Vol.08 No.03(2017), Article ID:74881,12 pages
Existence and Uniqueness for the Boundary Value Problems of Nonlinear Fractional Differential Equation
Yufeng Sun1, Zheng Zeng2, Jie Song1*
1School of Mathematics and Statistics, Shaoguan University, Shaoguan, China
2Office of Party Committee, Foshan University, Foshan, China
Copyright © 2017 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
Received: October 6, 2016; Accepted: March 21, 2017; Published: March 24, 2017
This paper studies the existence and uniqueness of solutions for a class of boundary value problems of nonlinear fractional order differential equations involving the Caputo fractional derivative by employing the Banach’s contraction principle and the Schauder’s fixed point theorem. In addition, an example is given to demonstrate the application of our main results.
Fractional Order Differential Equations, Boundary Value Problem, Caputo Fractional Derivative, Fractional Integral, Fixed Point
This paper considers the following boundary value problems of fractional order differential equations
where is the Caputo fractional derivative, is continuous function and are real constants.
Fractional order Differential equations have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Applications can be found in fields of control, porous media, eletromagnetic, etc. (see      ). There has been a significant progress in the investigation of fractional differential equations in recent years, The readers are referred to the monographs of Oldham and Spanier  , Miller and Ross  , Podlubny  , Hilfer  and the papers of Agarwal et al.  , El-Sayed     , Benchohra et al.   , Yu and Gao   , Zhang  , He  and the others references therein  -  .
Recently some basic theory for the initial value problems of fractional differential equations involving Riemann-Liouville differential operator () has been discussed by Lakshmikantham et al.    . In a series of papers (see   ), the authors considered some classes of boundary value problems for differential equations involving Riemann-Liouville and Caputo fractional derivatives of order and.
This paper generalizes the results of the papers above  and presents some existence theorems for the boundary value problems (BVP) (1.1). Two theorems are based on the Banach fixed point theorem, and the others are based on Schau- der’s fixed point theorem and Leray-Schauder type nonlinear alternative. An example is given to demonstrate the application of our main results.
Some notions and Lemmas are important in order to state our results. Denote by the Banach space of all continuous functions from J into R with the norm
Definition 2.1 (   ) The fractional order integral of the function is defined by
where is the gamma function.
Definition 2.2 (   ) For a function h given on the interval [a,b], the - th Caputo fractional-order derivative of is defined by
where and denotes the integer part of.
A solution of the problem (1.1) is defined as follows.
Definition 2.3 A function that satisfies (1.1) is called a solution of (1.1).
Lemma 2.1 (  ) Let, then the differential equation
Lemma 2.2 Let, then
In particular, when,
Proof. By (2.1), (2.2),
Lemma 2.3 (  ) The relation
is valid in following case
As a consequence of Lemmas 2.1, Lemmas 2.2 and Lemmas 2.3, the following result is useful in what follows.
Lemma 2.4 Let, , and let be continuous. A function is a solution of the fractional BVP
if and only if is a solution of the fractional integral equation
Proof. Assume satisfies (2.4), then Lemma 2.2 implies that
And the following simple calculation can be obtained by (2.4)
Hence Equation (2.5). Conversely, it is clear that if satisfies Equation (2.5), then Equations (2.4) hold.
3. Existence and Uniqueness of Solutions
In this section, Our first result is based on the Banach fixed point theorem (see  ).
Theorem 3.1 Assume that
(H1) There exists a function such that
Then the BVP (1.1) has a unique solution on J.
Proof. Transform the problem (1.1) into a fixed point problem. Consider the operator
The Banach contraction principle is used to prove that T has afixed point.
Consequently, by (3.1) T is a contraction operator. As a consequence of the Banach Fixed point theorem, T has a fixed point which is the unique solution of the problem (1.1). The proof is completed.
In Theorem 3.1, if the function is replaced by a constant L > 0, the second result follows.
Theorem 3.2 Assume that
(H2) There exists a constant L > 0 (i.e.), such that
Then the BVP (1.1) has a unique solution on J.
The third result is based on Schauder’s Fixed point theorem.
Theorem 3.3 Assume that
(H3) The function is continuous.
(H4) There exists a constant M > 0, such that
Then the BVP (1.1) has at least one solution on J.
Proof. Schauder’s Fixed point theorem is used to prove that T defined by (3.2) has a fixed point. The proof will be given in several steps.
Step 1: T is continuous.
Let be a sequence such that in. Then for each
Since f is a continuous function, it can be shown that
Step 2: T maps the bounded sets into the bounded sets in.
For any, it can be shown that there exists a positive constant such that.
In fact, , by (3.2) and (H4)
Step 3: Tmaps the bounded sets into the equicontinuous sets of.
Let, be abounded set of as above, and .
As, the right-hand side of the aboveinequality tends to zero. As a consequence of Steps 1 to 3 together with the Arzelá-Ascoli theorem, is completely continuous.
Step 4: A priori bounds.
Let, it shall be shown that the set is bounded.
Let, then for some. Thus,
By the condition (H4) and Step 2,
Thus for every,
This shows that the set is bounded. As a consequence of Schauder’s fixed point theorem, T has a fixed point which is a solution of the problem (1.1).
In Theorem 3.3, if the condition (H4) is weakened, the fourth result can be obtained, which is a more general existence result (see  ).
Theorem 3.4 Assume that (H3) and the following conditionshold.
(H5) There exist a functional and a continuous and nondecreasing, such that
(H6) There exists a number K > 0, such that
Then the BVP (1.1) has at least one solution on J.
Proof. Consider the operator T defined by (3.2), , let meets, then from (H5) and (H6),
By (H6), there exists K such that. Let, the operator is completely continuous. Through proper selection of D, there exists no such that for some.
Therefore, T is Leray-Schauder type operator (see  ), so that it has a fixed point in, which is a solution of the BVP (1.1).
4. An Example
For the boundary value problem
Hence the condition (H1) holds with. It can be checked that condition (3.2) is satisfied with. In fact,
For example, , then, ,
Then by Theorem 3.1 the boundary value problem (4.1) has a uniquesolution on for the values of.
The authors would like to thank the reviewers for their valuable suggestions and comments, which improved the completeness of the paper. Research of J. Song is funded by the High-level Talents Project of Guangdong Province Colleges and Universities (2013-178). Research of Z. Zeng is funded by the Natural Science Foundation of Guangdong Province of China (S2012010010069). These supports are greatly appreciated.
Cite this paper
Sun, Y.F., Zeng, Z. and Song, J. (2017) Existence and Uniqueness for the Boundary Value Problems of Nonlinear Fractional Differential Equation. Applied Mathematics, 8, 312-323. https://doi.org/10.4236/am.2017.83026
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