Method of search functionals and its applications in fixed point and coincidence theory

Abstract

The paper contains a survey of several results from the author’s papers and joint papers by the author and Yu. N. Zakharyan, both on the zero existence and approximation for single-valued and multivalued (α, β)-search functionals, and also on the zero existence preservation for parametric family of such functionals, under the parameter changing. Some corollaries of these results in the fixed point and coincidence theory of single-valued and multi-valued mappings of metric spaces are also given. The comparison is provided with some known results by other authors. In the concluding part of the paper, we investigate the problem on the existence of a parameter-continuous single-valued branch of zeros for a parametric family of search functionals. A theorem on the existence of solution of this problem is proved.