Oscillation theorems for perturbed nonlinear second order difference equations.(English)Zbl 0807.39002

The author establishes six theorems and two corollaries which yield sufficient conditions for the oscillation of all solutions to the difference equation (1) $$\Delta (a_{n - 1} \Delta y_{n - 1}) + Q(n,y_ n) = P(n, y_ n, \Delta y_ n)$$, $$n \in\mathbb{N}$$, where $$\mathbb{N} = \{1,2, \dots\}$$, $$\Delta y_ n = y_{n + 1} - y_ n$$, and $$\{a_ n\}$$ is a sequence of positive numbers, $$Q :\mathbb{N} \times \mathbb{R} \to \mathbb{R}$$ and $$P : \mathbb{N} \times \mathbb{R}^ 2 \to \mathbb{R}$$ are known functions.
It is assumed that, there exist real sequences $$\{q_ n\}$$, $$\{p_ n\}$$ and a function $$f : \mathbb{R} \to \mathbb{R}$$ such that, (i) $$uf(u) > 0$$ for $$u \neq 0$$; (ii) $$Q(n,u)/f(u) \geq q_ n$$ and $$P(n,u,v)/f(u) \leq p_ n$$ for all $$u,v \neq 0$$; and (iii) there is a nonnegative function $$g:\mathbb{R} \times \mathbb{R} \to [0,\infty)$$ such that $$f(u) - f(v) = g(u,v) (u-v)$$ for all $$u,v \neq 0$$. All sufficient conditions are given in terms of the sequences $$\{a_ n\}$$, $$\{q_ n\}$$, and $$\{p_ n\}$$ and some integrals involving the function $$f$$. The corollaries yield sufficient conditions ensuring the oscillation of all bounded solutions of (1). Three examples are inserted to illustrate the results.
Theorem 4.1 of J. W. Hooker and W. T. Patula [J. Math. Anal. Appl., 91, 9-29 (1983; Zbl 0508.39005)], Theorem 4.2 of M. R. Kolenovic and S. Budincevic [An. Ştiinţ. Univ. Al. I. Cuza Iaşi N. Ser., Secţ. Ia 30, No. 3, 39-52 (1984; Zbl 0572.39001)], and Theorem 1 of B. Szmanda [Atti Accad. Naz. Lincei, VIII. Ser., Rend. Cl. Sci. Fis. Mat. Nat. 69, 120-125 (1980; Zbl 0508.39004)] are generalized.
The paper, contains some misprints.