## Existence of positive nondecreasing solutions of nonlinear difference equations.(English)Zbl 0805.39004

The paper is concerned with the nonlinear difference equation (1) $$(\Delta y_ k)^{p-1} - (\Delta y_{k - 1})^{p - 1} + s_ ky_ k^{p - 1} = 0$$, $$k = 1,2, \dots$$ where $$s_ k \geq 0$$, $$p>1$$. The authors establish some sufficient and/or necessary conditions for equation (1) to have a positive nondecreasing solution. A nontrivial solution $$\{y_ k\}^ \infty_{k = 0}$$ of (1) is said to be oscillatory if for every positive integer $$N$$ there exists $$n \geq N$$ such that $$y_ n y_{n-1} \leq 0$$. Oscillation and nonoscillation criteria for solutions of (1) are given.