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Coincidence and common fixed point results on metric spaces endowed with an arbitrary binary relation and applications. (English) Zbl 1267.54039

Let \((X,d)\) be a complete metric space, let \({\mathcal R}\) be a relation over \(X\), and let \({\mathcal S}={\mathcal R}\cup {\mathcal R}^{-1}\) stand for the associated symmetric relation. Further, let \(\Phi\) be the class of all nondecreasing \(\phi:\mathbb R_+\to \mathbb R_+\) with \(\Sigma_n \phi^n(t)< \infty\), \(\forall t> 0\); and, given \(T,g:X\to X\), denote for each \(x,y\in X\): \(M_T(gx,gy)=\max\{d(gx,gy), (1/2)[d(gx,Tx)+d(gy,Ty)], (1/2)[d(gx,Ty)+d(gy,Tx)]\}\). The following is the main result of the author: {Theorem.} Suppose that \(T(X)\subseteq g(X)\), \(g(X)\) is closed, \(T\) is \(g\)-comparative, and there exists \(\phi\in \Phi\) such that \(d(Tx,Ty)\leq \phi(M_T(gx,gy))\), for all \(x,y\in X\) with \(gx{\mathcal S} gy\). Then,
i) If, in addition, \((X,d,{\mathcal S})\) is regular and there exists \(x_0\in X\) such that \(gx_0{\mathcal S}Tx_0\), then \({\mathcal C}(g,T):=\{x\in X; gx=Tx\}\) is nonempty,
ii) If, in addition to this, \({\mathcal C}(g,T)\) is \({\mathcal S}-g\)-directed and \(T\), \(g\) commute at their coincidence points, then \({\mathcal C}(g,T)\) is a singleton.
In particular, when \(g\)=identity, one gets the related fixed point result in B. Samet and M. Turinici [Commun. Math. Anal. 13, No. 2, 82–97 (2012; Zbl 1259.54024)]. An application of this result to coincidence and common fixed points for cyclic contractive maps is also given, to illustrate its usefulness.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
54E40 Special maps on metric spaces
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces

Citations:

Zbl 1259.54024
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References:

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