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Nonlocal nonlinear integrodifferential equations of fractional orders

Amar Debbouche1, Dumitru Baleanu23* and Ravi P Agarwal4

Author affiliations

1 Department of Mathematics, Faculty of Science, Guelma University, Guelma, Algeria

2 Department of Mathematics and Computer Science, Cankaya University, Ankara, Turkey

3 Institute of Space Sciences, P.O. Box MG-23, Magurele-Bucharest, RO, 76900, Romania

4 Department of Mathematics, Texas A&M University, Kingsville, TX, 78363, USA

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Citation and License

Boundary Value Problems 2012, 2012:78  doi:10.1186/1687-2770-2012-78

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/78


Received:11 May 2012
Accepted:9 July 2012
Published:24 July 2012

© 2012 Debbouche 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

In this paper, Schauder fixed point theorem, Gelfand-Shilov principles combined with semigroup theory are used to prove the existence of mild and strong solutions for nonlinear fractional integrodifferential equations of Sobolev type with nonlocal conditions in Banach spaces. To illustrate our abstract results, an example is given.

MSC: 35A05, 34G20, 34K05, 26A33.

Keywords:
fractional evolution equation; nonlocal condition; Schauder’s fixed point theorem; uniformly continuous semigroup

1 Introduction

We are concerned with the nonlocal nonlinear fractional problem

(1.1)

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M4">View MathML</a> is the Riemann-Liouville fractional derivative, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M6">View MathML</a> are real numbers, B and A are linear closed operators with domains contained in a Banach space X and ranges contained in a Banach space Y, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M8">View MathML</a> is a family of linear closed operators defined on dense sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M9">View MathML</a> respectively in X into X, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M11">View MathML</a> are given abstract functions. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M12">View MathML</a>.

Fractional differential equations have attracted many authors [1,6-8,21,24,25,30]. This is mostly because it efficiently describes many phenomena arising in engineering, physics, economics and science. In fact, we can find several applications in viscoelasticity, electrochemistry, electromagnetic, etc. For example, Machado [22] gave a novel method for the design of fractional order digital controllers.

Following Gelfand and Shilov [20], we define the fractional integral of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M13">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M14">View MathML</a>

also, the (Riemann-Liouville) fractional derivative of the function f of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M15">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M16">View MathML</a>

where f is an abstract continuous function on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M18">View MathML</a> is the Gamma function, see also [14,24].

The existence results to evolution equations with nonlocal conditions in a Banach space was studied first by Byszewski [9,10]; subsequently, many authors were pointed to the same field, see for instance [2-4,11-13,19,28].

Deng [15] indicated that using the nonlocal condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M19">View MathML</a> to describe, for instance, the diffusion phenomenon of a small amount of gas in a transparent tube can give a better result than using the usual local Cauchy problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M20">View MathML</a>. Let us observe also that since Deng’s papers, the function h is considered

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M21">View MathML</a>

(1.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M22">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M23">View MathML</a> are given constants and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M5">View MathML</a>.

However, among the previous research on nonlocal Cauchy problems, few authors have been concerned with mild solutions of fractional semilinear differential equations [23].

Recently, many authors have extended this work to various kinds of nonlinear evolution equations [2,3,5,11,12,18]. Balachandran and Uchiyama [3] proved the existence of mild and strong solutions of a nonlinear integrodifferential equation of Sobolev type with nonlocal condition.

In this paper, motivated by [3,13,17,19], we use Schauder fixed point theorem and the semigroup theory to investigate the existence and uniqueness of mild and strong solutions for nonlinear fractional integrodifferential equations of Sobolev type with nonlocal conditions in Banach spaces, the solutions were obtained by using Gelfand-Shilov approach in fractional calculus and are given in terms of some probability density functions such that their Laplace transforms are indicated [17].

Our paper is organized as follows. Section 2 is devoted to the review of some essential results. In Section 3, we state and prove our main results; the last section deals with giving an example to illustrate the abstract results.

2 Preliminary results

In this section, we mention some results obtained by Balachandran [3], El-Borai [19] and Pazy [26], which will be used to get our new results. Let X and Y be Banach spaces with norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M25">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M26">View MathML</a> respectively. The operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M27">View MathML</a> satisfies the following hypotheses: (H1) = B is bijective,; (H2) = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M28">View MathML</a> is compact.. The above fact and the closed graph theorem imply the boundedness of the linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M29">View MathML</a>. Further <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M30">View MathML</a> generates a uniformly continuous semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M31">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M32">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M33">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M34">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M35">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M36">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M37">View MathML</a>, see [29].

