论文标题:多滞量中立型微分系统正解的存在性和振动性及相应的脉冲系统的渐近性
The Existence of Positive Solution and Oscillation of Neutral Differential System with Multi-delays and Asymptotic Behavior of Corresponding Impulsive System
论文作者 陈春涛
论文导师 罗桂烈；杨启贵,论文学位 硕士,论文专业 基础数学
论文单位 广西师范大学,点击次数 76,论文页数 35页File Size215k
2005-03-01论文网 http://www.lw23.com/lunwen_105890772/ 中立型方程; 正解; 非线性; 振动性; 脉冲; 渐近性
Neutral equation; Positive solution;Nonlinear; Oscillation;Asymptotic behavior
近年来,对中立型时滞微分方程的研究受到诸多学者的关注,关于方程的振动性,渐近性,稳定性等等,都取得了大量的成果.但大多数研究的是线性中立型方程,对于非线性情形的研究并不多见,特别是含有多个时滞的,而且对方程的正解的存在性研究更是少见. 本文研究多滞量的中立型微分方程的正解的存在性,解的振动性及脉冲中立型微分方程的渐近性.文章的构成如下,在第一章中,我们使用文[4]的思想,运用Banach 压缩映象原理得到了这类方程正解存在性的充分条件.此外,在讨论方程的振动性的时候,我们分三种情形得到了方程振动的充分性判据.在第二章中,我们讨论了相应的脉冲中立型微分方程的渐近性,分别得到了方程的振动解和非振动解趋于零的充分条件.由于方程中函数f 的广泛性,所得的这些结果分别应用于线性的情形同样成立,从而使得方程的应用更为广泛. 第一章多滞量中立型时滞微分系统的正解存在性及振动性 在这一章中,我们讨论中立型时滞微分方程:
In recent years, delay differential equations of neutral type have been concerned by many authors. Many results have been gain about the equation’s oscillation, asymptotic behavior and stability, etc. But most of them are devoted to linear neutral equations, the study about nonlinear equations is not much.. Especially for the existence of positive solutions of equations with many delays. In this paper, we first study the existence of positive solution and oscillation of neutral differential equation with multi-delays. After that, we study asymptotic behavior of neutral delay differential equation with impulses. This paper is organized as follows. In Chapter one, we refer to paper [4] and obtain some sufficient conditions for the existence of positive solution of the multi-delays differential equation by the Banach contraction mapping principle. Furthermore, when we discuss the oscillation of above equation, we divide three situations to gain the result. In Chapter two, we discuss the asymptotic behavior of solutions of corresponding neutral delay differential equation with impulses and obtain some sufficient conditions about the nonoscillatory solutions and oscillatory solution tend to zero. With the extensively of the function f in the equation, all these results can also be utilized for linear equations, thus make the application of the equation more extensively. ChapterⅠThe Existence of Positive Solution and Oscillation of Neutral Delay Differential System with Multi-delays In this chapter, we discuss the existence of positive solution and oscillation of neutral differential equation: where p , r , qi (i = 1,2, …, n ) ∈C ([t_0, ∞),R),and p ,r is derivable; σ, ρ, τ_i (i = 1,2, L…n) ∈(O, ∞), O < τ_1 < τ_2<…<τ_n, f ∈C ( R , R), f (O) = 0,and let m = max{σ, ρ, τ_n}. Main results: Theorem1 Suppose p (t), p′(t), r(t ), r′(t), q_i (t)(i=1,2,…, n) is boundary on [t_0, ∞),furthermore, the following conditions hold ( ) ( )01( ) ( ) ( ) ( ) ( ) ,iniip t μp t e μσr t μr t e μρq t e μτμt t=′+ + ′+ + ∑≤≥(1.2) then equation(1.1)has positive solution. Theorem2 Assume p (t ), p ′( t ), r (t ), r ′( t ), qi (t )(i = 1,2, L , n) is boundary on [t 0, ∞),and condition ( A1 )hold.If there exist t ? ≥t 0, μ> 0,for any λ∈Λ= {λ: λ(t ) = 0, t ∈(t ? ? m, t ?), λ(t )is continuous on[t ? , ∞)and 0 ≤λ(t )≤μ}, the following inequalities hold: [ ] ( ) [ ] ( )( ) ( ( ))10 ( ) ( ) ( ) exp ( ) ( ) ( ) ( ) exp ( )( )exp ( ) exp ( ) ,it tt tn t tii t tp t p t t s ds r t r t t s dsq t s ds f s ds t tσρτλσλλρλλλμ? ?? ?? ?=≤? ′+ ? ? ? ′+ ? ?? ? ≤≥∫∫∑∫∫then equation(1.1)has positive and increase solution on [t ? , ∞). Suppose equation(1.1)satisfy: 1 1( B ) y R , f ( y)( y0)? ∈α≤y≠, α1 is a positive constant; ( B2 ) qi (t )(i = 1,2, L , n) satisfy: qi (t ) ≥0 Situation(Ⅰ) p (t ) ≤0, r (t ) ≤0, ? p (t ) ? r (t ) ≤1 Theorem3 Assume there exist some i1 ∈{1,2, L , n},such that 111liminf ( )1itt →∞∫t?ταqi s ds >e, then all the solution of equation(1.1)are oscillatory. Theorem4 Assume ( H 1) there exist i0 ∈{1,2, L , n},such that 00liminf ( ) 0itt →∞∫t?τqi s ds> if there exist a I , I is a subset of{1,2, L , n}and I is not empty,and i0 of ( H 1)is in I ,for some large enough T ≥t0,when t ≥T,we have ( )1 1limt →i∞n f ??? ??1 + mi∈iI n ? p (t ? τi ) ? r (t ? τi ) ??∫tt?τα∑i∈Iqi ( s )ds ???>1e, then all the solution of equation(1.1)are oscillatory.Theorem5 Assume ( H 1)and ( H 2)there exist T ≥t0, when t ≥Thave ? p (t ? τi ) qi (t ) ≥? p (t ) qi (t ? σ),? r (t ? τi ) qi (t ) ≥? r (t ) qi (t ? ρ), i = 1,2, L ,n hold.If one of the following conditions hold: (1) Suppose I is a subset of{1,2, L , n}and I is not empty, ω= min{τ1 + σ, τ1+ ρ},we have 1limt →i∞n f ∫tt? ωα[ ? p ( s ) ∑i∈I qi ( s ? σ) ? r ( s ) ∑i∈Iqi ( s ? ρ)]ds >1e. (2)1 1limt →i∞n f ∫tt? τα[ ? p ( s ) ∑i∈I qi ( s ? σ) ? r ( s ) ∑i∈I qi ( s ? ρ) + ∑i∈Iqi ( s )]ds >1e, hold, then all the solution of equation(1.1)are oscillatory. Situation(Ⅱ) p (
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