The method of coupled fixed points and coupled quasisolutions when working with ODE's with arguments of bounded variation

TL;DR

This paper introduces the coupled quasisolutions method for solving first-order ODEs with bounded variation arguments, utilizing Jordan decomposition and coupled fixed points of multivalued operators.

Contribution

It presents a novel application of coupled quasisolutions and fixed point theory to differential equations with functional arguments of bounded variation.

Findings

Demonstrates the effectiveness of the method on specific ODEs

Provides a framework for handling advanced arguments of bounded variation

Connects coupled fixed points with solutions of differential equations

Abstract

The aim of this paper is to show the use of the coupled quasisolutions method as a useful technique when treating with ordinary differential equations with functional arguments of bounded variation. We will do this by looking for solutions for a first-order ordinary differential equation with an advanced argument of bounded variation. The main trick is to use the Jordan decomposition of this argument in a nondecreasing part and a nonincreasing one. As a necessary step, we will also talk about coupled fixed points of multivalued operators.