Unb ounded Motions in Asymmetric Oscillators Dep ending on Derivatives

In this paper, we consider the unboundedness of solutions for the asymmetric equation x00+ax+?bx?+?(x)ψ(x0)+f(x)+g(x0)=p(t), where x+ = max{x, 0}, x? = max{?x, 0}, a and b are two different positive constants, f (x) is locally Lipschitz continuous and bounded,?(x), ψ(x), g(x) and p(t) are continuous functions, p(t) is a 2π-periodic function. We discuss the existence of unbounded solutions under two classes of conditions: the resonance case √1a+ √1b ∈Q and the nonresonance case√1a + √1b /∈Q.     

Quasi-periodic solutions for the general semilinear Duffing equations with asymmetric nonlinearity and oscillating potential

In this paper,we prove the existence of quasi-periodic solutions and the boundedness of all the solutions of the general semilinear quasi-periodic differential equation x′′+ax~+-bx~-=G_x(x,t)+f (t),where x~+=max{x,0},x~-=max{-x,0},a and b are two different positive constants,f(t) is C~(39) smooth in t,G(x,t)is C~(35) smooth in x and t,f (t) and G(x,t) are quasi-periodic in t with the Diophantine frequency ω=(ω_1,ω_2),and D_x~iD_t~jG(x,t) is bounded for 0≤i+j≤35.     

连续时间延滞的非自治系统的周期解（英）

We consider the nonautonomous cooperative system with continuous time delay (x)(t)=x(t)[a1(t)-a2(t)+a3(t)∫o-Tk1(s)x(t+s)ds-a4(t) ∫o-Tk2(s)y(t+s)ds] (y)(t)=y(t)[b1(t)-b2(t)y(t)+a3(t) ∫o-Tk3(s)y(t+s)ds-b4(t) ∫o-Tk4(s)x(t+s)ds](1)where ai(t),bi(t)(i=1,2,3,4) are assumed to be continuous, positive and ω-periodic functions; and x(t),y(t) are the density of species; ki(s) (i=1,2,3,4) denote nonnegative piecewise continuous defined in [-τ,0] (there 0<τ<+∞) and normalized such that ∫o-T ki(s)ds=1. Let fL=inf{f(t):t∈R}, fM=sup{f(t),t∈R}, for a continuous and bounded function f(t).     

MULTI-BUMP SOLUTIONS FOR NONLINEAR CHOQUARD EQUATION WITH POTENTIAL WELLS AND A GENERAL NONLINEARITY

In this article,we study the existence and asymptotic behavior of multi-bump solutions for nonlinear Choquard equation with a general nonlinearity-△u+(λa(x)+1)u=(1/|x|α*F(u))f(u) in R~N,where N≥3,0 αmin{N,4},λ is a positive parameter and the nonnegative potential function a(x) is continuous.Using variational methods,we prove that if the potential well int(a~(-1)(0)) consists of k disjoint components,then there exist at least 2~k-1 multi-bump solutions.The asymptotic behavior of these solutions is also analyzed as λ→+∞.     

ALMOST PERIODIC SOLUTIONS OF THE EQUATIONx = x~3 + λg (t) x + μf (t) AND THEIR STABILITY

By using the Liapunov function and the contraction mapping principle, the authorinvestigates the existence and stability of almost periodic solutions of the first ordernonlinear equations dx/dt=-h_1(x)+h_2(x)g(t)+f(t)and dx/dt=r(t)x~n+λg(t)x+μf(t),where r(t), g(t), f(t) are given almost periodic functions, n(≥2) integer, and λ, μ realparameters.     

Existence of Semiclassical States for a Quasilinear Schr?dinger Equation on R~N with Exponential Critical Growth

We study a quasilinear Schr?dinger equation{-ε~NΔNu+V(x)|u|~(N-2)u=Q(x)f(u) in R~N,0u∈W~(1,N)(R~N),u(x)|x|→∞→0,where V,Q are two continuous real functions on R~N and ε0 is a real parameter.Assume that the nonlinearity f is of exponential critical growth in the sense of Trudinger–Moser inequality,we are able to establish the existence and concentration of the semiclassical solutions by variational methods.     

