Existence and Uniqueness of Solutions for Coupled Systems of Higher-Order Nonlinear Fractional Differential Equations
© B. Ahmad and A. Alsaedi. 2010
Received: 14 May 2010
Accepted: 11 August 2010
Published: 18 August 2010
We study an initial value problem for a coupled Caputo type nonlinear fractional differential system of higher order. As a first problem, the nonhomogeneous terms in the coupled fractional differential system depend on the fractional derivatives of lower orders only. Then the nonhomogeneous terms in the fractional differential system are allowed to depend on the unknown functions together with the fractional derivative of lower orders. Our method of analysis is based on the reduction of the given system to an equivalent system of integral equations. Applying the nonlinear alternative of Leray-Schauder, we prove the existence of solutions of the fractional differential system. The uniqueness of solutions of the fractional differential system is established by using the Banach contraction principle. An illustrative example is also presented.
where are given functions, denotes the Caputo fractional derivative, , and are suitable real constants. We also discuss the case when the nonlinearities and in (1.1) are of the form and , that is, and depend on and in addition to and respectively.
Now we state a known result  which provides a relationship between (2.1) and (2.2).
For let and . Then
(iv)if there exist such that for each with and , then
In the sequel, will be understood in the sense of the limit, that is, . We also point out that the fractional order derivatives do not satisfy the relation of the form (in general).
For the sequel, we need the following results .
Applying the initial conditions (2.7) and the fact that , (2.12) transforms to (2.9).
which implies that . Also, it is easy to infer that . Hence we conclude that is a solution of (2.6) and (2.7).
3. Existence Result
For the forthcoming analysis, we introduce the following assumptions:
(A1)let be a continuously differentiable function with and on a compact subinterval of ;
(A2)let be a continuously differentiable function with and on a compact subinterval of ;
(A3)there exist nonnegative functions such that
Now we state a result which describes the nonlinear alternative of Leray and Schauder .
Let be a normed linear space, be a convex set, and be open in with Let be a continuous and compact mapping. Then either the mapping has a fixed point in or there exist and with .
where and are given by (3.3) and (3.5), respectively.
Let denote the space of all continuous functions defined on Let and be normed spaces with the sup-norm and respectively. Then, is a normed space endowed with the sup-norm defined by
Assume that are continuous functions. Then is a solution of (1.1)-(1.2) if and only if is a solution of (3.6).
Using the initial conditions of (1.1) together with and (3.9) becomes (3.3), and an application of Cauchy function yields the first equation of (3.6). The converse of the theorem follows by applying the arguments used to prove the converse of Lemma 2.7. Similarly, it can be shown that satisfying the second equation of (3.6) together with (3.5) is a solution of (1.2) and vice versa. Thus, satisfying (3.6) is a solution of the system (1.1)-(1.2) and vice versa.
and are given by (3.11). In view of ( )-( ), it follows that is well defined and continuous.
where , and let be such that .
Since the functions are uniformly continuous on , it follows from the above estimates that is an equicontinuous set. Also, it is uniformly bounded as Thus, we conclude that is a completely continuous operator.
which imply that Hence, by Theorem 3.1, has a fixed point in such that This completes the proof.
subject to the initial conditions given by (1.1)-(1.2), where are given functions.
In order to prove the existence of solution for the system (3.25), we need the following assumptions:
(let be a continuously differentiable function with and on a compact subinterval of ;
(let be a continuously differentiable function with and on a compact subinterval of ;
(there exist nonnegative functions such that
The following corollary presents the analogue form of Theorem 3.4 for the fractional differential system (3.25).
where and are given by (3.11).
The method of proof is similar to that of Theorem .
4. Uniqueness Result
To prove the uniqueness of solutions of (1.1)-(1.2), we need the following assumptions.
Then there exists a unique solution for the coupled integral equations (3.3) and (3.5).
Similarly, by using (4.3) and (4.5), it can be shown that . Thus, .
Since , , therefore is a contraction. Hence, by Banach contraction principle, has a unique fixed point in such that , which is a solution of the coupled integral equations (3.3) and (3.5). This completes the proof.
The following Corollary ensures the uniqueness of the solutions of (3.25). We do not provide the proof as it is similar to that of Theorem 4.1.
Assume that , and the following condition hold:
For each , there exist nonnegative functions , , and such that
Then there exists a unique solution for the coupled integral equation (3.27).
where . Thus, all the conditions of Theorem 3.4 are satisfied, and hence there exists a solution of (5.1).
As all the conditions of Theorem 4.1 hold, therefore the conclusion of Theorem 4.1 applies, and hence the coupled system of fractional differential equation (5.1) has a unique solution.
We have presented some existence and uniqueness results for an initial value problem of coupled fractional differential systems involving the Caputo type fractional derivative. The nonlinearities in the coupled fractional differential system depend on (i) the fractional derivatives of lower orders, (ii) the unknown functions together with the fractional derivative of lower orders. The proof of the existence results is based on the nonlinear alternative of Leray-Schauder, while the uniqueness of the solutions is proved by applying the Banach contraction principle. The present work can be extended to nonlocal coupled systems of nonlinear fractional differential equations.
The authors are grateful to the reviewers and Professor Juan J. Nieto for their suggestions. This paper was supported by Deanship of Scientific Research, King Abdulaziz University through Project no. 429/47-3.
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