Generalisations of monotony (Q2754559)

This paper builds on and extends work of \textit{C. Fiocchi} and \textit{V. Zanelli} [Atti Semin. Mat. Fis. Univ. Modena 46, No. 2, 413-434 (1998; Zbl 0911.40001)] on positive bounded sequences \(\{a_n\}\) satisfying the two conditions (I) there exists \(M>0\) such that \(0<a_n<M\) for all \(n\), and (II) there exist \(\{\lambda_n\}, \{\alpha_n\}\) with \(0\leq\lambda_n\leq 1\) and \(0\leq\alpha_n\leq 1\), such that \(a_n=\alpha_n[(1-\lambda_n)a_{n-2}+\lambda_na_{n-1}]\), for all \(n\geq 3\) and \(a_1\neq a_2\). NEWLINENEWLINENEWLINEIn general, convergence properties of the sequence \(\{a_n\}\) or the subsequences \(\{a_{2n}\}, \{a_{2n+1}\}\) are related in this paper to convergence of series or sequences involving the \(\lambda_n\) or \(\alpha_n\). Sample theorem: If \(\{a_n\}\) satisfies (I) and (II), and \(\lambda_{2n}\geq\lambda^*>0\) for all \(n\) and \(\sum_n (1-\alpha_{2n+1})=\infty\), then \(\lim a_n=0\).NEWLINENEWLINEFor the entire collection see [Zbl 0956.00027].