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By means of topological theory, a class of Liénard type equations with a deviating argument of the form \((\varphi_{p}(x'(t)))'+f(x(t-\tau))x'(t-\tau)+\beta(t)g(x(t-\tau))=e(t)\) is studied. It is notable that the coefficient \(\beta(t)\) in front of \(g(x(t-\tau))\) is allowed to change sign in this paper. Moreover, a numerical simulation is carried out to verify the validity of the obtained results. In addition, the generalized form of the above equation with time-varying delays is also discussed briefly.

Periodic solutions for Liénard type equation with time-variable coefficient