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The recurrent sequences of inequalities. (English) Zbl 0736.39007
Pr. Nauk. Uniw. Śląsk. Katowicach 1120, Ann. Math. Silesianae 3(15), 41-44 (1990).
Consider the sequence of inequalities \(a_{i,k+1}\leq\Gamma(\sum^ n_{j=1}s_{i,j}(a_{1,k},\dots,a_{n,k}))+b_{i,k}\), \(i=1,2,\ldots,n\), \(k=0,1,2,...\) where \(\Gamma\) is a concave and increasing function. Under certain assumptions on the functions \(s_{i,j}\), \(i,j=1,\ldots,n\) and on the sequences \(\{b_{i,k}\}\) the author proves the following:
(a) if the sequences \(\{b_{i,k}\}\) are bounded, then \(\{a_{i,k}\}\) are bounded; (b) if \(\lim_{k\to\infty}b_{i,k}=0\), then \(\lim_{k\to\infty}a_{i,k}=0\), \(i=1,2,\ldots,n\); (c) if \(\sum_{p=0}^ \infty \sum_{k=0}^ \infty \alpha^ p(b_{i,k})\), \(i=1,2,\ldots,n\), are convergent, then \(\sum_{k=0}^ \infty a_{i,k}\), \(i=1,2,\ldots,n\), are convergent, where \(\alpha(t)=\sup\{\Gamma(\beta(s)):0\leq s\leq t\}.\) The results obtained here generalize some of the results of J. Matkowski [Dissert. Math. 127 (1975; Zbl 0318.39005)].
39B72 Systems of functional equations and inequalities
40A05 Convergence and divergence of series and sequences