Local and global solutions of well-posed integrated Cauchy problems

Abstract

Abstract. We study the local well-posed integrated Cauchy problemwith κ > 0, α ≥ 0, and x ∈ X, where X is a Banach space and A a closed operator on X. We extend solutions increasing the regularity in α. The global case (κ = ∞) is also treated in detail. Growth of solutions is given in both cases.

Sharp extensions and algebraic properties for solution families of vector-valued differential equations

“…The following result extends [22, Theorem 2], because we obtain the sharp extension of (g α , g β+1 )-regularized resolvent families when 0 < α < 1 and β − α > −1, and when α → 1 − we recover the α-times integrated semigroup case, considered in [22]. More generally one could consider the case of K-convoluted resolvent families, i.e.…”

“…However, this property is not longer true in the general case of (a, k)regularized resolvent families, where loss of regularity is present. This phenomena has been observed for the case of k-convoluted semigroups [9] (in particular for α-times integrated semigroups in [2,22]) and k-convoluted cosine families [23].…”

Sharp Extensions for Convoluted Solutions of Abstract Cauchy Problems

“…cannot be extended to t ≥ αT , see [29,Example 1]. We may apply the Corollary 4.7 to define (S nα (t)) for t < nαT and Corollary 5.4 to define the map G α :…”

“…Originally they were the first example of convoluted semigroups. An extension formula for n-times integrated semigroups (for n ∈ N) was given in [2, Section IV, (4.2)] and for α-times integrated semigroup in [29,Formula (5)] with α > 0. Extensions of local α-times integrated C-semigroups were given in [24,Theorem 6.1] and automatic extension of local regularized semigroups appears in [36, Section 2].…”

Hille-yosida theorems for local convoluted C-semigroups and cosine functions

“…A large numbers of papers, starting presumably with [2] and [35], written over the last twenty years, have concerned local integrated C-semigroups and cosine functions. Standard references are [26]- [29], [31], [33]- [34] and [36]- [38].…”