obviously <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M3">View MathML</a> is a Banach space with norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M11">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M1">View MathML</a> := {<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M13">View MathML</a> exists and let it be continuous at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M6">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M16">View MathML</a> exist, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M9">View MathML</a>}, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M16">View MathML</a> represent the right and the left limits of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M20">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M7">View MathML</a>, respectively. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M22">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M23">View MathML</a>

So observing the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M16">View MathML</a> and taking limits as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M25">View MathML</a> in the above equality, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M26">View MathML</a> exists and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M27">View MathML</a>

Similarly, we can show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M28">View MathML</a> exists. In the same way, we get the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M29">View MathML</a>. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M30">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M9">View MathML</a>). Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M33">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M31">View MathML</a>), and, as is natural, in the following, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M35">View MathML</a> is understood as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M36">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M1">View MathML</a> is a Banach space with norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M38">View MathML</a>

Let P be a cone in E which defines a partial ordering in E by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M39">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M40">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/2014/1/128/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M41">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M42">View MathML</a>, where the smallest N is called the normal constant of P. For convenience, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M43">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M44">View MathML</a>, in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M45">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M46">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M47">View MathML</a>, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M48">View MathML</a>. We consider the following singular boundary value problem (SBVP for short) for an nth-order impulsive nonlinear integro-differential equation in E:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M49">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M50">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M51">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M52">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M9">View MathML</a>), and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M55">View MathML</a>

(2)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M56">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M57">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M58">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M59">View MathML</a> denotes the jump of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M60">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M7">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M62">View MathML</a>

and θ denotes the zero element of E.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M63">View MathML</a> is singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M64">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M66">View MathML</a> and/or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M67">View MathML</a> if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M68">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M70">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M71">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M72">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M73">View MathML</a>), and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M74">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M75">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M77">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M78">View MathML</a>).

Remark Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M79">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M80">View MathML</a> is a normal cone of E if P is a normal cone of E. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M80">View MathML</a> and P has the same normal constant N.

In the following, we assume that P is a normal cone. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M82">View MathML</a>. A map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M83">View MathML</a> is called a solution of SBVP (1) if it satisfies (1).

2 Several lemmas

To continue, let us formulate some lemmas.

Lemma 2.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M84">View MathML</a>is bounded and the elements of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M85">View MathML</a>are equicontinuous on each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M86">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M87">View MathML</a>), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M88">View MathML</a>

in whichαdenotes the Kuratowski measure of noncompactness, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M89">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53">View MathML</a>).

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53">View MathML</a>, it is easy to prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M92">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M93">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53">View MathML</a>), we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M95">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53">View MathML</a>). Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M97">View MathML</a>

(3)

Next, we check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M98">View MathML</a>

In fact, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M99">View MathML</a>, there is a division <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M100">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53">View MathML</a>) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M102">View MathML</a>

(4)

By hypothesis, the elements of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M85">View MathML</a> are equicontinuous on each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M86">View MathML</a> and there is a division:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M105">View MathML</a>

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M106">View MathML</a>

(5)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M107">View MathML</a>

(6)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M108">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M109">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M110">View MathML</a>). By virtue of (5) and (6), we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M111">View MathML</a>

(7)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M112">View MathML</a>. There is a division <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M113">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M114">View MathML</a>

(8)

Let F be the finite set of all maps <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M115">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M116">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M117">View MathML</a>). For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M118">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M119">View MathML</a>. It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M120">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M122">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M123">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M124">View MathML</a>, and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M125">View MathML</a>

(9)

Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M126">View MathML</a>

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M127">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M128">View MathML</a> is arbitrary, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M129">View MathML</a>

(10)

Finally, the conclusion follows from (3) and (10). For details of the Kuratowski measure of noncompactness, please see [10]. □

Lemma 2.2 (see [11])

Let us take a countable set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M130">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M131">View MathML</a>). For all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M132">View MathML</a>, there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M133">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M134">View MathML</a>, a.e, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M122">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M136">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M137">View MathML</a>

Lemma 2.3Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M138">View MathML</a>is bounded and equicontinuous on each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M86">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M87">View MathML</a>). Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M141">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M142">View MathML</a>

Proof By Theorem 1.2.2 of [10], the conclusion is obvious. □

Lemma 2.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M143">View MathML</a>be two countable sets. Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M144">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M145">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M146">View MathML</a>

Proof The conclusion is obvious by Lemma 6 of [12]. □

Lemma 2.5 (see [13]) (The Mönch fixed point theorem)

LetEbe a Banach space. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M147">View MathML</a>is close and convex. Assume also that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M148">View MathML</a>is continuous with the further property that for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M149">View MathML</a>, we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M150">View MathML</a>countable, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M151">View MathML</a>implies thatCis relatively compact. ThenAhas a fixed-point inD.

