Existence of bounded and unbounded nonoscillatory solutions of nonlinear partial difference equations (Q884370)

The author considers the higher order nonlinear partial difference equation of neutral type \[ \triangle_n^h\triangle_m^r(y(m,n)+cy(m-k,n-l))+F(m,n,y(m-\tau,n-\sigma))=0 \] where \(h,r\in \mathbb{N}(1)\), \(k,l,\tau,\sigma\in \mathbb{N}(0)\), \(c\in \mathbb{R}\) and \(F:\mathbb{N}\times \mathbb{N}\times \mathbb{R}\mapsto \mathbb{R}\) is continuous as a map between topological spaces with the discrete topology on \(R\). Existence of a nonoscillatory solution satisfying the inequalities \[ a\leq {{y(m,n)}\over{m^{r_1}n^{h_1}}}\leq b \] for some integers \(r_1\leq r\), \(h_1\leq h\) and some interval \(0<a<b\) is considered in two cases \smallskip \(1^o\) \(c\neq -1\) : the above \(r_1\), \(h_1\), \(a\), \(b\) are such that \[ \displaystyle{\sum_{m_0}^{\infty}\sum_{n_0}^{\infty}m^{r-1}n^{h-1}\sup_{w\in[a,b]} | F(m,n,w(m-r)^{r_1}(n-\sigma)^{h_1})| <\infty} \] \(2^o\) \(c=-1\) : the above \(r_1\), \(h_1\), \(a\), \(b\) are such that \[ \sum_{m_0}^{\infty}\sum_{n_0}^{\infty}m^rn^h\sup_{w\in[a,b]} | F(m,n,w(m-r)^{r_1}(n-\sigma)^{h_1})| <\infty \]