Existence and Uniqueness of Mild Solution for Fractional Integrodifferential Equations
© Fang Li and Gaston M. N'Guérékata. 2010
Received: 1 April 2010
Accepted: 17 June 2010
Published: 13 July 2010
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).
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.
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 .
The following well-known theorem will be used later.
Theorem 2.3 (Krasnosel'skii).
3. Main Results
We will require the following assumptions.
The conclusion follows by the contraction mapping principle.
We assume the following.
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.
By Krasnosel'skii's theorem, we have the conclusion of the theorem.
In Theorem 3.2, if we furthermore suppose that the hypothesis
holds, then we can obtain the uniqueness of the mild solution in Theorem 3.2.
then (1.1) has a unique mild solution by Theorem 3.2 and Remark 3.3.
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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