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Existence and Uniqueness of Mild Solution for Fractional Integrodifferential Equations
Advances in Difference Equations volume 2010, Article number: 158789 (2010)
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 wellknown 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:
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
for is called a mild solution of (1.1), where
and is a probability density function defined on such that its Laplace transform is given by
Remark 2.2.
Noting that (cf., [23]), we can see that
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 supnorm given by for . Set .
The following wellknown 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
(H2) The function , , satisfies
Theorem 3.1.
Let be the infinitesimal generator of a strongly continuous semigroup with , . If the maps and satisfy (H1), satisfies (H2), and
then (1.1) has a unique mild solution for every .
Proof.
Define the mapping by
Set , .
Choose such that
Let be the nonempty closed and convex set given by
Then for , we have
Noting that
we obtain
for . Hence .
Let and be two elements in . Then
So
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
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
then (1.1) has a mild solution for every .
Proof.
Define
Choose such that
where .
Let be the closed convex and nonempty subset of the space .
Letting , we have
Set .
According to the Hölder inequality, (H1), and (3.8), for , we have
Thus, .
For and , using (H1), we obtain
So, we know that is a contraction mapping.
Set .
Fix . For , set
Since is compact for each , the sets are relatively compact in for each , . Furthermore,
which implies that is relatively compact in .
Next, we prove that is equicontinuous.
For , we have
By (H3), we get
In view of the assumption of , we see that tends to 0 as , and one
Clearly, the last term tends to as . Hence as , and
The righthand 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 ArzelaAscoli theorem.
Next we show that is continuous.
Let be a sequence of such that in . By the continuity of on , we have
Noting the continuity of , we get
Thus, we have
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)
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
Let be another mild solution of (1.1). Then
which implies by Gronwall's inequality that (1.1) has a unique mild solution .
Example 3.4.
Let . Define
Then generates a compact, analytic semigroup of uniformly bounded linear operators.
Let , , and let , be positive constants. We set
, and .
It is not hard to see that and satisfy (H1), (H3), respectively, and if
then (1.1) has a unique mild solution by Theorem 3.2 and Remark 3.3.
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Acknowledgments
The authors are grateful to the referees for their valuable suggestions. The first author is supported by the NSF of Yunnan Province (2009ZC054M).
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Li, F., N'Guérékata, G. Existence and Uniqueness of Mild Solution for Fractional Integrodifferential Equations. Adv Differ Equ 2010, 158789 (2010). https://doi.org/10.1155/2010/158789
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Keywords
 Banach Space
 Probability Density Function
 Electric Circuit
 Nonempty Subset
 Complex Medium