Study of minimal invariant pairs for relatively nonexpansive mappings with respect to orbits

ABSTRACT

In this article, we survey the existence of best proximity pairs for noncyclic contractions with respect to orbits which are defined on a non-convex and weakly compact pair of subsets of a strictly convex Banach space. We then consider the class of relatively nonexpansive mappings with respect to orbits and present a characterization for proximal normal structure. Finally, the structure of minimal invariant pairs under relatively nonexpansive mappings with respect to orbits will be studied. Our conclusions improve and extend the well-known results in the literature.

Keywords:

• Best proximity pair
• T-regular pair
• noncyclic relatively mapping w.r.t. orbits
• strictly convex Banach space

Acknowledgments

The authors would like to thank the reviewer for his/her careful reading of the paper and many helpful comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author was partially supported by a grant from Ayatollah Boroujerdi University [grant number 15664-211960].