Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : A generalised fixed point theorem of Presic type in cone metric spaces and application to Markov process | George et al. Fixed Point Theory and Applications 2011 2011 85 http content 2011 1 85 Fixed Point Theory and Applications a SpringerOpen Journal RESEARCH Open Access A generalised fixed point theorem of Presic type in cone metric spaces and application to Markov process Reny George1 2 KP Reshma2 and R Rajagopalan 1 Correspondence renygeorge02@ 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 Springer Abstract A generalised common fixed point theorem of Presic type for two mappings f X X and T Xk X in a cone metric space is proved. Our result generalises many well-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 . Let X d be a metric space k a positive integer T Xk X be a mapping satisfying the following condition d T xi X2 . . Xk T X2 X3 . . . Xk 1 q d xi X2 q2 d X2 Xs - qk d xi Xk 1 where x1 x2 . xk 1 are arbitrary elements in X and q1 q2 . qk are non-negative constants such that q1 q2 qk 1. Then there exists some x e X such that x T x x . x . Moreover if x1 x2 . xk are arbitrary points in X and for n e N xn k T xn xn 1 . xn k-1 then the sequence xn is convergent and lim xn T lim xn lim xn . lim xn . Note that for k 1 the above theorem reduces to the well-known Banach Contraction Principle. Ciric and Presic 2 generalising the above theorem proved the following Theorem . Let X d be a metric space k a positive integer T Xk X be a mapping satisfying the following condition d T x1 X2 . Xk T x2 X3 . Xk 1 d x1 x2 d x2 x3 . d Xk Xk 1 where x1 x2 . xk 1 are arbitrary elements in X and l e 0 1 . Then there exists some x e X such that x T x x . x . Moreover if x1 x2 . xk are arbitrary points in X