Oscillation of sublinear difference equations with positive neutral term (Q816001)

The authors consider the oscillatory behavior of the solutions for the following equation \[ \Delta(x(n)+p(n)x(\tau(n))) + f(n,x(g_1(n)),\ldots,x(g_m(n)))=0 \] where \(\Delta x(n)=x(n+1)-x(n)\); \(p(n)\) is a sequence of real numbers, \(1<p_1\leq p(n)\leq p_2\), \(\tau(n)<n\) is a sequence of strictly increasing integers, \(\lim_{n\rightarrow\infty}\tau(n)=\infty\); \(g_i(n)\) is a sequence of positive integers, \(\lim_{n\rightarrow\infty}g_i(n)=\infty\), \(f\) is continuous in \(x_1\), \(\ldots\), \(x_m\), \(g(n)=\max_{1\leq i\leq m}\{g_i(n)\}\leq\tau(n)\). Let i) \(f\in C([n_o,\infty)\times \mathbb R^m,\mathbb R)\) be nondecreasing in each \(y_i\) and \(y_1f(n,y_1,\ldots,y_m)>0\) for \(y_1y_i>0\); ii) \(f\) be strongly sublinear; iii) \(\sum_n^\infty f(s,a,\ldots,a)=\infty\), \(\forall a>0\). Then every solution of the above equation is oscillatory. The equation has a bounded non-oscillatory solution which is bounded away from 0 if and only if \(\sum_n^\infty | f(s,d,\ldots,d)| <\infty\) for some \(d\neq 0\) provided the assumptions i)-iii) hold. Under the same conditions i)-iii) and if \(g(n)\leq\tau(n)\) then the solutions of the equations are oscillatory if and only if \(\sum_n^\infty | f(s,a,\ldots,a)| =\infty\) for every \(a\neq 0\).