3 Main theorem and example

For convenience, we list the following conditions:

(H1) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M152">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M153">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M154">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M155">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M157">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M154">View MathML</a>) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M159">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M160">View MathML</a> is nonincreasing, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M161">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M163">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M164">View MathML</a> are nondecreasing. And there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M165">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M166">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M167">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M9">View MathML</a>) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M169">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M170">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M171">View MathML</a>).

(H2) There exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M172">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M173">View MathML</a> denotes the dual cone of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M80">View MathML</a>) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M175">View MathML</a>. And for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M47">View MathML</a>, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M177">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M178">View MathML</a>

(H3) There exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M179">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M180">View MathML</a>

where b, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M181">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M154">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M160">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53">View MathML</a>), φ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M185">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M154">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M187">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M189">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M167">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M87">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M192">View MathML</a> are defined as in conditions (H1) and (H2), and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M193">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M194">View MathML</a>.

(H4) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M195">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M154">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M197">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M198">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M199">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M200">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M201">View MathML</a>. There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M202">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M167">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M9">View MathML</a>) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M205">View MathML</a>

Remark Obviously, condition (H4) is satisfied automatically when E is finite dimensional.

Lemma 3.1Suppose conditions (H1), (H2) and (H3) are satisfied. ThenQdefined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M206">View MathML</a>

is a nonempty, convex and closed subset of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M1">View MathML</a>.

Proof Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M208">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M171">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M210">View MathML</a>

(11)

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M211">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M46">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M171">View MathML</a>, by (11), one can see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M214">View MathML</a>

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M215">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53">View MathML</a>) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M122">View MathML</a>.

By conditions (H1), (H2), (H3), and (11), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M218">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M219">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M220">View MathML</a> and Q is a nonempty set.

Now, we check that Q is a convex subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M1">View MathML</a>. In fact, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M222">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M223">View MathML</a>, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M224">View MathML</a>, which means <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M225">View MathML</a>. It is clear that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M226">View MathML</a>

(12)

By virtue of the characters of elements of Q and the characters of φ, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M227">View MathML</a>

(13)

In the same way,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M228">View MathML</a>

(14)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M229">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M230">View MathML</a>. Thus, Q is a convex subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M1">View MathML</a>. It is clear that Q is a closed subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M1">View MathML</a>. So the conclusion holds. □

Lemma 3.2Assume that conditions (H1), (H2) and (H3) are satisfied. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M233">View MathML</a>, where the operatorAis defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M234">View MathML</a>

(15)

Proof For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M235">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M236">View MathML</a>

(16)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M237">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M235">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M239">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M240">View MathML</a>

(17)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M241">View MathML</a>

(18)

Because of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M242">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M175">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M244">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M245">View MathML</a>, we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M246">View MathML</a>

(19)

Analogously, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M247">View MathML</a>, it is easy to see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M248">View MathML</a>

(20)

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M249">View MathML</a>

(21)

Differentiating (15) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M250">View MathML</a> times, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M251">View MathML</a>

(22)

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M252">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M253">View MathML</a>) exist and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M254">View MathML</a>

(23)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M255">View MathML</a> is understood as θ for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M256">View MathML</a>. Similarly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M257">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M253">View MathML</a>) exist. Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M259">View MathML</a>

(24)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M260">View MathML</a>

(25)

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M261">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M262">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M9">View MathML</a>), it follows from (15), (16), (17) and (18) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M265">View MathML</a>

(26)

It is clear that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M266">View MathML</a>

(27)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M267">View MathML</a>, by (22), (26) and condition (H2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M268">View MathML</a>

(28)

Now, we show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M269">View MathML</a>

(29)

By (15), (19), (21), conditions (H1) and (H3) imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M270">View MathML</a>

(30)

which implies that (29) is true. By (24), (26) to (29), the conclusion holds. □

Lemma 3.3Suppose conditions (H1) to (H4) are satisfied. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M271">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M272">View MathML</a>

in which<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M122">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M274">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M275">View MathML</a>

(31)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M276">View MathML</a>

(32)

in which<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M277">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53">View MathML</a>).

Proof In order to avoid the singularity, given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M279">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M280">View MathML</a>

By conditions (H1), (H2) and (H3), for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M122">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M282">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M283">View MathML</a>

(33)

By virtue of absolute continuity of the Lebesgue integrable function, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M284">View MathML</a>

(34)

in which, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M285">View MathML</a> denotes the Hausdorff distance between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M286">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M287">View MathML</a>. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M288">View MathML</a>

(35)

Now, we show

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M289">View MathML</a>

In fact, by Lemma 2.2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M290">View MathML</a>

(36)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M291">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M292">View MathML</a>.

