A generalised fixed point theorem of presic type in cone metric spaces and application to markov process

George Reny, Reshma Kp, Rajagopalan, Rajagopalan R

Description

A generalised common fixed point theorem of Presic type for two mappings f : X → X and T: X k → X in a cone metric space is proved. Our result generalises many well-known results. 2000 Mathematics Subject Classification 47H10 A generalised common fixed point theorem of Presic type for two mappings f : X → X and T: X k → X in a cone metric space is proved. Our result generalises many well-known results. 2000 Mathematics Subject Classification 47H10

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• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 0
• Langue: English

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Georgeet al.Fixed Point Theory and Applications2011,2011:85 http://www.fixedpointtheoryandapplications.com/content/2011/1/85

R E S E A R C H

Open Access

A generalised fixed point theorem of Presic type in cone metric spaces and application to Markov process Reny George 1,2* , KP Reshma 2 and R Rajagopalan 1

* Correspondence: [e-mail] 1 Department of Mathematics, College of Science, Al-Kharj University, Al-Kharj, Kingdom of Saudi Arabia Full list of author information is available at the end of the article

Abstract A generalised common fixed point theorem of Presic type for two mappingsf:X® XandT: X k ®Xa cone metric space is proved. Our result generalises many well-in known results. 2000 Mathematics Subject Classification 47H10 Keywords:Coincidence and common fixed points, cone metric space; weakly compatible

1. Introduction Considering the convergence of certain sequences, Presic [1] proved the following: Theorem 1.1.Let(X,d)be a metric space,k a positive integer,T: X k ®X be a map- ping satisfying the following condition:

d T x 1 ,x 2 ,. . .,x k ,T x 2 ,x 3 ,. . .,x k+1 ≤q 1 ·d x 1 ,x 2 +q 2 ·d x 2 ,x 3 +· · ·+q k ·d x k ,x k+1 (1:1) where x 1 ,x 2 , ...,x k+1 arbitrary elements in X and qare 1 ,q 2 , ...,q k are non-negative constants such that q 1 +q 2 + ∙ ∙ ∙+ q k < 1.Then,there exists some xÎX such that x= T(x,x, ...,x).Moreover if x 1 ,x 2 , ...,x k arbitrary points in X and for nare ÎN x n+k =T (x n ,x n+1 , ...,x n+k-1 ),then the sequence<x n >is convergent and lim x n =T(lim x n ,lim x n , ...,lim x n ). Note that fork= 1 the above theorem reduces to the well-known Banach Contrac- tion Principle. Ciric and Presic [2] generalising the above theorem proved the following: Theorem 1.2.Let(X,d)be a metric space,k a positive integer,T: X k ®X be a map- ping satisfying the following condition:

d T x 1 ,x 2 ,. . .,x k ,T x 2 ,x 3 ,. . .,x k+1 ≤λ.max{d x 1 ,x 2 ,d x 2 ,x 3 ,. . .d x k ,x k+1 (1:2) where x 1 ,x 2 , ...,x k+1 are arbitrary elements in X andlÎ(0,1).Then,there exists some xÎX such that x=T(x,x, ...,x).Moreover if x 1 ,x 2 , ...,x k are arbitrary points in X and for nÎNx n+k =T(x n ,x n+1 , ...,x n+k-1 ),then the sequence<x n >is convergent and lim x n =T(lim x n ,lim x n , ...,lim x n ).If in addition T satisfies D(T(u,u, ...u),T(v,v, ... v)) <d(u,v),for all u,vÎX then x is the unique point satisfying x=T(x,x, ...,x).

© 2011 George et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.