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Nonexistence of monotone solutions of neutral partial difference equations. (English) Zbl 1092.39014

The authors consider the neutral partial difference equation \[ T(\Delta_m,\Delta_n)(y(m,n)-p(m,n)y(m-r,n-h))+q(m,n)y(m-k,n-l)=0, \] where \(T(\Delta_m,\Delta_n)=a\Delta_m\Delta_n +b\Delta_m+c\Delta_n+dI\), \(\Delta_{m}y(m,n)=y(m+1,n)-y(m,n)\), \(\Delta_{n}y(m,n)=y(m,n+1)-y(m,n)\), \(Iy(m,n)=y(m,n)\), \(a\), \(b\), \(c\) and \(d\) are nonnegative constants, \(0\leq p(m,n)\leq 1\), \(q(m,n)\geq 0\), the delays \(r\), \(h\), \(k\) and \(l\) are positive integers. The main results of the paper are four theorems in which the authors give sufficient conditions under which the equation has no eventually positive (negative) and nondecreasing (nonincreasing) solution.

MSC:

39A11 Stability of difference equations (MSC2000)
39A10 Additive difference equations
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