In this paper $\left( S(t)\right) _{t\geq 0}$ is an exponentially bounded integrated semigroup on a Banach space $X,$ with generator $A.$ We present some relations between an integrated semigroup and its generator $A,$ or its resolvent.   2000 Mathematics Subject Classification. 47D60, 47D62

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

In this paper we analyze the values and the properties of the function $S(n,l):=\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}(n-k)^{l}\ \(n,l\in \mathbb{N\cup }\left\{ 0\right\} ),$ for $n<l.$ At first, we obtain two recurrence relations. Namely, we prove that forevery $n\in \mathbb{N\cup } \left\{ 0\right\} $ and every $l\in \mathbb{N}$ such that $l>n,$ we have\begin{equation*}S(n+1,l)=\sum_{k=1}^{l-n}\binom{l}{k}S(n,l-k),\end{equation*}and also, for every $n\in \mathbb{N\cup }\left\{ 0\right\} $and every $l\in \mathbb{N}$, we have\begin{equation*}S(n+1,l)=(n+1)S(n,l-1)+(n+1)S(n+1,l-1).\end{equation*}Further, we conclude that for every $n\geq 2$ and every $l\geq n$ the following representation formula holds\begin{multline*}S(n,l) =\sum\limits_{k_{1}=1}^{l-(n-1)}\binom{l}{k_{1}}\sum\limits_{k_{2}=1}^{l-k_{1}-(n-2)}\binom{l-k_{1}}{k_{2}}\\\cdot\sum\limits_{k_{3}=1}^{l-k_{1}-k_{2}-(n-3)}\binom{l-k_{1}-k_{2}}{k_{3}}\dots\sum\limits_{k_{n-1}=1}^{l-\sum\limits_{i=1}^{n-2}k_{i}-1}\binom{l-\sum\limits_{i=1}^{n-2}k_{i}}{k_{n-1}}.\end{multline*}We obtain an explicit formula for the calculation $S(n,l),$ especially for $ l=n+1,\dots,n+5,$ and later we give a general result.   2000 Mathematics Subject Classification. 40B05, 11Y55, 05A10

Motivated with Hille's first exponential formula for $C_{0}$ semigroups, we prove a formula for $n-$times integrated semigroups. At first we prove a formula for twice integrated semigroup, and, later, we generalize this formula for $n-$times integrated semigroups.   2000 Mathematics Subject Classification. 47D60, 47D62

In this paper, we consider the topic from the theory of cosine operator functions in 2-dimensional real vector space, which is an interplay between functional analysis and matrix theory. For the various cases of a given real matrix A= [α , β; γ , δ] we find out the appropriate cosine operator function C(t)= [a(t), b(t); c(t), d(t)], (t \in R) in a real vector space R2 as the solutions of the Cauchy problem C''(t)=AC(t), C(0)=I, C'(0)=0.

. In this paper, we consider the topic from the theory of cosine operator functions in 2-dimensional real vector space, which is an interplay between functional analysis and matrix theory. For the various cases of a given real matrix A = , we find out the appropriate cosine operator function in a real vector space 2 , as the solutions of the

In this paper, we consider the nonlinear superposition operator F in lp spaces of sequences (1 ≤ p ≤ ∞), generated by the function f(s,u)=a(s) + arctan u or f(s,u) = a(s) - arctan u. We find out the Rhodius spectra σR(F) and the Neuberger spectra σN(F) of these operators and finally the radii of these spectra. The superposition operator generated by the function f(s,u) = a(s) ∓ arccot u appears to be a special case of above mentioned operator.

In this paper we consider the nonlinear superposition operator F in lp spaces of sequences, generated by the function f (s, u) = a (s) + u or f (s, u) = a (s) · u First we show that these operators are Fréchet differentiable. Then we find out the Neuberger spectra σN (F ) of these operators. We compare it with some other nonlinear spectra and indicate some possible applications.

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.