Periodic solutions of second‐order degenerate differential equations with infinite delay in Banach spaces

We consider the well‐posedness of the second‐order degenerate differential equations (Mu′)′(t)+Λu′(t)+∫−∞ta(t−s)u′(s)ds=Au(t)+∫−∞tb(t−s)Bu(s)ds+f(t)(P)$$\begin{eqnarray*} &&\hspace*{18pc} (Mu^{\prime })^{\prime }(t) +\Lambda u^{\prime }(t)+\int _{-\infty }^t a(t-s)u^{\prime }(s)ds\\ &&\hspace*{18pc}\quad = Au(t)+ \int _{-\infty }^t b(t-s) Bu(s) ds + f(t)\qquad\qquad\qquad \mathrm{{(P)}} \end{eqnarray*}$$with infinite delay on [0, 2π] in Lebesgue–Bochner spaces Lp(T;X)$L^p(\mathbb {T}; X)$ and periodic Besov spaces Bp,qs(T;X)$B_{p,q}^s (\mathbb {T}; X)$, where A,B,Λ$A,\ B,\ \Lambda$, and M are closed linear operators in a Banach space X satisfying D(A)∩D(B)⊂D(M)∩D(Λ)$D(A)\cap D(B)\subset D(M) \cap D(\Lambda )$ and the kernels a,b∈L1(R+)$ a, b\in L^1(\mathbb {R}_+)$. Using known operator‐valued Fourier multiplier theorems, we are able to give necessary and sufficient conditions for the well‐posedness of (P) in Lp(T;X)$L^p(\mathbb {T}; X)$ and Bp,qs$B_{p,q}^s$(T;X)$(\mathbb {T}; X)$. These results are applied to examine some concrete examples.