Abstract

Let A and B be two nonempty subsets of a normed linear space X. A mapping is said to be noncyclic if and In the current paper, we consider the problem of finding the best proximity pair for the noncyclic mapping T, that is, two fixed points of T which achieve the minimum distance between the sets A and B. We do it from some different approaches. The common condition on these results is relatively nonexpansivity of the mapping T. At the first conclusion, we obtain the existence of best proximity pairs in the setting of uniformly convex in every direction Banach spaces where the pair (A, B) is nonconvex. Then we conclude a similar result by replacing the geometric property of Opial’s property of the Banach space and adding another assumption on the mapping T, called condition We also show that the same result is true when X is a 2-uniformly convex Banach space. In the setting of k-uniformly convex Banach spaces, we prove that every nonempty, and convex pair of subsets has a geometric notion of proximal normal structure and then, we deduce the existence of best proximity pairs for relatively nonexpansive mappings in such spaces.

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.