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Existence of mild solutions for fractional evolution equations with nonlocal conditions

He Yang

Author Affiliations

Department of Mathematics, Northwest Normal University, Lanzhou, 730070, People’s Republic of China

Boundary Value Problems 2012, 2012:113  doi:10.1186/1687-2770-2012-113

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/113


Received:8 August 2012
Accepted:28 September 2012
Published:17 October 2012

© 2012 Yang; licensee Springer

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

Abstract

This paper deals with the existence and uniqueness of mild solutions for a class of fractional evolution equations with nonlocal initial conditions. We present some local growth conditions on a nonlinear part and a nonlocal term to guarantee the existence theorems. An example is given to illustrate the applicability of our results.

MSC: 34A12, 35F25.

Keywords:
fractional evolution equations; nonlocal initial conditions; existence; uniqueness

1 Introduction

The differential equations involving fractional derivatives in time have recently been proved to be valuable tools in the modeling of many phenomena in various fields of engineering and science. Indeed, we can find numerous applications in electrochemistry, control, porous media, electromagnetic processing etc. (see [1-5]). Hence, the research on fractional differential equations has become an object of extensive study during recent years; see [6-11] and references therein.

On the other hand, the nonlocal initial condition, in many cases, has much better effect in applications than the traditional initial condition. As remarked by Byszewski and Lakshmikantham (see [12,13]), the nonlocal initial value problems can be more useful than the standard initial value problems to describe many physical phenomena.

Let X be a Banach space with norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M1">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M2">View MathML</a> be a constant. Consider the existence and uniqueness of mild solutions of fractional evolution equation with nonlocal condition in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M3">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M4">View MathML</a> is the Caputo fractional derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M5">View MathML</a>, the linear operator −A is the infinitesimal generator of an analytic semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M6">View MathML</a> in X, the functions f, h and g will be specified later. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M7">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M9">View MathML</a>. Throughout this paper, we always assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M10">View MathML</a>.

In some existing articles, the fractional differential equations with nonlocal initial conditions were treated under the hypothesis that the nonlocal term is completely continuous or global Lipschitz continuous. It is obvious that these conditions are not easy to verify in many cases. To make the things more applicable, in [6] the authors studied the existence and uniqueness of mild solutions of Eq. (1) under the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M11">View MathML</a>. In their main results, they did not assume the complete continuity of the nonlocal term, but they needed the following assumptions:

(F1) there exist a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M13">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M14">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M15">View MathML</a> and almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M16">View MathML</a>;

(F2) there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M17">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M18">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M19">View MathML</a>;

and some other conditions.

In this paper, we will improve the conditions (F1) and (F2). We only assume that f and h satisfy local growth conditions (see assumption (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M20">View MathML</a>)) and g is local Lipschitz continuous (see assumption (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M21">View MathML</a>)). We will carry out our investigation in the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M23">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M24">View MathML</a> is the domain of the fractional power of A. Finally, an example is given to illustrate the applicability of our main results. We can see that the main results in [6] cannot be applied to our example.

The rest of this paper is organized as follows. In Section 2, some preliminaries are given on the fractional power of the generator of an analytic semigroup and on the mild solutions of Eq. (1). In Section 3, we study the existence and uniqueness of mild solutions of Eq. (1). In Section 4, we give an example to illustrate the applicability of our results.

2 Preliminaries

Let X be a Banach space with norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M1">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M26">View MathML</a> be the infinitesimal generator of an analytic semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M27">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28">View MathML</a>) of a uniformly bounded linear operator in X, that is, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M29">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M30">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28">View MathML</a>. Without loss of generality, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M32">View MathML</a>. Then for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M33">View MathML</a>, we can define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M34">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M35">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M36">View MathML</a> is injective, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M37">View MathML</a> can be defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M38">View MathML</a> with the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M39">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M40">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M41">View MathML</a>.

Lemma 1 ([14])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M37">View MathML</a>has the following properties:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M24">View MathML</a>is a Banach space with the norm<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M44">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M45">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M46">View MathML</a>for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M47">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M48">View MathML</a>for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M45">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28">View MathML</a>;

(iv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M34">View MathML</a>is a bounded linear operator onXwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M52">View MathML</a>;

(v) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M53">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M54">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M55">View MathML</a> be the Banach space of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M24">View MathML</a> endowed with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M57">View MathML</a>. Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M58">View MathML</a> the Banach space of all continuous functions from J into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M55">View MathML</a> with the supnorm given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M60">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M61">View MathML</a>. From Lemma 1(iv), since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M34">View MathML</a> is a bounded linear operator for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M33">View MathML</a>, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M64">View MathML</a> the operator norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M34">View MathML</a> in X, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M66">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28">View MathML</a>, denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M68">View MathML</a> the restriction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M27">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M55">View MathML</a>. From Lemma 1(ii) and (iii), for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M71">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M72">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M73">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M74">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M27">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28">View MathML</a>) is a strongly continuous semigroup in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M55">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M78">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28">View MathML</a>. To prove our main results, the following lemma is needed.

