In this work the essential reachability property of positive N -periodic discrete-time linear systems is studied. A characterization, by means of the colored union of directed graphs of the periodic matrices, and a canonical form for the esential reachability property is shown. Key–Words:Positive invariant systems, positive periodic systems, essential reachability, directed graphs. 1 Background...

We study the dynamics of a holomorphic self-map f of complex projective space of degree d > 1 by utilizing the notion of a Fatou map, introduced originally by Ueda (1997) and independently by the author (2000). A Fatou map is intuitively like an analytic subvariety on which the dynamics of f are a normal family (such as a local stable manifold of a hyperbolic periodic point). We show that globa...

We establish conditions for the existence of periodic solutions of nonlinear, second-order difference equations of the form y(t + 2) + by(t + 1) + cy(t) = f (y(t)), where c = 0 and f :R→ R is continuous. In our main result we assume that f exhibits sublinear growth and that there is a constant β > 0 such that u f (u) > 0 whenever |u| ≥ β. For such an equation we prove that ifN is an odd integer...

In this paper, we consider the higher dimensional nonlinear beam equations utt + u + σu + f (u) = 0, with periodic boundary conditions, where the nonlinearity f (u) is a real– analytic function nearu = 0 with f (0) = f ′(0) = 0 and σ is a real parameter in an interval I ≡ [σ1, σ2]. It is proved that for ‘most’ positive parameters σ lying in the finite interval I, the above equations admit a fam...

In this paper, by using the Hopf’s bifurcation theorem we will discuss the existence of small amplitude periodic solutions of the equation ẍ(t)+ f(x(t))ẋ(t) + g(x(t − r)) = 0, taking as bifurcation parameter c either d or r. We assume that r > 0, f ∈ C, f(0) = c > 0, g(0) = 0 and ġ(0) = d > 0.

We obtain the existence and multiplicity of positive T -periodic solutions for the difference equations ∆x(n) = a(n, x(n))− λb(n)f(x(n− τ(n))) and ∆x(n) + a(n, x(n)) = λb(n)f(x(n− τ(n))), where f(·) may be singular at x = 0. Using a fixed point theorem in cones, we extend recent results in the literature.

We express the Fourier coefficients of the Hilbert cusp form Lhf associated with mixed Hilbert cusp forms f and h in terms of the Fourier coefficients of a certain periodic function determined by f and h. We also obtain an expression of each Fourier coefficient of Lhf as an infinite series involving the Fourier coefficients of f and h.

Let $$f:X\rightarrow X$$ be a hereditarily locally connected continuum homeomorphism and denote respectively by P(f), AP(f) $$\Omega (f)$$ , the sets of periodic points, almost points non-wandering f. We show that any $$\omega $$ -limit set (resp. $$\alpha set) is minimal. Moreover, we (f)=AP(f)$$ . also prove if $$P(f)=\emptyset then there exists unique minimal set. On other hand, $$P(f)\ne \e...

Let F be the class of all 1-periodic real functions with absolutely convergent Fourier series expansion and let (xn)n≥0 be the van der Corput sequence. In this paper results on the boundedness of ∑N−1 n=0 f(xn) for f ∈ F are given. We give a criterion on the convergence rate of the Fourier coefficients of f such that the above sum is bounded independently of N . Further we show that our result ...

As a natural generalization of various investigations in different function spaces, we study the following problem. Let A be an arbitrary set, and Tj (j = 1, . . . , n) be arbitrary commuting mappings – transformations – from A into A. Under what conditions can we state that a function f : A → A is the sum of “periodic”, that is, Tj-invariant functions fj? (A function g is periodic, i.e., invar...