Homoclinic orbits for second order self-adjoint difference equations

In this paper we discuss how to use variational methods to study the existence of nontrivial homoclinic orbits of the following nonlinear difference equations

     

Homoclinic orbits and subharmonics for nonlinear second order difference equations

We show the existence of a nontrivial homoclinic orbit and subharmonic solutions for a class of second order difference equations by applying the “Mountain Pass” theorem relying on Ekeland’s variational principle and the diagonal method, and the homoclinic orbit as the limit of the subharmonics. A completely new way is provided for dealing with the existence of solutions for difference equations.     

Homoclinic orbits and subharmonics for second order p-Laplacian difference equations

In this paper, a second order p-Laplacian difference equation is considered. By using the critical point theory, we obtain the existence of a nontrivial homoclinic orbit and give some new results. The proof is based on the Mountain Pass Lemma in combination with periodic approximations.     

A note on homoclinic orbits for second order Hamiltonian system

Some existence and multiplicity of homoelinic orbits for second order Hamiltonian system x-a(t)x f(t,x)=0 are given by means of variational methods, where the function -1/2a（t）|s|^2∫^t0f（t,s）ds is asymptotically quadratic in s at infinity and subquadratic in s at zero, and the function a (t) mainly satisfies the growth condition limt→∞∫^t 1 t a（t）dt= ∞,VI∈R^1.A resonance case as well as a noncompact case is discussed too.     

HOMOCLINIC ORBITS FOR SECOND ORDER HAMILTONIAN SYSTEM WITH QUADRATIC GROWTH

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Existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems

In this paper we study the following nonperiodic second order Hamiltonian system

     

Group actions and orbits

Let A be a group acting via automorphisms on a group G, and let Ω be the set of orbits in this action. Then C = C _G (A) acts on Ω in a natural manner, and using this action, we deduce some divisibility information about |Ω|.     

二阶渐近周期奇异哈密顿系统同宿轨道

运用集中紧性方法和Ekeland变分原理研究R^2中二阶渐近周期奇异Hamilton系统ue (1 g(t))V‘u(t,u)=0的极小问题，并证明该系统具有两条非平凡同宿轨道。     

Sobolev orthogonal polynomials in two variables and second order partial differential equations

We consider polynomials in two variables which satisfy an admissible second order partial differential equation of the form

(∗)     

一类二阶渐近周期微分方程的正同宿轨道

§ 1　IntroductionIn this note we are concerned with the asymptotically periodic second order equation-u″+α( x) u =β( x) uq +γ( x) up,　 x∈ R,( 1 )where1 相似文献   

Infinitely many homoclinic orbits for the second order Hamiltonian systems with general potentials

In this paper we study the following nonperiodic second order Hamiltonian systems

     

Existence and multiplicity of homoclinic orbits for a class of asymptotically periodic second order Hamiltonian systems

We consider the Hamiltonian system q=L(t)q–V(t, q) in R ^m ,L and V being asymptotic, as t–, to certain periodic functions L_, V_. Under suitable assumptions on the functions L, L_, V, V_, we prove for any kN, the existence of infinitely many k- bump homoclinic solutions of the Hamiltonian system.     

A miraculously commuting family of orthogonal matrix polynomials satisfying second order differential equations

We find structural formulas for a family (P_n)_n of matrix polynomials of arbitrary size orthogonal with respect to the weight matrix e^−t2e^Ate^A∗t, where A is certain nilpotent matrix. It turns out that this family is a paradigmatic example of the many new phenomena that show the big differences between scalar and matrix orthogonality. Surprisingly, the polynomials P_n, n≥0, form a commuting family. This commuting property is a genuine and miraculous matrix setting because, in general, the coefficients of P_n do not commute with those of P_m, n≠m.     

Multiplicity-free actions and the geometry of nilpotent orbits

Let G be a reductive Lie group. Take a maximal compact subgroup K of G and denote their Lie algebras by and respectively. We get a Cartan decomposition . Let be the complexification of , and the complexified decomposition. The adjoint action restricted to K preserves the space , hence acts on , where denotes the complexification of K. In this paper, we consider a series of small nilpotent -orbits in which are obtained from the dual pair ([R. Howe, Transcending classical invariant theory. J. Amer. Math. Soc. 2 (1989), no. 3, 535–552]). We explain astonishing simple structures of these nilpotent orbits using generalized null cones. For example, these orbits have a linear ordering with respect to the closure relation, and acts on them in multiplicity-free manner. We clarify the -module structure of the regular function ring of the closure of these nilpotent orbits in detail, and prove the normality. All these results naturally comes from the analysis on the null cone in a matrix spaceW , and the double fibration of nilpotent orbits in and . The classical invariant theory assures that the regular functions on our nilpotent orbits are coming from harmonic polynomials on W with repspect to or . We also provide many interesting examples of multiplicity-free actions on conic algebraic varieties. Received November 1, 1999 / Published online October 30, 2000     

Multiplicity of periodic orbits for a class of second order Hamiltonian systems with superlinear and sublinear nonlinearity

This paper is concerned with a class of second order Hamiltonian systems with superlinear and sublinear nonlinearity(P)where b(t) is a real function defined on [0, T], μ > 2 and H : [0, T] × R^N → R is a Carathéodory function. Some new multiplicity results of periodic orbits for the problem (P) are obtained via some critical point theorems.