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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)].
##### MSC:
 39B72 Systems of functional equations and inequalities 40A05 Convergence and divergence of series and sequences