Periodic solutions for p(t)-Lienard equations with a singular nonlinearity of attractive type

TL;DR

This paper establishes the existence of periodic solutions for a class of p(t)-Lienard equations with singular nonlinearities, using continuation theorems, a priori estimates, and lower-upper solutions.

Contribution

It extends the theory of periodic solutions to nonlinear differential equations with singularities and variable exponents, employing a novel combination of continuation methods and a priori bounds.

Findings

Proves existence of T-periodic solutions under certain conditions.

Handles singular nonlinearities where g(x) tends to infinity as x approaches zero.

Utilizes a continuation theorem from recent literature and lower-upper solution techniques.

Abstract

We are concerned with the existence of $T$-periodic solutions to an equation of type $$\left (|u'(t))|^{p(t)-2} u'(t) \right )'+f(u(t))u'(t)+g(u(t))=h(t)\quad \mbox{ in }[0,T]$$ where $p:[0,T]\to(1,\infty)$ with $p(0)=p(T)$ and $h$ are continuous on $[0,T]$, $f,g$ are also continuous on $[0,\infty)$, respectively $(0,\infty)$. The mapping $g$ may have an attractive singularity (i.e. $g(x) \to +\infty$ as $x\to 0+$). Our approach relies on a continuation theorem obtained in the recent paper M. Garc\'{i}a-Huidobro, R. Man\'{a}sevich, J. Mawhin and S. Tanaka, J. Differential Equations (2024), a priori estimates and method of lower and upper solutions.