Periodic solutions of fractional degenerate differential equations with delay in Banach spaces (Q2317694)

The paper deals with the fractional degenerate differential equation with finite delay, \[D^{\alpha}(Mu)(t)= Au(t) +G u'_t + F u_t + f(t),\quad t\in[0,\pi],\] for some fixed \(\alpha>0\), closed linear operators \(A,M\) in a Banach space \(X\) whose domains have nonempty intersection, and bounded linear operators \(F,G\) from an \(X\)-valued \(L^p\)-space (resp. Bessov space) into \(X\). The authors characterize the well-posedness of the equation in \(L^p\) (resp. Bessov space) by the Rademacher boundedness (resp. norm boundedness) of the \(M\)-resolvent of \(A\). The main tools in this study are operator-valued Fourier multiplier theorems obtained by \textit{W. Arendt} and \textit{S. Bu} [Math. Z. 240, No. 2, 311--343 (2002; Zbl 1018.47008); Proc. Edinb. Math. Soc., II. Ser. 47, No. 1, 15--33 (2004; Zbl 1083.42009)]. They also present a concrete example to which the abstract results can be applied.