Nontrivial Solutions of Superquadratic Hamiltonian Systems with Lagrangian Boundary Conditionsand the L-index Theory

In this paper, the authors study the existence of nontrivial solutions for the Hamiltonian systems z(t) = JH(t,z(t)) with Lagrangian boundary conditions, where H(t,z)=1/2((B)(t)z,z) (H)(t,z),(B)(t) is a semipositive symmetric continuous matrix and (H) satisfies a superquadratic condition at infinity. We also obtain a result about the L-index.     

对称超二次二阶哈密尔顿系统的周期解

我们利用Ambrosetti-Rabinowitz对称形式的山路引理证明了给定周期T的对称超二次二阶哈密尔顿系统具有无穷多个反T/2-周期且奇的周期解.     

The iteration formula of the Maslov-type index theory with applications to nonlinear Hamiltonian systems

In this paper, the iteration formula of the Maslov-type index theory for linear Hamiltonian systems with continuous, periodic, and symmetric coefficients is established. This formula yields a new method to determine the minimality of the period for solutions of nonlinear autonomous Hamiltonian systems via their Maslov-type indices. Applications of this formula give new results on the existence of periodic solutions with prescribed minimal period for such systems, and unify known results under various convexity conditions.

     

Periodic solutions for a class of superquadratic Hamiltonian systems

We prove the existence of nontrivial critical points for a class of superquadratic nonautonomous second-order Hamiltonian systems by applying condition ^∗(C) to critical point theory, and some new solvability conditions of nontrivial periodic solutions are obtained.     

Minimal period estimates on P-symmetric periodic solutions of first-order mild superquadratic Hamiltonian systems

With the aid of P-index iteration theory,we consider the minimal period estimates on P-symmetric periodic solutions of nonlinear P-symmetric Hamiltonian systems with mild superquadratic growth.     

Periodic solutions with prescribed minimal period for convex autonomous hamiltonian systems

Clarke has shown that the problem of findingT-periodic solutions for a convex Hamiltonian system is equivalent to the problem of finding critical points to a certain functional, dual to the classical action functional. In this paper, we relate the Morse index of the critical point to the minimal period of the correspondingT-periodic solution. In particular, we show that if the critical point is obtained by the Ambrosetti-Rabinowitz mountain-pass theorem the corresponding solution has minimal periodT, that is, it cannot beT/k-periodic withk integer,k2. As a consequence, we prove that if the Hamiltonian is flat near an equilibrium and superquadratic near infinity, then for anyT>0, the corresponding Hamiltonian system has a periodic solution with minimal periodT.     

ON THE MINIMAL PERIOD FOR PERIODIC SOLUTION PROBLEM OF NONLINEAR HAMILTONIAN SYSTEMS

ＯＮＴＨＥＭＩＮＩＭＡＬＰＥＲＩＯＤＦＯＲＰＥＲＩＯＤＩＣＳＯＬＵＴＩＯＮＰＲＯＢＬＥＭＯＦＮＯＮＬＩＮＥＡＲＨＡＭＩＬＴＯＮＩＡＮＳＹＳＴＥＭＳ＊＊ＬＯＮＧＹＩＭＩＮＧ＊ＡｂｓｔｒａｃｔＴｈｅａｕｔｈｏｒｐｒｏｖｅｓａｓｈａｒｐｅｒｅｓｔｉｍａｔｅ...     

Periodic Solutions of Classical Hamiltonian Systems without Palais-Smale Condition

In this paper, we study the existence of periodic solutions for classical Hamiltonian systems without the Palais-Smale condition. We prove that the information of the potential function contained in a finite domain is sufficient for the existence of periodic solutions. Moreover, we establish the existence of infinitely many periodic solutions without any symmetric condition on the potential function V.     

Periodic solutions with prescribed minimal period for the second order Hamiltonian systems with even potentials

In this paper, under a similar but stronger condition than that of Ambrosetti and Rabinowitz we find a T-periodic solution of the autonomous superquadratic second order Hamiltonian system with even potential for any T 〉 0; moreover, such a solution has T as its minimal period.     

