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 (P2): (Mu′)′(t)+(Lu)′(t)=Au(t)+ ∫ −∞ta(t−s)Bu(s)ds+f(t), (0≤t≤2π) with periodic boundary conditions u(0)=u(2π), (Mu′)(0)=(Mu′)(2π), in Lebesgue–Bochner spaces Lp(𝕋;X) and periodic Besov spaces Bp,qs(𝕋;X), where A, B, L and M are closed linear operators in a Banach space X satisfying D(A)∩D(B)⊂D(M)∩D(L), D(A)∩D(B)≠{0} and a∈L1(ℝ+). We completely characterize the well-posedness of (P2) in the above function spaces by using known operator-valued Fourier multiplier theorems. We also give concrete examples to support our abstract results.