Journal of Fixed Point Theory and Applications最新文献

Abstract

Fix a suitable link on the disk. Recently, F. Morabito associates each Hamiltonian symplecticmorphism preserving the link to a braid type. Based on this construction, Morabito defines a family of pseudometrics on the braid groups by using the Hofer metric. In this paper, we show that two Hamiltonian symplecticmorphisms define the same braid type provided that their Hofer distance is sufficiently small. As a corollary, the pseudometrics defined by Morabito are nondegenerate.

In this paper, we use truncation argument combined with method of minimization, argument of comparison, topological degree arguments and sub-supersolutions method to show existence of multiple positive solutions (which are ordered in the (C(overline{Omega }))-norm) for the following class of problems:

where (Omega ) is a bounded smooth domain of (mathbb {R}^N) ((Nge 1), kappa ,mu ,lambda > 0,qge 1) are parameters, the nonlinearity (f: mathbb {R}rightarrow mathbb {R}) is a continuous function that can change sign and satisfies an area condition and (h: mathbb {R}rightarrow mathbb {R}) is a general nonlinearity.

We classify, up to fiberwise symplectomorphisms, a saturated neighborhood of a singular fiber of an integrable system (which is proper onto its image and has connected fibers) containing (k geqslant 1) focus-focus critical points. Our proof recovers the classification for (k=1) which was known prior to this paper. Our result shows that there is a one-to-one correspondence between such neighborhoods and k formal power series, up to a ((mathbb {Z}_2 times D_k))-action, where (D_k) is the kth dihedral group. The k formal power series determine the dynamical behavior of the Hamiltonian vector fields associated to the components of the momentum map on the symplectic manifold ((M,omega )) near the singular fiber containing the k focus-focus critical points. This proves a conjecture of San Vũ Ngọc from 2003.

We construct finite energy foliations and transverse foliations of neighbourhoods of the circular orbits in the rotating Kepler problem for all negative energies. This paper would be a first step towards our ultimate goal that is to recover and refine McGehee’s results on homoclinics [23] and to establish a theoretical foundation to the numerical demonstration of the existence of a homoclinic–heteroclinic chain in the planar circular restricted three-body problem [20], using pseudoholomorphic curves.

Abstract

We show that for a generic Riemannian or reversible Finsler metric on a compact differentiable manifold M of dimension at least three all closed geodesics are simple and do not intersect each other. Using results by Contreras (Ann Math 2(172):761–808, 2010; in: Proceedings of International Congress Mathematicians (ICM 2010) Hyderabad, India, pp 1729–1739, 2011) this shows that for a generic Riemannian metric on a compact and simply-connected manifold all closed geodesics are simple and the number N(t) of geometrically distinct closed geodesics of length (le t) grows exponentially.

We show how methods from Hamiltonian Floer theory can be used to establish lower bounds for the number of different time-periodic measures of time-periodic Hamiltonian systems with diffusion. After proving the existence of closed random periodic solutions and of the corresponding Floer curves for Hamiltonian systems with random walks with step width 1/n for every (nin mathbb {N}), we show that, after passing to a subsequence, they converge in probability distribution as (nrightarrow infty ). Besides using standard results from Hamiltonian Floer theory and about convergence of tame probability measures, we crucially use that sample paths of Brownian motion are almost surely Hölder continuous with Hölder exponent (0<alpha <frac{1}{2}).

In this paper, we prove the existence of non-radial solutions to the problem (-triangle u= f(x,u)), (u|_{partial Omega }=0) on the unit ball (Omega :={xin {mathbb {R}}^3: Vert xVert <1}) with (u(x)in {mathbb {R}}^s), where f is a sub-linear continuous function, differentiable with respect to u at zero and satisfying (f(gx,u) = f(x,u)) for all (gin O(3)), ( f(x,-u)=- f(x,u)). We investigate symmetric properties of the corresponding non-radial solutions. The abstract result is supported by a numerical example.