## Oscillation of certain second order nonlinear damped difference equations with continuous variable.(English)Zbl 1124.39014

The following second-order nonlinear damped difference equation with continuous variable is considered $\Delta^2_\tau x(t)+ p(t)\Delta_\tau x(t)+ q(t)\cdot f(t, x(t-\sigma))= 0,\tag{1}$ where $$\tau$$ and $$\sigma$$ are nonnegative constants and $$\Delta_\tau x(t)= x(t+\tau)- x(t)$$, $$t\in I= [t_0,\infty)$$. It is assumed that the following conditions hold
$$(\text{H}_1)$$ $$p\in C(I,(0,1))$$, $$q\in C(I,\mathbb{R}^+)$$ and $$q(t)\not\equiv 0$$ on any ray $$[t_\mu,\infty)$$ for some $$t_\mu\geq t_0$$;
$$(\text{H}_2)$$ $$f\in C(I\times \mathbb{R},\mathbb{R})$$, $$f(t,0)\equiv 0$$ and $${f(t,x)\over x}\geq K> 0$$ for all $$x\neq 0$$;
$$(\text{H}_3)$$ $$\limsup_{n\to\infty}\,\sum^n_{i=0} \prod^{i-1}_{j=0} [1- p(t+ j\tau)]= \infty$$.
For $$k\in C'(I,\mathbb{R}^+)$$, $$z\in C(I,\mathbb{R})$$ and $$t\geq T\geq t_0$$ the following two integral operators $$X$$, $$Y\in C(\mathbb{R},\mathbb{R})$$ are considered
$X^{H,k}_{T,t}= \int^t_T H(s, T)k(s) z(s)\,ds\text{ and }Y^{H,k}_{T,t}= \int^t_T H(t,s) k(s)z(s)\,ds,$
where $$H\in C(D,\mathbb{R}^+)$$ with $$D= \{(t, s)\mid-\infty< s\leq t< +\infty\}$$ and possesses the following properties: $$H(t,t)= 0$$, $$H(t,0)> 0$$ for $$t> s$$, $${\partial H\over\partial t}= h_1(t,s)\cdot H(t,s)$$ and $${\partial H\over\partial s}=-h_2(t, s)\cdot H(t,s)$$, $$h_1,h_2\in L_{\text{loc}}(D, \mathbb{R})$$.
It is shown that if for $$g\in C'(I,\mathbb{R})$$, $$\rho(t)= \exp[-2\int^t_T g(s)\,ds]$$, $$\varphi(t)= -\overline p(t)$$ and $$\theta(t)\rho(t)[-\overline p(t) g(t)+ K\overline q(t)+ g^2(t)- g'(t)]$$ we have $\limsup_{t\to\infty}\, Y^{H,k}_{l,t}\Biggl[\theta- {1\over 4}\rho\Biggl(h_1+ \varphi+ {k'\over k}\Biggr)^2\Biggr]> 0\text{ and } \limsup_{t\to\infty}\, Y^{H,k}_{l,t} \Biggl[\theta- {1\over 4}\rho\Biggl(h_2- \varphi- {k'\over k}\Biggr)^2\Biggr]> 0$ then (1) is oscillatory. Here $$\overline p(t)= \min_{t\leq s\leq t+2\tau} \{{p(s)\over 2\tau}\}$$, $$\overline q(t)= \min_{t\leq s\leq t+ 2\tau} \{{q(s)\over \tau^2}\}$$.

### MSC:

 39A11 Stability of difference equations (MSC2000) 39A10 Additive difference equations
Full Text:

### References:

 [1] Thandapani, E.; Pandian, S.; Lalli, B. S., Oscillatory and nonoscillatory behavior of second-order difference equations, Appl. Math. Comput., 70, 53-66 (1995) · Zbl 0823.39003 [2] Thandapani, E.; Lalli, B. S., Oscillation criteria for a second-order damped difference equation, Appl. Math. Lett., 8, 1-6 (1995) · Zbl 0813.39003 [3] Saker, S. H.; Cheng, S. S., Oscillation criteria for difference equations with damping terms, Appl. Math. Comput., 148, 421-442 (2004) · Zbl 1041.39007 [4] Zhang, Z. G.; Bi, P.; Chen, J. F., Oscillation of second order nonlinear difference equation with continuous variable, J. Math. Anal. Appl., 255, 349-357 (2001) · Zbl 0971.39007 [5] Li, W. T.; Agarwal, R. P., Interval oscillation criteria for second order nonlinear differential equations with damping, Comput. Math. Appl., 40, 217-230 (2000) · Zbl 0959.34026 [6] Li, W. T.; Agarwal, R. P., Interval oscillation criteria related to integral averaging technique for certain nonlinear differential equations, J. Math. Anal. Appl., 245, 171-188 (2000) · Zbl 0983.34020 [7] Kong, Q., Interval criteria for oscillation of second-order linear ordinary differential equations, J. Math. Anal. Appl., 229, 258-270 (1999) · Zbl 0924.34026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.