Oscillations of high order neutral differential equations with oscillatory coefficient

Consider the high order neutral differential equation(y(t) p(t)y(h(t)))~(n) q(t)y(g(t))=0, (1)where p(t), q(t), h(t) and g(t)∈ C[t_0, ∞);q(t)>0;h(t)→∞, g (t)→∞ as t→∞;n≥2.The author studies the oscillation of (1) when p(t) has an arbitrarily large zero, and obtainssome sufficient conditions.     

Forced oscillation of second order differential equations with mixed nonlinearities

This article is concerned with the oscillation of the forced second order differential equation with mixed nonlinearities a(t) x ′ (t) γ′ + p 0 (t) x γ (g 0 (t)) + n i =1 p i (t) | x (g i (t)) | α i sgn x (g i (t)) = e(t), where γ is a quotient of odd positive integers, α i > 0, i = 1, 2, ··· , n, a, e, and p i ∈ C ([0, ∞ ) , R), a (t) > 0, gi : R → R are positive continuous functions on R with lim t →∞ g i (t) = ∞ , i = 0, 1, ··· , n. Our results generalize and improve the results in a recent article by Sun and Wong [29].     

The Oscillation of the Solutions of Higher-order Functional Differential Equations

In this paper we study the oscillation of the solutions of higher-order functional differential equations, and have given the new integral conditions when equation (1) is oscillatory and extended some known results.x(n)(t) p(t)x(g(t)) = 0, (1)where n is even number, p(t) ∈ C(R, [0, ∞)), g(t) ∈ C(R; R), g(t) f55 t,.     

OSCILLATORY AND ASYMPTOTIC BEHAVIORS OF FIRST ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper the author discusses the following first order functional differentialequations: x'(t) +integral from n=a to b p(t, ξ)x[g(t, ξ)]dσ(ξ)=0, (1) x'(t) +integral from n=a to b f(t, ξ, x[g(t, ξ)])dσ(ξ)=0. (2)Some suffcient conditions of oscillation and nonoseillafion are obtained, and two asymptolioproperties and their criteria are given. These criferia are better than those in [1, 2], and canbe used to the following equations: x'(t) + sum from i=1 to n p_i(t)x[g_i(t)] =0, (3) x'(t) + sum from i=1 to n f_i(t, x[g_i(t)] =0. (4)     

一类非线性微分方程的可积性准则

本文考虑二阶线微分方程 y″+t~2f(t)g(y)=0 (1) 的可积性,设G(y)=integral from n=0 to y(g(s)ds),我们证明了在一定的条件下,方程(1)的一切解满足估计: integral from n=t_0 to ∞((G(y(t))/f(t))dt)〈+∞。     

小时间区间中的条件扩散过程

本文讨论具有一致连续系数条件扩散过程的大偏差性质。设X(t)是具有Dirichlet空间(ξ、H_0~1(P_0~d))的扩散过程,其中 ξ(f,g)=1/2 integral from n=R~d to (〈▽f,▽g〉(x)dx)。 P_a~e是过程x_6(t)=x(∈t)满足条件x_6(0)=x,x_6(1)=y的律。那么当∈→0时,(P_(?)~(?),y)具有大偏差性质,且具有速率函数 J_(x,y)(ω)=1/2 integral from n=0 to 1(〈(?)(t),a(-1)(ω(t)),(?)(t)〉dt-1/2 d~2(x,y)。     

偶数阶非线性滞后变元微分方程解的振荡性

将一类偶数阶非线性偏差变元微分方程 :x(n) ( t) + F{ t,x( t) ,x[g( t) ]} =0推广到非线性项 F中含有形如 x( t)及 x[g( t) ],x(n- 1) [g( t) ]项时解的振荡问题 .     

n阶非线性泛函微分不等式的强振动性质

本文考虑了非线性泛函微分不等式x(t)[x~(n)(t)+p(t)f(x(t),x(g(t)))+r(t)]≤0的振动性,其中n为偶数。给出了相应不等式x(t)[x~(n)(t)+λp(t)f(x(t),x(g(t)))+r(t)]≤0,λ>0,的所有解是振动的充分条件。本文的结果改进和推广了已知的一些结果。     

The Existence of Almost Periodic Solutions of Second Order Neutral Differential Equations with Piecewise Constant Argument

51.Intr0ducti0nInthispaper,weconsidertheexistenceofalmostperiodicsoluti0nsforthefoll0wingneutralfunctionaldifferentialebuationwithpiecewiseconstantargument:wherep,qarenonzeroconstants,p(t)lsanalmostper1odicfunctlon,g:RXRXR-Risanalm0stperiodicfunctionintanduniformlyonRXR,andL.jstandsforthegreatest-integerfunction.Notethatforanyintegern,if2n-10)andtheequations(1),(2)areadvanced(orretardedrespectively).Differentialequationswithpiecewiseconstantar…     

