In this paper $X$ is a Banach space, $\left( {S(t)}\right) _{t\geq 0}$ is non-dege\-ne\-ra\-te $\alpha -$times integrated, exponentially bounded semigroup on $X$ $(\alpha \in \mathbb{R}^{+}),$ $M\geq 0$ and $\omega _{0}\in \mathbb{R}$ are constants such that $\left\| {S(t)}\right\| \leqslant Me^{\omega _{0}t}$ for all $t\geq 0,$ $\gamma $ is any positive constant greater than $\omega _{0},$ $\Gamma $ is the Gamma-function, $(C,\beta )-\lim $ is the Ces\`{a}ro-$\beta $ limit. Here we prove that\begin{equation*}\mathop {\lim }\limits_{n\rightarrow \infty }\frac{1}{{\Gamma(\alpha )}} \int\limits_{0}^{T}{(T-s)^{\alpha -1}\left({\frac{{n+1}}{s}}\right) ^{n+1}R^{n+1}\left({\frac{{n+1}}{s},A}\right) x\,ds=S(T)x,}\end{equation*}for every $x\in X,$ and the limit is uniform in $T>0$ on any bounded interval. Also we prove that\begin{equation*}S(t)x=\frac{1}{{2\pi i}}(C,\beta )-\mathop {\lim }\limits_{\omega\rightarrow \infty }\int\limits_{\gamma -i\omega }^{\gamma+i\omega }{ e^{\lambda t}\frac{{R(\lambda ,A)x}}{{\lambda^{\alpha }}}\,d\lambda },\end{equation*}for every $x\in X,\,\,\beta >0$ and $t\geq 0.$   2000 Mathematics Subject Classification. 47D06, 47D60, 47D62

We investigate global dynamics of the equation\begin{equation*}x_{n+1}=\frac{x_{n-1}+F}{ax_{n}^2+f},\text{ \ }n=0,1,2,...,\end{equation*}where the parameters $a,F$ and $f$ are positive numbers and the initial conditions $x_{-1},x_{0}$ are arbitrary nonnegative numbers such that $x_{-1}+x_{0}>0$. The existence and local stability of the unique positive equilibrium are analyzed algebraically. We characterize the global dynamics of this equation with the basins of attraction of its equilibrium point and periodic solutions.

We give necessary and sucient conditions for the conti- nuity of the Hilbert transform on complex quasi-Hilbert spaces, i.e. on complex, reexive, strictly convex Banach spaces with G^ ateaux- dierentiable norm and with generalized inner product.

. We will give a necessary and sufficient condition for the infinitesimal generator of a strongly continuous cosine operator function C ( t ), such that (cid:107) C ( t ) (cid:107) ≤ 1 for all t ∈ R on a reflexive, strictly convex (complex) Banach space with a Gˆateaux differentiable norm to be a spectral scalar type operator with the spectral family of hermitian bounded linear projectors.

The main purpose of this paper is to study the fixed point property of non-metric tree-like continua. It is proved, using the inverse systems method, that if X is a non-metric tree-like continuum and if h : X → X is a periodic homeomorphism, then h has the fixed point property (Theorem 2.4). Some theorems concerning the fixed point property of arc-like non-metric continua are also given.

We study various classes of distribution semigroups on the spaces of functions Fr, r 2 R distinguished by their behavior at the origin.

We obtain a formula of decomposition for Φ(A)=A∫Rn S(f(x))φ(x)dx +∫Rnφ (x)dx using the method of stationary phase. Here (S(t))t∈R is once integrated, exponentially bounded group of operators in a Banach space X with generator A, which satisfies the condition: For every x∈X there exists δ=δ(x)>0 such that S(t)x t1/2+δ → 0 as t → 0. The function φ(x) is infinitely differentiable, defined on Rn, with values in X, with a compact support supp φ, the function f(x) is infinitely differentiable defined on Rn, with values in R, and f(x) on supp φ has exactly one nondegenerate stationary point x0.

In this paper we consider bounded cosine operator functions and their connection to Hilbert transforms. AMS Mathematics Subject Classification (1991): 47D09(47A35-47D03)

In this paper we prove that