Solvability of Nonautonomous Fractional Integrodifferential Equations with Infinite Delay
© Fang Li. 2011
Received: 4 September 2010
Accepted: 29 October 2010
Published: 31 October 2010
We study the existence and uniqueness of mild solution of a class of nonlinear nonautonomous fractional integrodifferential equations with infinite delay in a Banach space . The existence of mild solution is obtained by using the theory of the measure of noncompactness and Sadovskii's fixed point theorem. An application of the abstract results is also given.
In recent years, the fractional differential equations have been proved to be good tools in the investigation of many phenomena in engineering, physics, economy, chemistry, aerodynamics, electrodynamics of complex medium and they have been studied by many researchers (cf., e.g., [13, 14, 16, 17] and references therein). Moreover, many phenomena cannot be described through classical differential equations but the integral and integrodifferential equations in abstract spaces in fields like electronic, fluid dynamics, biological models, and chemical kinetics. So many significant works on this topic have been appeared (cf., e.g., [10, 15, 18–25] and references therein).
In this paper, we study the existence of mild solution of (1.1) and obtain the existence theorem based on the measures of noncompactness without the assumptions that the nonlinearity satisfies a Lipschitz type condition and the semigroup generated by is compact (see Theorem 3.1). An example is given to show an application of the abstract results.
Next, we recall the definition of the Riemann-Liouville integral.
Definition 2.1 (see ).
See the following definition about phase space according to Hale and Kato .
Next, we consider the properties of Kuratowski's measure of noncompactness.
From the definition we can get some properties of -measure immediately, see .
Lemma 2.6 (see ).
The following lemma will be needed.
For a proof refer to .
Lemma 2.8 (see ).
We need to use the following Sadovskii's fixed point theorem here, see .
Lemma 2.10 (Sadovskii's fixed point theorem ).
where , , (see ).
According to , a mild solution of (1.1) can be defined as follows.
To our purpose the following conclusions will be needed. For the proofs refer to .
Lemma 2.12 (see ).
3. Main Results
We need the hypotheses as follows:
Now we assume that
Assume that (H1') and (H2') are satisfied, then (1.1) has a unique mild solution.
and the result follows from the contraction mapping principle.
Then generates an analytic semigroup satisfying assumptions (A1)–(A3) (see ).
Then the above equation (3.37) can be written in the abstract form as (1.1).
Suppose further that
then (3.37) has a mild solution by Theorem 3.1.
The author is grateful to the referees for their valuable suggestions. This work is supported by the NSF of Yunnan Province (no. 2009ZC054M).
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