Cascade search for zeros of functionals in quasi-metric and cone metric spaces. Survey of the results

Abstract

In 2009--2013 the author introduced a concept of an \((\alpha,\beta)\)-search functional on a metric space. Zero existence theorem was proved for such functionals, and fixed points and coincidence theorems were obtained as  consequences. These results generalized several known theorems.
Then this idea was expanded by the author to the more general class of spaces. Here we consider a \((b_{1},b_{2})\)-quasimetric space and survey some topological properties of such spaces. It turns out that the zero existence problem for  \((\alpha,\beta)\)-search functionals is solved in a \((b_{1},b_{2})\)-quasimetric space rather similarly, though there are some features concerning the proof of the Cauchy property and convergence of correspondent sequences. Like as in usual metric space, we obtain in \((b_{1},b_{2})\)-quasimetric space  fixed point and coincidence theorems for multivalued mappings as consequences of the zero existence theorem for an \((\alpha,\beta)\)-search functional. These results also generalize some previous theorems of other authors.
As well, it is of an interest to consider analogous concepts and constructions on a cone-valued metric space, where the metric takes its values in a given cone in a normed space. We consider such cone metric space, investigate its properties and expand the idea of an \((\alpha,\beta)\)-search functional for such spaces. Moreover, we define a concept of a multivalued \((A,B)\)-search conic function on a cone metric space using positive linear operators \(A,B\), instead of number coefficients \((\alpha,\beta)\), for the characteristics of a search conic function. Zero existence theorem is proved for multivalued \((A,B)\)-search conic functions. As consequences of this theorem, coincidence and fixed point theorems are presented for multivalued mappings defined on a cone metric space.