Open Access

Existence and Uniqueness of Solutions for Coupled Systems of Higher-Order Nonlinear Fractional Differential Equations

Fixed Point Theory and Applications20102010:364560

DOI: 10.1155/2010/364560

Received: 14 May 2010

Accepted: 11 August 2010

Published: 18 August 2010

Abstract

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.

1. Introduction

In recent years, the applications of fractional calculus in physics, chemistry, electrochemistry, bioengineering, biophysics, electrodynamics of complex medium, polymer rheology, aerodynamics, continuum mechanics, signal processing, electromagnetics, and so forth are highlighted in the literature. The methods of fractional calculus, when defined as a Laplace, Sumudu, or Fourier convolution product, are suitable for solving many problems in emerging biomedical research. The electrical properties of nerve cell membranes and the propagation of electrical signals are well characterized by differential equations of fractional order. The fractional derivative accurately describes natural phenomena that occur in common engineering problems such as heat transfer, electrode/electrolyte behavior, and subthreshold nerve propagation. Application of fractional derivatives to viscoelastic materials establishes, in a natural way, hereditary integrals and the power law stress-strain relationship for modeling biomaterials. A systematic presentation of the applications of fractional differential equations can be found in the book of Oldham and Spanier [1]. For more details, see the monographs of Miller and Ross [2], Samko et al. [3], Podlubny [4], and Kilbas et al. [5]. In consequence, the subject of fractional differential equations is gaining much importance and attention; see [631] and the references therein. There has also been a surge in the study of the theory of fractional differential systems. The study of coupled systems involving fractional differential equations is quite important as such systems occur in various problems of applied nature; for instance, see [3235] and the references therein. Recently, Su [36] discussed a two-point boundary value problem for a coupled system of fractional differential equations. Ahmad and Nieto [37] studied a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Ahmad and Graef [38] proved the existence of solutions for nonlocal coupled systems of nonlinear fractional differential equations. For applications and examples of fractional order systems, we refer the reader to the papers in [3947]. Motivated by the recent work on coupled systems of fractional order, we consider an initial value problem for a coupled differential system of fractional order given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ1_HTML.gif
(1.1)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ2_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq1_HTML.gif are given functions, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq2_HTML.gif denotes the Caputo fractional derivative, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq3_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq4_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq5_HTML.gif are suitable real constants. We also discuss the case when the nonlinearities https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq6_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq7_HTML.gif in (1.1) are of the form https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq8_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq9_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq10_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq11_HTML.gif depend on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq12_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq13_HTML.gif in addition to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq14_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq15_HTML.gif respectively.

2. Preliminaries

First of all, we recall some basic definitions [35].

Definition 2.1.

For a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq16_HTML.gif , the Caputo derivative of fractional order https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq17_HTML.gif is defined as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ3_HTML.gif
(2.1)

Definition 2.2.

The Riemann-Liouville fractional integral of order https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq18_HTML.gif , inversion of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq19_HTML.gif is the expression given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ4_HTML.gif
(2.2)

Definition 2.3.

The Riemann-Liouville fractional derivative of order https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq20_HTML.gif for a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq21_HTML.gif is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ5_HTML.gif
(2.3)

Now we state a known result [48] which provides a relationship between (2.1) and (2.2).

Lemma 2.4.

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq22_HTML.gif let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq23_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq24_HTML.gif . Then

(i) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq25_HTML.gif ;

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq26_HTML.gif ;

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq27_HTML.gif ;

(iv)if there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq28_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq29_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq30_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq31_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq32_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq33_HTML.gif

Remark 2.5.

In the sequel, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq34_HTML.gif will be understood in the sense of the limit, that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq35_HTML.gif . We also point out that the fractional order derivatives do not satisfy the relation of the form https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq36_HTML.gif (in general).

For the sequel, we need the following results [26].

Lemma 2.6.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq37_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq38_HTML.gif . Then, for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq39_HTML.gif and for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq40_HTML.gif , the following relations hold:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ6_HTML.gif
(2.4)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ7_HTML.gif
(2.5)

Lemma 2.7.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq41_HTML.gif be a continuously differentiable function with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq42_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq43_HTML.gif on a compact subinterval of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq44_HTML.gif . Then, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq45_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq46_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq47_HTML.gif , a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq48_HTML.gif is a solution of the initial value problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ8_HTML.gif
(2.6)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ9_HTML.gif
(2.7)
if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ10_HTML.gif
(2.8)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq49_HTML.gif is a solution of the integral equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ11_HTML.gif
(2.9)

Proof.

For the sake of completeness and later use, we outline the proof. Using (2.4) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq50_HTML.gif yields
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ12_HTML.gif
(2.10)
On the other hand, in view of (2.1), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ13_HTML.gif
(2.11)
Using (2.2) and Lemma 2.4 (ii) together with the substitution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq51_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ14_HTML.gif
(2.12)

Applying the initial conditions (2.7) and the fact that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq52_HTML.gif , (2.12) transforms to (2.9).

