On some new N-independent-variable discrete inequalities of the Gronwall type
In the paper three theorems about discrete inequalities of Gronwall type for functions of many variables are considered. The initial inequalities are \[ u(x)\leq f(x)+\sum^{x,n}_{y,1}p(y)[u(y)+\delta \sum^{y,n}_{z,1}q(z)u(z)],\quad x\in N^ n_ 0, \] (\(\delta\) \(=0\) in Theorem 1, \(\delta =1\) in Theorem 2), and \[ u(x)\leq f(x)+\sum^{x,n}_{y,1}p(y)W(u(y)) \] (Theorem 3), with: f - positive, nondecreasing, having nonnegative mixed differences up to \(n-1\), and nonpositive mixed difference of n-th order; u, p, q - nonnegative; W - continuous, positive, and increasing, where for a function \(f: R\to R^ n\) \[ \sum^{x,n}_{y,1}f(y):=\sum^{x_ 1-1}_{y_ 1=0}\sum^{x_ 2-1}_{y_ 2=0}...\sum^{x_ n-1}_{y_ n=0}f(y_ 1,y_ 2,...,y_ n). \] The results and the method are similar to those presented by \textit{C.-C. Yeh} [J. Math. Anal. Appl. 105, 322-332 (1985; Zbl 0604.26012); ibid. 106, 282-285 (1985; Zbl 0604.26013)]. See also \textit{R. P. Agarwal} [An. Ştiinţ. Univ. Al. I. Cuza, Iaşi, N. Ser., Secţ. Ia 30, 65-68 (1984; Zbl 0585.26011)] and \textit{F.O. Mamikonyan} [Izv. Akad. Nauk. Arm. SSR, Mat. 20, No.5, 349-365 (1985; Zbl 0605.26016)].
discrete inequalities of Gronwall type for functions of many variables
scientific article; zbMATH DE number 3979417