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

Abstract

We consider the well‐posedness of the second‐order degenerate differential equations (Mu′)′false(tfalse)+Λu′false(tfalse)+∫−∞tafalse(tgoodbreak−sfalse)u′false(sfalse)ds=Aufalse(tfalse)+∫−∞tbfalse(tgoodbreak−sfalse)Bufalse(sfalse)ds+ffalse(tfalse)false(normalPfalse)$$\begin{eqnarray*} &&\hspace*{18pc} (Mu^{\prime })^{\prime }(t) +\Lambda u^{\prime }(t)+\int _{-\infty }^t a(t-s)u^{\prime }(s)ds\\ &&\hspace*{18pc}\quad = Au(t)+ \int _{-\infty }^t b(t-s) Bu(s) ds + f(t)\qquad\qquad\qquad \mathrm{{(P)}} \end{eqnarray*}$$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,0.33emB,0.33emnormalΛ$A,\ B,\ \Lambda$, and M are closed linear operators in a Banach space X satisfying Dfalse(Afalse)∩Dfalse(Bfalse)⊂Dfalse(Mfalse)∩Dfalse(normalΛfalse)$D(A)\cap D(B)\subset D(M) \cap D(\Lambda )$ and the kernels a,b∈L1(double-struckR+)$ 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$B_{p,q}^s$false(double-struckT;Xfalse)$(\mathbb {T}; X)$. These results are applied to examine some concrete examples.