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Zbl 1220.54017
Abbas, Mujahid; Cho, Yeol Je; Nazir, Talat
Common fixed point theorems for four mappings in TVS-valued cone metric spaces.
(English)
[J] J. Math. Inequal. 5, No. 2, 287-299 (2011). ISSN 1846-579X

Let $(X,d)$ be a TVS-valued cone metric space in the sense of [{\it L.-G. Huang} and {\it X. Zhang}, Math.\ Anal.\ Appl.\ 332, No.\ 2, 1468--1476 (2007; Zbl 1118.54022)] and [{\it I. Beg, A. Azam} and {\it M. Arshad}, Int.\ J. Math.\ Math.\ Sci.\ 2009, Article ID 560264, 8p. (2009; Zbl 1187.54032)] over a cone $P$ which need not be normal. \par Let $f, g, S, T$ be self-mappings in $X$ satisfying $f(X)\subset T(X)$, $g(X)\subset S(X)$ and suppose that there exists $h\in(0,1)$ such that $d(fx,gy)\preceq hu_{x,y}(f,g,S,T)$, where $u_{x,y}(f,g,S,T)\in\{d(Sx,Ty),d(fx,Sx),d(gy,Ty), \frac12(d(x,Ty)+d(gy,Sx))\}$ for all $x,y\in X$. \par The authors prove that, if $f(X)\cup g(X)$ or $S(X)\cup T(X)$ is complete, then the pairs $\{f,S\}$ and $\{g,T\}$ have a unique point of coincidence in $X$. Moreover, if the pairs $\{f,S\}$ and $\{g,T\}$ are weakly compatible, then $f, g, S, T$ have a unique common fixed point in $X$. The same conclusion is obtained when there are $p,q,r,t\in[0,1)$ satisfying $p+q+r+2t<1$ such that $d(fx,gy)\preceq pd(Sx,Ty)+qd(fx,Sx)+rd(gy,Ty)+t[d(fx,Ty)+d(gy,Sx)]$ for all $x,y\in X$.
MSC 2000:
*54H25 Fixed-point theorems in topological spaces

Keywords: coincidence point; common fixed point; TVS-valued cone metric space; weakly compatible maps

Citations: Zbl 1118.54022; Zbl 1187.54032

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