On the oscillation of the second order neutral delay differential equations

In this paper, we consider the oscillation of the second order neutral delay differential equations[x(t) cx(t-τ)]" p(t)x(t-σ)=0 (1)and obtain some sufficient conditions of the oscillation of (1) for the case c≥0, -1≤c<0 and c<-1.     

SOLUTION TO p(x)-LAPLACIAN PROBLEM WITH HARDY TERM

We study the existence of solutions to a class of p(x)-Laplacian equations involving singular Hardy terms for nonlinear terms of the type f(x, t) = ±(-λ|t|m(x)-2t +|t|q(x)-2t). First we show the existence of infinitely many weak solutions for anyλ > 0, and next prove that if λ is large enough then there exists a nontrivial weak solution. Our approach relies on direct variational methods and the theory of variable exponent Lebesgue-Sobolev spaces.     

OSCILLATION AND ASYMPTOTIC BEHAVIOR OF SOME SECOND-ORDER RETARDED DIFFERENTIAL EQUATIONS

In this paper, we consider the following second order retarded differential equations x″(t)+cx′(t)=qx(t-σ)-lx(t-δ) (1) x″(t)+p(t)x(t-τ)=0 (2) We give some sufficient conditions for the oscillation of all solutions of Eq. (1) in the case where q, ι, σ, δ are positive numbers and c is a real number. And also, we study the asymptotic behavior of the nonoscillatory solutions. If necessary, we give some examples to illustrate our results. At last, we study Eq. (2) with some conditions on p(t).     

Quasi-periodic solutions of a semilinear Li(?)nard equation at resonance

We are concerned with the existence of quasi-periodic solutions for the follow- ing equation x″ F_x(x,t)x′ ω~2x φ(x,t)=0, where F and φare smooth functions and 2π-periodic in t,ω>0 is a constant.Under some assumptions on the parities of F and φ,we show that the Dancer's function,which is used to study the existence of periodic solutions,also plays a role for the existence of quasi-periodic solutions and the Lagrangian stability (i.e.all solutions are bounded).     

Boundedness of solutions of second order nonlinear dynamic equations

This paper deals with the boundedness of the solutions of the following dynamic equations(r(t)x△(t))△+a(t)f(xσ(t))+b(t)g(xσ(t))=0and(r(t)x△(t))△+a(t)xσ(t)+b(t)f(x(t-τ(t)))=e(t)on a time scale T.By using the Bellman integral inequality,we establish some suffcient conditions for boundedness of solutions of the above equations.Our results not only unify the boundedness results for differential and difference equations but are also new for the q-difference equations.     

CLASSIFICATION AND EXISTENCE OF POSITIVE SOLUTIONS OF SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS WITH DELAY DEPENDING ON THE UNKNOWN FUNCTION

A class of second order nonlinear differential equations with delay depengingon the unknown function of the from(r(t)ψ(x(t))x' (t))' f (t, x(t), x (△ (t, x(t)))) = 0in the case where ∫∞0 ds/r(s) ＜∞ is studied. Various classifications of their eventually positive solutions are given in terms of their asymptotic magnitudes, and necessary as well as sufficient conditions for the existence of these solutions are also obtained.     

非线性二阶阻尼微分方程的振动定理

Several oscillation criteria are given for the second order nonlinear differential equation with damped term of the form [α(t)(y'(t))σ]' p(t)(y'(t))σ q(t)f(y(t)) = 0, where α∈C(R, (0,∞)), p(t) and q(t) are allowed to change sign on [t0, ∞), and f∈C1 (R, R) such that xf(x) > 0 for x ≠0. Our results improve and extend some known oscillation criteria. Examples are inserted to illustrate our results.     

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO SECOND-ORDER NONLINEAR DIFFRENTIAL EQUATIONS

In this paper the authors study the oscillation and the asymptotic behavior of solutiosnto the second-order nonlinear differential equations ±X(t,x,)=0 (X_±)and give necessary and sufficient conditions for Equation(X+)to have a boundednonoscillatory solution or to be oscillatory.There are four classes of solutions to Equation(X_) with different asymptotic behavior.For each class of solution,the necessary orsufficient condition of the existence is obtained.     

UNIFORM BLOWUP PROFILES FOR DIFFUSION EQUATIONS WITH NONLOCAL SOURCE AND NONLOCAL BOUNDARY

Long time behavior of solutions to semilinear parabolic equations with nonlocal nonlinear source ut - △u = ∫Ω g(u)dx inΩ× (0, T) and with nonlocal boundary condition u(x, t) = ∫Ω f(x, y)u(y, t)dy on(e) Ω× (0, T) is studied. The authors establish local existence, global existence and nonexistence of solutions and discuss the blowup properties of solutions. Moveover, they derive the uniform blowup estimates for g(s) = sp(p ＞ 1) and g(s) = es under the assumption fΩ f(x, y)dy ＜ 1 for x ∈(e)Ω.     

FORCED OSCILLATION FOR GENERAL DIFFERENTIAL EQUATIONS

New oscillation criteria for general differential equations of the form x(n)(t) Pn-1(t)x(n-1)(t) … p1(t)x'(t) p0(t)x(t) q1(t)xμ(t)=q2(t)xλ(t) e(t), whereλ,μare the ratios of positive odd integers, 0 <μ< 1 andλ> 1 are established.     

