- Research Article
- Open Access
Nonlocal Cauchy Problem for Nonautonomous Fractional Evolution Equations
© Fei Xiao. 2011
- Received: 28 November 2010
- Accepted: 29 January 2011
- Published: 24 February 2011
- Probability Density Function
- Fractional Order
- Fixed Point Theorem
- Mild Solution
- Fractional Differential Equation
During the past decades, the fractional differential equations have been proved to be valuable tools in the investigation of many phenomena in engineering and physics; they attracted many researchers (cf., e.g., [1–9]). On the other hand, the autonomous and nonautonomous evolution equations and related topics were studied in, for example, [6, 7, 10–20], and the nonlocal Cauchy problem was considered in, for example, [2, 5, 18, 21–26].
where , , ().
We study the existence of mild solution of (1.1) and obtain the existence theorem based on the measures of noncompactness. An example is given to show an application of the abstract results.
From the definition, we can get some properties of -measure immediately, see ().
Lemma 2.3 (see ).
The following lemma will be needed.
Lemma 2.4 (see ).
Lemma 2.5 (see ).
We need to use the following Sadovskii's fixed point theorem.
Definition 2.6 (see ).
Lemma 2.7 (Sadovskii's fixed point theorem ).
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 2.9 (see ).
Then we proceed in five steps.
which contradicts (B4).
Our next result is based on the Banach's fixed point theorem.
Assume that (G1), (G2) are satisfied, then (1.1) has a unique mild solution.
Then (4.1) has a mild solution by Theorem 3.1.
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