On the other hand, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M293">View MathML</a>, it follows from (19) and (21) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M294">View MathML</a>

(37)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M295">View MathML</a>

(38)

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M296">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M297">View MathML</a>, by (16), (17) and (18), one can see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M298">View MathML</a>

(39)

Therefore, by condition (H4) and (36), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M299">View MathML</a>, it is easy to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M300">View MathML</a>

(40)

Since B is a bounded set of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M1">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M302">View MathML</a> is a bounded set, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M303">View MathML</a> is equicontinuous on each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M86">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M9">View MathML</a>). By Lemma 2.3, it is easy to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M306">View MathML</a>

(41)

Substituting (41) into (40), we get (31).

Similarly, we obtain (32) and our conclusion holds. □

Lemma 3.4Let conditions (H1) to (H3) be satisfied. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M83">View MathML</a>is a solution of SBVP (1), if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M235">View MathML</a>is a fixed point of the operatorAdefined by (15).

Proof First of all, by mathematical induction, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M83">View MathML</a>, Taylor’s formula with the integral remainder term holds,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M310">View MathML</a>

(42)

In fact, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M311">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M312">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M313">View MathML</a>, it is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M314">View MathML</a>

Adding these together, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M315">View MathML</a>

that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M316">View MathML</a>

(43)

This proves that (42) is true for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M311">View MathML</a>.

Suppose (42) is true for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M250">View MathML</a>, i.e., for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M319">View MathML</a>, the next formula holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M320">View MathML</a>

(44)

Now we check that (42) is also true for n. In fact, suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M83">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M322">View MathML</a>, by (43), one can see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M323">View MathML</a>

(45)

Substituting the above equation into (44), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M324">View MathML</a>

(46)

So, (42) is also true for n. By mathematical induction, (42) holds.

Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M83">View MathML</a> is a solution of SBVP (1). By (42), we can see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M326">View MathML</a>

(47)

Substituting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M327">View MathML</a>

into (47), by (15), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M328">View MathML</a>. So u is a fixed point of the operator A defined by (15) in Q.

Conversely, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M235">View MathML</a> is a fixed point of the operator A, i.e., u is a solution of the following impulsive integro-differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M330">View MathML</a>

Then, by (15), similar to (26), from the derivative of both sides of the above equation one can draw the following conclusions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M331">View MathML</a>

(48)

So, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M332">View MathML</a>

(49)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M333">View MathML</a>

(50)

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M334">View MathML</a>

(51)

It follows from (48) and (50) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M83">View MathML</a>. By (48), (49) and (50), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M336">View MathML</a>

(52)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M337">View MathML</a>

(53)

It is easy to see by (48) and (50)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M338">View MathML</a>

(54)

By (51) to (54), u is a solution of SBVP (1). □

Theorem 3.1Let conditions (H1) to (H4) be satisfied. Assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M339">View MathML</a>

(55)

Then SBVP (1) has at least a solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M83">View MathML</a>.

Proof We will use Lemma 2.5 to prove our conclusion. By (H1)-(H3), from Lemma 3.2, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M341">View MathML</a>.

We affirm that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M342">View MathML</a> is continuous. In fact, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M343">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M344">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M345">View MathML</a> (as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M346">View MathML</a>). From the continuity of f and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M347">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M9">View MathML</a>) and the definition of A, by virtue of the Lebesgue dominated convergence theorem, we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M350">View MathML</a>

(56)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M351">View MathML</a> (fixed), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M352">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M53">View MathML</a>). We also see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M354">View MathML</a> is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M355">View MathML</a> is equicontinuous on each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M86">View MathML</a>. By Lemma 2.1, it is easy to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M357">View MathML</a>

(57)

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M358">View MathML</a> is a relatively compact set in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M359">View MathML</a>. The reduction to absurdity is used to prove that A is continuous. Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M360">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M361">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M362">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M363">View MathML</a>

(58)

On the other hand, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M358">View MathML</a> is a relatively compact set in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M359">View MathML</a>, there exists a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M366">View MathML</a> which converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M367">View MathML</a>. Without loss of generality, we may assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M368">View MathML</a> itself converges to y, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M369">View MathML</a>

(59)

By virtue of (56), we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M370">View MathML</a>. Obviously, this is in contradiction to (58). Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M371">View MathML</a>

(60)

Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M372">View MathML</a> is continuous.

