Qualitative Behaviours Of Solutions For Some Differential Equations

This paper includes three chapters.The first chapter is overview ,in which we introduce the some important results and developing situation about existence and uniqueness of limit cycles for Lienard systemand equation x+ f(x, x) x+ g(x) = 0 (E 3)Otherwise ,we introduce some study situation about oscillatory and asymptotic behavior of solutions of second order nonlinear differential equation(a(t)(y'(t) )' + q(t)f(y(t)) = 0, t t0 (E 4)In the second chapter ,the existence and uniqueness of limit cycle of equations (E1 ) and (E2) are discussed .we gave some new sufficient conditions about acertain negative Half-trajectory L- which did not cross with the line when inclning and a certain positive half-trajectory which turn around origin .Specially ,some sufficient conditions are given under the conditions forlim F(x) = + or lim F(x) = - and G(- )<+ or G(+ )<+ and so forth .Using comparison theorem of differential equation.we built a certain controlled equation and obtain an important conclusion .The many results of [3]~[12] are generalized and its conditions improved greatly.In thethird chapter .when 0< = even/odd the oscillatory and asymptotic behavior ofsolution of the equation (E4) are discussed .The major results of [15] , [16]are generalized and its conditions improved.Some new conclusions are obtained.