Existence and Uniqueness of Mild Solution for Fractional Integrodifferential Equations

Abstract

We study the existence and uniqueness of mild solution of a class of nonlinear fractional integrodifferential equations , , , in a Banach space , where . New results are obtained by fixed point theorem. An application of the abstract results is also given.

1. Introduction

An integrodifferential equation is an equation which involves both integrals and derivatives of an unknown function. It arises in many fields like electronic, fluid dynamics, biological models, and chemical kinetics. A well-known example is the equations of basic electric circuit analysis. In recent years, the theory of various integrodifferential equations in Banach spaces has been studied deeply due to their important values in sciences and technologies, and many significant results have been established (see, e.g., [1–11] and references therein).

On the other hand, many phenomena in Engineering, Physics, Economy, Chemistry, Aerodynamics, and Electrodynamics of complex medium can be modeled by fractional differential equations. During the past decades, such problem attracted many researchers (see [1, 12–21] and references therein).

However, among the previous researches on the fractional differential equations, few are concerned with mild solutions of the fractional integrodifferential equations as follows:

(1.1)

where , and the fractional derivative is understood in the Caputo sense.

In this paper, motivated by [1–27] (especially the estimating approaches given in [4, 6, 10, 24, 27]), we investigate the existence and uniqueness of mild solution of (1.1) in a Banach space : generates a compact semigroup of uniformly bounded linear operators on a Banach space . The function is real valued and locally integrable on , and the nonlinear maps and are defined on into . New existence and uniqueness results are given. An example is given to show an application of the abstract results.

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 semigroup of uniformly bounded linear operators. Then there exists such that for .

According to [22, 23], a mild solution of (1.1) can be defined as follows.

Definition 2.1.

A continuous function satisfying the equation

(2.1)

for is called a mild solution of (1.1), where

(2.2)

and is a probability density function defined on such that its Laplace transform is given by

(2.3)

Remark 2.2.

Noting that (cf., [23]), we can see that

(2.4)

In this paper, we use to denote the norm of whenever for some with . denotes the Banach space of all continuous functions endowed with the sup-norm given by for . Set .

The following well-known theorem will be used later.

Theorem 2.3 (Krasnosel'skii).

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

• (i) whenever ,

• (ii) is compact and continuous,

• (iii) is a contraction mapping.

Then there exists such that .

3. Main Results

We will require the following assumptions.

(H1) The function is continuous, and there exists such that

(3.1)

(H2) The function , , satisfies

(3.2)

Theorem 3.1.

Let be the infinitesimal generator of a strongly continuous semigroup with , . If the maps and satisfy (H1), satisfies (H2), and

(3.3)

then (1.1) has a unique mild solution for every .

Proof.

Define the mapping by

(3.4)

Set , .

Choose such that

(3.5)

Let be the nonempty closed and convex set given by

(3.6)

Then for , we have

(3.7)

Noting that

(3.8)

we obtain

(3.9)

for . Hence .

Let and be two elements in . Then

(3.10)

So

(3.11)

The conclusion follows by the contraction mapping principle.

We assume the following.

(H3) The function is continuous, and there exists a positive function () such that

(3.12)

and set

Let be the infinitesimal generator of a compact semigroup of uniformly bounded linear operators. Then there exists a constant such that for .

Theorem 3.2.

If the maps and satisfy (H1), (H3), respectively, and

(3.13)

then (1.1) has a mild solution for every .

Proof.

Define

(3.14)

Choose such that

(3.15)

where .

Let be the closed convex and nonempty subset of the space .

Letting , we have

(3.16)

Set .

According to the Hölder inequality, (H1), and (3.8), for , we have

(3.17)

Thus, .

For and , using (H1), we obtain

(3.18)

So, we know that is a contraction mapping.

Set .

Fix . For , set

(3.19)

Since is compact for each , the sets are relatively compact in for each , . Furthermore,

(3.20)

which implies that is relatively compact in .

Next, we prove that is equicontinuous.

For , we have

(3.21)

By (H3), we get

(3.22)

In view of the assumption of , we see that tends to 0 as , and one

(3.23)

Clearly, the last term tends to as . Hence as , and

(3.24)

The right-hand side of (3.24) tends to as as a consequence of the continuity of in the uniform operator topology for by the compactness of . So as . Thus, , as , which is independent of . Therefore is compact by the Arzela-Ascoli theorem.

Next we show that is continuous.

Let be a sequence of such that in . By the continuity of on , we have

(3.25)

Noting the continuity of , we get

(3.26)

Thus, we have

(3.27)

So is continuous.

By Krasnosel'skii's theorem, we have the conclusion of the theorem.

Remark 3.3.

In Theorem 3.2, if we furthermore suppose that the hypothesis

(H4)

(3.28)

holds, then we can obtain the uniqueness of the mild solution in Theorem 3.2.

Actually, from what we have just proved, (1.1) has a mild solution and

(3.29)

Let be another mild solution of (1.1). Then

(3.30)

which implies by Gronwall's inequality that (1.1) has a unique mild solution .

Example 3.4.

Let . Define

(3.31)

Then generates a compact, analytic semigroup of uniformly bounded linear operators.

Let , , and let , be positive constants. We set

(3.32)

, and .

It is not hard to see that and satisfy (H1), (H3), respectively, and if

(3.33)

then (1.1) has a unique mild solution by Theorem 3.2 and Remark 3.3.