Mild Solutions for Fractional Differential Equations with Nonlocal Conditions
© Fang Li. 2010
Received: 8 January 2010
Accepted: 21 January 2010
Published: 26 January 2010
This paper is concerned with the existence and uniqueness of mild solution of the fractional differential equations with nonlocal conditions , in a Banach space , where . General existence and uniqueness theorem, which extends many previous results, are given.
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
where , , and generates an analytic compact semigroup of uniformly bounded linear operators on a Banach space . The term which may be interpreted as a control on the system is defined by
where (the set of all positive function continuous on ) and
The functions and are continuous.
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 ).
is a Banach space with the norm for .
for each and .
For every and , .
for is called a mild solution of (1.1).
In this paper, we use to denote the norm of whenever for some with . We denote by the Banach space endowed with the sup norm given by
The following well-known theorem will be used later.
Theorem 2.3 (Krasnoselkii, see ).
Let be a closed convex and nonempty subset of a Banach space . Let be two operators such that
is compact and continuous,
is a contraction mapping.
Then there exists such that .
3. Main Results
- (H1)The function is continuous, and there exists a positive function such that(3.1)
- (H2)The function is continuous and there exists such that(3.2)
for any .
Let be the infinitesimal generator of an analytic compact semigroup with and . If the maps and satisfy (H1), (H2), respectively, and , then (1.1) has a mild solution for every .
Hence we obtain .
which means that is continuous.
we can see that is uniformly bounded on .
It follows from the assumption of that tends to 0 as . For , using the Hölder inequality, we can see that tends to 0 as and .
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 .
Thus, , as , which does not depend on .
So, is relatively compact. By the Arzela-Ascoli Theorem, is compact.
We now conclude the result of the theorem by Krasnoselkii's theorem.
Now we assume the following.
(H3) There exists a positive function such that
the function belongs and
Let be the infinitesimal generator of an analytic semigroup with and . If and (H2)–(H4) hold, then (1.1) has a unique mild solution .
Obviously, is well defined on .
and the result follows from the contraction mapping principle.
This work is supported by the NSF of Yunnan Province (2009ZC054M).
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