Remarks on contractive type mappings

Abstract

We present an analog of Banach's fixed point theorem for CJM contractions in preordered metric spaces. These results substantially extend theorems of Ran and Reurings' (Proc. Am. Math. Soc. 132(5): 1435-1443, 2003 and Nieto and Rodríguez-López (Order 22: 223-239, 2005). MSC: Primary 47H10; secondary 54H25Keywords: contractive map; fixed point; preordered space; G-metric space DefinitionsWe have to introduce many types of contractions.Definition . Let T be a mapping on a metric space (X, d).. T is said to be a (usual) contraction (C, for short) [], if there exists λ ∈ [, ) such that d(Tx, Ty) ≤ λd(x, y) for any x, y ∈ X.. T is said to be a Browder contraction (Bro, for short) [], if there exists a function ϕ from (, ∞) into itself satisfying the following:

Fixed point theorems in preordered sets

“…In this section we extend some fixed point theorems established by Ran and Reurings [16] and Nieto and Rodrígez-López [11] and O'Regan and Petruşel [12] and the author [4] to mappings with the control functions on preordered metric spaces, where a preordered binary relation is weaker than a partial order.…”

“…Next, Nieto and Rodríguez-López [11] established a similar result replacing the continuity of the nonlinear operator by monotonicity. These studies have been continued by many authors, see for example [4,12].…”

Remarks on asymptotic regularity and fixed points

“…In Refs. [36,37], we find an analogue of Banach theorem in partially ordered sets, further extensions are contained in [24,42,43]. In all these works, the mapping considered are monotone.…”

On New Type of F -Contractive Mapping for Quasipartial b -Metric Spaces and Some Results of Fixed-Point Theorem and Application

“…Wardowski [8] in 2012 Introduced the definition of contraction called F-contraction. After that, in 2017, Górnicki [9] called F-contraction to be Wardowski contraction and also extended fixed-point theorems established by Ran and Reurings [10] and Nieto and Rodríguez-López [11] to CJM contractions on preordered metric spaces, where a preordered binary relation is weaker than a partial order. Recently, in 2019, Goswami et al [12] defined F-contractive type mappings in b-metric space; they proved fixed-point theorems with suitable examples.…”