This article concerns the diffusion system ∂tu−∆xu + V (x)u = g(t, x, v), −∂tv −∆xv + V (x)v = f(t, x, u), where z = (u, v) : R × RN → R2, V (x) ∈ C(RN , R) is a general periodic function, g, f are periodic in t, x and asymptotically linear in u, v at infinity. We find a minimizing Cerami sequence of the energy functional outside the Nehari-Pankov manifold N and therefore obtain ground state so...

In order to obtain the conditions for the existence of periodic and almost periodic solutions of Volterra difference equations, x(n+1) = f(n, x(n))+ ∑n s=−∞ F (n, s, x(n+ s), x(n)), we consider certain stability properties, which are referred to as (K, ρ)-weakly uniformly-asymptotic stability and (K, ρ)-uniformly asymptotic stability. Moreover, we discuss the relationship between the ρ-separati...

with periodic boundary conditions are considered; V is a periodic smooth or analytic function and the nonlinearity f is an analytic function vanishing together with its derivative at u = 0. It is proved that for “most” potentials V (x), the above equation admits small-amplitude periodic or quasi-periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dim...

The existence of time periodic solutions of nonlinear wave equations utt −∆nu + ` n− 1 2 ́2 u = g(u)− f(t, x) on n-dimensional spheres is considered. The corresponding functional of the equation is studied by the convexity in suitable subspaces, minimax arguments for almost symmetric functional, comparison principles and Morse theory. The existence of infinitely many time periodic solutions is o...

We consider the Monge-Amp\`ere equation $\det(D^2u)=f$ in $\mathbb{R}^n$, where $f$ is a positive bounded periodic function. prove that $u$ must be sum of quadratic polynomial and For $f\equiv 1$, this classic result by J\"orgens, Calabi Pogorelov. $f\in C^\alpha$, was proved Caffarelli first named author.

In this paper we study the existence and asymptotic stability of periodic solutions of the differential equation ẍ+ f (x)ẋ+g(x) = h(t), where h(t) is T -periodic, f (x) is positive and g(x) is strictly monotonically increasing and has one or two weak singularities. The method of proof relies on the construction of a positively invariant region of the flux. Mathematics subject classification (20...

In this work, we show that if f is a uniformly continuous map defined over Polish metric space, then the set of f-invariant measures with zero entropy G δ $G_\delta$ (in weak topology). particular, generic f-periodic dense in measures. This settles conjecture posed by Sigmund (Trans. Amer. Math. Soc. 190 (1974), 285–299), which states an invariant measure topological dynamical system satisfies ...

Let θ be a Young function and consider the space Fθ(C) of all entire functions with θ-exponential growth. In this paper, we are interested in the solutions f ∈ Fθ(C) of the convolution equation T ⋆ f = 0, called mean-periodic functions, where T is in the topological dual of Fθ(C). We show that each mean-periodic function can be represented in an explicit way as a convergent series of exponentia...

fn = (k −mod(n, k))fbn/kc +mod(n, k)fdn/ke + an, n ≥ k, there is a unique continuous periodic function f∗ : R→ R with period 1 such that fn = nf(logk n)+o(n). If (an) is periodic with period k, ak = 0, and the initial conditions (fi : 1 ≤ i ≤ k − 1) are all zero, we obtain a specific iterated function system S, consisting of k continuous functions from [0, 1] × R into itself, such that the attr...

We discuss the number variance 2 (L) and the spectral form factor F() of the energy levels of bound quantum systems whose classical counterparts are strongly chaotic. Exact periodic{orbit representations of 2 (L) and F() are derived which explain the breakdown of universality, i. e., the deviations from the predictions of random{matrix theory. The relation of the exact spectral form factor F() ...