Existence and uniqueness of damped solutions of singular IVPs with $\phi$-Laplacian

Abstract

We study analytical properties of a singular nonlinear ordinary differential equation with a $\phi$-Laplacian. In particular we investigate solutions of the initial value problem $$ (p(t)\phi(u'(t)))'+ p(t)f(\phi(u(t)))=0, \quad u(0)=u_0 \in [L_0,L],\quad u'(0)=0 $$ on the half-line $[0,\infty)$. Here, $f$ is a continuous function with three zeros $\phi(L_0)<0<\phi(L)$, function $p$ is positive on $(0,\infty)$ and the problem is singular in the sense that $p(0)=0$ and $1/p(t)$ may not be integrable on $[0,1]$. The main goal of the paper is to prove the existence of damped solutions defined as solutions $u$ satisfying $\sup \{u(t), t\in [0,\infty)\}<L$. Moreover, we study the uniqueness of damped solutions. Since the standard approach based on the Lipschitz property is not applicable here in general, the problem is more challenging. We also discuss the uniqueness of other types of solutions.

Keywords

• second order ode
• time singularity
• existence and uniqueness
• $\phi$-laplacian
• damped solution
• half-line