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M39">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M40">View MathML</a>.

It is supposed that (H3) = f and g are continuous in t on I, Δ respectively, and there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M41">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M43">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M45">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M46">View MathML</a>..

Definition 2.1 By a strong solution of the nonlocal Cauchy problem (1.1), (1.2), we mean a function u with values in X such that

(i) u is a continuous function in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M44">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M48">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M49">View MathML</a> exists and is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M15">View MathML</a>, and u satisfies (1.1) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M50">View MathML</a> and (1.2).

Remark 2.1 Let us take in the considered problem B is the identity, the inhomogeneous part is equal to an abstract continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M53">View MathML</a>, and the nonlocal condition is reduced to the initial condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M54">View MathML</a>i.e.

(2.1)

(2.2)

According to El-Borai [17-19], we first apply the fractional integral on both sides of (2.1) and then using (2.2), we apply the Laplace transform on the new integral equations by considering a suitable one-sided stable probability density whose Laplace transform is given. Hence we can conclude that a solution of the problem (2.1)-(2.2) can be formally represented by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M57">View MathML</a>

(2.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M58">View MathML</a> is a probability density function defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M59">View MathML</a> such that its Laplace transform is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M60">View MathML</a>

For more details, we refer to Zhou et al.[27,31], see also [14,16].

Using Gelfand-Shilov principle [20], it is suitable to rewrite (1.1), (1.2) in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M61">View MathML</a>

(2.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M62">View MathML</a> is the Gamma function.

According to [17-19], the equation (2.4) is equivalent to the integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M63">View MathML</a>

(2.5)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M64">View MathML</a>

It is assumed that there exists an operator ψ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M65">View MathML</a> given by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M66">View MathML</a>

satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M67">View MathML</a> and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M68">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M69">View MathML</a>

also (H4) = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M70">View MathML</a>.. Further we assume (H5) = There is a number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M71">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M72">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M36">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M76">View MathML</a> is a positive constant, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M77">View MathML</a>.; (H6) = The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M78">View MathML</a> are uniformly Hölder continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M44">View MathML</a> for every element h in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M80">View MathML</a>.. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M81">View MathML</a> is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M82">View MathML</a>-semigroup of operators on X such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M83">View MathML</a>, where δ is a positive constant and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M84">View MathML</a>. Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M85">View MathML</a> (see [14], p.4]).

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M86">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M87">View MathML</a>, which achieves that ψ exists on X.

3 Main results

The following is different from [3,19,26] and represents the new result.

Lemma 3.1Ifuis a continuous solution of (2.5), thenusatisfies the integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M88">View MathML</a>

(3.1)

Proof Using (2.5) and (1.2), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M89">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M90">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M91">View MathML</a>

Hence the required result. □

Definition 3.1 A continuous solution of the integral equation (3.1) is called a mild solution of the nonlocal problem (1.1), (1.2) on I.

Theorem 3.2If the assumptions (H1)∼(H4) hold and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M92">View MathML</a>, then the problem (1.1), (1.2) has a mild solution onI.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M93">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M94">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M95">View MathML</a> is a bounded closed convex subset of Z. We define a mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M96">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M97">View MathML</a>

Noting also that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M98">View MathML</a> (see [14], p.4]), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M99">View MathML</a>

We deduce that φ is continuous and maps <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M95">View MathML</a> into itself. Moreover, φ maps <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M95">View MathML</a> into a precompact subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M95">View MathML</a>. Note that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M103">View MathML</a> is precompact in X, for every fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M44">View MathML</a>. We shall show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M105">View MathML</a> is an equicontinuous family of functions. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M106">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M107">View MathML</a>

The right-hand side of the above inequality is independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M108">View MathML</a> and tends to zero as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M109">View MathML</a> as a consequence of the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M110">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M111">View MathML</a> in the uniform operator topology for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M112">View MathML</a>. It is clear that S is bounded in Z. Thus by Arzela-Ascoli’s theorem, S is precompact. Hence by the Schauder fixed point theorem, φ has a fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M95">View MathML</a> and any fixed point of φ is a mild solution of (1.1), (1.2) on I such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M114">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M44">View MathML</a>. □

Theorem 3.3Assume that

(i) Conditions (H1)∼(H6) hold,

(ii) Yis a reflexive Banach space with norm<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M116">View MathML</a>,

(iii) there are numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M117">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M118">View MathML</a>such that

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M121">View MathML</a>and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M122">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M123">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M124">View MathML</a>.