Multibump homoclinic solutions for a class of second order, almost periodic Hamiltonian systems

We prove the existence of infinitely many homoclinic solutions for a class of second order Hamiltonian systems in R ^N of the form where is almost periodic and W is superquadratic. Received October 17, 1995     

THE STABILITY OF THE PERIODIC SOLUTIONS OF SECOND ORDER HAMILTONIAN SYSTEMS

51.IntroductionandMainResultsInthispaper,weconsiderthestabilityoftheperiodicsolutionsofthefollowingsecondorderHamiltoniansystemswherenisapositiveillteger.V:Rxac-RisafUnctionandforT>0itisT-periodicforthevariablet.VJdenotesitsgradientwithrespecttox.Wenowstatethemainresultsofthispaper.Forthesuperquadraticcase)wehavethefollowingtwotheoremsTheorem1.1.SmposeVsatiSfiesthefollowingconditions:(VI)VEC'(RxR",R)andV(t T,x)=V(t,x),V(t,x)ERxR".(V2)ThereexistconstantSp>2andco>0suchthat0相似文献   

Nontrivial Solutions of Superquadratic Hamiltonian systems with Lagrangian Boundary Conditions and the L-index Theory

In this paper, the authors study the existence of nontrivial solutions for the Hamiltonian systems z（t） = J△↓H（t, z（t）） with Lagrangian boundary conditions, where ^H（t,z）=1/2（^B（t）z, z） ＋ ^H（t, z）,^B（t） is a semipositive symmetric continuous matrix and ^H（t, z） = satisfies a superquadratic condition at infinity. We also obtain a result about the L-index.     

Computations of cohomology groups and nontrivial periodic solutions of Hamiltonian systems

By computing the E-critical groups at θ and infinity of the corresponding functional of Hamiltonian systems, we proved the existence of nontrivial periodic solutions for the systems which may be resonant at θ and infinity under some new conditions. Some results in the literature are extended and some new type of theorems are proved. The main tool is the E-Morse theory developed by Kryszewski and Szulkin.     

Periodic orbits of radially symmetric Keplerian-like systems: A topological degree approach

We are concerned with non-autonomous radially symmetric systems with a singularity, which are T-periodic in time. By the use of topological degree theory, we prove the existence of large-amplitude periodic solutions whose minimal period is an integer multiple of T. Precise estimates are then given in the case of Keplerian-like systems, showing some resemblance between the orbits of those solutions and the circular orbits of the corresponding classical autonomous system.     

Existence of even homoclinic orbits for second-order Hamiltonian systems

Some existence theorems for even homoclinic orbits are obtained for a class of second-order nonautonomous Hamiltonian systems with symmetric potentials under a class of new superquadratic conditions. A homoclinic orbit is obtained as a limit of solutions of a certain sequence of nil-boundary-value problems which are obtained by the minimax methods.     

SOME RESULTS ON THE MINIMAL PERIOD PROBLEM OF NONCONVEX SECOND ORDER HAMILTONIAN SYSTEMS

1.IntrodnctionandMainResultsInthispaper,weconsidertheminimalperiodproblemforthefollowingautonomoussecondorderHamiltoniansystemswhereNisapositiveinteger.andV'denotesitsgradient.Inthetextofthispaper,wedenotebya.bandtheusualinnerproductandnorminRNrespec...     

MULTIPLE SOLUTIONS FOR NONAUTONOMOUS SECOND ORDER PERIODIC SYSTEMS

We study nonautonomonus second order periodic systems with a nonslnooth potential. Using the nonsmooth critical theory, we establish the existence of at least two nontrivial solutions. Our framework incorporates large classes of both subquadratic and superquadratic potentials at infinity.     

Periodic solutions for a class of new superquadratic second order Hamiltonian systems

A new superquadratic growth condition is introduced, which is an extension of the well-known superquadratic growth condition due to P.H. Rabinowitz and the nonquadratic growth condition due to Gui-Hua Fei. An existence theorem is obtained for periodic solutions of a class of new superquadratic second order Hamiltonian systems by the minimax methods in critical point theory, specially, a local linking theorem.     

Periodic solutions of a class of superquadratic second order Hamiltonian systems

§1　IntroductionInthispaperwediscusstheexistenceofthesolutionforthefollowingsecondorderHamiltoniansystemx¨ Ax ΔF(x)=0,(1.1)whereAisann×nrealsymmetricmatrixandisnon-definite,F∈C1(Rn,R),andΔF(x)denotesthegradientofF.WhileworksforsecondorderHamiltonsystemshavemostlybeendoneundertheconditionA=0,westudythecasewhereA≠0andisnon-definiteinthepapers[1,2].DefineH=H1,2T([0,T],Rn)={x:R→Rn|xisabsolutelycontinuous,x∈L2([0,T],Rn),x(0)=x(T),x(0)=x(T)}and〈x,y〉=∫T0[(x(t),y(t)) (x…     

Existence of periodic solutions for a class of second-order discrete Hamiltonian systems

By using the variant fountain theorem, we study the existence of periodic solutions for a class of superquadratic non-autonomous second-order discrete Hamiltonian systems.