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

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 dTx , Ty, where x and y are arbitrary elements in X of a weighted combination of the distances dx,y , dx,Tx,dy,Ty,dx,Ty,dy,Tx, dx,Tx−dy,Ty and dx,Ty−dy,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