Zeros of a Functional Associated with a Family of Search Functionals. Corollaries for Coincidence and Fixed Points of Mappings of Metric Spaces

The study of the zero existence problem for a nonnegative set-valued functional on a metric space is continued. The zero existence problem for a functional related by a certain \(\theta\) -continuity condition to a parametric family of \((\alpha,\beta)\) -search functionals on an open subset of a metric space is examined. A theorem containing several sufficient conditions for this functional to have zeros is proved.

As corollaries of this result, theorems on the existence of coincidence and fixed points are also proved for set-valued mappings related by the \(\theta\) -continuity condition to families of set-valued mappings with the property that the existence of coincidence and fixed points in an open subset of a metric space is preserved under parameter variation. For uniformly convex metric spaces, analogs of M. Edelstein’s 1972 asymptotic center theorem and M. Frigon’s 1996 fixed point theorem for nonexpansive mappings of Banach spaces are obtained and compared with the main results of the paper.