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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 $$ (*)\quad x(n)\le p(n)+\sum\sp{q}\sb{j=1}\sum\sp{r\sb j}\sb{i=1}J\sb i\sp{(j)}(n,x)\quad (:=p(n)+A(x)),\quad n\in N, $$ where $$ J\sb i\sp{(j)}(n,x)=\sum\sp{n- 1}\sb{s\sb 1=n\sb 0}f\sb{i1}\sp{(j)}(n,s\sb 1)...\sum\sp{s\sb{j-1}- 1}\sb{s\sb j=n\sb 0}f\sb{ij}\sp{(j)}(s\sb{j-1},s\sb j)x(s\sb j), $$ all the functions x, p, $f\sb{ik}\sp{(j)}$ are real-valued and nonnegative, p - nondecreasing, $f\sb{ik}\sp{(j)} - nondecrea\sin g$ 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 $x(n)\le p(n)+g(n)H\sp{-1}(A(H(x)))$ with H nonnegative, strictly increasing, subadditive, $H(0)=0$, 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 {\it B. G. Pachpatte} [e.g. Indian J. Pure Appl. Math. 8, 1093-1107 (1977; Zbl 0402.26008)]. See also {\it R. P. Agarwal} and {\it 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

26D10Inequalities involving derivatives, differential and integral operators
39A12Discrete version of topics in analysis
Full Text: DOI
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