Conversely, suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq53_HTML.gif is a solution of (2.9). Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ15_HTML.gif
(2.13)
As https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq54_HTML.gif , it follows by Lemma 2.4 (i) and Lemma 2.6 that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ16_HTML.gif
(2.14)
Thus, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq55_HTML.gif is a solution of (2.6). Now, differentiating (2.9), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ17_HTML.gif
(2.15)
for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq56_HTML.gif Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq57_HTML.gif , the second term in the above expression becomes zero as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq58_HTML.gif . Thus, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ18_HTML.gif
(2.16)

which implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq59_HTML.gif . Also, it is easy to infer that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq60_HTML.gif . Hence we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq61_HTML.gif is a solution of (2.6) and (2.7).

3. Existence Result

For the forthcoming analysis, we introduce the following assumptions:

(A1)let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq62_HTML.gif be a continuously differentiable function with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq63_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq64_HTML.gif on a compact subinterval of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq65_HTML.gif ;

(A2)let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq66_HTML.gif be a continuously differentiable function with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq67_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq68_HTML.gif on a compact subinterval of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq69_HTML.gif ;

(A3)there exist nonnegative functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq70_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ19_HTML.gif
(3.1)

Now we state a result which describes the nonlinear alternative of Leray and Schauder [49].

Theorem 3.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq71_HTML.gif be a normed linear space, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq72_HTML.gif be a convex set, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq73_HTML.gif be open in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq74_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq75_HTML.gif Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq76_HTML.gif be a continuous and compact mapping. Then either the mapping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq77_HTML.gif has a fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq78_HTML.gif or there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq79_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq80_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq81_HTML.gif .

Lemma 3.2.

Suppose that the assumption https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq82_HTML.gif holds and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq83_HTML.gif . Then, a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq84_HTML.gif is a solution of the initial value problem (1.1) if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ20_HTML.gif
(3.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq85_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq86_HTML.gif is a solution of the integral equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ21_HTML.gif
(3.3)
and a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq87_HTML.gif is a solution of the initial value problem (1.2) if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ22_HTML.gif
(3.4)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq88_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq89_HTML.gif is a solution of the integral equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ23_HTML.gif
(3.5)
We do not provide the proof as it is similar to that of Lemma 2.7. Consider the coupled system of integral equations
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ24_HTML.gif
(3.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq90_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq91_HTML.gif are given by (3.3) and (3.5), respectively.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq92_HTML.gif denote the space of all continuous functions defined on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq93_HTML.gif Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq94_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq95_HTML.gif be normed spaces with the sup-norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq96_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq97_HTML.gif respectively. Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq98_HTML.gif is a normed space endowed with the sup-norm defined by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq99_HTML.gif

Lemma 3.3.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq100_HTML.gif are continuous functions. Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq101_HTML.gif is a solution of (1.1)-(1.2) if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq102_HTML.gif is a solution of (3.6).

Proof.

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq103_HTML.gif in (2.4), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ25_HTML.gif
(3.7)
Using the fact
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ26_HTML.gif
(3.8)
and making the substitutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq104_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq105_HTML.gif we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ27_HTML.gif
(3.9)

Using the initial conditions of (1.1) together with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq106_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq107_HTML.gif (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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq108_HTML.gif satisfying the second equation of (3.6) together with (3.5) is a solution of (1.2) and vice versa. Thus, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq109_HTML.gif satisfying (3.6) is a solution of the system (1.1)-(1.2) and vice versa.

Theorem 3.4.

Let the assumptions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq110_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq111_HTML.gif )–( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq112_HTML.gif ) hold. Then there exists a solution for the coupled integral equations (3.3) and (3.5) if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ28_HTML.gif
(3.10)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ29_HTML.gif
(3.11)

Proof.

Let us define an operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq113_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ30_HTML.gif
(3.12)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ31_HTML.gif
(3.13)

and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq114_HTML.gif are given by (3.11). In view of ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq115_HTML.gif )-( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq116_HTML.gif ), it follows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq117_HTML.gif is well defined and continuous.

Define a ball https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq118_HTML.gif in the normed space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq119_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ32_HTML.gif
(3.14)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq120_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq121_HTML.gif be such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq122_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq123_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq124_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ33_HTML.gif
(3.15)
Similarly, it can be shown that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ34_HTML.gif
(3.16)
Hence we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq125_HTML.gif This implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq126_HTML.gif Now we show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq127_HTML.gif is a completely continuous operator (continuous and compact). To do this, we first set
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ35_HTML.gif
(3.17)
For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq128_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq129_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq130_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ36_HTML.gif
(3.18)
Similarly,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ37_HTML.gif
(3.19)

Since the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq131_HTML.gif are uniformly continuous on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq132_HTML.gif , it follows from the above estimates that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq133_HTML.gif is an equicontinuous set. Also, it is uniformly bounded as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq134_HTML.gif Thus, we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq135_HTML.gif is a completely continuous operator.

