Two Exponential Formulas for $α$-Times Integrated Semigroups $\boldsymbol{(α} \boldsymbol{\in} \boldsymbol{\mathbb{R}^ + )}$ - Vajzović, Fikret; Vugdalić, Ramiz (Sarajevo Journal of Mathematics, 2024)
In this paper $X$ is a Banach space, $\left( {S(t)}\right) _{t≥ 0}$ is non-dege\-ne\-ra\-te $α -$times integrated, exponentially bounded semigroup on $X$ $(α \in \mathbb{R}^{+}),$ $M≥ 0$ and $ω _{0}\in \mathbb{R}$ are constants such that $\left\| {S(t)}\right\| \leqslant Me^{ω _{0}t}$ for all $t≥ 0,$ $γ $ is any positive constant greater than $ω _{0},$ $Γ $ is the Gamma-function, $(C,β )-\lim $ is the Cesàro-$β $ limit. Here we prove that \begin{equation*} \mathop {\lim }\limits_{n→ ∞ }\frac{1}{{Γ (α )}} \int\limits_{0}^{T}{(T-s)^{α -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}{{2π i}}(C,β )-\mathop {\lim }\limits_{ω → ∞ }\int\limits_{γ -iω }^{γ +iω }{ e^{λ t}\frac{{R(λ ,A)x}}{{λ ^{α }}}\,dλ }, \end{equation*} for every $x\in X,\,\,β >0$ and $t≥ 0.$ 2000 Mathematics Subject Classification. 47D06, 47D60, 47D62