POSITIVE ENTIRE SOLUTIONS TO SINGULAR SEMILINEAR ELLIPTIC EQUATIONS OF MIXED TYPE

In this paper, we are concerned with positive entire solutions to elliptic equations of the form Δu+ f(x,u)= 0 x∈ RN N ≥ 3 where u →f(x,u) is not assumed to be regular near u = 0 and f(x,u) may be more general involving both singular and sublinear terms. Some sufficient conditions are given with the aid of the barrier method and ODE approach, which guarantee the existence of positive entire solutions that tend to any sufficiently large constants arbitrarily prescribed in advance.

EXISTENCE OF BOUNDED SOLUTIONS ON THE REAL LINE FOR LINARD SYSTEM

The existence of monotone and non_monotone solutions of boundary value problem on the real line for Liénard equation is studied. Applying the theory of planar dynamical systems and the comparison method of vector fields defined by Liénard system and the system given by symmetric transformation or quasi_symmetric transformation, the invariant regions of the system are constructed. The existence of connecting orbits can be proved. A lot of sufficient conditions to guarantee the existence of solutions of the boundary value problem are obtained. Especially, when the source function is bi_stable, the existence of infinitely many monotone solusion is obtained.

Coexistence of unbounded solutions and periodic solutions of Liénard equations with asymmetric nonlinearities at resonance

In this paper, we deal with the existence of unbounded orbits of the mapping $$\left\{ \begin{gathered} \theta _1 = \theta + 2n\pi + \frac{1}{\rho }\mu (\theta ) + o(\rho ^{ - 1} ), \hfill \\ \rho _1 = \rho + c - \mu '(\theta ) + o(1), \rho \to \infty \hfill \\ \end{gathered} \right.$$ , where n is a positive integer, c is a constant and μ(θ) is a 2π-periodic function. We prove that if c > 0 and μ(θ) ≠ 0, θ, ∈ [0, 2?], then every orbit of the given mapping goes to infinity in the future for ρ large enough; if c < 0 and μ(θ) ≠ 0, θ ∈ [0, 2π], then every orbit of the given mapping goes to infinity in the past for ρ large enough. By using this result, we prove that the equation x″+f(x)x′+ax +?bx ?+?(x)=p(t) has unbounded solutions provided that a, b satisfy $1/\sqrt a + 1/\sqrt b = 2/n$ and ?(x) satisfies some limit conditions. At the same time, we obtain the existence of 2π-periodic solutions of this equation.

Quasi-periodic solutions of a semilinear Liénard equation at resonance

We are concerned with the existence of quasi-periodic solutions for the following equation x" + Fx (x, t)x' + ω2x + φ(x,t) = 0,where F and φ are 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).

PERIODIC SOLUTIONS FOR NEUTRAL LINARD EQUATION WITH A STATEMENT-DEPENDENT DEVIATION VARIABLE

Using the theroy of topological degree, the paper studies the periodic solutions to a type of neutral Liénard equation with state-dependent deviation variable. A sufficient condition for the existence of periodic solution is obtained.

LANDESMEN-LAZER CONDITIONS FOR PERIODIC SOLUTIONS OF ASYMMETRIC LINARD EQUATION

We prove the existence of periodic solutions for asymmetric Lienard equa-tions with Landesmen-Lazer condition.

NEW RESULTS ON THE EXISTENCE OF PERIODIC SOLUTIONS TO LINARD EQUATION

In this paper, a Liénard equation with a deviating argument has been studied by means of Mawhin’s continuation theorem. A new result guaranteeing the existence of periodic solutions is obtained.

On the Integrability of the Abel and of the Extended Liénard Equations

We present some exact integrability cases of the extended Liénard equation y′′+ f(y)(y′)n +k(y)(y′)m + g(y)y′+ h(y) = 0, with n > 0 and m > 0 arbitrary constants, while f(y), k(y), g(y), and h(y) are arbitrary functions. The solutions are obtained by transforming the equation Liénard equation to an equivalent first kind first order Abel type equation given bydv/dy= f(y)v3-n+ k(y)v3-m+ g(y)v2+ h(y)v3, with v = 1/y′.As a first step in our study we obtain three integrability cases of the extended quadratic-cubic Liénard equation,corresponding to n = 2 and m = 3, by assuming that particular solutions of the associated Abel equation are known. Under this assumption the general solutions of the Abel and Liénard equations with coefficients satisfying some differential conditions can be obtained in an exact closed form. With the use of the Chiellini integrability condition, we show that if a particular solution of the Abel equation is known, the general solution of the extended quadratic cubic Liénard equation can be obtained by quadratures. The Chiellini integrability condition is extended to generalized Abel equations with g(y) ≡ 0 and h(y) ≡ 0, and arbitrary n and m, thus allowing to obtain the general solution of the corresponding Liénard equation. The application of the generalized Chiellini condition to the case of the reduced Riccati equation is also considered.

T-PERIODIC SOLUTION TO A TIME DELAY LINARD’S EQUATION

In this paper, a time delay Liénard’s equation is considered, by the coincidence degree theory. Sufficient conditions for the existence of at least one T-periodic solution are obtained.

无周期解的Liénard系统

本文讨论了Ｌｉｎａｒｄ方程¨ｘ＋ｆ（ｘ）·ｘ＋ｇ（ｘ）＝０周期解的不存在性，给出了该方程在Ｆ（ｘ）＝∫ｘ０ｆ（ｕ）ｄｕ有无穷多个零点ｘｎ→０（ｎ→∞），或存在δ＞０使当０＜｜ｘ｜＜δ时Ｆ（ｘ）＞０（或＜０）这二种情况下无周期解的充分条件．

具有强迫项的Linard方程解的有界性

讨论了具有强迫项的Ｌｉｅｎａｒｄ方程解的有界性，得到了此方程的所有解及其导数有界的充要条件．所获结果改进和推广了Ｓｕｇｉｅ Ｊ（１９８７）和杨启贵（１９９９）的文章中的相应结果．

半线性椭圆方程径向正解的非存在性及解的振荡性和有界性的充分条件

设r=sum form i=1 to ∞(x_i^2),Ω={r|0≤r<a}是R^(?)中的开球。考虑无界域的二阶半线性椭圆方程边值问题: Δu+f(u,r)=0 r∈Ω′=R^n\(?) u|(?)Ω=0 r∈(?)Ω给出了问题径向正解的非存在性,解的振荡性及有界性的充分条件,并对有界解进行了估计。

Invariant tori in nonlinear oscillations

The boundedness of all the solutions for semilinear Duffing equationx″ + ω2 x + φ(x) =p(t), ω ∈ ?+? is proved, wherep (t) is a smooth 2π-periodic function and the perturbation ?(x) is bounded.