A Class With Deviating Arguments Asymptotic Behavior Of Higher Order Differential Equations And Integral Differential Equations

In this paper,we mainly discuss the asymptotic behavior of solutions of higher order differential equations with deviating argument and the asymptotic behavior of solutions of higher order integro-differential equations with deviating argument.This paper consists of three chapters.In the first chapter,the historical background and research status of these problem are introduced,in addition two lemmas-of proving problem in the paper are listed.In the second chapter,with suitable assumptions,we mainly discuss the asymptotic behavior of solutions of higher order differential equations with devi-ating argument, (2.1.1) the conclusions are as follows:Theorem 2.3.1 Assume that(ⅰ).for t∈R+,u1,u2,…,un∈R, v1, v2,…, vn∈R we have where el, bi, cj:R+→R+ are continuous,and ri, pi in (0,1] are constants.(ⅱ):The fuctions are in the class Ll(0,∞).Then for any initial functionθ(t) defined on [γ,0]∪R+ there is a solu-tion y(t) of (2.1.1),which can be written like (2.3.1) satisfying the interval condition y(t)=θ(t),t∈[γ,0],when t→∞,the limits of are existed,and satisfied(2.1.4).In the third chapter,with sujitable assumptions,we mainly discuss the asymp-totic behavior of solutions of higher order integro-differential equations with de-viating argument, (3.1.1) the consultion as followsTheorem 3.3.1 assume:(i).for t∈R+,u1,u2,…,un,v1,…,vn,vn+1∈R,we have where el and bi,cj:R+→R+are constious, and ri,pj in(0,1]are constants,i=1,2,…,n,j=1,2,…,n.(ii).for t,s∈R+,u1,u2,…,un∈R,we have where e2 and e3:R+} R+ are constious,ai(t,s):R+2→R+ are contious and nondecreasing,whhere s∈R+,qi are constants in(0,1].(iii):The fuctions are in the class L1(0,∞). (iv):as t∈∞,the following integrals are boundedThen for any initial functionθ(t) defined on [γ,0]∪R+ there is a solution y(t) of (3.1.1),which can be written like (3.3.1) satisfying the interval condition y(t)=θ(t),t∈[γ,0],where t→∞,the limits of n are existed,and satisfied (3.1.4).