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A Unified Topological Analysis of Variable Growth Kirchhoff-Type Equations

Christopher S. Goodrich, Gabriel Nakhl

Abstract

We consider a nonlocal differential equation of Kirchhoff type with a convolution coefficient involving variable growth. The novelty of our work lies in allowing a variable exponent in the nonlocal term. By relating the variable growth problem to a corresponding constant growth problem, we establish the existence of at least one positive solution subject to boundary conditions. Our approach relies on topological fixed point theory. The results treat convex, concave, and mixed growth regimes, providing a unified framework for one-dimensional Kirchhoff-type problems.

A Unified Topological Analysis of Variable Growth Kirchhoff-Type Equations

Abstract

We consider a nonlocal differential equation of Kirchhoff type with a convolution coefficient involving variable growth. The novelty of our work lies in allowing a variable exponent in the nonlocal term. By relating the variable growth problem to a corresponding constant growth problem, we establish the existence of at least one positive solution subject to boundary conditions. Our approach relies on topological fixed point theory. The results treat convex, concave, and mixed growth regimes, providing a unified framework for one-dimensional Kirchhoff-type problems.
Paper Structure (4 sections, 21 theorems, 138 equations)

This paper contains 4 sections, 21 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Existence Theory for \ref{['eq1.1']} in Case $p(x)>1$
  3. Existence Theory for \ref{['eq1.1']} in Case $0<p(x)\le1$
  4. Existence Theory for \ref{['eq1.1']} in Case $0<p^-< 1 < p^+$

Key Result

Lemma 2.1

Let $f\ : \ [0,1]\rightarrow[0,+\infty)$ be given. Suppose that $p\ : \ [0,1]\rightarrow(1,+\infty)$ satisfies condition (H2). Then, given any constant $q$ satisfying $1\le q<p^-$, for each $t\in[0,1]$ it holds that

Theorems & Definitions (42)

  • Lemma 2.1
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 32 more