Asymptotic Behavior of Solutions to Difference Equations of Neutral Type

• Magdalena Nockowska-Rosiak ^[1] ; Janusz Migda ^[2]
  1. [1] Lodz University of Technology

     Lodz University of Technology

     Łódź, Polonia

     
  2. [2] A. Mickiewicz University
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 2, 2024
• Idioma: inglés
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• Resumen

  • We present sufficient conditions for the existence of a solution x to an equation m(xn − un xn−k ) = an f (xn−τ ) + bn, which is “close” to a given solution y to the linear homogeneous equation of neutral type m(yn − λyn−k ) = 0, where λ is the limit of the sequence u. Closeness of solutions to above equations is understood as xn − yn = o(ωn), where ω is a given nonincreasing sequence with positive values. Moreover, we establish under which conditions for a given solution x to m(xn − un xn−k ) = an f (xn−τ ) + bn and a given nonincreasing sequence with positive values ω there exists a polynomial sequence ϕ of degree less than m such that xn = ϕ(n) + o(ωn). Presented conditions strongly depend on λ.