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Two abstract methods of lower and upper solutions with applications

Andrei Stan

TL;DR

The paper develops two abstract, monotone frameworks for constructing lower and upper solutions to fixed point problems: a Hammerstein-type method based on a linear operator $L$ with a positive eigenpair and a monotone nonlinearity $F$, and a Harnack-type method using weak inequalities to bound solutions. Each framework yields interval invariance and, under compactness assumptions, a fixed point within the constructed bounds. The authors then illustrate the approaches with two applications: positive solutions to the classical Hammerstein integral equation and to $p$-Laplace equations, providing explicit bounds and growth conditions. This work provides general, verifiable criteria for existence and localization of positive solutions in abstract operator settings with concrete PDE and integral equation applications.

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.

Two abstract methods of lower and upper solutions with applications

TL;DR

The paper develops two abstract, monotone frameworks for constructing lower and upper solutions to fixed point problems: a Hammerstein-type method based on a linear operator with a positive eigenpair and a monotone nonlinearity , and a Harnack-type method using weak inequalities to bound solutions. Each framework yields interval invariance and, under compactness assumptions, a fixed point within the constructed bounds. The authors then illustrate the approaches with two applications: positive solutions to the classical Hammerstein integral equation and to -Laplace equations, providing explicit bounds and growth conditions. This work provides general, verifiable criteria for existence and localization of positive solutions in abstract operator settings with concrete PDE and integral equation applications.

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.

Paper Structure

This paper contains 7 sections, 10 theorems, 71 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Main abstract results
  3. Abstract Hammerstein equations
  4. Fixed point equations via abstract Harnack inequality
  5. Applications
  6. Positive solution of the classical Hammerstein equation
  7. Positive solutions of $p$-Laplace equations

Key Result

Theorem 1

Let $X$ be a Banach space, $K \subset X$ a total cone, and $F$ a linear compact operator with $F(K) \subset K$ and the spectral radius $r(T)$ strictly positive. Then, $r(T)$ is an eigenvalue of $F$ and the corresponding eigenvector lies in the cone $K$.

Theorems & Definitions (23)

  • Theorem 1: Krein-Rutman
  • Theorem 2
  • Remark 1
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Remark 2
  • Theorem 5
  • proof
  • ...and 13 more