Research Of Oscillation Of Several Classes Of High Order Dynamic Equations

In this thesis,we study the oscillation of two classes of higher order dynamicequations and the existence of nonoscillatory solutions of a class of higher order dynamic equation,and some corresponding results are general-ized.Firstly,we investigate the following higher order dynamic equationand get some conditions under which every solution of this equation is either oscillatory or tends to zero,where f,n ≥ 2 are positive integers,γ,αi(0 ≤i≤k)are the quotient of two odd positive integers,S0(t)= x(t),Sl(t)-= al(t)SlΔ-1(t),r,qi,al ∈ Crd(T,(0,∞)),δi ∈ Crd(T,T),(0≤i ≤ k,1 ≤ l≤ n-1),Φp(u)= |u|p-1u(p>0 is a real).Secondly,we study the following higher order dynamic equation SnΔ(t,x(t))+f(f,x(δ(t))= 0,and get some conditions under which every solution of this equation is either oscillatory or tends to zero,where n>2 is an integer,αk{1 ≤ A ≤ n)are the quotient of two odd positive integers,S0(t,x(t))= x(t)-p(t)x(τ(t)),Sk(t,x(t))=ak(t)(SkΔ-1(t,x(t))αk,ak∈Crd(T,(0,∞)),(1≤k≤n),P∈ Crd(T,R),δ,τ∈Crd(T,T).At last,we study the following higher order equation SnΔ(t,x(t))+f(t,x(h(t))= 0,and obtain some conditions for the existence of non-oscillatory solutions of this equation,where n>2 is an integer,1≤ k ≤ n)are the quotien-t of two odd positive integers,S0(t,x(t))-x(t)+P(t)x(τ(t)),Sk(t,x(t))＝ak(t)Φak(SkΔ-1(t,x(f))),ak Crd(T,(0,∞)),(1<k<n),p∈Crd(T,R),h,τ∈Crd(T,T).