Journal of Fixed Point Theory and Applications最新文献

In Dekimpe and Dugardein (J Fixed Point Theory Appl 17:355–370, 2015), Fel’shtyn and Lee (Topol Appl 181:62–103, 2015), the Nielsen zeta function (N_f(z)) has been shown to be rational if f is a self-map of an infra-solvmanifold of type (R). It is, however, still unknown whether (N_f(z)) is rational for self-maps on solvmanifolds. In this paper, we prove that (N_f(z)) is rational if f is a self-map of a (compact) solvmanifold of dimension (le 5). In any dimension, we show additionally that (N_f(z)) is rational if f is a self-map of an (mathcal{N}mathcal{R})-solvmanifold or a solvmanifold with fundamental group of the form (mathbb {Z}^nrtimes mathbb {Z}).

We present an existence and localization result for periodic solutions of second-order non-linear coupled planar systems, without requiring periodicity for the non-linearities. The arguments for the existence tool are based on a variation of the Nagumo condition and the Topological Degree Theory. The localization tool is based on a technique of orderless upper and lower solutions, that involves functions with translations. We apply our result to a system of two coupled Van der Pol oscillators with a forcing component.

For each representative (mathfrak {B}) of a bordism class in the free loop space of a manifold, we associate a moduli space of finite length Floer cylinders in the cotangent bundle. The left end of the Floer cylinder is required to be a lift of one of the loops in (mathfrak {B}), and the right end is required to lie on the zero section. Under certain assumptions on the Hamiltonian functions, the length of the Floer cylinder is a smooth proper function, and evaluating the level sets at the right end produces a family of loops cobordant to (mathfrak {B}). The argument produces arbitrarily long Floer cylinders with certain properties. We apply this to prove an existence result for 1-periodic orbits of certain Hamiltonian systems in cotangent bundles, and also to estimate the relative Gromov width of starshaped domains in certain cotangent bundles. The moduli space is similar to moduli spaces considered in Abouzaid (J Symp Geom 10(1):27–79, 2012), Abbondandolo and Figalli (J Differ Equ 234:626–653, 2007) and Abbondandolo and Schwarz (Geom Topol 14:1569–1722, 2010) for Tonelli Hamiltonians. The Hamiltonians we consider are not Tonelli, but rather of “contact-type” in the symplectization end.

We prove that Lyapunov stability problem demonstrates pathologies even on the set-theoretical level. Namely, there exists an analytic one-parameter family of 5-jets of vector fields that crosses the set of stable jets by a countable union of disjoint intervals.

We show a Wolff–Denjoy type theorem in the case of a one-parameter continuous semigroup of nonexpansive mappings in which there is a compact mapping. Using the notion of attractor, we are also able to prove some specific properties directly related to the Karlsson–Nussbaum conjecture.

We use relative symplectic cohomology to detect heavy sets, with the help of index bounded contact forms. This establishes a relation between two notions SH-heaviness and heaviness, which partly answers a conjecture of Dickstein–Ganor–Polterovich–Zapolsky in the symplectically aspherical setting.

We study the set of invariant idempotent probabilities for place-dependent idempotent iterated function systems defined in compact metric spaces. Using well-known ideas from dynamical systems, such as the Mañé potential and the Aubry set, we provide a complete characterization of the densities of such idempotent probabilities. As an application, we provide an alternative formula for the attractor of a class of fuzzy iterated function systems.

We study time-periodic and spatially localised solutions (breathers) in general infinite conservative and dissipative nonlinear Klein–Gordon lattices. First, in the time-reversible (conservative) case, we give a concise proof of the existence of breathers not using the concept of the anticontinuous limit. The existence problem is converted into an operator equation for time-reversal initial conditions generating breather solutions. A nontrivial solution of this operator equation is established facilitating Schauder’s fixed point theorem. Afterwards, we prove the existence and uniqueness of breather solutions in damped and forced infinite nonlinear Klein–Gordon lattice systems utilising the contraction mapping principle.

In this article, we study the Hamiltonian dynamics on singular symplectic manifolds and prove the Arnold conjecture for a large class of (b^m)-symplectic manifolds. Novel techniques are introduced to associate smooth symplectic forms to the original singular symplectic structure, under some mild conditions. These techniques yield the validity of the Arnold conjecture for singular symplectic manifolds across multiple scenarios. More precisely, we prove a lower bound on the number of 1-periodic Hamiltonian orbits for (b^{2m})-symplectic manifolds depending only on the topology of the manifold. Moreover, for (b^m)-symplectic surfaces, we improve the lower bound depending on the topology of the pair (M, Z). We then venture into the study of Floer homology to this singular realm and we conclude with a list of open questions.

We introduce a notion of strong closing property of contact forms, inspired by the (C^infty ) closing lemma for Reeb flows in dimension three. We then prove a sufficient criterion for strong closing property, which is formulated by considering a monoidal functor from a category of manifolds with contact forms to a category of filtered vector spaces. As a potential application of this criterion, we propose a conjecture which says that a standard contact form on the boundary of any symplectic ellipsoid satisfies strong closing property.