Hölder Continuous Solutions of Degenerate Differential Equations with Finite Delay

Using known operator-valued Fourier multiplier results on vector-valued Hölder continuous function spaces Cα(ℝ; X), we completely characterize the Cα-well-posedness of the first order degenerate differential equations with finite delay (Mu)′(t) = Au(t) + Fut + f(t) for t ∊ ℝ by the boundedness of the (M, F)-resolvent of A under suitable assumption on the delay operator F, where A, M are closed linear operators on a Banach space X satisfying D(A) İ D(M) ≠ = ﹛0﹜, the delay operator F is a bounded linear operator from C([−r, 0]; X) to X, and r > 0 is fixed.