On the structure of solutions of operator-differential equations on the whole real axis. (English) Zbl 1340.34204

Gorbachuk, V. M.

Methods Funct. Anal. Topol. 21, No. 2, 170-178 (2015).

The author gives a description of solutions of the equation \[ \left( \frac{d^2}{dt^2}-B\right)^m y(t)=f(t), \] where \(B\) is a positive operator on a Banach space \(\mathcal B\), \(f\) is a bounded continuous \(\mathcal B\)-valued vector function on \(\mathbb R\). If \(f(t)\equiv 0\), every solution can be extended to a \(\mathcal B\)-valued entire function; an analog of the Phragmen-Lindelöf principle is proved. If \(f\) is periodic or almost periodic, then the solution has the same property.

Reviewer: Anatoly N. Kochubei (Kyïv)

MSC:

34G10

Linear differential equations in abstract spaces

Keywords:

operator-differential equations; positive operator; entire function