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

We are concerned with the existence of $T$-periodic solutions to an equation of type $${(\mathrm{\mid }{u}^{\mathrm{\prime }}(t)){\mathrm{\mid }}^{p(t)-2}{u}^{\mathrm{\prime }}(t))}^{\mathrm{\prime }}+f(u(t))u^{\mathrm{\prime }}(t)+g(u(t))=h(t)in[0,T]$$ where $p:[0,T]\to (1,\mathrm{\infty })$ with $p(0)=p(T)$ and $h$ are continuous on $[0,T]$, $f,g$ are also continuous on $[0,\mathrm{\infty })$, respectively $(0,\mathrm{\infty })$. The mapping $g$ may have an attractive singularity (i.e. $g(x)\to +\mathrm{\infty }$ as $x\to 0+$). Our approach relies on a continuation theorem obtained in the recent paper M. García-Huidobro, R. Manásevich, J. Mawhin and S. Tanaka, J. Differential Equations (2024), a priori estimates and method of lower and upper solutions.