非线性二阶中立型时滞微分方程正解的存在性

考虑非线性二阶中立型微分方程,[a(t)x(t)-∑ from i=1 to m (p_i(t)x(τi(t)))]″-∫from n=a to b (f(t,ξ,x[g(t,ξ)])dσ(ξ))=0,t≥t_0,和相应不等式[a(t)x(t)-∑ from i=1 to m (p_i(t)x(τi(t)))]″-∫from n=a to b (f(t,ξ,x[g(t,ξ)])dσ(ξ))≥0,t≥t_0.存在正解是相互等价的.其中a(t),pi(t)∈C([t0,∞),R+),a(t)>0,τi(t)∈C(R~+,R~+),τi(t)t,limt→∞τi(t)=∞(i=1,2,…,m).g(t,ξ)∈C([t_0,∞)×[a,b],R+).g(t,ξ)是分别关于t和ξ的增函数.g(t,ξ)t,ξ∈[a,b],limt→∞,ξ∈[a,b]g(t,ξ)=∞.f(t,ξ,x)∈C([t_0,∞)×[a,b]×R,R+).当x>0时,xf(t,ξ,x)>0.σ(ξ)∈C([a,b],R),且σ(ξ)非减.     

n阶非线性泛函微分方程的渐近性和非振动性

§1.引言本文讨论n阶非线性泛函微分方程 L_nx(t)+P(t)L_(n-1)x(t)+f(t,x(t),x(g(t)))=h(t) (1)解的渐近性和非振动性,其中L_0x(t)=x(t),L_kx(t)=a_k(t)(L_(k-1)x(t))′,k=1,2,…u,a,p,h,g∈C~0E[t_0,∞),且a_k(t)>0,k=1,2,…n-1,a_n(t)=1;t_0≤g(t)≤t,当t→∞时,g(t)→∞;f∈C~0([t_0,∞)×R_2,R)。我们给出了方程(1)的所有振动解和有界解具有渐近性态:L_kx(f)→0,k=0,1,2,…n-1,的若干充分性准则,并给出了它不存在有界振动解的几个保证性条件。所得定理和推论都分别推广了文[1]-[4]的相应结果。     

一个不等式的指数推广

贵刊文[1]给出了如下不等式:设a,b>0,λ≥3则aa λb λa b b≥12 λ(1)(见文[1](3)式)本文将把(1)式推广为:定理设a,b>0,n≥2且n∈N,λ≥2n-1则naa λb nbλa b≥n12 λ(2)证明令x1=ab,x2=ba,则x1,x2>0,且x1x2=1,于是(2)式等价于1n1 λx1 n11 λx2≥n12 λ(3)再令t1=n1 λx1,t2=n1 λx2,则t1,t2>0(3)式等价于1t1 t12≥n12 λn1 λ(t1 t2)≥2t1t2(1 λ)(t1 t2)n≥2nt1nt2n(1 λ)(t1n C1nt1n-1t2 C2nt1n-2t22 … Cnn-1t1t2n-1 t2n)≥2n(t1t2)n(1 λ)[2 λ(x1 x2) (C1ntn1-1t2 C2nt1n-2t22 … Cnn-1t1t2n-1]≥2n(t1t2)n(4)因为C1n C2n … C…     

THE GEOMETRY OF HYPERSURFACES IN A KAEHLER MANIFOLD

Let M be a complex analytic manifOld there is given a positive definite quadratic differentialform[1]dS2 = gjkdZJdZ* (1)where the Latin indices j, k take the values 1,2,' l n; 1, 2,' t n. Assume the Greek indices o, Ptake the value8 1,2,', n and.=. l cr if i= al = 1 cr if i= a, (2)t cr if i = cr.Assume now that the symmetric tensor gjk is selfadoint(see below Definitioll l), that is-- --gap = g95 = go0 = g9rr, (3)g.0 = gPcr = gap = g05. (4)and 8atisfiesgoP = gap = 0. (5)From the com…     

UPPER SEMICONTINUITY OF ATTRACTORS FORTHE REACTION DIFFUSION EQUATION

1IntroductionInthepresentpaper,weconsiderthefollowingreactiondiffusionequation:at~vAn f(u) A0u g(x)=0,V(x,t)ERxR .(1.1).u(x,0)=u000,VxER,(1.2)andforO=(--n,n)withnEN,otu.~aam. f(,u.) A0u,, g(x)=0,V(x,t)EfixR .(1.3)u.(x,0)=.no.(x),VxeO,(1.4)un(~n,f)=un(n,t)=0,(1.5)whereuandAcarepositivenumbers,g(x)EL'(R),f:R~Risasmoothfunctionwhichsatisfiesf(u)u20,VatER,(1.6)f(0)=0,f,(0)=0,f'(u)2~C,VatER,(1.7)If'(u)I5C(1 fi4lp),p>0,V.uER,(1.8)Inthefollowing,wedenotebyH=L'(R)witlltheusualillnerpro…     

