Abstract. The purpose of this paper is to give sufficient conditions under which an equivalent system to the equation (φp(ẋ))˙+ f(x)φp(ẋ) + g(x) = 0 has at least one stable limit cycle, where φp(·) is the one-dimensional p-Laplacian. The main results are proved by means of phase plane analysis with the Poincaré-Bendixson theorem. Sufficient conditions are also given for the origin (x,ẋ) = (0, 0) to be unstable and for all solutions to be bounded in the future.2000 Mathematics Subject Classification: 34C05, 34C07, 70K05, 70K42.

This paper is concerned with the asymptotic behavior of solutions of a class of second-order half-linear differential equations of the form ( ϕp ( ẋ )) . + a ( t ) ϕp ( ẋ ) + b ( t ) ϕp ( x ) = 0. The main purpose of this paper is to answer the question of how every solution approaches zero, under the assumption that the zero solution is globally asymptotically stable. Sufficient conditions are also given for the zero solution to be globally asymptotically stable. Moreover, an autonomous case is investigated in full detail and a geometrical classification is made based on the asymptotic behavior of solutions. The method used here is mainly phase plane analysis for a system equivalent to the half-linear differential equations. Some suitable examples are included to illustrate the main results. Global phase portraits are also attached for a deeper understanding.