Boundary-value problems and Cauchy problems for the second-order Euler operator differential equation. (English) Zbl 0624.34014
This paper deals with some linear differential equations on an abstract space, precisely the space of bounded operators on a Hilbert space. The model equation is a second order Euler equation with operator coefficients together with several boundary value conditions. Looking for solutions of the initial value problem of the form exp (tX) an algebraic equation is derived for X. Assuming the existence of such a solution, the question of existence of solutions to the boundary value problems is expressed in terms of algebraic conditions involving X and the operator coefficients connected to the equation. Some interesting general results related to functional operator equations are employed here, with appropriate references.
Reviewer: O.Arino
MSC:
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
| 34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |
| 34G10 | Linear differential equations in abstract spaces |
| 39B52 | Functional equations for functions with more general domains and/or ranges |
Keywords:
space of bounded operators on a Hilbert space; second order Euler equation with operator coefficientsReferences:
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