Solution of Some Impulsive Differential Equations via Coupled Fixed Point

Abstract

In this article, we employ the notion of coupled fixed points on a complete b-metric space endowed with a graph to give sufficient conditions to guarantee a solution of system of differential equations with impulse effects. We derive recisely some new coupled fixed point theorems under some conditions and then apply our results to achieve our goal.

Coupled Fixed Points for Hardy–Rogers Type of Maps and Their Applications in the Investigations of Market Equilibrium in Duopoly Markets for Non-Differentiable, Nonlinear Response Functions

“…The first result [1] deals with maps with the mixed monotone property in complete, partially ordered metric spaces. There are a great number of contemporary research papers on the theory of coupled fixed points with the mixed monotone property [2], without the mixed monotone property [3], in ordered probabilistic metric spaces [4], in modular metric spaces [5], in metric spaces endowed with a graph [6], fuzzy cone metric spaces [7], in b-metric spaces [8], and for multi-valued maps [9] . The idea of coupled fixed points was generalized for coupled best-proximity points [10].…”

“…The idea of coupled fixed points was generalized for coupled best-proximity points [10]. There are applications of coupled fixed points in different fields of mathematics-impulsive differential equations [6], integral equations [11], ordinary differential equations [12], periodic boundary value problems [13], fractional equations [14], and nonlinear matrix equations [15]-and in other scienceseconomics [16,17], aquatic ecosystems [18], and dynamic programming [19]-with these just being the most recent investigations dealing with coupled fixed points.…”

Solution of differential equations with infinite delay via coupled fixed point

“…Following this line, many coupled fixed point results in metric space endowed with a directed graph were considered by many authors, for example see [1,9,11,12,14].…”

Coupled Fixed Points for Hardy-Rogers Type of Maps and Their Applications in the Investigations of Market Equilibrium in Duopoly Markets for Non-Differentiable, Nonlinear Response Functions

“…The first result [13] deals with maps with the mixed monotone property in complete partially ordered metric spaces. There is a great number of conteporary research papers in the theory of coupled fixed points with the mixed monotone property [26], without the mixed monotone property [7], in ordered probabilistic metric spaces [30], in modular metric spaces [28], in a metric spaces endowed with a graph [3], fuzzy cone metric spaces [32], in b-metric spaces [19], for multi-valued maps [12] . The idea of coupled fixed points was generalized for coupled best proximity points [29].…”

“…The idea of coupled fixed points was generalized for coupled best proximity points [29]. There are applications of coupled fixed points in different fields of mathematics: impulsive differential equations [3], integral equations [21], ordinary differential equations [20], periodic boundary value problem [9], fractional equations [23], nonlinear matrix equations [27] in other sciences: economics [8,16], aquatic ecosystems [11], dynamic programming [5], which are just the most recent investigation that are dealing with coupled fixed points.…”