Differential equations in Banach spaces

Let H be a fixed Hilbert space and B ( H, H ) be the Banach space of bounded linear operators from H to H with the uniform operator topology. Oscillation criteria are obtained for the operator differential equation where the coefficients A, C are linear operators from B ( H, H ) to B ( H, H ), for each t ≤ 0. A solution Y : R + → B ( H, H ) is said to be oscillatory if there exists a sequence of points t i ∈ R + , so that t i → ∞ as i → ∞, and Y ( t i ) fails to have a bounded inverse. The main theorem states that a solution Y is oscillatory if an associated scalar differential equation is oscillatory.