On Stability And Oscillation Of Solutions Of Ordinary Differentiail Equation:

Nahed Abd El-fatah Mohamady

In this thesis, we discuss the oscillatory behavior of solutions of the second order neutral delay differential equation of the formr m(r(t)z’(t)) + q .(t)f (x(o- .(t))). , t ?to,J =1whereli n .z(t) = x(t) + E p i(t)x(r i(0), 0 5_ p i(t) 5 J . po < co, ,.,i =1(t) dt = co .We introduce new sufficient conditions for oscillation of solutions of the following second order nonlinear neutral differential equations:r , -,( kr (t) po(t)x(t)+ E p i(t)x(t — t i)i =1k+ E q .(t)f . (x(t — to)) = 0, t J =1andnr (t)(130(t)x(t)+ pi(t)x(o-(o) I q. (t) f (x(i- . (t))) = 0, t t„.i=1 1 kj =1Further, we discuss the oscillatory behavior of the second order quasilinear neutral delay differential equations(r(t)41 (x(t))I z’(t)la 1 zr(t)) + q (t) f (x (t))) = 0, t > t0i =1whereand the second order nonlinear neutral differential equations with deviating arguments of the form:(r(t)lz”(tr 0)1 ± f j (t ,x(c (t)))=0, t tJ =1wherez(t) = x(t) + E p i(t)x( (t)) and a ? O.i=1Moreover, we investigate the oscillation of the second order nonlinearneutral differential equations of the form:I r 11(r(t)tif (x(t))(x(t) + p(t)x(o-(t)))) +q (t) f (x(t),x(r (t)))=0, t r°.j =1Finally, we discuss the stability character of the second order nonlinear differential equation of the form:x”+ h(t , x’) + x + g(t,x)=0, t E R.The obtained