Qualitative Research For Several Kinds Of Differential Equations

This paper studies the existence of periodic solutions for a kind of second order neutral differential equation, the existence of positive solutions of a fourth-order boundary value problem and the oscillation for second order nonlinear 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 following nonlinear neutral delay equation[x(t) + kx(t -Ï„)]" = f(x(t -T₁),x'(t - Ï„₂)) +P(t).By using the Mawhin's continuation theorem, several sufficient conditions for the existence of the equation are given.In chapter three, we apply the degree theory to discuss the existence for positive solution of the nonlinear fourth-order two-point boundary value problemand some sufficient conditions are established.In the last chapter, we investigate oscillation for solutions of second order nonlinear differential equationand(r(t)Ψ(x(t))x'(t))' +p(t)x'(t) +F(t,x(t),x(T(t)),x'(t),x'(Ï„{t)) = 0.Some new oscillative criteria are obtained.