On The Asymptotic Behavior Of A Class Of Liénard Equation

In this paper, based on some previous results, we study a kind of Liénard Equation u′′(t ) +u′(t ) +c|u(t)|^1+δ=0 and the asymptotic behavior of the solutions of the equation with the initial state in the entire phase plane by phase plane analysis. First of all, we analyse the unique maximal global negative solution u ? (t )∈C2 (t ?*,?∞) and the unique global finally positive solution u + (t )∈C2 (t +*,+∞) and the properties of the solutions in phase plane. Further we study and prove the asymptotic behavior of an arbitrary solution with the initial state (u₀,u₁)∈R². Then we turn out an approximation process to find a function sequence {δ_n(u )}, which is convergent uniformly to a certain functionδ(u ), to find the approximation formulae ofΓ₁ ={( u_t, u_t′) t∈(t_<sup>*,+∞)} in the phase plane. By the conclusions and theorem proved above, we have a qualitative analysis of the asymptotic behavior of the solution of the equation u′′(t ) +u′(t ) +c|u(t)|^1+δ=0with the initial state p₀ = (u (0 ), u′(0 )) in phase plane, and come to a convergence of regional as well as offered more information for the asymptotic behaviors of a kind of Liénard Equation than that obtained in the previous papers. At last we have a simple study with the asymptotic behavior of the Liénard Equation like u′′(t ) ?u′(t ) +c|u(t)|^1+δ=0. By translating t to -Ï„, we can convert u′′(t ) -u′(t ) +c|u(t)|^1+δ=0 to u′′(t ) +u′(t ) +c|u(t)|^1+δ=0 to analyse and construction a similar function sequence. So we can get the results of u′′(t ) -u′(t ) +c|u(t)|^1+δ=0 by using the results of u′′(t ) +u′(t ) +c|u(t)|^1+δ=0.The results obtained in this paper is of some use in studying the open problem raised by A.Haraux and M.A.Jendoubi in paper [18]. Moreover, we believe that our results are important in searching for the ideal sliding modes for a certain control system with variable structure.