Oscillatory And Asymptotic Behaviors Of Two Classes Of Higher-order Nonlinear Functional Differential Equations With Pampink

Functional differential equations are widely used in the natural sciences. Researchon oscillatory and asymptotic behaviors of functional differential equations is animportant means of knowing the behaviors of the equations without solving theequations. It has been lots of literatures and documents on studying the behaviors ofthe lower-order functional differential equations, but papers on oscillation criteria forhigher-order functional differential equations are still rare.In this paper, we are concerned with the oscillation of higher-order nonlineardelay differential equations with damping of two forms. By using comparison method,integral averaging technique and generalized Riccati transformation, we establishsome new sufficient conditions which insure that any solution of this equationoscillates or converges to zero.This article contains two chapters. In the first chapter, we study a higher-ordernonlinear delay differential equation with damping of this form:y(n+2)(t)+p (t) y (n)(t)+q (t) f (y (g (t))=0,By combining the methods used in [27] and [58], we establish three theorems whichdetermine that any solution of this equation oscillates or converges to zero in differentconditions. Our results are new.In the second chapter, we study a higher-order nonlinear delay differentialequation with damping of this form:y (n+3)(t)+p (t) y(n)(t)+q(t)f(y(g(t))=0.we establish two theorems which determine that any solution of this equationoscillates or converges to zero. The first theorem contains the conclusions of [58], thesecond theorem gives a new oscillation criteria under philos conditions.In particular, several examples are given to illustrate the importance of ourresults.