On some new discrete generalizations of Gronwall's inequality
The main result of the paper (Theorem 3) concerns a linear discrete inequality of the type \[ (*)\quad x(n)\leq p(n)+\sum^{q}_{j=1}\sum^{r_ j}_{i=1}J_ i^{(j)}(n,x)\quad (:=p(n)+A(x)),\quad n\in N, \] where \[ J_ i^{(j)}(n,x)=\sum^{n- 1}_{s_ 1=n_ 0}f_{i1}^{(j)}(n,s_ 1)...\sum^{s_{j-1}- 1}_{s_ j=n_ 0}f_{ij}^{(j)}(s_{j-1},s_ j)x(s_ j), \] all the functions x, p, \(f_{ik}^{(j)}\) are real-valued and nonnegative, p - nondecreasing, \(f_{ik}^{(j)} - nondecrea\sin g\) 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 \(x(n)\leq p(n)+g(n)H^{-1}(A(H(x)))\) 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 \textit{B. G. Pachpatte} [e.g. Indian J. Pure Appl. Math. 8, 1093-1107 (1977; Zbl 0402.26008)]. See also \textit{R. P. Agarwal} and \textit{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)].
discrete generalization of the Gronwall-Bellman integral inequality
scientific article; zbMATH DE number 4047997
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