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Solutions to Fractional Differential Equations with Nonlocal Initial Condition in Banach Spaces
Advances in Difference Equations volume 2010, Article number: 340349 (2010)
Abstract
A new existence and uniqueness theorem is given for solutions to differential equations involving the Caputo fractional derivative with nonlocal initial condition in Banach spaces. An application is also given.
1. Introduction
Fractional differential equations have played a significant role in physics, mechanics, chemistry, engineering, and so forth. In recent years, there are many papers dealing with the existence of solutions to various fractional differential equations; see, for example, [1–6].
In this paper, we discuss the existence of solutions to the nonlocal Cauchy problem for the following fractional differential equations in a Banach space :
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ1_HTML.gif)
where is the standard Caputo's derivative of order
,
and
is a given
-valued function.
2. Basic Lemmas
Let be a real Banach space, and
the zero element of
. Denote by
the Banach space of all continuous functions
with norm
. Let
be the Banach space of measurable functions
which are Lebesgue integrable, equipped with the norm
. Let
,
function
is called a solution of (1.1) if it satisfies (1.1).
Recall the following defenition
Definition 2.1.
Let be a bounded subset of a Banach space
. The Kuratowski measure of noncompactness of
is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ2_HTML.gif)
Clearly, . For details on properties of the measure, the reader is referred to [2].
The fractional integral of order with the lower limit
for a function
is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ3_HTML.gif)
where is the gamma function.
Caputo's derivative of order with the lower limit
for a function
can be written as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ4_HTML.gif)
Remark.
Caputo's derivative of a constant is equal to .
Lemma (see [7]).
Let . Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ5_HTML.gif)
Lemma (see [7]).
Let and
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ6_HTML.gif)
Lemma(see [9]).
If is bounded and equicontinuous, then
(a);
(b) where
Lemma.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ7_HTML.gif)
where ,
.
Proof.
A direct computation shows
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ8_HTML.gif)
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ9_HTML.gif)
3. Main Results
, and there exist
such that
for
and each
For any and
,
is relatively compact in
, where
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ10_HTML.gif)
Lemma.
If holds, then the problem (1.1) is equivalent to the following equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ11_HTML.gif)
Proof.
By Lemma 2.6 and (1.1), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ12_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ13_HTML.gif)
So,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ14_HTML.gif)
and then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ15_HTML.gif)
Conversely, if is a solution of (3.2), then for every
, according to Remark 2.4 and Lemma 2.5, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ16_HTML.gif)
It is obvious that This completes the proof.
Theorem.
If (H1) and (H2) hold, then the initial value problem (1.1) has at least one solution.
Proof.
Define operator , by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ17_HTML.gif)
Clearly, the fixed points of the operator are solutions of problem (1.1).
It is obvious that is closed, bounded, and convex.
Step 1.
We prove that is continuous.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ18_HTML.gif)
Then and
For each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ19_HTML.gif)
It is clear that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ20_HTML.gif)
It follows from (3.11) and the dominated convergence theorem that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ21_HTML.gif)
Step 2.
We prove that .
Let . Then for each
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ22_HTML.gif)
Step 3.
We prove that is equicontinuous.
Let ,
and
. We deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ23_HTML.gif)
As , the right-hand side of the above inequality tends to zero.
Step 4.
We prove that is relatively compact.
Let be arbitrarily given. Using the formula
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ24_HTML.gif)
for and
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ25_HTML.gif)
It follows from (3.16) that This, together with Lemma 2.7, yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ26_HTML.gif)
From (3.17), we see that is relatively compact. Hence,
is completely continuous. Finally, the Schauder fixed point theorem guarantees that
has a fixed point in
.
Theorem.
Besides the hypotheses of Theorem 3.2, we suppose that there exists a constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ27_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ28_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ29_HTML.gif)
Then, the solution of (1.1) is unique in
.
Proof.
From Theorem 3.2, we know that there exists at least one solution in
. We suppose to the contrary that there exist two different solutions
and
in
. It follows from (3.8) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ30_HTML.gif)
Therefore, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ31_HTML.gif)
By (3.18), we obtain So, the two solutions are identical in
.
4. Example
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ32_HTML.gif)
with the norm Consider the following nonlocal Cauchy problem for the following fractional differential equation in
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ33_HTML.gif)
Conclusion 4.
Problem (4.2) has only one solution on
Proof.
Write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ34_HTML.gif)
Then it is clear that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ35_HTML.gif)
So, is satisfied.
In the same way as in Example in [9], we can prove that
is relatively compact in
.
By a direct computation, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ36_HTML.gif)
Hence, condition is also satisfied.
Moreover, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ37_HTML.gif)
so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ38_HTML.gif)
Clearly,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340349/MediaObjects/13662_2010_Article_1279_Equ39_HTML.gif)
Therefore, . Thus, our conclusion follows from Theorem 3.3.
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Acknowledgments
This work was supported partially by the NSF of China (10771202), the Research Fund for Shanghai Key Laboratory for Contemporary Applied Mathematics (08DZ2271900) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007035805).
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Lv, ZW., Liang, J. & Xiao, TJ. Solutions to Fractional Differential Equations with Nonlocal Initial Condition in Banach Spaces. Adv Differ Equ 2010, 340349 (2010). https://doi.org/10.1155/2010/340349
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DOI: https://doi.org/10.1155/2010/340349