Some Fixed Point Theorems for Compatible Mappings Satisfying an Implicit Relation

Abstract

IntroductionLet S and T be two self mappings of a metric space (X, d) Sessa [2] defines S and T to be weakly commuting if d (STx, TSx) < d(Tx, Sx) for all x in X Jungck [1] defines S and T to be compatible if limn_>00(5Txn,T5xn) = 0 whenever {xn} is a sequence in X such that limn_njo Sxn = lim^oo Txn = x for some x in X. By Lemma 1, we suppose that X contains at least three points. LEMMA 2 [1]. Let f and g be compatible self mappings on a metric space (X,d). If f{t) = g(t), then fg(t) = gf(t).The purpose of this paper is to prove some fixed point theorems for compatible mappings satisfying an implicit relation. Implicit relationsLet T be the set of all real continuous functions (.F a ): F(u, v, v,u,u + v, 0 F(u, v, u, v, (Fb): Let be u > 0 and F (u,viu,v,0,u+ vCzhU, where Ci > 0, C2, C3 > 0, ci + 2c2 < 1, and Ci + C3 < 1. F\\ Obviously.(Fa): Let be u > 0 and F (u, v, v,u,u + v (FB) Let be u > 0 and F (u, v,u, v,0,u+v (F0); Let be u > 0 and F (u,v,v,u,u + vwhere hi = f±| < 1.(Fb): Let be u > 0 and F(u, v, u, v,0,u + v F X : Obviously.

A general common fixed point theorem of Meir and Keeler type for noncontinuous weak compatible mappings

“…By [19] S and T are not compatible of type (A) and noncompatible of type (P). S and T are not compatible of type (B).…”

Unified implicit common fixed point theorems under non-negative complex valued functions satisfying the identity of indiscernible

“…Popa [13,14] initiated this idea of unification which had produced thus far a consistent literature on fixed point, common fixed point and coincidence point for both single valued and multi-valued mappings in various ambient spaces. Motivated by Popa [15,16], Ali and Imdad [2] and Imdad et al [8,9,10], we consider an implicit function under minimal requirement and utilize the same to prove some common fixed point theorems.…”

Some common fixed point theorems for hybrid pairs of maps without the completeness assumption

“…The following example supports this view point. In order to prove our main result, we employ a slightly different implicit function than the one employed by Popa [18,19]. In order to describe it, let Ψ be the family of real lower semi-continuous functions F : [0, ∞) 6 → satisfying the following conditions.…”

“…In recent years, the idea of implicit functions has been utilized very effectively to prove unified fixed point theorems enabling one to deduce several known results in one go besides yielding new results whose demonstration is available in [6,18,19]. Following [18], let Φ be the family of real lower semi-continuous functions F : [0, ∞) 6 → satisfying the following conditions:…”