Open Access

Mild Solutions for Fractional Differential Equations with Nonlocal Conditions

Advances in Difference Equations20102010:287861

DOI: 10.1155/2010/287861

Received: 8 January 2010

Accepted: 21 January 2010

Published: 26 January 2010

Abstract

This paper is concerned with the existence and uniqueness of mild solution of the fractional differential equations with nonlocal conditions , in a Banach space , where . General existence and uniqueness theorem, which extends many previous results, are given.

1. Introduction

The fractional differential equations can be used to describe many phenomena arising in engineering, physics, economy, and science, so they have been studied extensively (see, e.g., [18] and references therein).

In this paper, we discuss the existence and uniqueness of mild solution for

(11)

where ,   ,  and generates an analytic compact semigroup of uniformly bounded linear operators on a Banach space . The term which may be interpreted as a control on the system is defined by

(12)

where (the set of all positive function continuous on ) and

(13)

The functions and are continuous.

The nonlocal condition can be applied in physics with better effect than that of the classical initial condition . There have been many significant developments in the study of nonlocal Cauchy problems (see, e.g., [6, 7, 914] and references cited there).

In this paper, motivated by [17, 915] (especially the estimating approach given by Xiao and Liang [14]), we study the semilinear fractional differential equations with nonlocal condition (1.1) in a Banach space , assuming that the nonlinear map is defined on and is defined on where , for , the domain of the fractional power of . New and general existence and uniqueness theorem, which extends many previous results, are given.

2. Preliminaries

In this paper, we set , a compact interval in . We denote by a Banach space with norm . Let be the infinitesimal generator of a compact analytic semigroup of uniformly bounded linear operators , that is, there exists such that ; and without loss of generality, we assume that . So we can define the fractional power for , as a closed linear operator on its domain with inverse , and one has the following known result.

Lemma 2.1 (see [15]).

is a Banach space with the norm for .

   for each and .

For every and , .

For every , is bounded on and there exists such that
(2.1)

Definition 2.2.

A continuous function satisfying the equation
(2.2)

for is called a mild solution of (1.1).

In this paper, we use to denote the norm of whenever for some with . We denote by the Banach space endowed with the sup norm given by

(2.3)

for .

The following well-known theorem will be used later.

Theorem 2.3 (Krasnoselkii, see [16]).

Let be a closed convex and nonempty subset of a Banach space . Let be two operators such that

whenever .

is compact and continuous,

is a contraction mapping.

Then there exists such that .

3. Main Results

We require the following assumptions.
  1. (H1)
    The function is continuous, and there exists a positive function such that
    (3.1)

    where .

     
  2. (H2)
    The function is continuous and there exists such that
    (3.2)
     

for any .

Theorem 3.1.

Let be the infinitesimal generator of an analytic compact semigroup with and . If the maps and satisfy (H1), (H2), respectively, and , then (1.1) has a mild solution for every .

Proof.

Set and choose such that
(3.3)

where .

Let .

Define
(3.4)
Let , then for we have the estimates
(3.5)

Hence we obtain .

Now we show that is continuous. Let be a sequence of such that in . Then
(3.6)
since the function is continuous on . For , using (2.1), we have
(3.7)
In view of the fact that
(3.8)
and the function is integrable on , then the Lebesgue Dominated Convergence Theorem ensures that
(3.9)
Therefore, we can see that
(3.10)

which means that is continuous.

Noting that
(3.11)

we can see that is uniformly bounded on .

Next, we prove that is equicontinuous. Let , and let be small enough, then we have
(3.12)
Using (2.1) and (H1), we have
(3.13)

It follows from the assumption of that tends to 0 as . For , using the Hölder inequality, we can see that tends to 0 as and .

For , using (2.1), (H1), and the Hölder inequality, we have
(3.14)
Moreover,
(3.15)

Using the compactness of in implies the continuity of for integrating with , we see that tends to , as . For , from the assumption of and the Hölder inequality, it is easy to see that tends to 0 as and .

Thus, , as , which does not depend on .

So, is relatively compact. By the Arzela-Ascoli Theorem, is compact.

Now, let us prove that is a contraction mapping. For and , we have
(3.16)
So, we obtain
(3.17)

We now conclude the result of the theorem by Krasnoselkii's theorem.

Now we assume the following.

(H3) There exists a positive function such that

(3.18)

the  function  belongs and

(3.19)
(H4) The function ,   satisfies
(3.20)

Theorem 3.2.

Let be the infinitesimal generator of an analytic semigroup with and . If and (H2)–(H4) hold, then (1.1) has a unique mild solution .

Proof.

Define the mapping by
(3.21)

Obviously, is well defined on .

Now take , then we have
(3.22)
Therefore, we obtain
(3.23)

and the result follows from the contraction mapping principle.

Declarations

Acknowledgment

This work is supported by the NSF of Yunnan Province (2009ZC054M).

Authors’ Affiliations

(1)
School of Mathematics, Yunnan Normal University

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© Fang Li. 2010

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