Some new dimensional approach to the fixed point theories in complex valued metric spaces

Abstract

SOME NEW DIMENSIONAL APPROACH TO THE FIXED POINT newlineTHEORIES IN COMPLEX VALUED METRIC SPACES newlineDuring the last fifty years, fixed point theories in metric spaces are emerging areas of works newlinein the field of the complex as well as functional analysis. The concept of fixed point theorem newlinewas first introduced by Poincare and Miranda[14] in 1883. Since then the following results have newlinebeen developed accordingly. The theorem to establish fixed point introduced by Poincare and newlineMiranda[14] in 1883 states that newlineTheorem 0.0.1 Let f : In newlineÑ Rn, f p f1, f2, ... fnq be a continuous map such that for each newlinei and#271; n, fi newline` newlineI´ newlinei newlineand#728; newlineand#258; p´8, 0s and fi newline` newlineI` newlinei newlineand#728; newlineand#258; r0,`8q , then there exist a point c P In such that newlinef pcq 0. newlineIn 1912, Brouwer[2] published his famous fixed point theorem. The theorem states that newlineTheorem 0.0.2 If B is a closed unit ball in Rn and if T : B Ñ B is continuous then T has a newlinefixed point in B. newlineBanach s fixed point theorem plays a major role in the fixed point theory. It has applications newlinein many branches of mathematics. Banach s fixed point theorem plays a major role in the fixed newlinepoint theory. The famous Banach s theorem[3] states that newline