Hausdorff quasi-distances, periodic and fixed points for Nadler type set-valued contractions in quasi-gauge spaces

Abstract

In a quasi-gauge space (X,P) with quasi-gauge , using the left (right) -families of generalized quasi-pseudodistances on X (-families on X generalize quasi-gauge ), the left (right) quasi-distances DL−Jη ( DR−Jη) of Hausdorff type on 2X are defined, η∈{1,2,3}, the three kinds of left (right) set-valued contractions of Nadler type are constructed, and, for such contractions, the left (right) -convergence of dynamic processes starting at each point w0∈X is studied and the existence and localization of periodic and fixed point results are proved. As implications, two kinds of left (right) single-valued contractions of Banach type are defined, and, for such contractions, the left (right) -convergence of Picard iterations starting at each point w0∈X is studied, and existence, localization, periodic point, fixed point and uniqueness results are established. Appropriate tools and ideas of studying based on -families and also presented examples showed that the results: are new in quasi-gauge, topological, gauge, quasi-uniform and quasi-metric spaces; are new even in uniform and metric spaces; do not require completeness and Hausdorff properties of the spaces (X,P), continuity of contractions, closedness of values of set-valued contractions and properties DL−Jη(U,V)=DL−Jη(V,U) ( DR−Jη(U,V)=DR−Jη(V,U)) and DL−Jη(U,U)=0 ( DR−Jη(U,U)=0), η∈{1,2,3}, U,V∈2X; provide information concerning localizations of periodic and fixed points; and substantially generalize the well-known theorems of Nadler and Banach types.