Generalized Cyclic p-Contractions and p-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 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.This research was funded by Basq ue Government, Grant number IT1555-22

On Some Properties of a Class of Eventually Locally Mixed Cyclic/Acyclic Multivalued Self-Mappings with Application Examples

In this paper, a multivalued self-mapping is defined on the union of a finite number of subsets p(≥2) of a metric space which is, in general, of a mixed cyclic and acyclic nature in the sense that it can perform some iterations within each of the subsets before executing a switching action to its right adjacent one when generating orbits. The self-mapping can have combinations of locally contractive, non-contractive/non-expansive and locally expansive properties for some of the switching between different pairs of adjacent subsets. The properties of the asymptotic boundedness of the distances associated with the elements of the orbits are achieved under certain conditions of the global dominance of the contractivity of groups of consecutive iterations of the self-mapping, with each of those groups being of non-necessarily fixed size. If the metric space is a uniformly convex Banach one and the subsets are closed and convex, then some particular results on the convergence of the sequences of iterates to the best proximity points of the adjacent subsets are obtained in the absence of eventual local expansivity for switches between all the pairs of adjacent subsets. An application of the stabilization of a discrete dynamic system subject to impulsive effects in its dynamics due to finite discontinuity jumps in its state is also discussed.Basque Government, Grant IT1555-22