- 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
We investigate the mild solutions of a nonlocal Cauchy problem for nonautonomous fractional evolution equations , , in Banach spaces, where , . New results are obtained by using Sadovskii's fixed point theorem and the Banach contraction mapping principle. An example is also given.
- 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].
In this paper, we denote that is a positive constant and assume that a family of closed linear satisfying
(A1) the domain of is dense in the Banach space and independent of ,
(A2) the operator exists in for any with and
(A3) There exists constant and such that
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.
Throughout this work, we set . We denote by a Banach space, the space of all linear and bounded operators on , and the space of all -valued continuous functions on .
where is a Beta function.
From the definition, we can get some properties of -measure immediately, see ().
Lemma 2.3 (see ).
, if .
, where denotes the closure of .
if and only if is precompact.
, where .
, for any .
for , where .
The following lemma will be needed.
Lemma 2.4 (see ).
, for .
Lemma 2.5 (see ).
We need to use the following Sadovskii's fixed point theorem.
Definition 2.6 (see ).
Let be an operator in Banach space . If is continuous and takes bounded, sets into bounded sets, and for every bounded set of with , then is said to be a condensing operator on .
Lemma 2.7 (Sadovskii's fixed point theorem ).
Let be a condensing operator on Banach space . If for a convex, closed, and bounded set of , then has a fixed point in .
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 ).
and set .
where is a nonnegative function, and ,
where , and .
Suppose that (B1)–(B4) are satisfied, and if , then (1.1) has a mild solution on .
Then we proceed in five steps.
We show that is continuous.
We show that maps bounded sets of into bounded sets in .
This means .
We show that there exists such that .
which contradicts (B4).
We show that is equicontinuous.
It follows from Lemma 2.9, (B1), and (3.20) that .
In summary, we have proven that has a fixed point . Consequently, (1.1) has at least one mild solution.
Our next result is based on the Banach's fixed point theorem.
(G1) There exists a positive function and a constant such that
(G2) There exists a constant such that the function defined by
Assume that (G1), (G2) are satisfied, then (1.1) has a unique mild solution.
By the Banach contraction mapping principle, has a unique fixed point, which is a mild solution of (1.1).
Then generates an analytic semigroup .
Then (4.1) has a mild solution by Theorem 3.1.
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