Two Exponential Formulas for α-Times Integrated Semigroups (α R^ + )
In this paper X is a Banach space, ( S(t)) _t≥ 0 is non-degenerate α -times integrated, exponentially bounded semigroup on X (α R^+), M≥ 0 and ω _0R are constants such that \| S(t)\| Me^ω _0t for all t≥ 0, γ is any positive constant greater than ω _0, Γ is the Gamma-function, (C,β )- is the Cesàro-β limit. Here we prove that equation* _n→ ∞ 1Γ (α ) _0^T(T-s)^α -1( n+1 s ) ^n+1R^n+1( n+1 s,A) x ds=S(T)x, equation* for every xX, and the limit is uniform in T>0 on any bounded interval. Also we prove that equation* S(t)x=12π i (C,β )-_ω → ∞ _γ -iω ^γ +iω e^λ tR(λ ,A)x λ ^α dλ , equation* for every xX, β >0 and t≥ 0. 2000 Mathematics Subject Classification. 47D06, 47D60, 47D62 , number=1, journal=Sarajevo Journal of Mathematics, author=Vajzović, Fikret and Vugdalić, Ramiz, year=2024, month=Jun., pages=93–115