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A few remarks on Poincaré-Perron solutions and regularly varying solutions

  • Pavel Řehák EMAIL logo
Published/Copyright: December 30, 2016
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 66 Issue 6

Abstract

We establish conditions guaranteeing the existence of (generalized) regularly varying solutions to nth order linear differential and q-difference equations. The proofs are based mainly on classical Poincaré’s and Perron’s theorems and certain transformations. In some special cases, our results reduce to the existing ones, thus actually we offer an alternative approach to some parts of the asymptotic theory of differential equations made in the framework of regular variation. For higher order cases, our statements are essentially new. Another important feature of this paper is that it reveals connections among various results and somehow revises some of them.

MSC 2010: Primary 26A12; 34C11; 34C41; 34Exx; 39A12; 39A13; 39A22
Keywords: Poincaré theorem; Perron theorem; regularly varying function; linear differential equation; linear difference equation; linear q-difference equation; nonoscillation

This work was supported by the grant 201/10/1032 of the Czech Science Foundation and by RVO 67985840

(Communicated by Michal Fečkan)

Acknowledgements

The author thanks the both referees for their careful reading of the manuscript and helpful comments.

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Received: 2014-2-23
Accepted: 2014-9-17
Published Online: 2016-12-30
Published in Print: 2016-12-1

© 2016 Mathematical Institute Slovak Academy of Sciences

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  4. Periods of morgan-voyce sequences and elliptic curves
  6. Multiplicative generalized derivations on ideals in semiprime rings
  8. A few remarks on Poincaré-Perron solutions and regularly varying solutions
  10. Applications of extremal theorem and radius equation for a class of analytic functions
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  14. New criteria for global exponential stability of linear time-varying volterra difference equations
  16. On approximation of functions by some hump matrix means of Fourier series
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  20. Poisson kernels on semi-direct products of abelian groups
  22. Locally convex projective limit cones
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  32. On the internal approach to differential equations 3. Infinitesimal symmetries
  34. A Characterization of the discontinuity point set of strongly separately continuous functions on products
  36. Wick differential and Poisson equations associated to the 𝚀𝚆𝙽-Euler operator acting on generalized operators
  38. Multivariate EIV models
  40. On codes over 𝓡k, m and constructions for new binary self-dual codes
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https://doi.org/10.1515/ms-2016-0224
Keywords for this article
Poincaré theorem; Perron theorem; regularly varying function; linear differential equation; linear difference equation; linear q-difference equation; nonoscillation

Articles in the same Issue

  2. A triple representation of lattice effect algebras
  4. Periods of morgan-voyce sequences and elliptic curves
  6. Multiplicative generalized derivations on ideals in semiprime rings
  8. A few remarks on Poincaré-Perron solutions and regularly varying solutions
  10. Applications of extremal theorem and radius equation for a class of analytic functions
  12. On the “bang-bang” principle for a class of Riemann-Liouville fractional semilinear evolution inclusions
  14. New criteria for global exponential stability of linear time-varying volterra difference equations
  16. On approximation of functions by some hump matrix means of Fourier series
  18. Pseudo-amenability and pseudo-contractibility for certain products of Banach algebras
  20. Poisson kernels on semi-direct products of abelian groups
  22. Locally convex projective limit cones
  24. On the properties (wL) and (wV)
  26. On the existence of solutions for quadratic integral equations in Orlicz spaces
  28. Existence and uniqueness of best proximity points under rational contractivity conditions
  30. Almost Weyl structures on null geometry in indefinite Kenmotsu manifolds
  32. On the internal approach to differential equations 3. Infinitesimal symmetries
  34. A Characterization of the discontinuity point set of strongly separately continuous functions on products
  36. Wick differential and Poisson equations associated to the 𝚀𝚆𝙽-Euler operator acting on generalized operators
  38. Multivariate EIV models
  40. On codes over 𝓡k, m and constructions for new binary self-dual codes
  42. Domination number of total graphs
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