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On common fixed points of pairs of a single and a multivalued coincidentally commuting mappings in $$D$$-metric spaces. (English) Zbl 1060.47053
This article deals with common fixed points for a pair of singlevalued and multivalued mappings $$g$$ and $$F$$ in a $$D$$-metric space $$(X,\rho)$$ satisfying a contraction condition of type $\begin{split}\delta^r(Fx,Fy,Fz) \leq \phi(\max\{\rho^r(gx,gy,gz),\delta^r(Fx,Fy,gz),\delta^r(gx,Fx,gz),\\ \delta^r(gy,Fy,gz),\delta^r(gx,Fy,gz),\delta^r(gy,Fx,gz)\})\end{split}$ ($$\phi: \;[0,\infty) \to [0,\infty)$$ is continuous, nondecreasing, $$\phi(t)$$ for all $$t > 0$$, $$\sum_{n=1}^\infty \phi^n(t) < \infty$$ for all $$t \in [0,\infty)$$) or of type $\begin{split}\delta^r(Fx,Fy,Fz) < \max \, \{\rho^r(gx,gy,gz),\delta^r(Fx,Fy,gz),\delta^r(gx,Fx,gz),\\ \delta^r(gy,Fy,gz), \delta^r(gx,Fy,gz),\delta^r(gy,Fx,gz)\}\end{split}$ with some $$r > 0$$ (note that $$D$$-metric space $$(X,\rho)$$ is a set $$X$$ and a function ($$D$$-metric) $$\rho: \;X \times X \times X \to [0,\infty)$$ satisfying the following properties: (i) $$\rho(x,y,z) = 0$$ if and only if $$x = y = z$$, (ii) $$\rho(x,y,z)$$ is symmetric function with respect to the permutation of arguments, (iii) $$\rho(x,y,z) \leq \rho(x,y,a) + \rho(x,a,z) + \rho(a,y,z)$$; $$\delta$$ is the corresponding Hausdorff $$D$$-metric on the set of nonempty closed and bounded sets in $$X$$).
Four theorems about the existence of the unique common fixed point $$u \in X$$ ($$Fu = \{u\} = g(u)$$) are proved; the authors state that these theorems “generalize more than a dozen known fixed-point theorems in $$D$$-metric spaces including those of Dhage (2000) and Rhoades (1996)”.

##### MSC:
 47H10 Fixed-point theorems 54H25 Fixed-point and coincidence theorems (topological aspects)
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