A Fixed Point Technique for Set-Valued Contractions with Supportive Applications

Hasanen A Hammad,Manuel De La Sen

doi:10.1155/2021/6880478

Abstract

In this manuscript, exciting fixed point results for a pair of multivalued mappings justifying rational Gupta-Saxena type Ω -contractions in the setting of extended b -metric-like spaces are established. The theoretical results have also been strengthened by some nontrivial examples. Finally, the theoretical results are used to study the existence of the solution of Fredholm integral equation which arises from the damped harmonic oscillator, to study initial value problem which arises from Newton’s law of cooling and to study infinite systems of fractional ordinary differential equations (ODEs).

Highlights

The importance of fixed-point (FP) theory increased afterBanach introduced his principle [1]
We suggest the reader to read the books of Kilbas et al [2], Samko et al [3], Wang et al [4], and Atangana and Baleanu [5]. This methodology is widely used in finding solutions of integral equations, fractional differential equations, and boundary value problems, for example, see [6,7,8,9,10,11]
The theoretical results are applied to study the existence of the solution to the Fredholm integral equation which arises from the damped harmonic oscillator, to study the initial value problem (IVP) which arises from Newton’s law of cooling and to study infinite systems of fractional ordinary differential equations (ODEs)

Summary

Introduction

Banach introduced his principle [1]. It became an essential tool in nonlinear analysis. We suggest the reader to read the books of Kilbas et al [2], Samko et al [3], Wang et al [4], and Atangana and Baleanu [5] This methodology is widely used in finding solutions of integral equations, fractional differential equations, and boundary value problems, for example, see [6,7,8,9,10,11]. In 2012, Wardowski [13] generalized the old principle of Banach to a broader kind He called it Ω-contraction and presented it in the following definition: Definition 1. The theoretical results are applied to study the existence of the solution to the Fredholm integral equation which arises from the damped harmonic oscillator, to study the initial value problem (IVP) which arises from Newton’s law of cooling and to study infinite systems of fractional ODEs

Preliminaries

Main Theorem

Illustrative Examples

Newton’s Law of Cooling