International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 04 Issue: 07 | July -2017
p-ISSN: 2395-0072
www.irjet.net
A generalized metric space and related fixed point theorems Dr C Vijender Dept of Mathematics, Sreenidhi Institute of Sciece and Technology, Hyderabad. ------------------------------------------------------------------***-----------------------------------------------------------------
Abstract:
We introduce a new concept of generalized metric spaces for which we extend some well-known fixed point results including Banach contraction principle, Ćirić’s fixed point theorem, a fixed point result due to Ran and Reurings, and a fixed point result due to Nieto and Rodríguez-López. This new concept of generalized metric spaces recover various topological spaces including standard metric spaces, b-metric spaces, dislocated metric spaces, and modular spaces. Keywords:generalized metric, b-metric, dislocated metric, modular spacefixed point, partial order. 1. Introduction: The concept of standard metric spaces is a fundamental tool in topology, functional analysis and nonlinear analysis. This structure has attracted a considerable attention from mathematicians because of the development of the fixed point theory in standard metric spaces. In this work, we present a new generalization of metric spaces that recovers a large class of topological spaces including standard metric spaces, b-metric spaces, dislocated metric spaces, and modular spaces. In such spaces, we establish new versions of some known fixed point theorems in standard metric spaces including Banach contraction principle, Ćirić’s fixed point theorem, a fixed point result due to Ran and Reurings, and a fixed point result due to Nieto and Rodíguez -López. 2. A generalized metric space: Let X be a nonempty set and D:X×X→[0,+∞] be a given mapping. For every x∈X, let us define the set C(D,X,x)={{xn}⊂X:limn→∞D(xn,x)=0}. 2.1 General definition Definition 2.1 We say that D is a generalized metric on X if it satisfies the following conditions: (D1)for every (x,y)∈X×X, we have D(x,y)=0⟹x=y; (D2)for every (x,y)∈X×X, we have D(x,y)=D(y,x); (D3)there exists C>0 such that if (x,y)∈X×X,{xn}∈C(D,X,x),then D(x,y)≤Clim n→∞ supD(xn,y). In this case, we say the pair (X,D) is a generalized metric space. Remark 2.2 Obviously, if the set C(D,X,x) is empty for every x∈X, then (X,D) is a generalized metric space if and only if (D1) and (D2) are satisfied.
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