C~n中复超球上的一类奇异积分方程的解

设α(t),g(t)和K(t,u)分别是复超球面S和S×S上满足Lipschitz连续条件,且K(t,U)/{α(u)-b(u)}是B×B上的解析函数在S上的边界值,在S上有α~2(t)±b~2(t)≠0, 则方程α(t)f(t)+2/w integral from n=s ((K(t, u)f(u)du)/((1-tu′)~n))=g(t) (*) 当且仅当g(t)使函数 (b(t)g(t))/(b(t)+a(t))+(b(t)-a(t))/(b(t)+a(t)) integral from n=s ((2K(t, u)g(u)du)/(w{b(u)-a(u)}(1-tu′)~n)) 是复超球B上的解析函数的边界值函数时,方程(*)有唯一解: f(t)=(a(t)g(t))/(a~2(t)-b~2(t))+2/(w{a(t)+b(t)}) integral from n=s ((K(t, u)g(u)du)/({b(u)-a(u)}(1-tu′)~n)) 这里b(t)=K(t, t)。     

组合数集的毕达哥斯定理

本文得到两个组合数集的毕达哥斯定理的推广。(ⅰ )当n为奇数时∑[(n+ 3 ) / 2 ]t=0n +3-tt2 - ∑[(n+ 1) / 2 ]t=0n +1-tt2 2 +2 ∑[n/ 2 ]t=0n -tt · ∑[(n+ 4 ) / 2 ]t=0n +4-tt2 +4∑[(n+ 2 ) / 2 ]t=0n +2 -tt2= ∑[(n+ 1) /2 ]t=0n+ 1 -tt2 + ∑[(n+ 3) /2 ]t=0n+ 3-tt2 2 。(ⅱ )当n为偶数时∑[(n+ 4 ) / 2 ]t=0n+4-tt2 - ∑[n/ 2 ]t=0n-tt2 2 +2 ∑[(n+ 1) / 2 ]t=0n+1-tt · ∑[(n+ 3 ) / 2 ]t=0n+3-tt2 +4∑[(n+ 2 ) / 2 ]t=0n+2 -tt2= ∑[n/2 ]t=0n -tt2 + ∑[(n+ 4) /2 ]t=0n + 4 -tt2 2 。     

MODIFIED NEWTON AND SECANT METHODS FOR SOLVING AN ORDER O(h~4) FINITE-DIFFERENCE PROBLEM

1IntroductionSolution0fn0nlineartwo-pointb0undaryvaIuepr0blems(NBVP)canoftenbefoundbythefinite-differenceappr0ach,wheref(t,y)isaconti-nuousfunction.Collatz[1]firstpresentedanapproximation0ffourthorderfwherey=(y1,''tyN)',g=(g1,'-,gN)'andtherelativepaperscanals0beseenin[2].Toestablishthesolutionof(1.l),thef0llowingmethodscanbeusedfnonlinearsuccessiverelaxati0n(NSOR)method[3],thedifferenceNewt0nmethod(0rNewtonmethod)[4],therelativesparsenonlinearequationpr0blemscanals0beseenin[5-8]-lnthisp…     

一类连续最优控制问题的有限维近似

1　引　　言本文考虑具有状态终端约束、控制受限的非线性连续最优控制问题min　h0(x(0))+∫T0f0(x(t),u(t))dt+g0(x(T))(1.1)s.t.　x(t)=f(x(t),u(t)),　　t∈[0,T](1.2)D(x(0))=0,(1.3)E(x(T))=0,(1.4)S(u(t))≤0,　　t∈[0,T](1.5)其中,h0:Rn→R,f0:Rn×Rm→R,f:Rn×Rm→Rn,g0:Rn→R,D:Rn→Rp,E:Rn→Rq,S:Rm→Rr均为二次连续可微函数.T为终端时间(固定),p,q≤n,x(t)∈W1,∞[0,T]n,u(t)∈L∞[0,T]m分别为状态函数和控制函数.U(t)={u:S(u(t))≤0}为紧凸集.问题(1.1)—(1.5)要求寻找最佳控制u(t)使得目标函数(1.1)达到极小.…     

方程w″-w+f(t,w)=0的Dirichlet边值问题的正解存在性与多解性(英文)

考察了下列常微分方程的 Dirichlet边值问题的正解w″(t) -w(t) +f (t,w(t) ) =0 ,　 0 t 1w(0 ) =w(1 ) =0建立了 n正解的存在性 ,其中 n是一个任意的自然数 .     

偶阶非线性中立型时滞微分方程的振动性

本文研究一类新的偶阶非线性中立型时滞微分方程(r(t)|z~((n-1))(t)|~(a-1)z~((n-1))(t))'+F(t,x(g(t)))=0,t≥t_0(其中z(t)=x(t)+p(t)x(T(t)),α 0为常数,n为偶数)的振动性.利用广义Riccati不等式和积分平均技巧得到方程一切解均为振动的若干新的振动准则,推广和改进了一些文献中的结果.