Periodic solutions for a Rayleigh type equation with a variable coefficient ahead of the nonlinear term

Using the Manásevich–Mawhin continuation theorem, a class of Rayleigh type equations with a deviating argument of the form (φp(x′(t)))′+f(x′(t))+β(t)g(x(t−τ(t)))=e(t)(φp(x′(t)))′+f(x′(t))+β(t)g(x(t−τ(t)))=e(t) is studied. It is notable that the coefficient β(t)β(t) ahead of g(x(t−τ(t)))g(x(t−τ(t))) is allowed to change sign in this paper, which could be achieved infrequently in previous papers.