We show that if A is a closed linear operator defined in a Banach space X and there exist t_{0} geq 0 and M>0 such that {(im)^{alpha }}_{|m|> t_{0}} subset rho (A), the resolvent set of A, and ∥(im)α(A+(im)αI)−1∥≤M for all |m|>t0,m∈Z,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\bigl\\Vert (im)^{\\alpha }\\bigl(A+(im)^{\\alpha }I \\bigr)^{-1} \\bigr\\Vert \\leq M \\quad \\text{ for all } \\vert m \\vert > t_{0}, m \\in \\mathbb{Z}, $$\\end{document} then, for each frac{1}{p}<alpha leq frac{2}{p} and 1< p < 2, the abstract Cauchy problem with periodic boundary conditions {DtαGLu(t)+Au(t)=f(t),t∈(0,2π);u(0)=u(2π),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} _{GL}D^{\\alpha }_{t} u(t) + Au(t) = f(t), & t \\in (0,2\\pi ); \\\\ u(0)=u(2\\pi ), \\end{cases} $$\\end{document} where _{GL}D^{alpha } denotes the Grünwald–Letnikov derivative, admits a normal 2π-periodic solution for each fin L^{p}_{2pi }(mathbb{R}, X) that satisfies appropriate conditions. In particular, this happens if A is a sectorial operator with spectral angle phi _{A} in (0, alpha pi /2) and int _{0}^{2pi } f(t),dt in operatorname{Ran}(A).