Two abstract methods of lower and upper solutions with applications

TL;DR

This paper introduces two abstract methods for constructing lower and upper solutions to fixed point equations, applicable under different conditions on the nonlinear operator, with practical applications demonstrated for each method.

Contribution

The paper presents novel abstract methods for lower and upper solutions applicable to fixed point equations with specific operator structures.

Findings

Methods successfully construct bounds for fixed point solutions.

Applications demonstrate practical utility of the methods.

Methods extend existing techniques for nonlinear operator equations.

Abstract

In this paper, we present two abstract methods for constructing a lower and an upper solution for a fixed point equation. The first method applies when the nonlinear operator is a composition of a linear and a nonlinear mapping, while the second method applies when the nonlinear operator satisfies an inequality of Harnack type. An application is provided for each method.