Distribution groups

Abstract

We introduce distribution groups and [B0,... ,Bn,C0,...,Cn-1]- groups with not necessarily densely defined generators and systematically analyze relations between them.

Local K-Convoluted C-groups and Abstract Cauchy Problem

“…Here Γ(•) denotes the Gamma function, a (local) α-times integrated C-group on X is called a (local) α-times integrated group on X if C = I; and a (local) C-group on X is called a c 0 -group on X if C = I (see [1,5]). Some basic properites of a nondegenerate (local) α-times integrated C-semigroup on X have been established by many authors (in [2,3,[26][27][28] for α = 0, and in [6, 10, 17-20, 22, 23, 25, 29, 30] for α > 0), which can be extended to the case of local K-convoluted C-semigroup just as results in [7][8][9][10][13][14][15][16]. Some equivalence relations between the generation of a nondegenerate (local) K-convoluted C-semigroup on X with subgenerator A and the unique existence of solutions of the abstract Cauchy problem ACP(A, f, x) are also discussed in [2,26,27] for the case K = j α−1 with α = 0 and in [11-13, 30, 31] with α > 0, and in [8,13,16] for the general case.…”

“…Here S(t)z = t 0 S(s)zds. In general, a local Kconvoluted C-group on X is called a K-convoluted C-group on X if T 0 = ∞; a (local) K-convoluted C-group on X is called a (local) K-convoluted group on X if C = I (the identity operator on X) or a (local) α-times integrated C-group on X if K is equal to the function j α−1 for some α ≥ 0, defined by j α (t) = t α Γ(α+1) (see [4,7,21]). Here Γ(•) denotes the Gamma function, a (local) α-times integrated C-group on X is called a (local) α-times integrated group on X if C = I; and a (local) C-group on X is called a c 0 -group on X if C = I (see [1,5]).…”