Existence and nonexistence of limit cycles for Liénard-type equations with bounded nonlinearities and φ-Laplacian

This paper deals with an equivalent system to the nonlinear differential equation of Lienard type (phi(x over dot))(.) + f(x)phi(x over dot)+ g(x) = 0, where the range of the function phi is bounded. Sufficient conditions are obtained for the system to have at least one limit cycle. The proofs of our results are based on phase plane analysis of the system with the Poincare-Bendixon theorem. Moreover, to show that these sufficient conditions are suitable in some sense, we also establish the results that the system has no limit cycles. Finally, some examples are given to illustrate our results.