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

Abstract

Abstract In a quasi-gauge space ( X , P ) with quasi-gauge P , using the left (right) J -families of generalized quasi-pseudodistances on X ( J -families on X generalize quasi-gauge P ), the left (right) quasi-distances D η L − J ( D η R − J ) of Hausdorff type on 2 X 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) P -convergence of dynamic processes starting at each point w 0 ∈ 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) P -convergence of Picard iterations starting at each point w 0 ∈ X is studied, and existence, localization, periodic point, fixed point and uniqueness results are established. Appropriate tools and ideas of studying based on J -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 D η L − J ( U , V ) = D η L − J ( V , U ) ( D η R − J ( U , V ) = D η R − J ( V , U ) ) and D η L − J ( U , U ) = 0 ( D η R − J ( U , U ) = 0 ), η ∈ { 1 , 2 , 3 } , U , V ∈ 2 X ; provide information concerning localizations of periodic and fixed points; and substantially generalize the well-known theorems of Nadler and Banach types. MSC:54A05, 54C60, 47H09, 37C25, 54H20, 54H25, 54E15.