The dynamical system (f,R) is introduced and some of its properties are investigated. It is proven that there is an invariant set Λ on which the periodic points of f are dense.

In this paper the periodic solutions of fourth order delay differential equation of the form $ddddot{x}(t)+adddot{x}(t)+f(ddot{x}(t-tau(t)))+g(dot{x}(t-tau(t)))+h({x}(t-tau(t)))=p(t)$  is investigated. Some new positive periodic criteria are given.  

in this paper the periodic solutions of fourth order delay differential equation of the form $ddddot{x}(t)+adddot{x}(t)+f(ddot{x}(t-tau(t)))+g(dot{x}(t-tau(t)))+h({x}(t-tau(t)))=p(t)$  is investigated. some new positive periodic criteria are given.

By counting the numbers of periodic points of all periods for some interval maps, we obtain infinitely many new congruence identities in number theory. Let S be a nonempty set and let f be a map from S into itself. For every positive integer n, we define the n iterate of f by letting f 1 = f and f = f ◦ f for n ≥ 2. For y ∈ S, we call the set { f(y) : k ≥ 0 } the orbit of y under f . If f(y) = ...

Let f be a continuous map of the circle into itself . The main purpose of this paper is to study the properties of the unstable manifold associated to a periodic point of f . Let 2(f) denote the nonwandering set of f . Suppose f has finitely many periodic points . Then, using the unstable manifolds associated to periodic points of f, three theorems are proved providing complete answers to the f...

On the other hand, let S be a subset of the real numbers and let / be a function from S into itself. For every positive integer n, we let f denote the n iterate of f : f =f and f=f<> f~ for n > 2. For every #0 G s > w e call the set {f(xQ)\k > 0} the orbit of xQ under /. If XQ satisfies f{x$) = XQ for some positive integer 77?, then we call XQ a periodic point of / and call the smallest such po...

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

this paper is concerned with the following elliptic system:$$ left{ begin{array}{ll} -triangle u + b(x)nabla u + v(x)u=g(x, v), -triangle v - b(x)nabla v + v(x)v=f(x, u), end{array} right. $$ for $x in {r}^{n}$, where $v $, $b$ and $w$ are 1-periodic in $x$, and $f(x,t)$, $g(x,t)$ are super-quadratic. in this paper, we give a new technique to show the boundedness of cerami sequences and establi...