Mild Solutions for Fractional Differential Equations with Nonlocal Conditions
© Fang Li. 2010
Received: 8 January 2010
Accepted: 21 January 2010
Published: 26 January 2010
The fractional differential equations can be used to describe many phenomena arising in engineering, physics, economy, and science, so they have been studied extensively (see, e.g., [1–8] and references therein).
In this paper, we discuss the existence and uniqueness of mild solution for
The nonlocal condition can be applied in physics with better effect than that of the classical initial condition . There have been many significant developments in the study of nonlocal Cauchy problems (see, e.g., [6, 7, 9–14] and references cited there).
In this paper, motivated by [1–7, 9–15] (especially the estimating approach given by Xiao and Liang ), we study the semilinear fractional differential equations with nonlocal condition (1.1) in a Banach space , assuming that the nonlinear map is defined on and is defined on where , for , the domain of the fractional power of . New and general existence and uniqueness theorem, which extends many previous results, are given.
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 analytic semigroup of uniformly bounded linear operators , that is, there exists such that ; and without loss of generality, we assume that . So we can define the fractional power for , as a closed linear operator on its domain with inverse , and one has the following known result.
Lemma 2.1 (see ).
The following well-known theorem will be used later.
Theorem 2.3 (Krasnoselkii, see ).
3. Main Results
Using the compactness of in implies the continuity of for integrating with , we see that tends to , as . For , from the assumption of and the Hölder inequality, it is easy to see that tends to 0 as and .
We now conclude the result of the theorem by Krasnoselkii's theorem.
Now we assume the following.
and the result follows from the contraction mapping principle.
This work is supported by the NSF of Yunnan Province (2009ZC054M).
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