Periodic Solutions of Third‐Order Mixed Degenerate Differential Equations With Two Infinite Delays in Vector‐Valued Function Spaces

The aim of this paper is to consider the well‐posedness of third‐order mixed degenerate differential equation with two infinite delays (P3)$(P_3)$: (Mu′)′′(t)+(Lu′)′(t)=Au(t)+Bu′(t)+∫−∞td(t−s)Au(s)ds+∫−∞te(t−s)Bu(s)ds+f(t),(t∈[0,2π])$(Mu^{\prime })^{\prime \prime }(t) + (Lu^{\prime })^{\prime }(t) = Au(t) + Bu^{\prime }(t) + \int _{-\infty }^t d(t-s)Au(s)ds + \int _{-\infty }^t e(t-s)Bu(s)ds + f(t),(t\in [0,2\pi])$ 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$A$, B$B$, L$L$, and M$M$ are closed linear operators on a Banach space X$X$ satisfying D(A)∩D(B)⊂D(M)∩D(L)$D(A)\cap D(B)\subset D(M)\cap D(L)$ and d,e∈L1(R+)$d,e\in L^1(\mathbb {R}_+)$. Necessary and sufficient conditions for (P3)$(P_3)$ to be Lp$L^p$‐well‐posed (or Bp,qs$B_{p,q}^s$‐well‐posed) are given by using known operator‐valued Fourier multiplier theorems.