Abstract
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).
Highlights
Existence of periodic solutions for differential equations of fractional order is a very desirable property for analyzing cyclic processes, see [28]
In recent years many papers have appeared on this topic [1, 2, 17, 23], and there are different methods that allow periodic solutions, the Fourier transform being the most common
It is well known that we cannot expect the existence of periodic solutions in time-invariant systems with each definition of fractional order derivative, see e.g. [24, 26, 27]
Summary
Existence of periodic solutions for differential equations of fractional order is a very desirable property for analyzing cyclic (e.g. biological) processes, see [28]. We note that Haraux [16, Chapter B, I] gave a similar approach in the case that X is a Hilbert space, which has been the main motivation in this work Using this approach, we can capture the minimum requirements that are needed in a system of 2π -periodic solutions of (1) in the sense that any solution of (1) can be represented by a normally convergent series formed by functions of the following set: umeimt |m|≤t0 ∪ A + (im)αI –1fmeimt |m|>t0 , where we assume {(im)α}|m|>t0 ⊆ ρ(A) and fm = ((A + (im)αI)–1)um for |m| ≤ t0, where fm are the Fourier coefficients of f. The second takes the additive perturbation again but in the scenario of a Banach space In this case, we assume that A can be represented as a sum of a sectorial operator with an angle depending on the fractional parameter α and a bounded linear operator. We can guarantee the existence of normal 2π -periodic solutions
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