Fixed Point Theorems for Nonexpansive Mappings under Binary Relations

Abstract

In the present article, we establish relation-theoretic fixed point theorems in a Banach space, satisfying the Opial condition, using the R-Krasnoselskii sequence. We observe that graphical versions (Fixed Point Theory Appl. 2015:49 (2015) 6 pp.) and order-theoretic versions (Fixed Point Theory Appl. 2015:110 (2015) 7 pp.) of such results can be extended to a transitive binary relation.

Solving a Boundary Value Problem via Fixed-Point Theorem on ®-Metric Space

“…Ali, Imdad and Sessa [7] proved fixed-point theorems on -complete regular symmetric spaces. Alam, George, Imdad and Hasanuzzaman [8] proved fixed-point theorems for nonexpansive mappings under binary relations. Javed, Arshad, Baazeem and Nabil [9] proved fixed-point theorems on -complete metric spaces.…”