Well-posedness of second order differential equations in Hölder continuous function spaces

Abstract

By using operator-valued C˙ α -Fourier multiplier results on vector-valued H¨older continuous function spaces C α (R; X) proved by Arendt, Batty and Bu, we obtain a necessary and sufficient condition for the C α -well-posedness for the following second order differential equations: u ′′(t) = Au(t) + Bu′ (t) + f(t), (t ∈ R), where A and B are closed linear operators on a Banach space X satisfying D(A) ⊂ D(B). We give a concrete example that our abstract result may be applied.

Key words: C α -well-posedness, second order differential equations, C˙ α -Fourier multiplier, H¨older continuous function spaces.