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.
Highlights
1 Introduction There are in the literature many different versions of the well-known theorems due to Banach [ ] and Nadler [ ] concerning fixed points for single-valued and set-valued dynamic systems, respectively, in complete metric spaces
Recall that a set-valued dynamic system is defined as a pair (X, T), where X is a certain space and T is a set-valued map T : X → X ; here X denotes the family of all nonempty subsets of a space X
By Fix(T) and Per(T) we denote the sets of all fixed points and periodic points of T, respectively, i.e., Fix(T) = {w ∈ X : w ∈ T(w)} and Per(T) = {w ∈ X : w ∈ T[s](w) for some s ∈ N}
Summary
There are in the literature many different versions of the well-known theorems due to Banach [ ] and Nadler [ ] concerning fixed points for single-valued and set-valued dynamic systems, respectively, in complete metric spaces. That there exist a quasi-gauge space (X, P), a left (right) J -family on X, w ∈ X and r = {rα}α∈A ∈ ( ; ∞)A such that w ∈/ BL–J (w , r) (w ∈/ BR–J (w , r))
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
