A function or a real variable f is said to be periodic with period P if f(x+ P ) = f(x) holds for all x. Hence, if we know the values of f on an interval of length P , we know its values everywhere. If f is a function defined on an interval [a, b), we can extend f to a function defined for all x which is periodic of period b− a. We simply define f(x) to be f(x+ n(b− a)), where n is the integer ...

For all μ > 0, a locally Lipschitz continuous map f with xf (x) > 0, x ∈ R\{0}, is constructed, such that the scalar equation ẋ (t) = −μx (t)−f (x (t− 1)) with delayed negative feedback has an infinite number of periodic orbits. All periodic solutions defining these orbits oscillate slowly around 0 in the sense that they admit at most one sign change in each interval of length of 1. Moreover, i...

Let f be an orientation preserving homeomorphism of the disc D which possesses a periodic point of period 3. Then either f is isotopic, relative the periodic orbit, to a homeomorphism g which is conjugate to a rotation by 2π/3 or 4π/3, or J has a periodic point of least period n for each n ∈ N∗.

1 1. Introduction This paper is devoted to the study of asymptotic almost periodicity of bounded solutions for the following time almost periodic one dimensional scalar parabolic equation:    u t = u xx + f (t, x, u, u x), t > 0, 0 < x < 1, u(t, 0) = u(t, 1) = 0, t > 0, (1.1) where f : IR 1 × [0, 1] × IR 1 × IR 1 → IR 1 is a C 2 function, and f (t, x, u, p) with all its partial derivatives (...

This is a brief survey of up-to-date results on holomorphic almost periodic functions and mappings in one and several complex variables, mainly due to the Kharkov mathematical school. While the notion of almost periodic function on R (or R m) seems to be quite understood, it is not the case for holomorphic almost periodic functions on a strip in C (or, more generally, on a tube domain in C m). ...

The existence of common periodic points for a family of continuous commuting self-mappings on an interval is proved and two illustrative examples are given in support of our theorem and definition. 1. Introduction and preliminaries. All mappings considered here are assumed to be continuous from the interval I = [u, v] to itself. Let F(f) and P (f) be the set of fixed and periodic points of f , ...

This paper is concerned with some extensions of the classical Liouville theorem for bounded harmonic functions to solutions of more general equations. We deal with entire solutions of periodic and almost periodic parabolic equations including the elliptic framework as a particular case. We derive a Liouville type result for periodic operators as a consequence of a result for operators periodic ...

Let f be a homeomorphism of the torus isotopic to the identity and suppose that there exists a periodic orbit with a non-zero rotation vector ( q , r q ). Then f has a topologically monotone periodic orbit with the same

The temperature inversion symmetry, for a non interacting super-symmetric ensemble, at finite volume, is studied. It is found that, the scaled free energy, f (ξ), is antisymmetric under temperature inversion transformation, i.e. f (ξ) = −ξ d f (1 ξ). This occurs for antiperiodic bosons and periodic fermions, in the compact dimension. On the contrary , for periodic bosons and antiperiodic fermio...

We obtain some results regarding the problem of periodicity entire functions f(z) when differential polynomials P(z, f) with constant coefficients generated by are periodic. provide sufficient conditions that ensure , and we discuss properties periodic functions. Our generalize improve earlier ones have an importance concerning solutions equations form $$ P(z,f) =h(z)$$ where h(z) is a function.