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Boundary value problem classification of complete second-order equation in a Banach space. (English. Russian original) Zbl 0729.34037

Sov. Math. 34, No. 6, 45-52 (1990); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1990, No. 6(337), 39-45 (1990).
One considers the Cauchy problem and Dirichlet problem for \((1)\quad d^ 2u/dt^ 2=Adu/dt+Bu.\) The authors show that the sufficient condition imposed on the resolvent of the operators A,B; \(R(\lambda^ 2,A,B)=R(\lambda^ 2)=(B+\lambda A-\lambda^ 2I)^{-1}\) in the case of the commutativity of A,B, may be extended also to the case of noncommutativity of A,B and the fact that the separation of this condition over the resolvent of one of the operators can be realized easier using simpler comparison by comparing their definition domains.
In this sense, for (1) with comparable operators A,B the classification with respect to the Cauchy problem takes the form:
- D(A)\(\subseteq D(B)\); A is the principal operator;
- D(B)\(\subset D(A)\neq E\); there is no principal operator;
- \(D(A)=E\); B is the principal operator.
Some considerations concerning the correctness of the Dirichlet problem, respectively the classification with respect to the Dirichlet problem are given, too.

MSC:

34G10 Linear differential equations in abstract spaces
34B05 Linear boundary value problems for ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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