Generalized Cyclic p-Contractions and p-Contraction Pairs Some Properties of Asymptotic Regularity Best Proximity Points, Fixed Points

Abstract

This paper studies a general p-contractive condition of a self-mapping T on X, where (X,d) is either a metric space or a dislocated metric space, which combines the contribution to the upper-bound of d(Tx, Ty), where x and y are arbitrary elements in X of a weighted combination of the distances d(x,y), d(x, Tx), d(y, Ty), d(x, Ty), d(y, Tx), ｜d(x,Tx) – d(y, Ty)｜ and ｜d(x,Ty) – d(y, Tx)｜. The asymptotic regularity of the self-mapping T on X and the convergence of Cauchy sequences to a unique fixed point are also discussed if (X,d) is complete. Subsequently, (T,S) generalized cyclic p-contraction pairs are discussed on a pair of non-empty, in general, disjoint subsets of X. The proposed contraction involves a combination of several distances associated with the (T,S)-pair. Some properties demonstrated are: (a) the asymptotic convergence of the relevant sequences to best proximity points of both sets is proved; (b) the best proximity points are unique if the involved subsets are closed and convex, the metric is norm induced, or the metric space is a uniformly convex Banach space. It can be pointed out that both metric and a metric-like (or dislocated metric) possess the symmetry property since their respective distance values for any given pair of elements of the corresponding space are identical after exchanging the roles of both elements.

Best Proximity Point Results for n-Cyclic and Regular-n-Noncyclic Fisher Quasi-Contractions in Metric Spaces

“…In fact, the researchers mentioned above have fused cyclical and noncyclic concepts of mappings with the bpp theory to solve some problems in the approximation and optimization theories. Hence, many authors are working on finding the bpp for cyclic and noncyclic mappings in various spaces in [14][15][16][17] and the references therein. Ultimately, Safari-Hafshjani et al [18] defined a Fisher quasi-contraction and studied the existence of fp(s) and bpp(s) for noncyclic and cyclic Fisher quasi-contraction mappings.…”