Asymptotic behavior of solutions of second order nonlinear difference equations (Q1816451)

The authors consider the nonlinear second order difference equation \[ \Delta \bigl(r(n) \Delta x(n) \bigr)+ f\bigl(n,x(n)\bigr) =0, \quad n\in N(n_0), \tag{1} \] where \(N(n_0) =\{n_0, n_0+1, \dots\}\), \(n_0\in \{1,2, \dots\}\), \(r:N(n_0) \to(0,\infty)\), and \(f:N(n_0) \times (-\infty, \infty)\to (-\infty, \infty)\) satisfying \[ \bigl|f(n,x)\bigr|\leq\sum^n_{i=1} b_i (n)g_i \bigl(|x|\bigr) + b_{m+1}(n), \quad (n,x)\in N(n_0) \times(-\infty, \infty), \] herein \(b_j: N(n_0) \to [0,\infty)\), \(j=1,2, \dots, m+1\), and \(g_i\in {\mathcal J}\) for \(i=1,\dots,m\). A function \(g:(0,\infty) \to(-\infty, \infty)\) is said to belong to the class \({\mathcal J}\) if \(g\) is nondecreasing and continuous such that \(g(u)/v \leq g(u/v)\) holds for \(u\geq 0\) and \(v\geq 1\). By using a discrete inequality proved herein, the boundedness of solutions to equation (1) is proved under the additional conditions: \[ \sum^{n-1}_{s=n_0} b_j(s) R(n,s+1) \text{ is bounded for } n\in N(n_0),\;1\leq j \leq m+1; \tag{i} \] and \[ \lim R(n,n_0)< \infty \text{ as } n\to\infty, \text{ here } R(n,s): =\sum^{n-1}_{k=s} {1\over r(k)}. \tag{ii} \] The last main result of the paper implies discrete analogues of some known theorems for nonlinear differential equations due to \textit{L. Hatvani} [Publ. Math. 19(1972), 225-237 (1973; Zbl 0271.34061)] and the reviewer [Nonlinear Anal., Theory Methods Appl. 8, 541-547 (1984; Zbl 0537.34030)]. A sufficient condition is also given to guarantee that \(|x(n) |= O(R(n,n_0))\) and \(|r(n) \Delta x(n) |=O(1)\) when \(n\to \infty\). Some illustrating examples are also indicated.