On the structure of solutions of operator-differential equations on the whole real axis
We consider differential equations of the form (d^2 dt^2 - B)^my(t) = f(t), m N, t (-∞, ∞), where B is a positive operator in a Banach space B, f(t) is a bounded continuous vector-valued function on (-∞, ∞) with values in B, and describe all their solutions. In the case, where f(t) 0, we prove that every solution of such an equation can be extended to an entire B-valued function for which the Phragmen-Lindelöf principle is fulfilled. It is also shown that there always exists a unique bounded on R^1 solution, and if f(t) is periodic or almost periodic, then this solution is the same as f(t)., number=2, journal=Methods of Functional Analysis and Topology, author=Gorbachuk, V. M., year=2015, month=Jun., pages=170–178