EXISTENCE, UNIQUENESS AND STABILITY SOLUTIONS FOR NEW NONLINEAR SYSTEM OF INTEGRO-DIFFERENTIAL EQUATIONS OF VOLTERRA TYPE

In this article, we established the existence, uniqueness and stability solutions for a nonlinear system of integro-differential equations of Volterra type in Banach spaces. Krasnoselskii Fixed point theorems and Picard approximation method are the main tool used here to establish the existence and uniqueness results.


INTRODUCTION
The integral equations form an important part of applied mathematics, with links with many theoretical fields, especially with practical fields (K. E. Atkinson, 1997;P. Linz, 1985 ;E. Babolian and J. Biazar, 2000 ) The Volterra integral equations were introduced by Vito Volterra and then studied by Traian Lalescu in his 1908 thesis. Volterra integral equations find application in demography, the study of Viscoelastic materials, and in insurance mathematics through the renewal equation. Fredholm equations arise naturally in the theory of signal processing, most notably as the famous spectral concentration problem popularized by David Slepian. They also commonly arise in linear forward modelling and inverse problems. Throughout the last decade, physicists and mathematicians have paid attention to the concept of fractional calculus. Actually, real problems in scientific fields such as groundwater problems, physics, mechanics, chemistry, and biology are described by partial differential equations or integral equations. Many scholars have shown with great success the applications of fractional calculus to groundwater pollution and groundwater flow problems, acoustic wave problems, and others (F. Mainardi, 1997;P. Zhuang, F. Liu, V. Anh, and I. Turner, 2009;S. B. Yuste and L. Acedo, 2005;C.-M. Chen, F. Liu, I. Turner, and V. Anh, 2007).
The Picard iteration method, or the successive approximations method, is a direct and convenient technique for the resolution of differential equations. This method solves any problem by finding successive approximations to the solution by starting with the zeroth approximation. The symbolic computation applied to the Picard iteration is considered in (Parker, G.E. and Sochacki, J.S. 1996;Bailey, P.B., Shampine, L.F. and Waltman, P.E. 1968) and the Picard iteration can be used to generate the Taylor series solution for an ordinary differential equation (Butris, R. N. 1994;Butris R.N., 2015;Butris, R. N. and Ghada, Sh. J. 2006) studied existence and unique solution for different kind of equations and (Raad N. Butris and Hojeen M. Haji 2019) studied existence and unique solution of Volltera-friedholm: In this work our aim is to show the existence solutions of the system of integrodifferential equations: ) ∫ ) ( ) )) Where are closed and bounded, let ( ) )), ( ) )) is defined on the domain: Assume that the vector functions ( ) )), ( ) )), and kernels ) ) are satisfying the following inequalities: Where [ ] and ‖ ‖ [ ] | |, suppose that A = ( ) ) are nonnegative square matrices of order (n), we defined non empty sets as: As well as, we suppose the maximum value of the following matrix: Define a sequence of functions ) ) as: With ) ) , m=0,1,2…

Definition 2.1 (Syed Abbas, 2011 ). Assume that
) is defined on the set ) ) is said to satisfy Lipschitz condition with respect to the second variable, if for all ) and for any where > 0 does not depend on ). converges to x .
Theorem 2.5: (Azhar H.Sallo , 2006 ). If the function ) satisfy the existence and uniqueness theorem for IVP(1), then the successive approximation ) converges to the unique solution ) of the IVP (1):

MAIN RESULTS: EXISTENCE
Theorem 3.1: Let the right side of system (1) are defined and continuous on domain ). Suppose that the vector functions ( ) )) , ( ) )) are satisfying the inequalities (3)-(5) and the conditions (6)-(9).Then there exist a sequences of functions (10) converges uniformly as on domain (2) to the limit functions which satisfying the following integral equations: Provided that: ) for all

Proof:
By using the sequence of function (10) when m=0, we get: And by the same we have ‖ ) ‖ That is: ], by mathematical induction we conclude that: To prove that the sequences (10) convergence uniformly in domain (2): And by the same Ishak, Faraj Y.
Existence, Uniqueness and Stability Solutions for New Nonlinear System… By the mathematical induction the following inequalities hold: Rewrite (12) with vector form: That is: Where: Take the maximum value for both sides of (13): where [ ] ) By repletion of (14) we obtain: Since the matrix has eigenvalue , ) , the series (15) is uniformly convergent, i.e.

∑ ∑ ) )
Thus the limiting relation (16) signifies uniform convergence of sequences: , that is: By all conditions and inequalities of the theorem the estimate Is hold for all m = 0,1,2, … To prove that ) ) we prove that: We have: ‖ And for the function y(t, ) we have And since the sequences: ) ) uniformly convergence to ) ) respectively on the interval [0,T], that is (17)  ) are unique solution for system (1) on domain (2).

Proof: let
be another solution for system (1) then: Rewrite in vector form: By take the maximum value for both sides of (19) and reputation it we get: From (19) and condition (9) we have when that is: ) is a unique solution for system (1).

Stability
Rewrite (21), (22) in victor form we get: By condition (9) and for we have: By the definition of stability, we find that ̅ ) ̅ ) is stable solution for system (1)
Thus using (25) we conclude that is a contraction mapping.

For
calculating the norm of the function we have: moreover for we obtain: To prove the continuity of ,let us consider a sequence ) converging to y, and taking the norm: ‖ ) ) ) )‖ ∫ ∫ ) )‖ )) ))‖

‖ ‖
And hence whenever ,this prove the continuity of . For compactness of , let and taking the norm: The right-hand side of above expression does not depend on y, thus we conclude that is relatively compact and hence is compact by Arzela -Ascoli theorem. Using Krasnoselskii fixed point theorem we obtain that exist such that: Which is a fixed point of F. E-Jurnal Matematika Vol. 9(2), Mei 2020, pp. 109-116 ISSN: 2303-1751DOI: https://doi.org/10.24843/MTK.2020 By the same steps we can prove that Hence system (1.1) has at least one solution in . Example (3.1): consider the following system of integro-differential equations: For t,s I = [0,1] here ) ) = 0.988 , = 1.33 , )) . Thus, by Theorem (

CONCLUSIONS
Based on the results and discussion it can be concluded that: (1) The proof of existence solution for the non-linear system proposed in this paper using the existence and uniqueness theorem needs some hypotheses and conditions, (2). Using the idea of Krasnoselskii fixed point theorem was very effective for the proposed non-linear system of equations. The present work can be extended to boundary value problem.

ACKNOWLEDGEMENTS:
The author is grateful to the reviewers for their suggestions and advices.