Coupled system of a fractional order differential equations with weighted initial conditions

Abstract Here, a coupled system of nonlinear weighted Cauchy-type problem of a diffre-integral equations of fractional order will be considered. We study the existence of at least one integrable solution of this system by using Schauder fixed point Theorem. The continuous dependence of the uniqueness of the solution is proved.

In comparison with earlier results, we study the coupled system of weighted Cauchy-type problems of di reintegral equations of fractional order where t ∈ I = [ , ] and α, β ∈ ( , ) with the initial conditions such that the functions f i and g i , i = , satisfy the following assumptions: (i) f i : I × R × R → R be a function with the following properties: (a) for each t ∈ I, f i (t, ·, ·) is continuous, there exist a real function t → a(t), a ∈ L (I) and a positive constants b and b such that (ii) g i : I × I × R → R be a function with the following properties: (a) for each (t, s) ∈ I × I, g i (t, s, ·) is continuous, (b) for each u ∈ R, g i (·, ·, u) is measurable, (c) there exist a real function (t, s) → k(t, s), k ∈ L (I) and a positive constant b such that , then problem (2) will take the form which is the generalization of problem (1).

Main results . Integral representation
Proof. Let u(t) be a solution of Integrate both sides, we get Operating by I α on both sides of the last equation, we get Di erentiate both sides, we get From the initial condition, we nd that C = k , then Similarly, we can get Therefore, the solution (u, v) of system (2) can be represented by system (3).

. Existence of solution
Let L (I) be a class of Lebesgue integrable functions on the interval I, with the norm ||x|| = I |x(t)|dt.
De ne the operator T by It is clear that the xed point of the operator T is the solution of system (3).
Theorem 2.1. Assume that f i and g i satisfy the assumptions (i-iii). Then the coupled system of weighted Cauchytype problems (2) has at least one solution (u, v) ∈ L × L .
Similarly, we get Then, Therefore, for (u, v) ∈ X, we get T(u, v) ∈ X and hence TX ∈ X. Now, from the assumptions (i-a) -(ii-a), we deduce that T maps X into L × L continuously. Moreover, we have This estimation shows that f i in L (I). Now, we will use Kolmogorov compactness criterion (see [8]) to show that T is compact. So, let ℵ be a bounded subset of L . Then T (ℵ) is bounded in L (I). Now we show that (T v) h → T v in L (I) as h → , uniformly with respect to T v ∈ T ℵ. Indeed: s, v(s))ds ds dt, since f ∈ L (I) we get that I α f (.) ∈ L (I). Moreover t α− ∈ L (I). So, we have (see [10]) for a.e. t ∈ I. Therefore, T (ℵ) is relatively compact, that is, T is a compact operator, similarly T is a compact operator. Hence T is a compact operator Therefore, Schauder xed point Theorem (see [9]) implies that T has a xed point (u, v) which is a solution of the coupled system (3).
To complete the proof, let (u(t), v(t)) be a solution of Operating on both sides of the rst and second equations in (4) by I −α and I −β respectively, we get s, v(s))ds , s, u(s))ds .
Di erentiate both sides, we obtain s, u(s))ds .

. Uniqueness of the solution
For the uniqueness of the solution we have the following theorem:

Theorem 2.2. Suppose that the functions f i and g i satisfy conditions (i-b), (ii-b) and (iii) of Theorem 2.1 in addition to the following assumptions:
and Then the coupled system of weighted Cauchy-type problems (2) has a unique solution.
Proof. From assumption (5), we get Let (u , v ) and (u , v ) be two solutions of (3). Then Therefore Therefore This completes the proof.

. Continuous dependence on initial data
Now we show that the solution of the coupled system (2) is depending continuously on initial data.

Theorem 2.3. Let the assumptions of Theorem 2.2 be satis ed. Then the solution of the weighted Cauchy-type problem (2) is depending continuously on initial data,
Proof. Let (u(t), v(t)) be a solution of the couple and let ( u(t), v(t)) be a solution of the above coupled system such that t −α u(t)| t= = k and t −β v(t)| t= = k . Then Therefore where M * = max α , β and N * = max Γ( +α) , Γ( +β) .

Solution in C −α ([ , T])
Now Therefore {T(u, v)(t)} is equi-continuous. By Arzela-Ascoli Theorem then {T(u, v)(t)} is relatively compact. Therefore, the conditions of the Schauder xed point Theorem hold, which implies that T has a xed point in Qr. Then (2)