On Unique and Nonunique Fixed Points in Metric Spaces and Application to Chemical Sciences

Meena Joshi,Anita Tomar,Calogero Vetro

doi:10.1155/2021/5525472

Meena Joshi, Anita Tomar + Show 1 more

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https://doi.org/10.1155/2021/5525472

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Abstract

We introduce the notions of a generalized Θ -contraction, a generalized Θ E -weak contraction, a Ψ E -weak JS-contraction, an integral-type Θ E -weak contraction, and an integral-type Ψ E -weak JS-contraction to establish the fixed point, fixed ellipse, and fixed elliptic disc theorems. Further, we verify these by illustrative examples with geometric interpretations to demonstrate the authenticity of the postulates. The motivation of this work is the fact that the set of nonunique fixed points may include a geometric figure like a circle, an ellipse, a disc, or an elliptic disc. Towards the end, we provide an application of Θ -contraction to chemical sciences.

Highlights

Introduction and PreliminariesThe study of the geometry of the set of nonunique fixed points of a map is a significant area of research
Consider a self-map M on the metric space ð U, dÞ with the usual metric defined on the two-dimensional plane R2 as ðu, vÞ, ðu, vÞ ∈ u2 + v2 = 1, Mðu, vÞ =
We are dealing with maps satisfying some novel contractions which fix one element of the space or more than one element of the space under suitable conditions and a set of nonunique fixed points, including some geometrical shapes, may be either an ellipse or an elliptic disc

Summary

Introduction

Introduction and PreliminariesThe study of the geometry of the set of nonunique fixed points of a map is a significant area of research. There are numerous examples of a map where the set of nonunique fixed points of the self-map includes some geometric shapes. Ð1Þ ð1, 0Þ, otherwise: Noticeably, the set of nonunique fixed points fðcos nθ, sin nθÞ: n ∈ Z, θ ∈ 1⁄20, 2πÞg includes the circle Cðð0, 0Þ, 1Þ centered at ð0, 0Þ having radius 1; that is, Cðð0, 0Þ, 1Þ is a fixed circle of M. Let M be a self-map on the two-dimensional plane R2 defined by Mðu, vÞ = u2 u + v2 u2 v + v2 , u, v ∈ R: ð2Þ. A geometric figure (a circle, a disc, an ellipse, and so on) included in the set of nonunique fixed points is called a fixed figure (a fixed circle, a fixed disc, a fixed ellipse, and so on) of the self-map [15]

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