α-Times Integrated Semigroups (α∈R+)

Abstract

The ␣-times integrated semigroups, ␣ ) 0, are introduced and analyzed. It is shown that suitable differential operators generate ␣-times integrated semigroups Ž . for ␣ g 1r2, 1 . ᮊ 1997 Academic Press ␣ in Proposition 1, Corollary 1, Theorem 2, and Corollary 2 the properties of the generator A of an ␣-times integrated semigroup S and apply the ␣ 790

a-times integrated semigroups and fractional derivation

“…being a pseudoresolvent, i.e., Rl À Rm m À lRlRm for`lY`m b o [18], [29]. Moreover, the operator A de®ned by the previous pseudoresolvent is called the in®nitesimal generator of the semigroup fS tg t0 .…”

“…In an informal way, the``formal solution'' of the Cauchy Problem is smoothed integrating it times in the fractional sense of Riemann-Liouville. Some works about these semigroups were published later on, see for instance [29], [8] (about spectral mapping theorem), [37] (in locally convex spaces).…”

Integrated Fractional Resolvent Operator Function and Fractional Abstract Cauchy Problem

“…If is a closed linear operator, ( ) denotes the resolvent set of and ( , ) = ( − ) −1 denotes the resolvent operator of . 1…”

Hille–Yosida type theorems for -times integrated semigroups

“…O(1 + t k ) once integrated semigroups were investigated in [8,9]. It should be pointed out that Mijatović, Pilipović and Vajzović established Hille-Yosida type theorems for α-times integrated semigroups (α ∈ R + ) in [7], Xiao and Liang established approximation theorems for α-times integrated semigroups under weaker conditions in [14]. α-times integrated C semigroups were also extensively studied, for example, by Li and Shaw in [13] for α = n, by Bachar, Desch and Mardiyana in [15].…”