Problem for the Sturm-Liouville differential equation with an operator coefficient (Q1088838)

The author establishes sufficient conditions for existence and uniquenes of weak solutions of \[ -y''(t)+A^ 2y(t)=f(t),\quad y(0)=\phi_ 1,\quad y(\tau)=\phi_ 2,\quad \| y'(0)\| =v, \] where y and f map [0,\(\tau\) ] into a Hilbert space and A is a self-adjoint positive- definite operator. The analysis uses results of \textit{V. I. Gorbachuk} and \textit{M. L. Gorbachuk} [Mat. Zametki 24, 801-807 (1978; Zbl 0423.34082)].