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The oscillation of an \(m\)th order perturbed nonlinear difference equation. (English) Zbl 0870.39001
The authors consider the equation \(|\Delta ^my(k) |^{\alpha - 1}\Delta ^my(k)+Q(k, y(k-\sigma _k))= P(k,y(k-\sigma _k))\), \(k\geq k_0\), where \(\alpha >0\), \(\Delta \) is the forward difference operator defined by \(\Delta y(k)=y(k+1)-y(k)\), \(Q(k,z(j))\) and \(P(k, z(j))\) are functions depending on \(k\), and \(\Delta ^i(z(j))\) for \(0\leq i\leq m-2\), or \(0\leq i\leq m-1\), respectively, and \(\sigma _k\) is an integer such that \(\lim (k-\sigma _k)=\infty \).
They provide some criteria for oscillation of the solutions. A typical result: Assume that \(f\) is a real function, \(uf(u)>0\) for \(u\neq 0\), and \(\{ q(k)\} , \{ p(k)\}\) are real sequences such that \(Q(k, x(k-\sigma _k))/f(x(k-\sigma _k)\geq q(k)\), \(P(k, x(k-\sigma _k))/f(x(k-\sigma _k)\leq p(k)\), and \(\lim |q(k)-p(k) |\geq 0\). If \(m=1\) or \(m\) is even, \(f\) is continuous, \(\liminf _{|u |\rightarrow \infty }f(u)>0\) and \(\sum _k[q(k)-p(k)]^{1/\alpha } =\infty \), then all solutions are oscillatory.

39A10 Additive difference equations
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