EXISTENCE AND UNIQUENESS RESULTS FOR VOLTERRA-FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS

This paper establishes a study on some important latest innovations in the existence and uniqueness results by means of Banach contraction fixed point theorem for Caputo fractional Volterra-Fredholm integro-differential equations with boundary condition. New conditions on the nonlinear terms are given to pledge the equivalence. Finally, an illustrative example is also presented.


Introduction
In recent years, there has been a growing interest in the linear and nonlinear integro-differential equations which are a combination of differential and integral equations [4,6,7,18,20,25]. The nonlinear integrodifferential equations play an important role in many branches of nonlinear functional analysis and their applications in the theory of engineering, mechanics, physics, electrostatics, biology, chemistry and economics [15] and signal processing [27].
The challenging work is to find the solution while dealing with Volterra-Fredholm fractional integrodifferential equations. Therefore, many researchers have tried their best to use different techniques to find the analytical and numerical solutions of these problems [1,2,3,5,9,10,11,16,23,24,30].
The study of iterative differential and integro-differential equations is linked to the wide applications of calculus in mathematical sciences. These equations are vital in the study of infection models. Many papers have dealt with the existence, uniqueness and other properties of solutions of special forms of the iterative differential equations and integro-differential equations [17,18,21,22].
Recently, Cheng et al. [8,22], investigated analytic and exact solutions of an iterative functional differential equation of the type u (x) = f (x, u(h(x) + g(u(x)))), Lauran [21], studied the existence and uniqueness results for first order differential and iterative differential equations with deviating argument of the type Kendre et al. [18], studied the existence of solution for iterative integro-differential equations of the type In [17], Ibrahim investigated the existence and uniqueness of solution for iterative differential equations of the type Unhale and Kendre [29], established the existence and uniqueness of solution for iterative integro-differential equations of the type Motivated by these problems, in this paper, we discuss new existence and uniqueness results for nonlinear fractional Volterra-Fredholm integro-differential equation with deviating argument of the type with the boundary condition where D α (.), 0 < α < 1, is the Caputo fractional derivative, f (t), h(x, s) and k(x, s) are given continuous functions, u(x) is the unknown function to be determined. The main objective of the present paper is to study the new existence and uniqueness results for iterative nonlinear fractional Volterra-Fredholm integro-differential equation with deviating argument.
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 nonlinear fractional Volterra-Fredholm integro-differential equation have been proved. In Section 5, focuses on an example to illustrate the theory. Finally, we will give a report on our paper and a brief conclusion is given in Section 6.

Preliminaries
In this segment, we first survey some fundamental definitions of the fractional calculus theory which are required for building up our outcomes. For more details, see [9,11,12,13,19,20,26,28,30].
Definition 2.1. [18] The Riemann-Liouville fractional integral of order α > 0 of a function f is defined as where R + is the set of positive real numbers.
[18] The Riemann-Liouville derivative of order α with the lower limit zero for a function f : [0, 1) −→ R can be written as Definition 2.3. [14] The Caputo derivative of order α for a function f : [0, 1) −→ R can be written as Definition 2.4. [14] The fractional derivative of f (x) in the Caputo sense is defined by where the parameter α is the order of the derivative, in general it is real or even complex.
Definition 2.5. [14] The Riemann-Liouville fractional derivative of order α > 0 is normally defined as Lemma 2.1. [14] (Gronwall-Bellman's Inequality). Let u(x) and f (x) be nonnegative continuous functions defined on J = [α, α + h] and c be a nonnegative constant. If Theorem 2.1. [28] (Banach contraction principle). Let (X, d) be a complete metric space, then each contraction mapping T : X −→ X has a unique fixed point x of T in X i.e. T x = x.

Main Results
In this section, we shall give the existence and uniqueness results of Eq.(1), with the condition (2). Let B = C(J, J) be the Banach space equipped with the norm u = max x∈[0,T ] |u(x)|. For convenience, we are listing the following hypotheses used in our further discussion: (A1) There exists constants β h and β k such that , then the problems (1)- (2) are equivalent to the problem of finding a continuous solution of the integral equation Theorem 3.1. Suppose that the hypotheses (A1)-(A4) are satisfied and Then there is a unique solution to the problems (1)- (2).
To apply Banach contraction principle, we define an operator Ψ : So, we have Hence, This shows that Ψ maps from B(ρ) −→ B(ρ), Now, for all u, v ∈ B(ρ) we have by the Banach contraction principle, Ψ has a unique fixed point. This means that the problems (1)-(2) has unique solution.
The above theorem shows that there exists a unique solution to the problems (1)-(2). However, it does not tell us how to find this solution. To find the solution of the problems (1)-(2), we will define the following sequence u n+1 (x) h(t, s)u n (u n (s))ds + T 0 k(t, s)u n (u n (s))ds dt (7) h(t, s)u n (u n (s))ds + T 0 k(t, s)u n (u n (s))ds dt − c .
Theorem 3.2. If the assumptions of the Theorem 3.1 are satisfied then the sequences defined in (7) converges uniformly to the unique solution of the problems (1)- (2). Then Assume that result is true for n i.e. U n ≤ T U n−1 ≤ T n . Now, we show that result holds for n + 1 h(t, s)u 1 (u n (s))ds + T 0 k(t, s)u n (u n (s))ds dt h(t, s)u n−1 (u n−1 (s))ds h(t, s) u n (u n (s)) − u n−1 (u n−1 (s)) ds Thus by induction, we have U k ≤ T k . Since Hence U k tends to zero as k tends to infinity. Since the family {U k } is the Arzelà-Ascoli family thus for every subsequence {u kj } of {U k } there exists a subsequence {u kj } uniformly convergent and the limit needs to be a solution of the problem (1)- (2). Thus, the sequence {U k } tends uniformly to the unique solution of the problem (1)-(2).
Proof. let > 0 and let function v ∈ C 1 (J, J) which satisfies the inequality and let u ∈ C(J, J) be the unique solution of the following problem from Lemma 3.1, we obtain h(t, s)u(u(s))ds + T 0 k(t, s)u(u(s))ds dt, h(t, s)u(u(s))ds + T 0 k(t, s)u(u(s))ds dt . Let On the other hand, if From inequality (12) we have If we integrate each term of the above inequality and appling the boundary conditions, then we have .

An Example
We consider the nonlinear iterative fractional integro-differential equation (1)  Since all the hypotheses of Theorem 3.1 are fulfilled, then there exists a unique solution of the given equation.

Conclusion
The main purpose of this paper was to present new existence and uniqueness results as well as the Ulam-Hyers stability and generalized Ulam-Hyers stability results of the solution for Caputo fractional iterative Volterra-Fredholm integro-differential. The techniques used to prove our results are a variety of tools such as the Gronwall-Bellman's inequality, some properties of fractional calculus and the Banach contraction fixed point theorem. Moreover, the results of references [17,18,29] appear as a special case of our results.