Research On Oscillation For Several Classes Of Functional Differential Equations

This paper studies the oscillation of three classes of differential equations.In chapter one, we introduce the history and development of the subject and some necessary preliminary knowledge of this paper.In chapter two, we discuss the nonlinear higher order functional equationsX⁽ⁿ⁾(t)+p(t)f(t,x(t),x^(n-1)(t)x)^(n-1)(t)-q(t)x(s)^λsgn x(t)=m(t).Some new oscillation criterions of the equation are established. Using the same technique, we establish oscillation criterion for a certain forced nonlinear neutral equation[x(t)+cx(t-Ï„)]⁽ⁿ⁾+a(t)x(t)+b(t)x(t-Ï„)=m(t)+q(t)∣x(t)∣^λsgnx(t)+α(t)∣x(t-Ï„)∣^σsgnx(t-Ï„).In chapter three, the oscillation criterion for the solutions of the following neutral functional differential equations with positive and negative coefficients are investigated:[x(t)-R(t)x(t-r)]'+P(t)x(t-σ)-Q(t)x(t-Ï„)=0Some new sufficient conditions for all solutions of the equations to be oscillatory are obtained.In chapter four, we consider certain second order nonlinear neutral differential equation[x(t)+p(t)x(t-Ï„)]" +q(t)f(x[g₁(t)],x[g₂(t)],A ,x[g_m(t)])=0.Some sufficient conditions for all solutions of the equations to be oscillatory are obtained.