一类二阶泛函微分方程的多重周期解

利用变分原理和Z_2不变群指标,得出二阶泛函微分方程x″(t-τ)+c(x(t)-x(t-2τ))′-x(t-τ)+λf(t,x(t),x(t-τ),x(t-2τ))=0的多重周期解的存在性质.     

关于一个二阶非线性中立时滞微分方程的非振荡解的存在性

A new second-order nonlinear neutral delay differential equation r（t） x（t） ＋ P（t）x（t-τ） ＋ cr（t） x（t）-x（t-τ） ＋ F t,x（t-σ1）,x（t-σ2）,...,x（t-σn） = G（t）,t ≥ t0,where τ 0,σ1,σ2,...,σn ≥ 0,P,r ∈ C（[t0,＋∞）,R）,F ∈ C（[t0,＋∞）×Rn,R）,G ∈ C（[t0,＋∞）,R） and c is a constant,is studied in this paper,and some sufficient conditions for existence of nonoscillatory solutions for this equation are established and expatiated through five theorems according to the range of value of function P（t）.Two examples are presented to illustrate that our works are proper generalizations of the other corresponding results.Furthermore,our results omit the restriction of Q1（t） dominating Q2（t）（See condition C in the text）.     

多时滞中立型微分积分方程的周期解

研究了一类具有多个时滞的中立型微分积分方程x'(t)=∫tt-σ9(t,s,x(s))ds f(t,x(t-τ0),x'(t-τ1))周期解的存在性,得到了方程周期解存在的充分条件.所得结果体现了滞量σ对周期解存在性的影响.     

具有偏差变元的Li\''{e}nard型方程的周期解

本文利用重合度理论研究了一类具偏差变元的Li\'{e}nard型方程$x'(t)+f_1(t,x(t))|x'(t)|^2+f_2(t,x(t),x(t-\tau_{0}(t)))x'(t)+g(t,x(t-\tau_{1} (t)))=p(t).$获得了该方程存在$\omega$-周期解的若干新结论, 改进和推广了已有文献中的相关结果.     

ASYMPTOTIC STABILITY FOR A CLASS OF NONAUTONOMOUS NEUTRAL DIFFERENTIAL EQUATIONS

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一类具复杂偏差变元的中立型微分方程的周期解

本文研究了一类具复杂偏差变元的中立型泛函微分方程■(t)=θ■(t-r) α(t)f(x(t)) β(t)g(x(x(t))) p(t)的周期解的存在性,得到了周期解存在的充分条件,并给出了所得结论的几个简单应用．     

二阶线性中立时滞方程非振动解的存在性

考虑具有正负系数的中立时滞微分方程这里P∈R和τ∈(0,∞),σ1,σ2∈[0,∞)且Q1,Q2∈C([t0,∞),R+).对于上面方程非振动解的存在性,得到一个用,∫sQids <∞,i=1,2,来表达的充分条件。这个结果去掉了M.R.S.Kulenovic和S.Hadziomerspahic文中一个相当强的假设,改进了其中的相关定理.     

An Inverse Coefficient Problem for an Integro-differential Equation

In this article we consider the inverse coefficient problem of recovering the function { ( x ) system of partial differential equations that can be reduced to a second order integro-differential equation $ -u_{xx} + c(x)u_{x} + d\phi (x)u-\gamma d\phi (x)\int _{0}^{t}e^{-\gamma (t-\tau )}u(x,\tau )\, d\tau = 0 $ with boundary conditions. We prove the existence and uniqueness of solutions to the inverse problem and develop a numerical algorithm for solving this problem. Computational results for some examples are presented.     

Oscillation and Global Attractivity in a Nonlinear Delay Periodic Model of Population Dynamics

In this article we shall consider the following nonlinear delay differential equation $$x'(t) + p(t)x(t)-\frac {q(t)x(t)}{r + x^{n}(t-m\omega )} = 0\eqno (*)$$ where m and n are positive integers, p ( t ) and q ( t ) are positive periodic functions of period y . In the nondelay case we shall show that (*) has a unique positive periodic solution $ \overline {x}(t), $ and provide sufficient conditions for the global attractivity of $ \overline {x}(t) $ . In the delay case we shall present sufficient conditions for the oscillation of all positive solutions of (*) about $ \overline {x}(t), $ and establish sufficient conditions for the global attractivity of $ \overline {x}(t). $     

Periodic solutions for a type of neutral system with variable parameters

In this paper, we firstly analyze some properties of the linear difference operator $[Ax](t)=x(t)-C(t)x(t-\tau),$ where $C(t)$ is a $n\times n$ matrix function, and then using Mawhin''s continuation theorem, a first-order neutral functional differential system is studied. Some new results on the existence and stability of periodic solutions are obtained. The results are related to the deviating arguments $\tau$ and $\mu$. Meanwhile, the approaches to estimate a $prior$ bounds of periodic solutions are different from the corresponding ones of the known literature.