A Dirichlet problem on the half-line for nonlinear equations with indefinite weight

TL;DR

The paper addresses the Dirichlet problem on the half-line for the nonlinear equation $(a(t)x')'+b(t)F(x)=0$ with $x(0)=0$ and $x(t)\to0$ as $t\to\infty$, allowing $b$ to change sign and $F$ to be asymptotically linear at both ends. The authors develop a two-step strategy: first solve auxiliary BVPs on $[0,1]$ (with $b\ge0$) via a shooting/continuation framework, then solve an auxiliary BVP on $[1,\infty)$ using a Fréchet-space fixed-point approach informed by principal solutions and disconjugacy of the linear comparison equation $v''+\frac{B}{a(t)}v=0$. They prove the existence of a globally positive, decreasing solution on $[1,\infty)$ that decays to zero and can be pasted to a solution on $[0,1]$ to yield a global solution on $[0,\infty)$ satisfying the boundary conditions, under explicit technical conditions involving $A(t)=\int_0^t\frac{1}{a(s)}ds$, $A(1)$, $|b|_{L^1[0,1]}$, and the asymptotic ratios $k_0,k_\infty$ of $F$. The results accommodate periodic or unbounded weights and are illustrated by concrete examples, contributing a robust half-line framework that extends compact-interval sign-indefinite-weight results to the unbounded domain via disconjugacy and a Schauder-type fixed-point construction.

Abstract

We study the existence of positive solutions on the half-line $[0,\infty)$ for the nonlinear second order differential equation \[ \bigl(a(t)x^{\prime}\bigr)^{\prime}+b(t)F(x)=0,\quad t\geq0, \] satisfying Dirichlet type conditions, say $x(0)=0$, $\lim_{t\rightarrow\infty}x(t)=0$. The function $b$ is allowed to change sign and the nonlinearity $F$ is assumed to be asymptotically linear in a neighborhood of zero and infinity. Our results cover also the cases in which $b$ is a periodic function for large $t$ or it is unbounded from below.

Theorems & Definitions (13)

• Theorem 1
• Lemma 1
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• Proposition 1
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• Lemma 5
• Theorem 2
• Theorem 3
• Theorem 4