Carisit's theorem and It's Restrition Dependibg on zermelo's theorem

Abstract

      The purpose of this paper is to establish the relation between fixed point theorem's  of  Zermelo and Caristi ,the equivalent between them, and show the restriction of Caristi's theorem to continuons  function  can be derived directly from the Zermelo theorem.

1.Introduction

       Let X be a non empty set and T be a self-map of X . Let  Fix(T) denote the set of all Fixed point of T the converse to Zermelo's fixed point theorem said that if

 Fix(T) ϕ, then there exists apartial ordering  such that every chain in (X,) has  a supremum and for all x X. xTx.This result is a converse of Zermelo's fixed point theorem .we also show the equivalent between fixed point theorems of Zermelo and Caristi . Finally ,We discuss relation between Caristi's theorem and it's restriction to mappings satisfying Caristi's condition with a continuous real function .