C-cosine functions and the abstract cauchy problem .2.

[摘要] For a bounded linear injection C on a Banach space X and a closed linear operator A : D(A) subset of X --> X which commutes with C we prove that (1) the abstract Cauchy problem, u ''(t) = Au(t), t is an element of R, u(0) = Cx, u'(0) = Cy, has a unique strong solution for every x, y is an element of D(A) if and only if (2) A(1) = A \ D(A(2)) generates a C-1-cosine function on X-1 (D(A) with the graph norm), if (and only if, in case A has nonempty resolvent set) (3) A generates a C-cosine function on X. Here C-1 = C \ X-1. Under the assumption that A is densely defined and C(-1)AC = A, statement (3) is also equivalent to each of the following statements: (4) the problem upsilon ''(t) = A upsilon(t) + C(x + ty) + integral(0)(t) Cg(r) dr, t is an element of R, upsilon(0) = upsilon'(0) = 0, has a unique strong solution for every g is an element of L-loc(1) and x, y is an element of X; (5) the problem w ''(t) = Aw(t) + Cg(t), t is an element of R, w(0) = Cx, w'(0) = Cy, has a unique weak solution for every g is an element of L-loc(1) and x, y is an element of X. Finally, as an application, it is shown that for any bounded operator B which commutes with C and has range contained in the range of C, A + B is also a generator. (C) 1997 Academic Press.

[发布日期] 1997-06-15