Hölder continuous solutions for second order integro-differential equations in banach spaces

We study Hölder continuous solutions for the second order integro-differential equations with infinite delay on the line ℝ, where 0 < α < 1, A is a closed operator in a complex Banach space X, c ∈ ℂ is a constant, f∈ℂα(ℝ,X) and β,γ,δ∈L1(ℝ+). Under suitable assumptions on the kernels β, γ and δ, we completely characterize the Cα-well-posedness of (P1) by using operator-valued Ċα-Fourier multipliers.