C-Cosine Functions and the Abstract Cauchy Problem, II |
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Authors: | Chung-Cheng Kuo Sen-Yen Shaw |
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Institution: | Department of Mathematics, National Central University, Chung-Li, Taiwan |
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Abstract: | For a bounded linear injectionCon a Banach spaceXand a closed linear operatorA : D(A) X → Xwhich commutes withCwe prove that (1) the abstract Cauchy problem,u″(t) = Au(t),t R,u(0) = Cx,u′(0) = Cy, has a unique strong solution for everyx,y D(A) if and only if (2)A1 = AD(A2) generates aC1-cosine function onX1(D(A) with the graph norm), if (and only if, in caseAhas nonempty resolvent set) (3)Agenerates aC-cosine function onX. HereC1 = CX1. Under the assumption thatAis densely defined andC−1AC = A, statement (3) is also equivalent to each of the following statements: (4) the problemv″(t) = Av(t) + C(x + ty) + ∫t0 Cg(r) dr,t R,v(0) = v′(0) = 0, has a unique strong solution for everyg L1locandx, y X; (5) the problemw″(t) = Aw(t) + Cg(t),t R,w(0) = Cx,w′(0) = Cy, has a unique weak solution for everyg L1locandx, y X. Finally, as an application, it is shown that for any bounded operatorBwhich commutes withCand has range contained in the range ofC,A + Bis also a generator. |
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