NEW RESULTS ON NONLOCAL FRACTIONAL VOLTERRA-FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. This paper considers nonlinear fractional mixed Volterra-Fredholm integro-differential equation with a nonlocal initial condition. We propose a fixed-point approach to investigate the existence and uniqueness of solutions. Results of this paper are based on nonstandard assumptions and hypotheses and provide a supplementary result concerning the regularity of solutions.


INTRODUCTION
The theory of fractional differential and partial differential have become an important area of investigation in the past two decades because of their applications to various problems arising in communications, control technology, impact mechanics, electrical engineering, medicine, and biology. Fractional systems has the advantages to describe the processes associate with previous history. Due to its tremendous applications many researchers are working in this field. Initially it was developed by several authors in their monograph [21,22,26,27,28,29].
In the fractional calculus the various integral inequalities plays an important role in the study of qualitative and quantitative properties of solution of differential and integral equations.
The work on nonlocal Cauchy problem was first initiated by Byszewski [4] in Banach spaces.
He proved the existence and uniqueness of mild, strong and classical solutions of nonlocal Cauchy problem using the method of semigroups and the Banach fixed point theorem. The nonlocal condition can be applied to physics with better effect than the classical condition, since the nonlocal condition is usually more precise for physical measurements than the classical one [1,5,6,7,8,21,23].
Volterra-Fredholm integro-differential equations arise in many engineering and scientific disciplines, often as approximation to partial differential equations, which represent much of the continuum phenomena. Many forms of these equations are possible. Some of the applications are unsteady aerodynamics and aero elastic phenomena, visco elasticity, visco elastic panel in super sonic gas flow, fluid dynamics, electro dynamics of complex medium, many models of population growth [22].
In recent years, many authors focus on the development of techniques for discussing the solutions of fractional differential and integro-differential equations. For instance, we can remember the following works: Ibrahim and Momani [19] studied the existence and uniqueness of solutions of a class of fractional order differential equations, Karthikeyan and Trujillo [20] proved existence and uniqueness of solutions for fractional integro-differential equations with boundary value conditions, Bahuguna and Dabas [2] applied the method of lines to establish the existence and uniqueness of a strong solution for the partial integro-differential equations, Matar [23] deliberated the existence of solutions for nonlocal fractional semilinear integro-differential equations in Banach spaces via Banach fixed point theorem. Momani et al. [25] proved the Local and global uniqueness result by using Bihari's inequality for the fractional integro-differential equation, Z(t, s, u(s))ds, 0 < α ≤ 1 with the initial condition u(0) = u 0 , Ahmad and Sivasundaram [1] studied some existence and uniqueness results in a Banach space for the fractional integro-differential equation, with the nonlocal condition Recently, many authors [3,11,25,32,33] obtained the result on existence and uniqueness of solutions for fractional integro-differential equations with nonlocal conditions using the fixed point theorem of Banach space couple with contraction mapping principle.
Inspired by the discussion, we consider Caputo fractional Volterra-Fredholm integrodifferential equation: with the nonlocal condition −→ X are appropriate functions satisfying some conditions which will be stated later.
The main objective of the present paper is to study the new existence and uniqueness results by means of Krasnoselskii's fixed point and the Banach fixed point theorems for Caputo fractional Volterra-Fredholm integro-differential equations.
The rest of the paper is organized as follows: In Section 2, some essential notations, definitions and Lemmas related to fractional calculus are recalled. In Section 3, the new existence and uniqueness results of the solution for Caputo fractional Volterra-Fredholm integro-differential equation have been proved. Finally, we will give a report on our paper and a brief conclusion is given in Section 4.
where R + is the set of positive real numbers.
where the parameter α is the order of the derivative and is allowed to be real or even complex.
In this paper, only real and positive α will be considered.
Hence, we have the following properties: 3. B is a contraction mapping.
Then there exists z ∈ M such that z = Az + Bz.

MAIN RESULTS
In this section, we shall give an existence and uniqueness results of Eq.(1), with the condition (2). Before starting and proving the main results, we introduce the following hypotheses: (A1) g : C(J, X) −→ X is continuous, bounded and there exists 0 < b < 1 2 such that (A2) f : J × X −→ X is continuous and there exist µ ∈ L 1 ([0, T ], R + ) such that First, we will state the following axiom lemma.
Now, we will prove the uniqueness result.
and if Then the fractional integro-differential equation (1)-(2) has a unique solution continuous on J.
Proof. Let the operator F : C(J, X) −→ C(J, X) be defined as define the operator F : C(J, X) −→ C(J, X) by and .
Also, we define B r = {u ∈ C(J, X) : u ≤ r}, for some r > 0, choosing Step 1. We show that FB r ⊂ B r . By the hypotheses, then for any u ∈ B r and for each t ∈ J, we have It follows that Fu ≤ r, which means that F : B r −→ B r .
Step 2. We shall show that F : B r −→ B r is a contraction mapping. Indeed, through the assumptions, then for any u, v ∈ B r , and for t ∈ J, we can write As Θ < 1, so the conclusion of the theorem follows by the contraction mapping principle. This implies that F is contraction mapping. As consequence of Lemma 2.1, there exists a fixed point u ∈ C(J, R) such that Fu = u which is the unique solution of (1)-(2) on J.
Next, we will prove the existence of solution for the problem (1)- (2) in the space C(J, X) by means of Krasnoselskii's fixed point theorem.
Proof. We transform the Cauchy problem (1)-(2) to be applicable to fixed point problem.
Let the operator F : C(J, X) −→ C(J, X) be defined as in Theorem 3.1.
Before move ahead, we need to analyze the operator F into sum of two operators P and Q as follows For any function u ∈ C(J, X), we define the norm u L 1 = sup u(t) C : t ∈ J}. Note that the norms u L 1 and u c are equivalent for u ∈ C(J, X).
We prove that Pu + Qv ∈ B r ⊂ C(J, X), for every u ∈ B r . Since f , Z 1 and Z 2 are respectively bounded on the compact sets ∆ 1 = J × X and ∆ 2 = J × J × X.
and r with Now, we show that operator P is uniformly bounded on B r .

CONCLUSION
The main purpose of this paper was to present new existence and uniqueness results of the solution for Caputo fractional Volterra-Fredholm integro-differential. The techniques used to prove our results are a variety of tools such as Krasnoselskii's fixed point, some properties of fractional calculus and Banach contraction mapping principle. Moreover, the results of references [1,25] appear as a special case of our results.