Lemma 2 ([15])

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M27">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28">View MathML</a>) is a compact semigroup inX, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M68">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28">View MathML</a>) is an immediately compact semigroup in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M55">View MathML</a>, and hence it is immediately norm-continuous.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M15">View MathML</a>, define two families <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M86">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M87">View MathML</a> of operators by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M88">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M89">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M90">View MathML</a> is the probability density function defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M91">View MathML</a>, which has properties <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M92">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M93">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M94">View MathML</a>

(2)

The following lemma follows from the results in [6-8,10].

Lemma 3The following properties are valid:

(i) For fixed<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28">View MathML</a>and any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M71">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M97">View MathML</a>

(ii) The operators<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M98">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M99">View MathML</a>are strongly continuous for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28">View MathML</a>.

(iii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M27">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28">View MathML</a>) is a compact semigroup inX, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M98">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M99">View MathML</a>are norm-continuous inXfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M47">View MathML</a>.

(iv) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M27">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28">View MathML</a>) is a compact semigroup inX, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M98">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M99">View MathML</a>are compact operators inXfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M47">View MathML</a>.

In this paper, we adopt the following definition of a mild solution of Eq. (1).

Definition 1 By a mild solution of Eq. (1), we mean a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M61">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M112">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M16">View MathML</a>.

To prove our main results, we also need the following two lemmas.

Lemma 4A measurable function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M114">View MathML</a>is Bochner integrable if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M115">View MathML</a>is Lebesgue integrable.

Lemma 5 (Krasnoselskii’s fixed point theorem)

LetXbe a Banach space, letBbe a bounded closed and convex subset ofXand let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M116">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M117">View MathML</a>be mappings fromBintoXsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M118">View MathML</a>for every pair<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M119">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M116">View MathML</a>is a contraction and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M117">View MathML</a>is completely continuous, then the operator equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M122">View MathML</a>has a solution onB.

Lemmas 4 and 5, which can be found in many books, are classical.

The following are the basic assumptions of this paper.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M123">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M27">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28">View MathML</a>) is a compact operator semigroup in X.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M126">View MathML</a>) There exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M127">View MathML</a> such that the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M128">View MathML</a> satisfy the following conditions:

(i) For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M71">View MathML</a>, the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M131">View MathML</a> are measurable.

(ii) For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M16">View MathML</a>, the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M133">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M134">View MathML</a> are continuous.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M20">View MathML</a>) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M16">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M137">View MathML</a>, there exist positive functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M138">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M139">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M140">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M141">View MathML</a>

and there are positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M142">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M143">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M144">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M21">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M146">View MathML</a> and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M137">View MathML</a>, there exists a positive constant L such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M148">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M149">View MathML</a>.

3 Main results

In this section, we introduce the existence and uniqueness theorems of mild solutions of Eq. (1). The discussions are based on fractional calculus and Krasnoselskii’s fixed point theorem. Our main results are as follows.

Theorem 1Suppose that the assumptions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M123">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M21">View MathML</a>) hold. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M152">View MathML</a>and the following inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M153">View MathML</a>

(3)

then Eq. (1) has at least one mild solution onJ.

Proof Define two operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M116">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M117">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M156">View MathML</a>

Obviously, u is a mild solution of Eq. (1) if and only if u is a solution of the operator equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M157">View MathML</a> on J. To prove the operator equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M157">View MathML</a> has solutions, we first show that there is a positive number r such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M159">View MathML</a> for every pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M160">View MathML</a>. If this were not the case, then for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M137">View MathML</a>, there would exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M162">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M163">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M164">View MathML</a>. Thus, from Lemma 3, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M20">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M21">View MathML</a>), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M167">View MathML</a>

Dividing on both sides by r and taking the lower limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M168">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M169">View MathML</a>

which contradicts (3). Hence, for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M159">View MathML</a> for every pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M160">View MathML</a>.

The next proof will be given in two steps.

Step 1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M116">View MathML</a> is a contraction on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M174">View MathML</a>.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M160">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M16">View MathML</a>, according to Lemma 3 and assumption (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M21">View MathML</a>), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M178">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M179">View MathML</a>. It follows from (3) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M180">View MathML</a>, hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M116">View MathML</a> is a contraction on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M174">View MathML</a>.

Step 2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M117">View MathML</a> is a completely continuous operator on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M174">View MathML</a>.

We first prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M117">View MathML</a> is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M174">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M187">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M188">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M189">View MathML</a>. Then for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M191">View MathML</a>, by assumption (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M126">View MathML</a>), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M193">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M189">View MathML</a>, and from assumption (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M20">View MathML</a>), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M196">View MathML</a>

This together with the Lebesgue dominated convergence theorem gives that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M197">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M189">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M199">View MathML</a>. This means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M117">View MathML</a> is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M174">View MathML</a>.

