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Sturm-Liouville; operator differential equation; operator algebraic equation; Cauchy problems; boundary value problems
Cauchy problem, boundary value problems with a boundary value condition and Sturm-Liouville problems related to the operator differential equation $X^{(2)}-AX=0$ are studied for the general case, even when the algebraic equation $X^2-A=0$ is unsolvable. Explicit expressions for the solutions in terms of data problem are given and computable expressions of the solutions for the finite-dimensional case are made available.
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