EXISTENCE AND UNIQUENESS OF SOLUTIONS TO SYSTEM OF LINEAR EQUATIONS AND INTEGRAL EQUATIONS USING BANACH FIXED-POINT THEOREM.

1. Department of Mathematics, University of Dhaka, Dhaka-1000. 2. Department of Mathematics, Jessore University of Science & Technology, Jessore-7408. 3. Department of General Educational Development, Daffodil International University, Dhaka-1000. 4. Department of Computer Science and Engineering, Dhaka International University, Dhaka-1000. ...................................................................................................................... Manuscript Info Abstract ......................... ........................................................................ Manuscript History

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 Ax b  , where A is a nn  co-efficient matrix and b is a constant matrix.In section 4, we study the existence and uniqueness of the solution of Fredholm integral equation where the kernel ( , ) K x t and () x  are known functions and  is a real parameter.Here, we seek the unknown function () ISSN: 2320-5407 Int.J. Adv.Res.6(3), 494-500 495 Preliminaries:-To prove the existence results of solutions for system of linear equations and integral equations, we need the definitions and theorems of the following paragraph.Definition 2.1:-Let X be a non-empty set and : T X X  be a mapping.A fixed point of the mapping

Tx x
 In other words, a fixed point of T is a solution of the functional equation Tx x Xd be a metric space.A mapping : T X X  is said to be contraction mapping if there exists a constant [0, 1) ,. X be a normed linear space.Then a complete normed linear space is called a Banach space [1,3].Every Banach space   ,. X also is a complete metric space ( , ) Xd under ( , ) .d x y x y  A Banach space is chosen in such a way that the existence problem is converted into a fixed-point problem for an operator over this Banach space.
We define an operator : Thus, a solution of integral equation (1.1) is a fixed point of the operator T over the Banach space X .

Existence and Uniqueness Results for a System of Linear Equations:-
In this section, we study the existence as well as the uniqueness of the solution of a system of linear equations.Under some conditions, the following theorem ensures the existence and uniqueness of a solution of the system of linear equations [6,9].(1 ) ( (1 )   So the above system of equations can be written as  ( , , , ) , We define a metric ( , ) sup sup ( )

L 
This shows that T is a contraction mapping.An application of Theorem 2.1 completes the proof.

Ax
where the kernel ( , ) ,  is a real parameter and the function () Thus, a solution of Fredholm linear integral equation (4.1) is a fixed point of the operator T [5].If in the Fredholm integral equation (4.1), we replace the upper integration limit b by the variable , x we obtain a Volterra integral equation.Under some conditions on the parameter ,  the following theorem ensures the existence and uniqueness of a solution of the Fredholm linear integral equation (4.1).T is a contraction mapping.By Definition 2.1, there exists a unique solution * f such that ** .Tf f  exists a unique solution if I A L  for (0, 1).L Proof: Given system of equationsAx b can be re-written as The Theorem 3.1 decreases the computation burden of determining the existence and uniqueness of the solutions to a system of linear equations.Existence and Uniqueness Results for Integral Equations:-In this section, we are interested in the study of the existence of continuous solutions of the Fredholm linear and non-linear integral equations over a Banach space.For this purpose, we have chosen the Banach space 2 [ , ] L a b of Lebesgue measurable functions.Consider, the following Fredholm linear integral equation of second kind The main tool in the existence result of a solution is the Banach fixed-point theorem.It is based on the complete metric space.