# On Application of the Fixed-Point Theorem to the Solution of Ordinary Differential Equations On Application of the Fixed-Point Theorem to the Solution of Ordinary Differential Equations

## Main Article Content

## Abstract

We know that a large number of problems in differential equations can be reduced to finding the solution x to an equation of the form Tx=y. The operator T maps a subset of a Banach space X into another Banach space Y and y is a known element of Y. If y=0 and Tx=Uxâˆ’x, for another operator U, the equation Tx=y is equivalent to the equation Ux=x. Naturally, to solve Ux=x, we must assume that the range R (U) and the domain D (U) have points in common. Points x for which Ux=x are called fixed points of the operator U. In this work, we state the main fixed-point theorems that are most widely used in the field of differential equations. These are the Banach contraction principle, the Schauderâ€“Tychonoff theorem, and the Lerayâ€“Schauder theorem. We will only prove the first theorem and then proceed.

## Article Details

*Asian Journal of Mathematical Sciences(AJMS)*,

*3*(2). Retrieved from http://ajms.in/index.php/ajms/article/view/204

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