Solutions to Fractional Differential Equations with Nonlocal Initial Condition in Banach Spaces
Advances in Difference Equations volume 2010, Article number: 340349 (2010)
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.
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 :
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
Let be a bounded subset of a Banach space . The Kuratowski measure of noncompactness of is defined as
Clearly, . For details on properties of the measure, the reader is referred to .
The fractional integral of order with the lower limit for a function is defined as
where is the gamma function.
Caputo's derivative of order with the lower limit for a function can be written as
Caputo's derivative of a constant is equal to .
Lemma (see ).
Let . Then we have
Lemma (see ).
Let and . Then
If is bounded and equicontinuous, then
where , .
A direct computation shows
3. Main Results
, and there exist such that for and each
For any and , is relatively compact in , where and
If holds, then the problem (1.1) is equivalent to the following equation:
By Lemma 2.6 and (1.1), we have
Conversely, if is a solution of (3.2), then for every , according to Remark 2.4 and Lemma 2.5, we have
It is obvious that This completes the proof.
If (H1) and (H2) hold, then the initial value problem (1.1) has at least one solution.
Define operator , by
Clearly, the fixed points of the operator are solutions of problem (1.1).
It is obvious that is closed, bounded, and convex.
We prove that is continuous.
Then and For each ,
It is clear that
It follows from (3.11) and the dominated convergence theorem that
We prove that .
Let . Then for each , we have
We prove that is equicontinuous.
Let , and . We deduce that
As , the right-hand side of the above inequality tends to zero.
We prove that is relatively compact.
Let be arbitrarily given. Using the formula
for and , we obtain
It follows from (3.16) that This, together with Lemma 2.7, yields that
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 .
Besides the hypotheses of Theorem 3.2, we suppose that there exists a constant such that
Then, the solution of (1.1) is unique in .
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
Therefore, we get
By (3.18), we obtain So, the two solutions are identical in .
with the norm Consider the following nonlocal Cauchy problem for the following fractional differential equation in :
Problem (4.2) has only one solution on
Then it is clear that
So, is satisfied.
In the same way as in Example in , we can prove that is relatively compact in .
By a direct computation, we get
Hence, condition is also satisfied.
Moreover, we have
Therefore, . Thus, our conclusion follows from Theorem 3.3.
Abbas S, Benchohra M: Darboux problem for perturbed partial differential equations of fractional order with finite delay. Nonlinear Analysis: Hybrid Systems 2009,3(4):597-604. 10.1016/j.nahs.2009.05.001
Henderson J, Ouahab A: Fractional functional differential inclusions with finite delay. Nonlinear Analysis: Theory, Methods & Applications 2009,70(5):2091-2105. 10.1016/j.na.2008.02.111
Lakshmikantham V: Theory of fractional functional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008,69(10):3337-3343. 10.1016/j.na.2007.09.025
Lakshmikantham V, Leela S: Nagumo-type uniqueness result for fractional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2009,71(7-8):2886-2889. 10.1016/j.na.2009.01.169
Mophou GM, N'Guérékata GM: Existence of the mild solution for some fractional differential equations with nonlocal conditions. Semigroup Forum 2009,79(2):315-322. 10.1007/s00233-008-9117-x
Zhu X-X: A Cauchy problem for abstract fractional differential equations with infinite delay. Communications in Mathematical Analysis 2009,6(1):94-100.
Kilbas AA, Srivastava HM, Trujjllo JJ: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam, The Netherlands; 2006.
Podlubny I: Fractional Differential Equations. Academic Press, San Diego, Calif, USA; 1993.
Guo DJ, Lakshmikantham V, Liu XZ: Nonlinear Integral Equations in Abstract Spaces. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1996.
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