• Title/Summary/Keyword: linear operator with periodic coefficients

Search Result 2, Processing Time 0.017 seconds

NOTE ON SPECTRUM OF LINEAR DIFFERENTIAL OPERATORS WITH PERIODIC COEFFICIENTS

  • Jung, Soyeun
    • Journal of the Chungcheong Mathematical Society
    • /
    • v.30 no.3
    • /
    • pp.323-329
    • /
    • 2017
  • In this paper, by rigorous calculations, we consider $L^2({\mathbb{R}})-spectrum$ of linear differential operators with periodic coefficients. These operators are usually seen in linearization of nonlinear partial differential equations about spatially periodic traveling wave solutions. Here, by using the solution operator obtained from Floquet theory, we prove rigorously that $L^2({\mathbb{R}})-spectrum$ of the linear operator is determined by the eigenvalues of Floquet matrix.

SOLUTIONS OF HIGHER ORDER INHOMOGENEOUS PERIODIC EVOLUTIONARY PROCESS

  • Kim, Dohan;Miyazaki, Rinko;Naito, Toshiki;Shin, Jong Son
    • Journal of the Korean Mathematical Society
    • /
    • v.54 no.6
    • /
    • pp.1853-1878
    • /
    • 2017
  • Let $\{U(t,s)\}_{t{\geq}s}$ be a periodic evolutionary process with period ${\tau}$ > 0 on a Banach space X. Also, let L be the generator of the evolution semigroup associated with $\{U(t,s)\}_{t{\geq}s}$ on the phase space $P_{\tau}(X)$ of all ${\tau}$-periodic continuous X-valued functions. Some kind of variation-of-constants formula for the solution u of the equation $({\alpha}I-L)^nu=f$ will be given together with the conditions on $f{\in}P_{\tau}(X)$ for the existence of coefficients in the formula involving the monodromy operator $U(0,-{\tau})$. Also, examples of ODEs and PDEs are presented as its application.