Asymptotic properties of solutions to difference equations of Sturm–Liouville type

We consider the discrete Sturm–Liouville type equation of the formΔ(rnΔxn)=anf(xσ(n))+bn.Assume s is a given nonpositive real number. We present sufficient conditions for the existence of solution x with the asymptotic behaviorxn=c(r1−1+⋯+rn−1−1)+d+o(ns)where c, d are given real numbers. Moreover, w...

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Bibliographic Details
Published in:Applied mathematics and computation 2019-01, Vol.340, p.126-137
Main Authors: Migda, Janusz, Nockowska-Rosiak, Magdalena
Format: Article
Language:English
Subjects:
Approximation of solution
Bounded solution
Convergent solution
Prescribed asymptotic behavior
Sturm–Liouville difference equation
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Summary:We consider the discrete Sturm–Liouville type equation of the formΔ(rnΔxn)=anf(xσ(n))+bn.Assume s is a given nonpositive real number. We present sufficient conditions for the existence of solution x with the asymptotic behaviorxn=c(r1−1+⋯+rn−1−1)+d+o(ns)where c, d are given real numbers. Moreover, we establish conditions under which for a given solution x there exist real numbers c, d such that x has the above asymptotic behavior.
ISSN:0096-3003
1873-5649
DOI:10.1016/j.amc.2018.08.001