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

We are concerned with the existence of T-periodic solutions to an equation of type (|u'(t))|ᵖ⁽ᵗ⁾⁻² u'(t) )'+f(u(t))u'(t)+g(u(t))=h(t) in [0,T] where p:[0,T]→(1,∞) with $p(0)=p(T)$ and h are continuous on $[0,T]$, $f,g$ are also continuous on [0,∞), respectively (0,∞). The mapping g may have an attractive singularity (i.e. g(x) → +∞ as x→ 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.