Best proximity points for p-cyclic summing iterated contractions

Abstract

We generalize the p-summing contractions maps. We found sufficient conditions for these new type of maps, that ensure the existence and uniqueness of best proximity points in uniformly convex Banach spaces. We apply the result for Kannan and Chatterjea type cyclic contractions and we obtain sufficient conditions for these maps, that ensure the existence and uniqueness of best proximity points in uniformly convex Banach spaces.

On Best Proximity Point Results for Some Type of Mappings

“…A convenience attention has been recently devoted to the research on existence and uniqueness of best proximity points of self-mappings, as well as, to the investigation of associated relevant properties, for instance, stability of the iterations. The various related performed researches include the cases of cyclic ϕ-contractions [6,7], cyclic Meir-Keeler contractions [8], weak cyclic Bianchini contractions [9], weak cyclic Kannan contractions [10], p-cyclic summing iterated contractions [11], and MF-cyclic contractions with Property UC [12]. Some contractive conditions and related properties under general contractive conditions including some proximal rational types have been also investigated [13].…”

Best Proximity Points for p–Cyclic Infimum Summing Contractions

“…It is interesting that whenever p-cyclic maps have been considered, using the ideas from [30] the distances between consecutive sets are equal [11,30,34,35]. A new type of condition that warrants the existence and the uniqueness of the best proximity points for sets with different distances between them is presented in [36][37][38][39]. This new type of a map has been called a p-summing map.…”

On Coupled Best Proximity Points in Reflexive Banach Spaces

“…A disadvantage of the classical theory of best proximity points in Banach spaces is the requirement that the Banach space should be uniformly convex [3]. This restriction was overcome in [26,27], where the underlying space was just a reflexive Banach space. There are applications in the theory of market equilibrium in the non-competitive market [28][29][30], where the underlying space is a uniformly convex Banach space.…”