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

Abstract

Using operator-valued ${Ċ}^{\alpha }$-Fourier multiplier results on vector-valued Hölder continuous function spaces and the Carleman transform, we characterize the $C^{\alpha }$-well-posedness of second order degenerate differential equations with infinite delay: ${\left(Mu\right)}^{\prime \prime }\left(t\right)=Au\left(t\right)+{\int }_{-\infty }^{t}a\left(t-s\right)Au\left(s\right)ds+f\left(t\right)$ and ${\left(M{u}^{\prime }\right)}^{\prime }\left(t\right)=Au\left(t\right)+{\int }_{-\infty }^{t}a\left(t-s\right)Au\left(s\right)ds+f\left(t\right)$ on $ℝ$, where $A:D\left(A\right)\to X$ and $M:D\left(M\right)\to X$ are closed linear operators in a complex Banach space $X$, and $a\in L^1\left({ℝ}_{+}\right)\cap L^1\left({ℝ}_{+};t^{\alpha }dt\right)$.