EXISTENCE AND UNIQUENESS OF MILD AND STRONG SOLUTIONS OF NONLINEAR FRACTIONAL INTEGRODIFFERENTIAL EQUATION

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we will discuss some results on the existence and uniqueness of mild and strong solution of initial value problem of fractional order subjected to non-local conditions, by using the Banach fixed point theorem and the theory of strongly continuous cosine family under Caputo sense. Furthermore, we also prove that solution of Nonlinear Fractional Volterra Integrodifferential Equations and Nonlinear Fractional Mixed Integrodifferential Equations With Nonlocal Conditions is unique. Moreover, examples demonstrate the validity of the obtained main result and we obtain the solution for an equation, and proved that this solution is unique.


INTRODUCTION
Some results on the problem of existence and uniqueness of solution of differential equations of fractional order have been discussed by some authors which can be found in [1,2,3,4].
The purpose of this paper is to discuss the existence and uniqueness of solution of differential equation of fractional order, by using the Banach fixed point theorem and the theory of strongly continuous cosine family. Now consider the fractional order non-linear differential equations with non-local conditions as follows: (1.1) ϑ α+1 (η) + β ϑ (η) = χ(η, ϑ (η), For the shake of simplicity let where β is the infinitesimal generator of a C 0 semigroup T (η), η ≥ 0, on a Banach space X and the nonlinear operators χ : Moreover, we consider the nonlinear fractional mixed Volterra -Fredholm integrodifferential , η ≥ 0, in a Banach space X and the nonlinear functions χ : Riemann-Liouville definition [5,6]: Caputo definition [5,6]: Let us denote T (η) , For shake of our convenience or for further use we list the following hypothesis.
Proof Let Φ : ϒ → ϒ be an operator defined by By our assumptions, we have where 0 < q < 1. From this it is clear that Φ is a contraction on ϒ. By the Banach fixed point theorem, the function Φ has a unique fixed point in the space ϒ and this point is the mild solution Next we prove that the problem (1.1) -(1.2) has a strong solution.
Theorem 2.2.2 As following assumptions hold (iv) χ : [η 0 , η 0 + ξ ] × X × X → X is continuous and there exist constants L i χ > 0, i = 1, 2 and L η > 0 such that Proof If all the assumptions of Theorem 2.2 are satisfied then the problem (1.1) -(1.2) has a unique mild solution belonging to ϒ which is denoted by ρ.
where I is the identity operator.

Preliminaries. Definition A continuous solution ϑ (η) of the integral equation
.

Main Results. Theorem 3.2.1 Assume that
and there exists a constant L > 0 such that there exist positive constants Ψ, H such that Then problem (1.1) -(1.2) has a unique mild solution on [η 0 , η 0 + ξ ].
By using hypotheses (H 2 ) − (H 4 ) and assumptions (ii), (iii), we have where q = M Λ + M L ξ + M L Ψξ 2 + M L H ξ 2 and hence, we obtain where 0 < q < 1. Hence the operator Φ is a contraction on ϒ. By using the Banach fixed point theorem,we observe that the function Φ has a unique fixed point in the space ϒ and this point is

CONCLUSIONS
The purpose of this paper is to discuss the existence and uniqueness of solution of differential equation of fractional order, by using the Banach fixed point theorem and the theory of strongly continuous cosine family. Moreover we also discuss the existence and uniqueness of mild and strong solution of initial value problem of fractional order subjected to non-local conditions, by using the Banach fixed point theorem and the theory of strongly continuous cosine family.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.