Common Fixed Points of(α,η)−(θ,ϝ)Rational Contractions with Applications

Mujahid Abbas,Hira Iqbal,Manuel De La Sen

doi:10.1155/2019/8509494

Abstract

We present the notion of set valued(α,η)-(θ,ϝ)rational contraction mappings and then some common fixed point results of such mappings in the setting of metric spaces are established. Some examples are presented to support the concepts introduced and the results proved in this paper. These results unify, extend, and refine various results in the literature. Some fixed point results for both single and multivalued(θ,ϝ)rational contractions are also obtained in the framework of a space endowed with partial order. As application, we establish the existence of solutions of nonlinear elastic beam equations and first-order periodic problem.

Highlights

Introduction and PreliminariesLet (X, d) be a metric space
The widely known Banach contraction theorem [1] states that a contraction mapping on a complete metric space X has a unique fixed point; that is, there exists a point x in X such that x = Tx
Jleli and Samet [2] presented a new type of contractive mapping, namely θ-contraction mapping and established an interesting fixed point theorem for such mappings in a generalized metric space

Summary

Introduction and Preliminaries

Holds whenever H(Tx, Ty) > 0 where 0 ≤ c < 1 They established the following fixed point results for multivalued θ-contraction mappings on complete metric spaces. Parvaneh et al [15] introduced the concept of α − HΘcontraction with respect to a family of functions H and obtained some θ−contraction fixed point results in metric and ordered metric spaces They introduced the following family of functions: Let denote the set of functions F : R4 󳨀→ R+ satisfying condition (A∗): For all s1, s2, s3, s4 ∈ R+ with s1.s2.s3.s4 = 0 there exists c ∈ [0, 1) such that. We introduce multivalued (α, η)-(θ, ) rational contraction pair of multivalued mappings and prove the existence of common fixed points of the pair in a metric space. We establish the existence of solutions of nonlinear elastic beam equations and first-order periodic problem

Common Fixed Point Results

Application to Nonlinear Elastic Beam Equations

An Application to First-Order Periodic Problem