Abstract:
Some existence and multiplicity of periodic solutions are obtained for nonautonomous second order Hamiltonian systems with sublinear nonlinearity by using the least action principle and minimax methods in critical point theory. Mathematics Subject Classification (2000): 34C25, 37J45, 58E50.

Abstract:
We give necessary and sufficient conditions for the existence of positive solutions for sublinear Dirichlet periodic parabolic problems Lu=g(x,t,u) in Ω×ℝ (where Ω⊂ℝN is a smooth bounded domain) for a wide class of Carathéodory functionsg:Ω×ℝ×[0,∞)→ℝ satisfying some integrability and positivity conditions.

Abstract:
Two sequences of distinct periodic solutions for second-order Hamiltonian systems with sublinear nonlinearity are obtained by using the minimax methods. One sequence of solutions is local minimum points of functional, and the other is minimax type critical points of functional. We do not assume any symmetry condition on nonlinearity. 1. Introduction and Main Result We are interested in the following second order Hamiltonian systems: where and satisfies the following assumption. is measurable in for each and continuously differentiable in for a.e. for and there exist such that for all and . Then the corresponding functional on given by is continuously differentiable and weakly lower semicontinuous on , where is a Hilbert space with the norm defined by for (see [1]). Moreover, for all . It is well known that the solutions of problem (1.1) correspond to the critical points of . There are large number of papers that deal with multiplicity results for this problem. Infinitely many solutions for problem (1.1) are obtained in [2–4], where the symmetry assumption on the nonlinearity has played an important role. In recent years, many authors have paid much attention to weaken the symmetry condition, and some existence results on periodic and subharmonic solutions have been obtained without the symmetry condition (see [5–7]). Particularly, Ma and Zhang [6] got the existence of a sequence of distinct periodic solutions under some superquadratic and asymptotic quadratic cases. Faraci and Livrea [7] studied the existence of infinitely many periodic solutions under the assumption that is a suitable oscillating behaviour either at infinity or at zero. In this paper, we suppose that the nonlinearity is sublinear, that is, there exist and such that for all and . We establish some multiplicity results for problem (1.1) under different assumptions on the potential . Roughly speaking, we assume that has a suitable oscillating behaviour at infinity. Two sequences of distinct periodic solutions are obtained by using the minimax methods. One sequence of solutions is local minimum points of functional, and the other is minimax type critical points of functional. In particular, we do not assume any symmetry condition at all. Our main result is the following theorem. Theorem 1.1. Suppose that satisfies assumptions (A) and (1.7). Assume that Then,(i)there exists a sequence of periodic solutions which are minimax type critical points of functional , and , as ;(ii)there exists another sequence of periodic solutions which are local minimum points of functional , and , as . 2. Proof

Abstract:
We prove several generic existence results for infinitely many periodic orbits of Hamiltonian diffeomorphisms or Reeb flows. For instance, we show that a Hamiltonian diffeomorphism of a complex projective space or Grassmannian generically has infinitely many periodic orbits. We also consider symplectomorphisms of the two-torus with irrational flux. We show that such a symplectomorphism necessarily has infinitely many periodic orbits whenever it has one and all periodic points are non-degenerate.

Abstract:
By applying minimax methods in critical point theory, we prove the existence of periodic solutions for the following discrete Hamiltonian systems Δ2(？1)

Abstract:
We give a necessary and sufficient condition for the existence of a positive principal eigenvalue for a periodic-parabolic problem with indefinite weight function. The condition was originally established by Beltramo and Hess [extit{frenchspacing Comm. Part. Diff. Eq.}, extbf{9} (1984), 919--941] in the framework of the Schauder theory of classical solutions. In the present paper, the problem is considered in the framework of variational evolution equations on arbitrary bounded domains, assuming that the coefficients of the operator and the weight function are only bounded and measurable. We also establish a general perturbation theorem for the principal eigenvalue, which in particular allows quite singular perturbations of the domain. Motivation for the problem comes from population dynamics taking into account seasonal effects.

Abstract:
This paper studies the perturbation systems. Under som e suitable conditions the proof of the existence of almost periodic solution and bounded solution of perturbation systems is given.

Abstract:
Using the component Lyapunov functions and critical point theory, the theorem of exisence and uniqueness of hyperbolic periodic solution for a class of Hamiltonian systems is given.

Abstract:
Aim of the paper is to provide a method to analyze the behavior of $T$-periodic solutions $x_\eps, \eps>0$, of a perturbed planar Hamiltonian system near a cycle $x_0$, of smallest period $T$, of the unperturbed system. The perturbation is represented by a $T$-periodic multivalued map which vanishes as $\eps\to0$. In several problems from nonsmooth mechanical systems this multivalued perturbation comes from the Filippov regularization of a nonlinear discontinuous $T$-periodic term. \noindent Through the paper, assuming the existence of a $T$-periodic solution $x_\eps$ for $\eps>0$ small, under the condition that $x_0$ is a nondegenerate cycle of the linearized unperturbed Hamiltonian system we provide a formula for the distance between any point $x_0(t)$ and the trajectories $x_\eps([0,T])$ along a transversal direction to $x_0(t).$

Abstract:
We present an illustrative application of the two famous mathematical theorems in differential topology in order to show the existence of periodic orbits with arbitrary given period for a class of hamiltonians .This result point out for a mathematical answer for the long standing problem of existence of Planetary Sistems around stars.