The Boundedness And The Asymptotic Of A Class Of Differential Equations

In this paper, we mainly discuss the boundedness of a class of second order differential equations and the asymptotic behavior of solutions of higher order integro-differential equations with deviating argument.The thesis is divided into three sections according to the contents.In the first chapter, the historical background and research status of these problem are introduced, in addition three lemmas-of proving problem in the paper are listed.In the second chapter, with the aid of the new integral inequality, with suitable assumptions, we mainly discuss the boundedness of solutions of second order differential equations, the conclusions are as follows:Theorem2.2.1Assume that:(i) f∈(R×R3,R), we have where0＜p≤1, e1∈(R+, R+),ai∈(R+,R+),i=1,2,3;(ii) where0×q≤1,e2, e3∈(R+, R+);ki∈(R2+, R+),i=1,2are continuous and nondecreasing in t for each fixed s∈R+;(iii) and are bounded in R+;(ⅳ)(?)R(t)＜∞, then each solution x(t) of the equation(2.1.1)is exist and bounded in R+Theorem2.2.2Assume that:(ⅰ) f∈(R×R3,R), where0＜p≤1,e1∈(R+,R+),ai∈(R+,R+),i=1,2,3；(ⅱ) where0＜q≤1,e2,e3∈(R+,R+),ki∈(R+2,R+),i=1,2are continuous and nondecreasing in t for each fixed s∈R+(ⅲ) and are bounded in R+are bounded in R+(ⅳ)limt→∞R(t)=∞, Then we have|x(t)|=O(R(t)),|(r(t)ψ(x(t))(x(t)))’|=O(1), t∈R+ In the third chapter,with Suitable assumptions,we mainly discuss the asymp-totic behavior of solutions of higher order integro-differential equations with de-viating argument, t∈[0,∞),the consultion as follows:Theorem3.3.1Assume:(i).f∈R+,u1,u2,…,un,v1,v2,…,vn,vn+1∈R,we have where e1,ai,bj:R+→R+are constant,and ri,pj are constant in(0,1],1,2,…,n,j=1,2,…,n；(ii)for t,s∈R+,u1,u2,…,un,v1,v2,…,vn∈R,we have where e2,e3:R+→R+are constious, ci(t,s),dj(t,s):R2+→R+are constious and nondecreasing in t for each fixed s∈R+； qi,sj are constant in(0,1]；(iii)The functions are differentiable in the class L1[0,∞)(iv)ast→∞,the following integrals are bounded:Then for any intial function θ(t)defined on [r,0]∪R+,there is a solu-tion x(t)of(3.1.1),which can be written like satisfying the interval condition x(t)=θ(t),f∈[r,0],where t→the limits0f∑in:1Ai(t)(i=1,2,…,n)are existed,and satisfieded(3.1.7).