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On linear relations generated by nonnegative operator function and degenerate elliptic differential-operator expression. (English) Zbl 1203.47024
The author considers spectral problems for equations on vector-functions with values in a Hilbert space, of the form $$-y''(t)+t^\alpha A_1(t)y(t)-\lambda A_2(t)y(t)=f(t),\;0<t<b$$, where $$\alpha \geq 0$$, $$A_1(t)$$ is a positive definite selfadjoint operator-function, and $$A_2(t)$$ is a nonnegative operator function. Such an equation, with appropriate boundary conditions, generates a linear relation, not necessarily an operator. As in the conventional theory of differential operators, maximal and minimal relations are defined. Under certain assumptions, the operators inverse to the invertible restrictions of the maximal relations are shown to be integral. The minimal relation is symmetric; its generalized resolvents are described.

##### MSC:
 47E99 Ordinary differential operators 34G10 Linear differential equations in abstract spaces 34L99 Ordinary differential operators 47A06 Linear relations (multivalued linear operators) 47B25 Linear symmetric and selfadjoint operators (unbounded)