Asymptotic behavior of solutions of discrete Volterra equations (Q2806531)

The following form of nonlinear discrete Volterra equations of non-convolution type is considered: NEWLINE\[NEWLINE\Delta^{m}x_{n}=b_{n}+\sum_{i=1}^{n}K(n,i)f(i,x_{i}), \,\, n\geq 1, \tag{\(*\)}NEWLINE\]NEWLINE where \(f:\mathbb N\times \mathbb R\rightarrow \mathbb R\), \(K:\mathbb N\times \mathbb N\rightarrow \mathbb R\) such that \(K(n,i)=0\) for \(n<i\), and \(b:\mathbb N\rightarrow \mathbb R\).NEWLINENEWLINEA sequence \(x:\mathbb N\rightarrow \mathbb R\) is said to be a solution of \((*)\) if it satisfies \((*)\) for all large \(n\). The sequence \( x \) is said to be a full solution \((*)\) if \((*)\) is satisfied for all \(n\geq 1\). If \(p\in \mathbb N\) and if \((*)\) is satisfied for all \(n\geq p\), then \(x\) is called a \(p\)-solution of \((*)\).NEWLINENEWLINEThe authors obtain sufficient conditions for the existence of solutions of \((*)\) with prescribed asymptotic behaviour. For example, under certain conditions, \((*)\) admits a solution \(x\) of the form \(x_{n}=c_{m-1}n^{m-1}+c_{m-2}n^{m-2}+\dots+c_{1}n+c_{0} +o(n^{s})\), where \(c_{0},c_{1},\dots,c_{m-1}\) are real constants and \(s\in (-\infty ,0]\). Some results are obtained concerning the existence of full solutions of \((*)\). Also sufficient conditions are obtained under which all solutions of \((*)\) are asymptotically polynomial.NEWLINENEWLINEIn the following, the main result is stated:NEWLINENEWLINETheorem. Suppose that (i) \(s\in (-\infty,0]\), \(k\in \mathbb N(0,m-1)\), \(g: [0,\infty ) \rightarrow [0,\infty )\), (ii) \(f\) is continuous such that \(|f(n,t)|\leq g(\frac{|t|}{n^{k}})\) for \((n,t)\in \mathbb N\times \mathbb R\) and (iii) \(\sum_{n=1}^{\infty}n^{m-1-s}\sum_{i=1}^{n}|K(n,i)|<\infty\).NEWLINENEWLINEFurther, assume that \(p\in \mathbb N\), \(Q,L,M > 0\), \(g(t)\leq M\) for \(t\leq L\) and \(M\sum_{n=p}^{\infty}n^{m-1}\sum_{i=1}^{n}|K(n,i)|\leq Q\).NEWLINENEWLINEThen, for every solution \(y\) of the equation \(\Delta^{m}y=b\) such that \(|y_{n}|\leq L n^{k}-Q\) for any \(n\), there exists a \(p\)-solution \(x\) of \((*)\) such that \( x=y+o(n^{s})\).NEWLINENEWLINEExamples are given to illustrate the results obtained.