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On Parametric Gevrey Asymptotics for Some Cauchy Problems in Quasiperiodic Function Spaces

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  • A. Lastra
  • S. Malek

Abstract

We investigate Gevrey asymptotics for solutions to nonlinear parameter depending Cauchy problems with 2π‐periodic coefficients, for initial data living in a space of quasiperiodic functions. By means of the Borel‐Laplace summation procedure, we construct sectorial holomorphic solutions which are shown to share the same formal power series as asymptotic expansion in the perturbation parameter. We observe a small divisor phenomenon which emerges from the quasiperiodic nature of the solutions space and which is the origin of the Gevrey type divergence of this formal series. Our result rests on the classical Ramis‐Sibuya theorem which asks to prove that the difference of any two neighboring constructed solutions satisfies some exponential decay. This is done by an asymptotic study of a Dirichlet‐like series whose exponents are positive real numbers which accumulate to the origin.

Suggested Citation

  • A. Lastra & S. Malek, 2014. "On Parametric Gevrey Asymptotics for Some Cauchy Problems in Quasiperiodic Function Spaces," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:153169
    DOI: 10.1155/2014/153169
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    References listed on IDEAS

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    1. Constantin Corduneanu, 2009. "Almost Periodic Oscillations and Waves," Springer Books, Springer, number 978-0-387-09819-7, March.
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    Cited by:

    1. David W. Pravica & Njinasoa Randriampiry & Michael J. Spurr, 2022. "Solutions of a Class of Multiplicatively Advanced Differential Equations II: Fourier Transforms," Abstract and Applied Analysis, John Wiley & Sons, vol. 2022(1).
    2. David W. Pravica & Njinasoa Randriampiry & Michael J. Spurr, 2024. "Solutions of Inhomogeneous Multiplicatively Advanced ODEs and PDEs with a q‐Fredholm Theory and Applications to a q‐Advanced Schrödinger Equation," Abstract and Applied Analysis, John Wiley & Sons, vol. 2024(1).

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