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.
Highlights
An analogue of the classical Banach contraction principle employing a partial ordering on underlying complete metric space was initiated by Ran and Reurings [1], which was further refined by Nieto and Rodríguez-López [2]
The involved contraction conditions remain relatively weaker than the usual contraction conditions, as these are required to hold merely for those elements which are related in the underlying binary relation
Alam et al [47] initiated the concept of R-nonexpansive mappings and utilized the same to extend the results of Bin Dehaish and Khamsi [42] up to the transitive binary relation, obtaining a relation-theoretic analogue of the classical Browder-G’ohde fixed point theorem
Summary
An analogue of the classical Banach contraction principle employing a partial ordering on underlying complete metric space was initiated by Ran and Reurings [1], which was further refined by Nieto and Rodríguez-López [2]. It is worth mentioning that Alfuraidan [37] used the transitivity of graph G to construct the Krasnoselskii sequence but failed to mention it In this continuation, Bachar and Khamsi [39] obtained a natural version of Theorem 1 by assuming a partial order instead of a graph and utilized the same to obtain nonnegative and nonpositive solutions of an integral equation. The idea of a monotone nonexpansive mapping in the context of graph as well as partial ordering is generalized and extended by several authors, such as [40–46]. Alam et al [47] initiated the concept of R-nonexpansive mappings and utilized the same to extend the results of Bin Dehaish and Khamsi [42] up to the transitive binary relation, obtaining a relation-theoretic analogue of the classical Browder-G’ohde fixed point theorem. The boundedness of whole set K must be replaced by the relatively weaker assumption
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