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

Abstract

We study Hölder continuous solutions for the second order integro-differential equations with infinite delay ( P 1 ) : u " ( t ) + c u ' ( t ) + ∫ − ∞ t β ( t − s ) u ' ( s ) d s + ∫ − ∞ t γ ( t − s ) u ( s ) d s = A u ( t ) − ∫ − ∞ t δ ( t − s ) A u ( s ) d s + f ( t ) 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 ∈ ℂ α ( ℝ , X ) and β , γ , δ ∈ L 1 ( ℝ + ) . Under suitable assumptions on the kernels β, γ and δ, we completely characterize the C α-well-posedness of (P 1) by using operator-valued Ċα-Fourier multipliers.