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On some new discrete generalizations of Gronwall’s inequality. (English) Zbl 0643.26013

The main result of the paper (Theorem 3) concerns a linear discrete inequality of the type

$\left(*\right)\phantom{\rule{1.em}{0ex}}x\left(n\right)\le p\left(n\right)+\sum _{j=1}^{q}\sum _{i=1}^{{r}_{j}}{J}_{i}^{\left(j\right)}\left(n,x\right)\phantom{\rule{1.em}{0ex}}\left(:=p\left(n\right)+A\left(x\right)\right),\phantom{\rule{1.em}{0ex}}n\in N,$

where

${J}_{i}^{\left(j\right)}\left(n,x\right)=\sum _{{s}_{1}={n}_{0}}^{n-1}{f}_{i1}^{\left(j\right)}\left(n,{s}_{1}\right)···\sum _{{s}_{j}={n}_{0}}^{{s}_{j-1}-1}{f}_{ij}^{\left(j\right)}\left({s}_{j-1},{s}_{j}\right)x\left({s}_{j}\right),$

all the functions x, p, ${f}_{ik}^{\left(j\right)}$ are real-valued and nonnegative, p - nondecreasing, ${f}_{ik}^{\left(j\right)}-nondecreasing$ in n for every $s\in N$ fixed. In the first two theorems some special cases of (*) are considered. Theorems 3, 4 concern nonlinear inequalities $x\left(n\right)\le p\left(n\right)+g\left(n\right){H}^{-1}\left(A\left(H\left(x\right)\right)\right)$ with H nonnegative, strictly increasing, subadditive, $H\left(0\right)=0$, and furthermore $g\equiv 1$ (Theorem 3); H - submultiplicative, g - nonnegative (Theorem 4). Linear inequalities are discrete analogies of those proved by the author in J. Math. Anal. Appl. 103, 184-197 (1984; Zbl 0573.26008) and extend many results proved by B. G. Pachpatte [e.g. Indian J. Pure Appl. Math. 8, 1093-1107 (1977; Zbl 0402.26008)]. See also R. P. Agarwal and E. Thandapani [Bull. Inst. Math., Acad. Sin. 9, 235-248 (1981; Zbl 0474.26009); An. Ştiinţ. Univ. Al. I. Cuza Iaşi, N. Ser., Secţ. Ia 28, 71-75 (1982; Zbl 0553.26004)].

Reviewer: J.Popenda
##### MSC:
 26D10 Inequalities involving derivatives, differential and integral operators 39A12 Discrete version of topics in analysis