On common fixed points of pairs of a single and a multivalued coincidentally commuting mappings in \(D\)-metric spaces (Q1415107)

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 \[ \begin{multlined}\delta^r(Fx,Fy,Fz) \leq \phi(\max\{\rho^r(gx,gy,gz),\delta^r(Fx,Fy,gz),\delta^r(gx,Fx,gz),\\ \delta^r(gy,Fy,gz),\delta^r(gx,Fy,gz),\delta^r(gy,Fx,gz)\})\end{multlined} \] (\(\phi: \;[0,\infty) \to [0,\infty)\) 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 \[ \begin{multlined}\delta^r(Fx,Fy,Fz) < \max \, \{\rho^r(gx,gy,gz),\delta^r(Fx,Fy,gz),\delta^r(gx,Fx,gz),\\ \delta^r(gy,Fy,gz), \delta^r(gx,Fy,gz),\delta^r(gy,Fx,gz)\}\end{multlined} \] 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)''.