Solutions of second-order degenerate equations with infinite delay in Banach spaces

Abstract

We consider the well-posedness of the second-order degenerate differential equations with infinite delay $\left(P_2\right)$: ${\left(M{u}^{\prime }\right)}^{\prime }\left(t\right)+{\left(Lu\right)}^{\prime }\left(t\right)=Au\left(t\right)+{\int }_{-\infty }^{t}a\left(t-s\right)Bu\left(s\right)ds+f\left(t\right)$, $\left(0\le t\le 2\pi \right)$ with periodic boundary conditions $u\left(0\right)=u\left(2\pi \right)$, $\left(M{u}^{\prime }\right)\left(0\right)=\left(M{u}^{\prime }\right)\left(2\pi \right)$, in Lebesgue–Bochner spaces $L^p\left(\mathbb{𝕋};X\right)$ and periodic Besov spaces $B_{p,q}^s\left(\mathbb{𝕋};X\right)$, where $A$, $B$, $L$ and $M$ are closed linear operators in a Banach space $X$ satisfying $D\left(A\right)\cap D\left(B\right)\subset D\left(M\right)\cap D\left(L\right)$, $D\left(A\right)\cap D\left(B\right)\ne \left\{0\right\}$ and $a\in L^1\left({ℝ}_{+}\right)$. We completely characterize the well-posedness of $\left(P_2\right)$ in the above function spaces by using known operator-valued Fourier multiplier theorems. We also give concrete examples to support our abstract results.