By Lemma 2.4, for any countable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M373">View MathML</a>, which satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M374">View MathML</a>, one can see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M375">View MathML</a>

By virtue of the character of noncompactness, it is easy to get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M376">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M377">View MathML</a>

(61)

For any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M122">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M299">View MathML</a>, by condition (H4) and Lemma 3.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M380">View MathML</a>

(62)

Similarly, by Lemma 3.3, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M381">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M382">View MathML</a>

(63)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M383">View MathML</a>. It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M384">View MathML</a>. By (61) and (62), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M299">View MathML</a>, it is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M386">View MathML</a>

(64)

Similarly, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M381">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M388">View MathML</a>

(65)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M389">View MathML</a>, by (64) and (65), we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M390">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M391">View MathML</a> is bounded and the elements of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M392">View MathML</a> are equicontinuous on each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M86">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M9">View MathML</a>). It follows from (61) and Lemma 2.1 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M395">View MathML</a>

(66)

Hence, B is a relatively compact set. By Lemma 2.5 (the Mönch fixed point theorem), A has at least a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M396">View MathML</a>, and by Lemma 3.4, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M397">View MathML</a> is the solution of SBVP (1) which means conclusion holds. □

An application of Theorem 3.1 is as follows.

Example Consider the following infinite system of scalar nonlinear second order impulsive integro-differential equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M398">View MathML</a>

(67)

Conclusion. The infinite system (67) has at least a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M399">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M400">View MathML</a>) solution, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M401">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M402">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M403">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M400">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M405">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M406">View MathML</a> with norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M407">View MathML</a>. We have the cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M408">View MathML</a>. Obviously P is a normal cone in E. Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M409">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M410">View MathML</a>), it is easy to see <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M45">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M412">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M413">View MathML</a>. The infinite system (67) can be regarded as a SBVP of the form (1) in E. In this situation,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M414">View MathML</a>

in which

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M415">View MathML</a>

(68)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M416">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M417">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M418">View MathML</a>), where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M419">View MathML</a>

(69)

Obviously, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M420">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M421">View MathML</a>

(70)

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M422">View MathML</a>

(71)

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M423">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M424">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M425">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M426">View MathML</a>

That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M427">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M428">View MathML</a>. By (71), we can see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M429">View MathML</a>

(72)

On the other hand, from (68) and (70), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M430">View MathML</a>

(73)

It is easy to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M431">View MathML</a>

(74)

It follows from (72), (73) and (74) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M432">View MathML</a>

(75)

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M433">View MathML</a>. Similarly, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M434">View MathML</a>.

Taking

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M435">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M436">View MathML</a>

by (69) and (72), condition (H1) holds.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M437">View MathML</a>, define φ by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M438">View MathML</a>. It is easy to see <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M172">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M175">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M441">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M47">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M443">View MathML</a>

Therefore, condition (H2) is satisfied. It follows from (68) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M444">View MathML</a>

(76)

Now, we check that condition (H3) is true. In fact, it is easy to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M445">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M446">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M447">View MathML</a>. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M448">View MathML</a>

there exists a sufficient large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M449">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M450">View MathML</a>

(77)

which implies that condition (H3) is satisfied.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M451">View MathML</a>

in which

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M452">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M453">View MathML</a>

(78)

For any

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M454">View MathML</a>

by (71) and (78), it is easy to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M455">View MathML</a>

(79)

Hence, the relative compactness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M456">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M457">View MathML</a> follows directly from a known result (see [14]): a bounded set X of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M457">View MathML</a> is relatively compact if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M459">View MathML</a>

That is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M460">View MathML</a>

(80)

For any

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M461">View MathML</a>

by (78), one can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M462">View MathML</a>

(81)

in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M463">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M464">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M465">View MathML</a>, by (81), it is easy to see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M466">View MathML</a>

(82)

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M467">View MathML</a>

(83)

Similarly, by (78),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M468">View MathML</a>

(84)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M469">View MathML</a>

(85)

By (80), (82), (83) and (84), it is easy to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M470">View MathML</a>

(86)

In the same way,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M471">View MathML</a>

(87)

Taking

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M472">View MathML</a>

the condition (H4) follows from (86) and (87). We can calculate and get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/128/mathml/M473">View MathML</a>

(88)

Therefore, by Theorem 3.1, the conclusion holds. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors typed, read and approved the final manuscript.

Acknowledgements

The authors would like to thank the reviewers for carefully reading this article and making valuable comments and suggestions. The project is supported by the National Natural Science Foundation of P.R. China (71272119), Social Science Foundation of Shandong Province (13CJRJ07) and Teaching and Research Projects of Qi Lu Normal University (201306).

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