## 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

Zbl 1259.54024
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### References:

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