Oscillation of certain second order nonlinear damped difference equations with continuous variable
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 \(\overline p(t)= \min_{t\leq s\leq t+2\tau} \{{p(s)\over 2\tau}\}\), \(\overline q(t)= \min_{t\leq s\leq t+ 2\tau} \{{q(s)\over \tau^2}\}\).
second-order nonlinear damped difference equation
scientific article; zbMATH DE number 5149466