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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.