## Abstract

We study the oscillation of Euler type half-linear difference equations with asymptotically periodic and bounded coefficients. Applying the adapted Riccati technique, we prove a non-oscillation criterion for the treated equations. Since the obtained criterion is new in several cases, we formulate many consequences explicitly. In particular, the criterion is new for intensively studied linear equations (e.g., with periodic coefficients).

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Petr Hasil and Michal Veselý are supported by Grant GA20-11846S of the Czech Science Foundation.

Michal Pospíšil is supported by the Grants VEGA-SAV 2/0127/20, VEGA 1/0358/20 and by the Slovak Research and Development Agency under the contract No. APVV-18-0308.

Jiřina Šišoláková is supported from Operational Programme Research, Development and Education, “Project Internal Grant Agency of Masaryk University”, No. CZ.02.2.69/0.0/0.0/19 073/0016943.

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Hasil, P., Pospíšil, M., Šišoláková, J. *et al.* Non-oscillation criterion for Euler type half-linear difference equations with consequences in linear case.
*Acta Math. Hungar.* **166**, 624–649 (2022). https://doi.org/10.1007/s10474-022-01218-1

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DOI: https://doi.org/10.1007/s10474-022-01218-1

### Keywords and phrases

- half-linear equation
- linear equation
- difference equation
- oscillation theory
- non-oscillation
- Euler type
- Riccati technique
- Riccati equation