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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 124954, 7 pages
Remark on Existence and Uniqueness of Solutions for a Coupled System of Multiterm Nonlinear Fractional Differential Equations
School of Mathematics and Statistics Science, Ludong University, Yantai, Shandong 264025, China
Received 29 July 2013; Revised 27 November 2013; Accepted 5 December 2013
Academic Editor: Yong Hong Wu
Copyright © 2013 Huichao Zou and Yonghong Fan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The aim of this paper is to extend the work of Sun et al. (2012) to a more general case for a wider range of function classes of and . Our results include the case of the previous work, which are essential improvement of the work of Sun et al. (2012), especially.
Fractional calculus can give a more vivid and accurate description of problems in various fields of sciences than the traditional calculus [1–3]. Recently many complicated dynamic phenomena were modeled by fractional order calculus system and have received more and more attention; see [4–16].
In recent work , Sun et al. studied the existence and uniqueness of solutions for a coupled system of multiterm nonlinear fractional differential equations with an initial value condition where , , , , for and , for , for any , is the standard Riemann-Liouville derivative, and are given functions. In order to obtain the existence and uniqueness of solutions of (1), the following growth conditions are introduced in .(H1)There exist two nonnegative functions such that where , , for .(H2)The functions and satisfy where , for .
However, there are many functions which cannot satisfy conditions (H1) and (H2); for example, Hence the results of  are limited only to a small subset of functions which satisfy (H1) and (H2). This paper thus aims to extend the work of Sun et al.  to a more general case with more general conditions on and . Our major contributions of this paper include three aspects.(1)We extend the function classes to more general case; that is, the power growth assumptions (H1) and (H2) are replaced by a very general assumption where the functions and are only required to be nondecreasing function classes (see (A1)), which implies that the function classes are extended to more general case and also include the case of  as a special case. In mathematics and applied science, this generalization is important and interesting.(2)In , the weight functions considered constants . But in physics, the influence of weight functions for the whole system is important, so in this work, we improve the weight functions to general Lebesgue integral functions , which is also an essential improvement.(3)In this paper, the nonlinearities and are allowed to be exponential growth. However, in , the nonlinearities and are only allowed to be power growth. It is known that in most cases exponential growth is faster than power growth. From this aspect, this is also a major contribution of this paper.
The remaining part of the paper is organized as follows. In Section 2, some preliminary results including definitions, notations, and lemmas are given. Section 3 presents the main results and the proof of the results. In addition, an example is given to illustrate the application of the main results.
2. Preliminaries and Lemmas
Definition 2 (see [1–3]). The Riemann-Liouville fractional derivative of order of a function : is given by where , denotes the integer part of the number , and , provided that the right-hand side is defined on .
Lemma 3 (see ). Assume that with a fractional derivative of order ; then where , , .
Lemma 4 (see ). Suppose that . Then the initial value problem has a unique solution where for and for .
Let and let be the space of all continuous functions defined on . We define the space endowed with the norm , where Then is a Banach space with norm .
3. Main Result
Theorem 5. Let be continuous. Assume that(A1)there exist nonnegative functions and nonnegative nondecreasing functions with respect to each variable , , such that (A2)there exists a constant such that where Then the coupled system (1) has a solution.
Proof. Define a closed ball of Banach space
We will prove that . In fact, for any , by (A1), we have
Thus it follows from (18) and (A2) that
In the same way, we also have Consequently, and , and then for any ; that is, .
By , we know that the operator is completely continuous. Therefore, the Schauder fixed point theorem implies that coupled system (1) has a solution in . The proof is completed.
From Theorem 5, we easily obtain the following corollaries.
Corollary 6. Let be continuous. Assume that(A1)there exist nonnegative functions and nonnegative nondecreasing functions with respect to each variable , , such that (A2)there exists a positive constant such that where Then the coupled system (1) has a solution.
Corollary 7. Let be continuous. Assume thatthere exist nonnegative functions such that Then the coupled system (1) has a solution.
Proof. In fact, let us choose , where and construct a closed ball of Banach space The rest of proof is similar to Theorem 5.
Remark 8. The condition (A1) is weaker than (H1) and (H2). Clearly, and include and , as special cases. Moreover (A1) also includes the case or/and , but (H1) and (H2) do not be allowed.
In the following, we focus on the uniqueness of the solution of the system (1).
Theorem 10. Let be continuous. Assume that(B1)there exist nonnegative functions and nonnegative nondecreasing functions with respect to each variable , , such that (B2)for any , and , where Then coupled system (1) has a unique solution.
Proof. We prove that the operator is contraction. To do this, let , ; we have
Thus it follows from (30) and (B2) that
Similarly, we can get
Hence, for the Euclidean distance on , we get Thus is a contraction since .
By Banach contraction principle, has a unique fixed point, which is a solution of the coupled system (1). The proof is completed.
An Example. Consider the existence of solutions for the following coupled system of multiterm nonlinear fractional differential equations: where .
Let and choose Then consequently, (A1) holds.
In the following, we check the condition (A1). Since take ; we have which implies that (A2) is satisfied. Hence, by Theorem 5, the coupled system of fractional differential equation (34) has a solution.
Remark 11. In the coupled system of fractional differential equation (34), the nonlinear function involves exponential growth, but the results of  are only allowed to be power growth; that is, (34) cannot be solved by using the results of . So the results obtained in this paper give a significant improvement of the previous work in .
The authors would like to express their sincere gratitude to the anonymous reviewers and academic editor for a number of valuable comments and suggestions. The authors were supported financially by the Natural Science Foundation of Shandong Province of China (ZR2010AM022).
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