Behaviour of solutions of a class of 2nd order 2D neutral delay difference systems

Abstract The present work deals with the study of the dynamic behaviour of all vector solutions of a second order neutral delay difference system with constant coefficients of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:msup> <m:mi mathvariant="normal">Δ</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>[</m:mo> <m:mtable displaystyle="true" rowspacing="0pt"> <m:mtr> <m:mtd columnalign="center"> <m:mrow> <m:mrow> <m:mi>ϕ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>m</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mi>q</m:mi> <m:mo>⁢</m:mo> <m:mi>ϕ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>m</m:mi> <m:mo>-</m:mo> <m:mi>l</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="center"> <m:mrow> <m:mrow> <m:mi>ψ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>m</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mi>q</m:mi> <m:mo>⁢</m:mo> <m:mi>ψ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>m</m:mi> <m:mo>-</m:mo> <m:mi>l</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mtd> </m:mtr> </m:mtable> <m:mo>]</m:mo> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mrow> <m:mo>[</m:mo> <m:mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"> <m:mtr> <m:mtd columnalign="center"> <m:msub> <m:mi>b</m:mi> <m:mn>1</m:mn> </m:msub> </m:mtd> <m:mtd columnalign="center"> <m:msub> <m:mi>b</m:mi> <m:mn>2</m:mn> </m:msub> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="center"> <m:msub> <m:mi>b</m:mi> <m:mn>3</m:mn> </m:msub> </m:mtd> <m:mtd columnalign="center"> <m:msub> <m:mi>b</m:mi> <m:mn>4</m:mn> </m:msub> </m:mtd> </m:mtr> </m:mtable> <m:mo>]</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo>[</m:mo> <m:mtable displaystyle="true" rowspacing="0pt"> <m:mtr> <m:mtd columnalign="center"> <m:mrow> <m:mi>ϕ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>m</m:mi> <m:mo>-</m:mo> <m:mi>τ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="center"> <m:mrow> <m:mi>ψ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>m</m:mi> <m:mo>-</m:mo> <m:mi>γ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mtd>