Extended Reich type mappings with multiple fixed points

doi:10.1007/s11784-026-01299-3

Extended Reich type mappings with multiple fixed points

2026-05-13

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Bisht, Ravindra K. *

DOI：10.1007/s11784-026-01299-3

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In 1971, Reich (Some remarks concerning contraction mappings. Can Math Bull 14:121-124, 1971) unified the Banach contraction principle and the Kannan contraction theorem by introducing a new class of contractions, now known as Reich contractions, which ensure the existence and uniqueness of a fixed point in a metric space. His condition permits discontinuity in the domain of definition but requires the self-mapping to be continuous at the fixed point. In this paper, we extend Reich's fixed point theorem to include both contractive and non-expansive mappings in metric spaces. This extension admits the possibility of multiple fixed points, even when a fixed point is itself a point of discontinuity. The resulting structured fixed-point sets and domains exhibit rich algebraic, geometric, and dynamical structures. Our theorem thus provides a broad generalization of several well-known results on contractive mappings. As a byproduct, we present a new solution-distinct from the known one-to the problem posed by Rhoades (Contractive definitions and continuity. Contemp Math 72:233-245, 1988) concerning the existence of contractive mappings that admit a fixed point which is also a point of discontinuity.

Keywords:

Fixed point

non-expansive mapping

n-th roots of unity

Extended Reich type mappings with multiple fixed points

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