Analysis of a relational fixed point problem

Binayak S Choudhury; Chaitali Bandyopadhyay; Priyam Chakraborty

doi:10.25587/2411-9326-2024-2-59-69

Choudhury Binayak S, binayak12@yahoo.co.in Indian Institute of Engineering
Bandyopadhyay Chaitali, chaitali.math@gmail.com Indian Institute of Engineering
Chakraborty Priyam, priyam.math123@gmail.com Indian Institute of Information Technology Ranchi

DOI: https://doi.org/10.25587/2411-9326-2024-2-59-69

Keywords: fixed point, MT -function, Kannan type contraction, binary relation, Hyers– Ulam–Rassias stability, data dependence

Abstract

We obtain a result by combining three prevalent trends of the ﬁxed point theory, namely (i) replacement of the Lipschitz constants in contraction inequality by functions, (ii) considerations of functions without continuity assumption and (iii) use of binary relations in the space. Speciﬁcally, we deﬁne a Mizoguchi–Takahashi–Kannan type contraction, which is shown to have ﬁxed points in a metric space with an appro- priate binary relation. The issue of the uniqueness of ﬁxed point is separately considered. There are two illustrative examples, in one of which the discontinuity of the function occurs at a ﬁxed point. We discuss Hyers–Ulam–Rassias stability of the ﬁxed point problem and also establish a data dependence result.

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