Next, we will show that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M202">View MathML</a> is relatively compact. It suffices to show that the family of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M202">View MathML</a> is uniformly bounded and equicontinuous, and for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M16">View MathML</a>, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M205">View MathML</a> is relatively compact.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M206">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M207">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M137">View MathML</a>, which means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M202">View MathML</a> is uniformly bounded. In what follows, we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M202">View MathML</a> is a family of equicontinuous functions. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M211">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M212">View MathML</a>

Hence, it is only necessary to consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M47">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M214">View MathML</a>, from Lemma 3 and assumption (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M20">View MathML</a>), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M216">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M217">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M218">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M219">View MathML</a>

It follows from Lemma 3 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M220">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M221">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M222">View MathML</a> independently of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M206">View MathML</a>. From the expressions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M224">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M225">View MathML</a>, it is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M226">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M227">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M221">View MathML</a> independently of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M206">View MathML</a>. Therefore, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M202">View MathML</a> is a family of equicontinuous functions.

It remains to prove that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M16">View MathML</a>, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M232">View MathML</a> is relatively compact.

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M233">View MathML</a> is relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M55">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M235">View MathML</a> be fixed. For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M237">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M206">View MathML</a>, we define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M239">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M240">View MathML</a>

Then the sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M241">View MathML</a> are relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M55">View MathML</a> since by Lemma 2, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M68">View MathML</a> is compact for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M47">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M55">View MathML</a>. Moreover, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M206">View MathML</a>, from Lemma 3 and assumption (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M20">View MathML</a>), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M248">View MathML</a>

Therefore, there are relatively compact sets arbitrarily close to the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M249">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M250">View MathML</a> and since it is compact at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M251">View MathML</a>, we have the relative compactness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M249">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M55">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M16">View MathML</a>.

Therefore, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M202">View MathML</a> is relatively compact by the Ascoli-Arzela theorem. Thus, the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M117">View MathML</a> and relative compactness of the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M202">View MathML</a> imply that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M117">View MathML</a> is a completely continuous operator. Hence, Krasnoselskii’s fixed point theorem shows that the operator equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M157">View MathML</a> has a solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M174">View MathML</a>. Therefore, Eq. (1) has at least one mild solution. The proof is completed. □

The following existence and uniqueness theorem for Eq. (1) is based on the Banach contraction principle. We will also need the following assumptions.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M261">View MathML</a>) There exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M262">View MathML</a> such that the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M263">View MathML</a> are strongly measurable.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M264">View MathML</a>) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M137">View MathML</a>, there exist functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M266">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M267">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M268">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M16">View MathML</a>.

Theorem 2Let the assumptions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M20">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M264">View MathML</a>) be satisfied. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M152">View MathML</a>and the inequalities (3) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M273">View MathML</a>

(4)

hold, then Eq. (1) has a unique mild solution.

Proof From Lemma 4 and assumption (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M261">View MathML</a>), it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M275">View MathML</a> is Bochner integrable with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M191">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M16">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M206">View MathML</a>, we define an operator Q by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M279">View MathML</a>

According to the proof of Theorem 1, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M280">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M137">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M160">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M16">View MathML</a>, from Lemma 3, assumptions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M21">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M264">View MathML</a>), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M286">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M287">View MathML</a>

which means that Q is a contraction according to (4). By applying the Banach contraction principle, we know that Q has a unique fixed point on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M174">View MathML</a>, which is the unique mild solution of Eq. (1). This completes the proof. □

4 An example

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M289">View MathML</a> equip with its natural norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M290">View MathML</a> and inner product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M291">View MathML</a>. Consider the following system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M292">View MathML</a>

(5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M2">View MathML</a> is a constant, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M294">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M295">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M296">View MathML</a> will be specified later.

Let the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M297">View MathML</a> be defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M298">View MathML</a>

Then −A generates a compact analytic semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M299">View MathML</a> of uniformly bounded linear operators and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M300">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M28">View MathML</a>. Hence, we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M302">View MathML</a>. Moreover, the eigenvalues of A are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M303">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M304">View MathML</a> and the corresponding normalized eigenvectors are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M305">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M306">View MathML</a> . The operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M307">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M308">View MathML</a>

for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M309">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M310">View MathML</a>.

Lemma 6 ([16])

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M311">View MathML</a>, thenξis absolutely continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M312">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M313">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M314">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M315">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M311">View MathML</a>. Assume that

(P1) The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M317">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M318">View MathML</a>, and the partial derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M319">View MathML</a> belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M320">View MathML</a>.

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M321">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M322">View MathML</a>, it follows from

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M323">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M324">View MathML</a>

that f and h are functions from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M325">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M326">View MathML</a> and they are continuous. Moreover, a similar computation of [17] together with Lemma 6 and assumption (P1) shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M327">View MathML</a> whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M328">View MathML</a>.

Then for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M137">View MathML</a>, we see that the assumptions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M20">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M264">View MathML</a>) are satisfied with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M332">View MathML</a>

Thus, the system (5) has at least one mild solution due to Theorem 1 provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M333">View MathML</a>. And by Theorem 2, this mild solution of the system (5) is unique on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/113/mathml/M334">View MathML</a>.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

Research was supported by the Fundamental Research Funds for the Gansu Universities.

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