Then the problem (1.1), (1.2) has a unique strong solution onI.

Proof Applying Theorem 3.2, the problem (1.1), (1.2) has a mild solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M125">View MathML</a>. Now, we shall show that u is a unique strong solution of the considered problem on I.

According to (H6), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M126">View MathML</a> is uniformly Hölder continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M44">View MathML</a> for every element u in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M80">View MathML</a> combined with (iii), which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M129">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M130">View MathML</a> are uniformly Hölder continuous on I.

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M131">View MathML</a>

From (3.1), the solution u of the considered problem can be written in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M132">View MathML</a>

Noting that Ψ and ψ are bounded, using our assumptions, we observe that there exists a unique function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M133">View MathML</a> which satisfies the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M134">View MathML</a>

Also as in [19], p.409], we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M135">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M44">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M137">View MathML</a>. It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M138">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M44">View MathML</a>. □

4 Example

Consider the nonlinear integro-partial differential equation of fractional order

(4.1)

with nonlocal condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M141">View MathML</a>

(4.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M144">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M145">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M146">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M147">View MathML</a> is an n-dimensional multi-index, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M148">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M149">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M150">View MathML</a>

and Ω is an open subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M151">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M152">View MathML</a> be the set of all square integrable functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M151">View MathML</a>. We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M154">View MathML</a> the set of all continuous real-valued functions defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M151">View MathML</a> which have continuous partial derivatives of order less than or equal to m. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M156">View MathML</a> we denote the set of all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M157">View MathML</a> with compact supports. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M158">View MathML</a> be the completion of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M156">View MathML</a> with respect to the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M160">View MathML</a>

It is supposed that

(i) The operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M161">View MathML</a> is uniformly elliptic on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M151">View MathML</a>. In other words, all the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M163">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M164">View MathML</a> are continuous and bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M151">View MathML</a>, and there is a positive number c such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M166">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M144">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M168">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M169">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M170">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M171">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M172">View MathML</a>.

(ii) All the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M163">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M164">View MathML</a>, satisfy a uniform Hölder condition on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M151">View MathML</a>. Under these conditions, the operator E with the domain of definition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M176">View MathML</a> generates an analytic semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M31">View MathML</a> defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M152">View MathML</a>, and it is well known that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M179">View MathML</a> is dense in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M180">View MathML</a>, see [17], p.438].

Lemma 4.1The solution representation of (4.1), (4.2) can be written explicitly.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M181">View MathML</a> be a family of deterministic square matrices of order k and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M182">View MathML</a>. We assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M183">View MathML</a>

has roots which satisfy the inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M184">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M185">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M144">View MathML</a> and for any real vector σ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M187">View MathML</a>. If Θ is a matrix of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M188">View MathML</a>, then we introduce <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M189">View MathML</a>.

It is well known that there exists a fundamental matrix solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M190">View MathML</a> which satisfies the system

This fundamental matrix also satisfies the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M192">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M193">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M194">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M195">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M196">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M76">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M198">View MathML</a> are positive constants. From [13], p.58], if the nonlocal function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M199">View MathML</a> is an element in Hilbert space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M179">View MathML</a>, then we can write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M201">View MathML</a>

It can be proved that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M202">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M203">View MathML</a>M is a positive constant, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M204">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M112">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M206">View MathML</a>.

([17]) The nonlocal Cauchy problems (4.1), (4.2) are equivalent to the integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M207">View MathML</a>

where the explicit form of Q is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M208">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M209">View MathML</a>. Applying Theorem 3.2, we achieve the proof of the existence of mild solutions of the problems (4.1), (4.2). In addition, if the operators F and G satisfy the following:

(iii) There are numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M210">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M211">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M212">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M213">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M214">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M215">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M122">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/78/mathml/M144">View MathML</a>. Then applying Theorem 3.3, we deduce that (4.1), (4.2) has a unique strong solution. □

5 Conclusion

In this article, a new solution representation for Sobolev type fractional evolution equation has been proved using Deng’s nonlocal condition, a suitable explicit form of the semigroup has been discussed. Moreover, the existence result of mild solutions for nonlinear fractional integrodifferential equations of Sobolev type with nonlocal conditions in Banach spaces has been established by using Arzela-Ascoli’s theorem and Schauder fixed point theorem. Further, the uniformly Hölder continuous condition has been applied for the existence of strong solution.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

AD wrote the first draft, DB corrected and improved it and RPA prepared the final version. All authors read and approved the final draft.

Acknowledgements

The authors would like to thank the referees for their valuable comments and remarks.

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