Opuscula Math. 37, no. 6 (2017), 887-898
http://dx.doi.org/10.7494/OpMath.2017.37.6.887

Opuscula Mathematica

Oscillation of solutions to non-linear difference equations with several advanced arguments

Sandra Pinelas
Julio G. Dix

Abstract. This work concerns the oscillation and asymptotic properties of solutions to the non-linear difference equation with advanced arguments $x_{n+1}- x_n =\sum_{i=1}^m f_{i,n}( x_{n+h_{i,n}}).$ We establish sufficient conditions for the existence of positive, and negative solutions. Then we obtain conditions for solutions to be bounded, convergent to positive infinity and to negative infinity and to zero. Also we obtain conditions for all solutions to be oscillatory.

Keywords: advanced difference equation, non-oscillatory solution.

Mathematics Subject Classification: 39A11.

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• Sandra Pinelas
• RUDN University, 6 Miklukho-Maklay St., Moscow, 117198, Russia
• Julio G. Dix
• Department of Mathematics, Texas State University, 601 University Drive, San Marcos, TX 78666, USA
• Communicated by Stevo Stević.
• Revised: 2017-03-11.
• Accepted: 2017-03-12.
• Published online: 2017-09-28.