On some geometrical techniques for the existence of best proximity pairs and applications

Abstract

A class of mappings, called relatively continuous, is introduced and incorporated to elicit best proximity pair theorems for a non-self mapping in the setting of reflexive Banach space. As a consequence we obtain a generalization of Carath´eodory extension theorem for an initial value problem with L1 functions on the right hand side. Next, to address an open problem given in [52] [MR3785465, Kirk, W. A.; Shahzad, Naseer, Normal structure and orbital fixed point conditions, J. Math. Anal. Appl. Vol. 463, no. 2, 461–476 (2018).], we give a characterization of weak normal structure using best proximity pair property. We also introduce a notion of pointwise cyclic contraction with respect to orbits and therein prove the existence of a best proximity pair in the setting of reflexive Banach spaces. For x 2 A[B; define dn = d 􀀀 Tnx;Tn􀀀1x

;n 1 where A and B are subsets of a metric space and T is a cyclic map on A[B. In this paper we introduce a new class of mappings called cyclic uniform Lipschitzian mappings for which fdng is not necessarily a non-increasing sequence and therein prove the existence of a best proximity pair. We also introduce a notion called proximal uniform normal structure and using the same we prove the existence of a best proximity pair for such mappings. Some open problems in this direction are also discussed. Further, we develop a new approach in which weak compactness is redundant to prove the best proximity pair theorems for a relatively nonexpansive mapping. We also, introduce a geometrical notion, called property wUC and use this to prove the existence of a best proximity point. Finally, we prove the existence of best proximity points for a cyclic contractive type mapping on a metric space endowed with a graph. We also provide a relation between the number of such points and the number of connected subgraphs. Furthermore, a fixed point theorem in a similar setting is proved to obtain the existence of a common solution for a system of periodic boundary value problems with the right hand side functions measurable.