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Mild Solutions for Fractional Differential Equations with Nonlocal Conditions
Advances in Difference Equations volume 2010, Article number: 287861 (2010)
Abstract
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.
1. Introduction
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F287861/MediaObjects/13662_2010_Article_1272_Equ1_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F287861/MediaObjects/13662_2010_Article_1272_Equ2_HTML.gif)
where (the set of all positive function continuous on
) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F287861/MediaObjects/13662_2010_Article_1272_Equ3_HTML.gif)
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 [14]), 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.
2. Preliminaries
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 [15]).
is a Banach space with the norm
for
.
  
for each
and
.
For every
and
,
.
For every
,
is bounded on
and there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F287861/MediaObjects/13662_2010_Article_1272_Equ4_HTML.gif)
Definition 2.2.
A continuous function satisfying the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F287861/MediaObjects/13662_2010_Article_1272_Equ5_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F287861/MediaObjects/13662_2010_Article_1272_Equ6_HTML.gif)
for .
The following well-known theorem will be used later.
Theorem 2.3 (Krasnoselkii, see [16]).
Let be a closed convex and nonempty subset of a Banach space
. Let
be two operators such that
whenever
.
is compact and continuous,
is a contraction mapping.
Then there exists such that
.
3. Main Results
We require the following assumptions.
-
(H1)
The function
is continuous, and there exists a positive function
such that
(3.1)where
.
-
(H2)
The function
is continuous and there exists
such that
(3.2)
for any .
Theorem 3.1.
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
.
Proof.
Set and choose
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F287861/MediaObjects/13662_2010_Article_1272_Equ9_HTML.gif)
where .
Let .
Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F287861/MediaObjects/13662_2010_Article_1272_Equ10_HTML.gif)
Let , then for
we have the estimates
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F287861/MediaObjects/13662_2010_Article_1272_Equ11_HTML.gif)
Hence we obtain .
Now we show that is continuous. Let
be a sequence of
such that
in
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F287861/MediaObjects/13662_2010_Article_1272_Equ12_HTML.gif)
since the function is continuous on
. For
, using (2.1), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F287861/MediaObjects/13662_2010_Article_1272_Equ13_HTML.gif)
In view of the fact that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F287861/MediaObjects/13662_2010_Article_1272_Equ14_HTML.gif)
and the function is integrable on
, then the Lebesgue Dominated Convergence Theorem ensures that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F287861/MediaObjects/13662_2010_Article_1272_Equ15_HTML.gif)
Therefore, we can see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F287861/MediaObjects/13662_2010_Article_1272_Equ16_HTML.gif)
which means that is continuous.
Noting that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F287861/MediaObjects/13662_2010_Article_1272_Equ17_HTML.gif)
we can see that is uniformly bounded on
.
Next, we prove that is equicontinuous. Let
, and let
be small enough, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F287861/MediaObjects/13662_2010_Article_1272_Equ18_HTML.gif)
Using (2.1) and (H1), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F287861/MediaObjects/13662_2010_Article_1272_Equ19_HTML.gif)
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
.
For , using (2.1), (H1), and the Hölder inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F287861/MediaObjects/13662_2010_Article_1272_Equ20_HTML.gif)
Moreover,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F287861/MediaObjects/13662_2010_Article_1272_Equ21_HTML.gif)
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.
Now, let us prove that is a contraction mapping. For
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F287861/MediaObjects/13662_2010_Article_1272_Equ22_HTML.gif)
So, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F287861/MediaObjects/13662_2010_Article_1272_Equ23_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F287861/MediaObjects/13662_2010_Article_1272_Equ24_HTML.gif)
the  function  belongs
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F287861/MediaObjects/13662_2010_Article_1272_Equ25_HTML.gif)
(H4) The function ,  
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F287861/MediaObjects/13662_2010_Article_1272_Equ26_HTML.gif)
Theorem 3.2.
Let be the infinitesimal generator of an analytic semigroup
with
and
. If
and (H2)–(H4) hold, then (1.1) has a unique mild solution
.
Proof.
Define the mapping by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F287861/MediaObjects/13662_2010_Article_1272_Equ27_HTML.gif)
Obviously, is well defined on
.
Now take , then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F287861/MediaObjects/13662_2010_Article_1272_Equ28_HTML.gif)
Therefore, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F287861/MediaObjects/13662_2010_Article_1272_Equ29_HTML.gif)
and the result follows from the contraction mapping principle.
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Acknowledgment
This work is supported by the NSF of Yunnan Province (2009ZC054M).
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Li, F. Mild Solutions for Fractional Differential Equations with Nonlocal Conditions. Adv Differ Equ 2010, 287861 (2010). https://doi.org/10.1155/2010/287861
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DOI: https://doi.org/10.1155/2010/287861