The current paper generalizes the Edelstein fixed point theorem for digital ( , )-chainable metric spaces. In order to generalize Edelstein fixed point theorem, we study the digital topological properties of digital images. Further, we establish the Banach fixed point theorem for digital images. We give the notion of digital ( , , )-uniformly locally contraction mapping on digital ( , )-chainable metric spaces. Finally, we generalize the Banach fixed point theorem to digital ( , )-chainable metric spaces which is known as the Edelstein fixed point theorem for digital images on digital ( , )-chainable metric spaces.

Fixed point, Banach fixed-point theorem, System of linear equations, Fredholm integral equation.In this paper, using Banach fixed-point theorem, we study the existence and uniqueness of solution for a system of linear equations. Further, we prove the existence and uniqueness of the continuous solutions of linear and non-linear Fredholm integral equations over the Banach space …………………………………………………………………………………………………….... Introduction:-Over the last few decades, fixed-point theory has become an important field of study in science. It provides a powerful tool for proving the existence of solutions of problems originating from various branches of mathematics. It has long been used in analysis to solve various kinds of differential and integral equations [1,8]. Existence theorem for differential equation was first given by Cauchy [8]. Applications of fixed point results to integral equations have been studied in [7,8]. In metric space, this theory begins with Banach fixed-point theorem (also known as Banach contraction principle) [2,8]. Banach fixed-point theorem has many applications to linear and nonlinear equations, to ordinary and semi-linear partial differential equations and to linear and non-linear integral equations [1,4,5,8]. In this paper, we study the applications of Banach fixed-point theorem for proving existence results to solutions of system of linear equations and integral equations. This paper is organized as follows. In section 2, we review some required background materials. In section 3, we investigate an existence and uniqueness result of the solution of a system of linear equations

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