SOLVABILITY OF COUPLED SYSTEMS OF FRACTIONAL ORDER INTEGRO-DIFFERENTIAL EQUATIONS

We present existence theorems for coupled systems of quadratic integral equations of fractional order. As applications we deduce existence theorems for two coupled systems of Cauchy problems. Also, an example illustrating the existence theorem is given.


Introduction
Systems occur in various problems of applied nature, for instance, see (Bashir Ahmad, Juan Nieto [9]-Y. Chen, H. An [11], El-Sayed and Hashem [22], Gafiychuk, Datsko, Meleshko [24], Gejji [25] and Lazarevich [27] ). Recently, X. Su [32] discussed a two-point boundary value problem for a coupled system of fractional differential equations. Gafiychuk et al. [33] analyzed the solutions of coupled nonlinear fractional reaction-diffusion equations. The solvability of the coupled systems of integral equations in reflexive Banach space proved in El-Sayed and Hashem [18]-El-Sayed and Hashem [20]. Also, a comparison between the classical method of successive approximations (Picard) method and Adomian decomposition method of coupled system of quadratic integral equations proved in El-Sayed, Hashem and Ziada [21]. Let R be the set of real numbers whereas I = [0, 1], L 1 = L 1 [I] be the space of Lebesgue integrable functions on I.
Firstly, we prove the existence of at least one continuous solution for the coupled system of quadratic functional integral equation of fractional order x(t) = a 1 (t) + g 1 (t, y(ψ 1 (t))) Quadratic integral equations are often applicable in the theory of radiative transfer, the kinetic theory of gases, the theory of neutron transport, the queuing theory and the traffic theory. Many authors studied the existence of solutions for several classes of nonlinear quadratic integral equations (see e.g. Argyros [1]-Banaś, Rzepka [8] and El-Sayed, Hashem [13]-El-Sayed, Rzepka [23]). However, in most of the above literature, the main results are realized with the help of the technique associated with the measure of noncompactness. Instead of using the technique of measure of noncompactness we use Tychonoff fixed point theorem. The existence of continuous solutions for some quadratic integral equations was proved by using Schauder-Tychonoff fixed point theorem Salem [31]. Also, the existence of solutions of the two Cauchy problems R D α x(t) = f 1 (t, y(φ 1 (t))), t ∈ (0, 1) and x(0) = 0, α ∈ (0, 1) (2) R D β y(t) = f 2 (t, x(φ 2 (t))), t ∈ (0, 1) and y(0) = 0, β ∈ (0, 1) (where R D α is the Riemann-Liouville fractional order derivative) and x(φ 2 (t))), t ∈ (0, 1), y(0) = y 0 , will be proved.
The proof of the main result will be based on the following fixed-point theorem.

Existence of Continuous Solutions
Now, the coupled system (1) will be investigated under the assumptions: (iii) There exist constants h i , l i , i = 1, 2 respectively satisfying for all t, s ∈ I and x, y ∈ R. (iv) f i : I × R → R, i = 1, 2 satisfy Carathèodory condition (i.e. measurable in t for all x : I → R and continuous in x for all t ∈ I ). (v) There exist two functions m i ∈ L 1 and positive constants b i such that Let C(I) be the class of all real functions defined and continuous on I with the norm Define the operator T by Proof. Define Also, from assumption (v) we obtain Then From the last estimate we deduce that r 1 = (M 1 + N1 k1 By a similar way as done above we have From the last estimate we can choose then, for every u = (x, y) ∈ U we have T u ∈ U and hence T U ⊂ U. It is clear that the set U is closed and convex. Assumptions (ii) and (iv) imply that T : U → C(I) × C(I) is a continuous operator. Now, for u = (x, y) ∈ U, and for each t 1 , t 2 ∈ I (without loss of generality assume that t 1 < t 2 ), we get Then |I β f 2 (t 2 , x(φ 2 (t 2 ))) − I β f 2 (t 1 , x(φ 2 (t 1 )))| Then we get i.e., As done above we can obtain Now, from the definition of the operator T, we get This means that the functions of T U are equi-continuous on I. Then by the Arzela-Ascoli Theorem Curtain and Pritchard [12] the closure of T U is compact. Since all conditions of the Schauder Fixed-point Theorem hold, then T has a fixed point in U which completes the proof.
Corollary 3.2. Let the assumptions of Theorem 2.1 be satisfied (with g i (t, x) = 1, i = 1, 2 ), then the coupled system of the fractional-order integral equations f 1 (s, y(φ 1 (s))) ds, t ∈ I, α > 0 (7) has at least one solution in X × Y . Now, letting α, β → 1, we obtain Corollary 3.3. Let the assumptions of Theorem 2.1 be satisfied (with g i (t, x) = 1, a 1 (t) = x 0 , a 2 (t) = y 0 and letting α, β → 1 ), then the coupled system of the integral equations has at least one solution in X × Y which is equivalent to the coupled system of the initial value problems (3).
4. The coupled system of the fractional order functional differential equations For the coupled system of the initial value problems of the nonlinear fractionalorder differential equations (2) we have the following theorem.
Theorem 4.1. Let the assumptions of Theorem 2.1 be satisfied (with a i (t) = 0 and g i (t, x(t)) = 1, i = 1, 2), then the coupled system of the Cauchy problems (2) has at least one solution in X × Y .
Operating with R D α on the first equation of the coupled system (8) and with R D β on the second equation of the coupled system (8) we obtain the coupled system of the initial value problems (2). So the equivalence between the coupled system of the initial value problems (2) and the coupled system of the integral equations (8) is proved and then the results follow from Theorem 2.1.