Existence, Uniqueness and Stability Theorems for Certain Functional Fractional Initial Value Problem

59 Existence, Uniqueness and Stability Theorems for Certain Functional Fractional Initial Value Problem Joseph G. Abulahad Shayma A. Murad College of Education University of Dohuk, Iraq Received on: 21 / 10 / 2009 Accepted on: 11 / 4 / 2010 ABSTRACT In this paper, we deal with non-linear functional fractional differential equation with initial condition in 1 L space. We will study the existence, uniqueness and stability


Preliminaries
Let (I) L 1 be the class of Lebesgue integrable functions on the interval I=[a,b] , where 0, ab ≤<<∞ and let (.) Γ be the gamma function.Recall that the operator T is compact if it is continuous and maps bounded sets into relatively compact ones .The set of all compact operators from the subspace UX ⊂ into the Banach space X is denoted by C(U,X).Moreover, we set Provided that this integral exists, where Γ is gamma function.

Definition 2: [9]
For a function f defined on the interval [a,b], the α th Riemann-liouville fractional order derivative of f is defined by where n= ] [α +1 and ] [α denotes the integer part of α .

Remark 1: [1]
Let the assumptions of the Lemma(1),be satisfied ,then we define 0 furthermore, the equality holds everywhere on (a, b], if, in addition, f (x) is continuous on (a, b].

Existence of Solution
We begin this section by proving the equivalence of the initial value problems (3) and ( 4) with the corresponding integral equation, for proof see [1] 1 1 (1)() First, we prove that y(x) satisfies the differential equation( 3) almost every where, by definition (1) equation ( 5) can be written as Operating both sides of equation ( 6) by From lemma (2), we have  5) ,we have By lemma (2), we have (1) Now using (8), remark(1) and definition (3), we have Thus y(x) satisfies the initial condition (4).Now define the operator T as (1)() It is necessary to find a fixed point of the operator T and here we present the main results to prove some local and global existence theorem for equations ( 3)-( 4) in the space 1 (0,1) L .Let us state the following assumptions: a.e on (0,1).

L
continuously into itself, 0 (,(())) x Iftyt α φ is continuous in y, since y is an arbitrary element in r B , then T maps r B continuously into 1 (0,1) L .Now we claim that T is compact .Let Ω be a bounded subset of , and from theorem (1), T has a fixed point.This complete the proof.

Theorem 5:
Let the condition (i) and (ii) of theorem (4) be satisfied.(iii) assume that every solution , then by using the assumption (iii), the condition (A2) of theorem (2) does not hold.Therefore from theorem (2) T has a fixed point .This complete the proof.

Theorem 6: (Uniqueness of The Solution)
Let the assumption (ii) of theorem (4) be satisfied .Let the right hand side f(x, y) of the fractional differential equation satisfies the Lipschitz condition that is Now by using the Lipschitz condition, we have integrating both sides from 0 to 1 with respect to x ,we get , then (1)1 t α −< , we obtain Contradiction, therefore 21 yy = .This complete the proof.

Theorem 7: (Stability of The Solution)
Suppose that the assumption of theorem (4)be satisfied, then the solution of the initial value problems (3) and ( 4) is uniformly stable.

Conclusion
We proved the existence of the solution for certain fractional differential equation by using Roth fixed point theorem in 1 L space, then we use the Laray-Schauder theorem we proved that the fractional differential equation has at least one solution then we prove the uniqueness theorem by using the Lipschitz condition.Also we discussed the stability of the solution and we proved that the solution is uniformly stable.

5 ]
Let U be an open and bounded subset of a Banach space E, let (,) TCUE ∈.Then T has a fixed point if the following condition holds () TUU ∂⊆.Theorem 2: (Nonlinear Alternative of Laray-Schauder Type) [5]Let U be an open subset of convex set D in a Banach space E, assume 0 U ∈ and (,) TCUE ∈ prove that y(x) satisfies the initial condition(4
Therefore the solution of the initial value problem is uniformly stable.