Blow-up of p-Laplacian evolution equations with variable source power

We study the blow-up and/or global existence of the following p-Laplacian evolution equation with variable source power ut(x,t)=div(|?u|~(p-2)?u)+u~(q(x)) in?×(0,T),where ? is either a bounded domain or the whole space R~N,and q(x) is a positive and continuous function defined in ? with 0q_-=inf q(x)=q(x)=sup q(x)=q_+∞.It is demonstrated that the equation with variable source power has much richer dynamics with interesting phenomena which depends on the interplay of q(x) and the structure of spatial domain ?,compared with the case of constant source power.For the case that ? is a bounded domain,the exponent p-1 plays a crucial role.If q_+p-1,there exist blow-up solutions,while if q_+p-1,all the solutions are global.If q_-p-1,there exist global solutions,while for given q_-p-1q_+,there exist some function q(x) and ? such that all nontrivial solutions will blow up,which is called the Fujita phenomenon.For the case ?=R~N,the Fujita phenomenon occurs if 1q_-=q_+=p-1+p/N,while if q_-p-1+p/N,there exist global solutions.     

Some oscillation criteria for second order nonlinear functional ordinary differential equations

The main objective of this article is to study the oscillatory behavior of the solutions of the following nonlinear functional differential equations (a(t)x'(t))' δ1p(t)x'(t) δ2q(t)f(x(g(t))) = 0,for 0 ≤ t0 ≤ t, where δ1 = ±1 and δ2 = ±1. The functions p,q,g : [t0, ∞) → R, f :R → R are continuous, a(t) ＞ 0, p(t) ≥ 0,q(t) ≥ 0 for t ≥ t0, limt→∞ g(t) = ∞, and q is not identically zero on any subinterval of [t0, ∞). Moreover, the functions q(t),g(t), and a(t) are continuously differentiable.     

半拟线性p-Laplacian问题结点解的存在性

In this paper, we study the existence of nodal solutions for the following problem:-(φ_p(x′))′= α(t)φ_p(x~+) + β(t)φ_p(x~-) + ra(t)f(x), 0 t 1,x(0) = x(1) = 0,where φ_p(s) = |s|~(p-2)s, a ∈ C([0, 1],(0, ∞)), x~+= max{x, 0}, x~-=- min{x, 0}, α(t), β(t) ∈C[0, 1]; f ∈ C(R, R), sf(s) 0 for s ≠ 0, and f_0, f_∞∈(0, ∞), where f_0 = lim_|s|→0f(s)/φ_p(s), f_∞ = lim|s|→+∞f(s)/φ_p(s).We use bifurcation techniques and the approximation of connected components to prove our main results.     

ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF NEUTRAL DIFFERENTIAL EQUATIONS WITH POSITIVE AND NEGATIVE COEFFICIENTS

1 IlltroductionConsider the neutral de1ay differential equations with positive and negativecoefficients of the fOrm[x(t) Ac(t)x(t -- a)]' p(t)x(t -- T) -- Q(t)x(t -- 6) = 0, t 2 to, (1)where A = {--1, 1}, a > 0, T? b 2 0, c,p, Q E C([to, oo), R ), and assume thatthere exists a constant A > 0 such thatQ(t b -- T) 5 Ap*(t), p*(t) = p(t) -- Q(t 6 -- T), for t 2 max{to, to T -- b}.The study of asymptotic behavior of so1ution of (1) has had some work, seefOr example [l-9]. However,…     

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.     

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)     

Periodic Solutions for a Class of n-dimensional Prescribed Mean Curvature Equations

In this paper,by using an extension of Mawhin’s continuation theorem and some analysis methods,we study the existence of periodic solutions for the following prescribed mean curvature system d/dtφ(x')+▽W(x)=p(t), where x∈R^n,W∈C^1(R^n,R),p∈C(R,R^n)is T-periodic and φ(x)=x/√1+|x|^2.     

跳跃非线性项依赖于导数的二阶微分方程周期解的存在性

本文研究二阶微分方程x"+ax+-bx-+f(x)g(x')=p(t)周期解的存在性,这里x+=max{x,0},x-=max{-x,0},a,b是正常数并且点(a,b)位于某一条Fucik谱曲线上.当g(x)的极限limx→∞(x)=g(+∞),limx→∞g(x)=g(-∞)和f(x)的极限limx→∞f(z)=f(+∞),limx→∞f(z)=f(-∞)都存在且有限时,给出了此方程存在周期解的充分条件.     

ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF NONLINEAR NEUTRAL DIFFERENTIAL EQUATION

The boundedness of the every solution and the asymptotic behavior of all solutions of the nonlinear neutral delay differential equation [x(t) - P(t)x(t - t)]' Q1 (t)f(x{t-σ1))-Q2(t)f(x(t -σ2))=0,t≥t0 are investigated, whereτ,σ1,σ2∈(0,∞), P∈C([t0,∞),R), and Q1,Q2∈C([t0,∞),R), f∈C(R,R). The sufficient conditions obtained improve the existing results in the literatures.     

PERIODIC SOLUTIONS FOR GENERALIZED PREDATOR-PREY SYSTEMS WITH TIME DELAY AND DIFFUSION

A set of easily verifiable sufficient conditions are derived for the existence of positive periodic solutions for delayed generalized predator-prey dispersion system x'1 (t) = x1 (t)g1 (t, x1 (t) ) - a1 (t)y(t)p1 (x1 (t) ) D1 (t)(x2(t) - x1 (t) ),x'2 (t) = x2 (t)g2 (t, x2 (t) ) - a2 (t)y(t)p2 (x2 (t) ) D2(t)(x1 (t) - x2 (t)),y' (t) = y(t) {-h(t, y(t) ) b1 (t)p1 (x1 (t - τ1 ) ) b2(t)p2(x2(t - τ2))],where ai(t), bi(t) and Di(t)(i = 1, 2) are positive continuous T-periodic functions, gi(t, xi)(i = 1,2) and h(t,y) are continuous and T-periodic with respect to t and h(t,y) ＞ 0 for y ＞ 0, t, y ∈ R, pi(x)(i = 1, 2) are continuous and monotonously increasing functions, and pi(xi) ＞ 0 for xi ＞ 0.     

Boundedness of Solutions of Quasi-periodic p-Laplacian Equations with Jumping Nonlinearity

In this article, we prove the existence of quasi-periodic solutions and the boundedness of all solutions of the p-Laplacian equation(φ_p(x’))’ + aφp(x⁺)-bφp(x^-) = g(x, t) + f(t), where g(x, t)and f(t) are quasi-periodic in t with Diophantine frequency. A new method is presented to obtain the generating function to construct canonical transformation by solving a quasi-periodic homological equation.     

一类具偏差变元的二阶微分方程周期解

By using the theory of coincidence degree,we study a kind of periodic solutions to second order differential equation with a deviating argument such as x"(t) f(x'(t)) h(x(t))x'(t) g(x(t-τ(t)))=p(t),some sufficient conditions on the existence of periodic solutions are obtained.     

A RESULT ONPERIODIC SOLUTIONS OF SECOND ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS

We obtain sufficient condition for the existence of periodic solutions of thefollowing second order functional differential equationsx"(t) + ax'~α(t) + bf(x(t)) + g(x(t-T_1), x'(t-T_2))=p(t)=p(t+2π).Our approach is based on the continuation theorem of coincidence degree, andthe α-priori: estimate of periodic solutions.     

CLASSIFICATION OF POSITIVE SOLUTIONS TO A SYSTEM OF HARDY-SOBOLEV TYPE EQUATIONS

In this paper, we are concerned with the following Hardy-Sobolev type system{(-?)~(α/2) u(x) =v~q(x)/|y|~(t_2) (-?)α/2 v(x) =u~p(x)/|y|~(t_1),x =(y, z) ∈(R ~k\{0}) × R~(n-k),(0.1)where 0 α n, 0 t_1, t_2 min{α, k}, and 1 p ≤τ_1 :=(n+α-2t_1)/( n-α), 1 q ≤τ_2 :=(n+α-2 t_2)/( n-α).We first establish the equivalence of classical and weak solutions between PDE system(0.1)and the following integral equations(IE) system{u(x) =∫_( R~n) G_α(x, ξ)v~q(ξ)/|η|t~2 dξ v(x) =∫_(R~n) G_α(x, ξ)(u~p(ξ))/|η|~(t_1) dξ,(0.2)where Gα(x, ξ) =(c n,α)/(|x-ξ|~(n-α))is the Green's function of(-?)~(α/2) in R~n. Then, by the method of moving planes in the integral forms, in the critical case p = τ_1 and q = τ_2, we prove that each pair of nonnegative solutions(u, v) of(0.1) is radially symmetric and monotone decreasing about the origin in R~k and some point z0 in R~(n-k). In the subcritical case (n-t_1)/(p+1)+(n-t_2)/(q+1) n-α,1 p ≤τ_1 and 1 q ≤τ_2, we derive the nonexistence of nontrivial nonnegative solutions for(0.1).