Well-posedness of second order degenerate differential equations with infinite delay in Hölder continuous function spaces

Abstract

Using operator-valued Ċα-Fourier multiplier results on vector-valued Hölder continuous function spaces and the Carleman transform, we characterize the Cα-well-posedness of second order degenerate differential equations with infinite delay: (Mu)′′(t)=Au(t)+∫−∞ta(t−s)Au(s)ds+f(t) and (Mu′)′(t)=Au(t)+∫−∞ta(t−s)Au(s)ds+f(t) on ℝ, where A:D(A)→X and M:D(M)→X are closed linear operators in a complex Banach space X, and a∈L1(ℝ+)∩L1(ℝ+;tαdt).