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Research report: "The general property of weak compatible mappings in metric spaces O"

Collection of research reports best university in 2008 honored authors: 4. Dinh Huy Hoang, Le Khanh Hung, Bui Thi Thuy Vinh, The common property of weak compatible mappings in space, the O-Physics can Metric.Cu scientific study of the laws of natural movement, from a macroscopic (particles that make up matter) to larger scales (planets, galaxies and the universe). In English, from physics (physics) is derived from Greek φύσις (phusis) means natural and φυσικός (phusikos) is of the nature. The main object of study now include physical matter, energy, space and time | COMMON FIXED POINTS FOR WEAKLy COMPATIBLE MAPS IN O-METRIZABLE SPACES DINH Huy HOANG a LE KHANH HUNG a BUI THI THUY VINH b Abstract. In this paper we prove some common fixed point theorems for weakly compatible selfmappings of o-metrizable spaces. I. INTRODUCTION It is well known that Banach contraction principle is a fundamental resul in fixed point theory which has been used and extended in many different directions. However it has been observed in 3 that some of the defining properties of the metric are not needed in the proofs of certain metric theorems. Motivated by this fact Hicks in 3 Aamiri and Moutawakil in 1 established some common fixed point theorems in symmetric spaces for weakly compatible maps. The main purpose of this paper is to give some common fixed point theorems for weakly compatible selfmappings of o-metric spaces. We begin at some basic definitions used in this paper. 1.1. Definition. Let X be a topological space and d be a nonnegative realvalued funtion defined on X X X such that d x y 0 if and only if x y. Such a function d is called an o-metric and X is called an o-metrizable space provided that a subset U of X is open if and only if d X X n U 0 for each x 2 U where d X X n U inf d x y y 2 X n U . 1.2. Definition. Let X d be an o-metrizable space and xn be a sequence in X. xn is called a Cauchy sequence if for each 0 there exists n0 2 N satisfying d xn xn m for all n no and m 2 N. X d is called a complete space if for every Cauchy sequence xn in X there exists x 2 X such that xn converges to x. A sequence xn is said to be d-converging to x if lim d x xn 0. An o-nn metrizable space X d is called to have property W4 if for any two sequences xn yn and x in X lim d x xn 0 and lim d yn xn 0 imply that lim d x yn 0. n 1 n 1 n 1 1.3. Definition. Let A and B be two selfmappings of an o-metrizable space X d . A and B are said to be compatible if lim d ABxn BAxn 0 whenever nn xn is a sequence in X such that lim d t Axn lim d t Bxn 0 for some t 2 X .