A theorem of A. and C. Rényi on periodic entire functions states that an function f(z) must be if P(f(z)) is periodic, where P(z) a nonconstant polynomial. By extending this theorem, we can answer some open questions related to the conjecture Yang concerning periodicity functions. Moreover, give more general forms for prove, in particular, either P(f (z))f(k)(z) or P(f(z))/f(k)(z) provided has ...

We consider the bifurcation of periodic solutions from an equilibrium point of the given equation: x =F(x,?) , where x ? R , ? is a vector of real parameters ? , ? , ... , ? and F:R x R ->R has at least second continuous derivations in variables

We present a general spectral decomposition technique for bounded solutions to inhomogeneous linear periodic evolution equations of the form _ x = A(t)x+f(t) (), with f having precompact range, which will be then applied to nd new spectral criteria for the existence of almost periodic solutions with speciic spectral properties in the resonnant case where e isp(f) may intersect the spectrum of t...

Let T be a tree with n vertices. Let f : T → T be continuous and suppose that the n vertices form a periodic orbit under f . We show: (1) (a) If n is not a divisor of 2k then f has a periodic point with period 2k. (b) If n = 2pq, where q > 1 is odd and p ≥ 0, then f has a periodic point with period 2pr for any r ≥ q. (c) The map f also has periodic orbits of any period m where m can be obtained...

In this paper, we prove the existence and uniqueness of fixed points for cyclic (α,β)-admissible type F-contraction and F−weak contraction under the setting of modular spaces, where the modular is convex and satisfying the ∆2-condition. Later, we prove some periodic point results for self-mappings on a modular space. We also give some examples to s...

Let X be a smooth projective surface over a number field K(⊂ C), and f : X → X an automorphism of positive topological entropy. In this paper, we show that there are only finitely many f -periodic curves on X . Then we define a height function ĥD corresponding to a certain nef and big R-divisor D on X and transforming well relative to f , and deduce some arithmetic properties of f -periodic poi...

We consider the existence, multiplicity and nonexistence of positive o-periodic solutions for the periodic equation x0ðtÞ 1⁄4 aðtÞgðxÞxðtÞ lbðtÞf ðxðt tðtÞÞÞ; where a; bACðR; 1⁄20;NÞÞ are o-periodic, Ro 0 aðtÞ dt40; Ro 0 bðtÞ dt40; f ; gACð1⁄20;NÞ; 1⁄20;NÞÞ; and f ðuÞ40 for u40; gðxÞ is bounded, tðtÞ is a continuous o-periodic function. Define f0 1⁄4 limu-0þ f ðuÞ u ; fN 1⁄4 limu-N f ðuÞ u ; i0...

Given a dynamical system and a function f from the state space to the real numbers, an optimal orbit for f is an orbit over which the time average of f is maximal. In this paper we consider some basic mathematical properties of optimal orbits: existence, sensitivity to perturbations of f , and approximability by periodic orbits with low period. For hyperbolic systems, we conjecture that for (to...

We consider the existence of positive ω-periodic solutions for the periodic equation x (t) = a(t)e x(t) x(t) − λb(t)f (x(t − τ (t))), where a, b ∈ C (R, [0, ∞)) are ω-periodic, ω 0 a(t)dt > 0, ω 0 b(t)dt > 0, f ∈ C ([0, ∞), [0, ∞)), and f (u) > 0 for u > 0, τ (t) is a continuous ω-periodic function.