Common Fixed Point Theorems for Compatible and Weakly Compatible Maps in G-Metric Spaces

Abstract

In this paper first we prove common fixed point theorems for compatible and weakly compatible maps. Secondly, we prove common fixed point theorems for weakly compatible maps along with property (E.A.) and (CLRg) property respectively.

Generalized Fixed Point Theorems in G-Metric Space Under an Implicit Relation

“…With these ideas, the following was proved in [3]: Theorem 1.1. Let f and r be self-maps on X such that…”

“…Then (1.7) follows from (2.19) and hence Corollary 1.1 is a particular case of Corollary 2.1. In view of the choice of k, it may be noted from the proof of Corollary 1.1 given in [3], that the symmetry of X can be dropped when the inequality (2.10) is condensed as (1.7) . If (f, r) satisfies the CLR r -property and r is weakly compatible with f, then f and r will have a coincidence point u.…”

Common Fixed Point Results for Compatible Map in Digital Metric Spaces

“…Fixed point theory leads to lots of applications in mathematics , computer science, engineering, game theory, fuzzy theory, image processing and so forth [ 4,7,14]. In metric spaces, this theory begins with the Banach fixed-point theorem which provides a constructive method of finding fixed points and an essential tool for solution of some problems in mathematics and engineering and consequently has been generalized in many ways.…”

“…In metric spaces, this theory begins with the Banach fixed-point theorem which provides a constructive method of finding fixed points and an essential tool for solution of some problems in mathematics and engineering and consequently has been generalized in many ways. Up to now, several developments have occurred in this area [ see 7,10,14,17,18]. A major shift in the arena of fixed point theory came in 1976 when Jungck [11], defined the concept of commutative maps and proved the common fixed point results for such maps.…”

“…After which, Sessa [16] gave the concept of weakly compatible, and Jungck ( [12,13])gave the concepts of compatibility and weak compatibility. Certain altercations of commutativity and compatibility can also be found in [1,6,14,19]. This paper is organized as follows.…”