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 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