MATHEMATICA BOHEMICA, Vol. 141, No. 2, pp. 169-182, 2016

# Oscillation properties for a scalar linear difference equation of mixed type

## Leonid Berezansky, Sandra Pinelas

Leonid Berezansky, Department of Mathematics, Ben Gurion University of the Negev, P.O.Box 653, Beer Sheva 8410501, Israel, e-mail: brznsky@cs.bgu.ac.il; Sandra Pinelas, Academia Militar, Departamento de Ciencias Exactas e Naturais, Avenide Conde Castro Guimaraes, 2720-113 Amadora, Portugal, e-mail: sandra.pinelas@gmail.com

Abstract: The aim of this work is to study oscillation properties for a scalar linear difference equation of mixed type
$$\Delta x(n)+\sum_{k=-p}^qa_k(n)x(n+k)=0,\quad n>n_0,$$
where $\Delta x(n)=x(n+1)-x(n)$ is the difference operator and $\{a_k(n)\}$ are sequences of real numbers for $k=-p,\ldots,q$, and $p>0$, $q\geq0$. We obtain sufficient conditions for the existence of oscillatory and nonoscillatory solutions. Some asymptotic properties are introduced.

Keywords: oscillation; difference equation; mixed type; asymptotic behavior

Classification (MSC 2010): 39A21, 39A99

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