Periodic and almost periodic results for some differential equations in Banach spaces. Ait Dads, E., Ezzinbi, K., & Arino, O. Nonlinear Anal., 31(1-2):163–170, 1998.
abstract   bibtex   
The main result is an extension of recent results of Kato and Imai [same journal, 24 (1995), 1183-1192 and 25 (1995), 409-411]. It is shown that under certain hypotheses, an equation of the form $x'(t) = A(t, x(t)) + f(t)$ has a unique bounded solution that is almost periodic. The hypotheses include regularity assumptions on $A$ and $f$ and a dissipativity condition on $A$. The techniques used allow the proof to work for $x(t)$ taking on values in a Banach space (the earlier results required a Hilbert space), and the required dissipativity condition is less restrictive than with the earlier results.
@Article{AitDadsEzzinbiArino1998,
  author     = {Ait Dads, E. and Ezzinbi, K. and Arino, Ovide},
  title      = {Periodic and almost periodic results for some differential equations in {B}anach spaces},
  journal    = {Nonlinear Anal.},
  year       = {1998},
  volume     = {31},
  number     = {1-2},
  pages      = {163--170},
  issn       = {0362-546X},
  abstract   = {The main result is an extension of recent results of
                  Kato and Imai [same journal, 24 (1995), 1183-1192
                  and 25 (1995), 409-411]. It is shown that under
                  certain hypotheses, an equation of the form $x'(t) =
                  A(t, x(t)) + f(t)$ has a unique bounded solution
                  that is almost periodic. The hypotheses include
                  regularity assumptions on $A$ and $f$ and a
                  dissipativity condition on $A$. The techniques used
                  allow the proof to work for $x(t)$ taking on values
                  in a Banach space (the earlier results required a
                  Hilbert space), and the required dissipativity
                  condition is less restrictive than with the earlier
                  results.},
  classmath  = {*34G99 ODE in abstract spaces },
  coden      = {NOANDD},
  fjournal   = {Nonlinear Analysis. Theory, Methods \& Applications. An International Multidisciplinary Journal},
  mrclass    = {34G20 (34C25 34C27)},
  mrnumber   = {98j:34116},
  mrreviewer = {A. I. Perov},
  pdf        = {AitDadsEzzinbiArino-1998-NATMA31.pdf},
  reviewer   = {Mike Hurley (Cleveland)},
}

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