Now, let us consider the eigenvalue problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ38_HTML.gif
(3.20)
Assuming that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq136_HTML.gif is a solution of (3.20) for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq137_HTML.gif , we find that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ39_HTML.gif
(3.21)
and, in a similar manner,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ40_HTML.gif
(3.22)

which imply that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq138_HTML.gif Hence, by Theorem 3.1, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq139_HTML.gif has a fixed point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq140_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq141_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq142_HTML.gif This completes the proof.

Thus, by Lemma 3.2 and Theorem 3.4, the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq143_HTML.gif of (1.1)-(1.2) is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ41_HTML.gif
(3.23)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ42_HTML.gif
(3.24)
Now we allow the nonlinear https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq144_HTML.gif in (1.1) to depend on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq145_HTML.gif in addition to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq146_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq147_HTML.gif in (1.2) to depend on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq148_HTML.gif together with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq149_HTML.gif . Precisely, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq150_HTML.gif we consider the following fractional differential system:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ43_HTML.gif
(3.25)

subject to the initial conditions given by (1.1)-(1.2), where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq151_HTML.gif are given functions.

In order to prove the existence of solution for the system (3.25), we need the following assumptions:

(let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq153_HTML.gif be a continuously differentiable function with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq154_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq155_HTML.gif on a compact subinterval of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq156_HTML.gif ;

(let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq158_HTML.gif be a continuously differentiable function with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq159_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq160_HTML.gif on a compact subinterval of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq161_HTML.gif ;

(there exist nonnegative functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq163_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ44_HTML.gif
(3.26)
In this case, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq164_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq165_HTML.gif involved in the coupled system of integral equations (3.6) modify to the following form:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ45_HTML.gif
(3.27)

The following corollary presents the analogue form of Theorem 3.4 for the fractional differential system (3.25).

Corollary 3.5.

Suppose that the assumptions ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq166_HTML.gif )–( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq167_HTML.gif ) hold. Then there exists a solution for the coupled integral equation (3.27) if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ46_HTML.gif
(3.28)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq168_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq169_HTML.gif are given by (3.11).

The method of proof is similar to that of Theorem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq170_HTML.gif .

4. Uniqueness Result

To prove the uniqueness of solutions of (1.1)-(1.2), we need the following assumptions.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq171_HTML.gif For each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq172_HTML.gif , there exist nonnegative functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq173_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq174_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ47_HTML.gif
(4.1)

Theorem 4.1.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq175_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq176_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq177_HTML.gif hold. Furthermore,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ48_HTML.gif
(4.2)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ49_HTML.gif
(4.3)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ50_HTML.gif
(4.4)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ51_HTML.gif
(4.5)

Then there exists a unique solution for the coupled integral equations (3.3) and (3.5).

Proof.

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq178_HTML.gif , we define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ52_HTML.gif
(4.6)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ53_HTML.gif
(4.7)
As before, we define the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq179_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq180_HTML.gif which is well defined and continuous. For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq181_HTML.gif , using (4.2) and (4.4), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ54_HTML.gif
(4.8)

Similarly, by using (4.3) and (4.5), it can be shown that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq182_HTML.gif . Thus, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq183_HTML.gif .

Now, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq184_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ55_HTML.gif
(4.9)
In a similar manner, we find that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ56_HTML.gif
(4.10)

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq185_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq186_HTML.gif , therefore https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq187_HTML.gif is a contraction. Hence, by Banach contraction principle, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq188_HTML.gif has a unique fixed point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq189_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq190_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq191_HTML.gif , 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.

Corollary 4.2.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq192_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq193_HTML.gif and the following condition hold:

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq195_HTML.gif For each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq196_HTML.gif , there exist nonnegative functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq197_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq198_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq199_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq200_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ57_HTML.gif
(4.11)
Furthermore,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ58_HTML.gif
(4.12)

Then there exists a unique solution for the coupled integral equation (3.27).

5. Example

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq201_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq202_HTML.gif , we consider the following coupled system of fractional differential equations:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ59_HTML.gif
(5.1)
Here https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq203_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq204_HTML.gif Clearly, the assumptions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq205_HTML.gif are satisfied with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq206_HTML.gif In this case
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ60_HTML.gif
(5.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq207_HTML.gif . Thus, all the conditions of Theorem 3.4 are satisfied, and hence there exists a solution of (5.1).

To prove the uniqueness of solutions of (5.1), we just need to verify the assumption https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq208_HTML.gif . With https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq209_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_IEq210_HTML.gif , we find that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F364560/MediaObjects/13663_2010_Article_1267_Equ61_HTML.gif
(5.3)

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.

6. Conclusions

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.

Declarations

Acknowledgments

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.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Abdulaziz University

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© B. Ahmad and A. Alsaedi. 2010

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