\(N\)-independent-variable discrete inequalities of Gronwall-Ou-Iang type (Q2717970)

The author discusses three \(n\)-variable discrete inequalities which are multidimensional discrete extensions of Gronwall's and Ou-Iang's integral inequalities. The typical result embodied in the Theorem 1 yields a priori bound on solutions to the nonlinear discrete inequality NEWLINE\[NEWLINE\phi(u(x))\leq m(x)+ S\{x;\phi'(u(s))[f(s) W(u(s))+ g(s) u(s)+ h(s)]\},\;x= (x_1,\dots, x_n)\in N^n,\tag{\(*\)}NEWLINE\]NEWLINE herein NEWLINE\[NEWLINES\{x; q(s)\}:= \sum_1 \sum_2\cdots\sum_n q(s_1, s_2,\dots, s_n),NEWLINE\]NEWLINE NEWLINE\[NEWLINE\sum_i= \sum^{x_i- 1}_{t_{1,i}= 0} \sum^{t_{1,i}- 1}_{t_{2,i}= 0}\cdots \sum^{t_{k_i- 2,i}-1}_{t_{k_{i-1},i}= 0} \sum^{t_{k_i-1,i}-1}_{s_i= 0}\qquad (i= 1,\dots, n),NEWLINE\]NEWLINE where \(k,k_1,\dots, k_n\in N\), \(q: N^n\to R_+\), \(u,f,g,h,m: N^n\to R_+\), \(N:= \{0,1,2,\dots\}\), \(R_+= [0,\infty)\), with \(m\) nondecreasing, i.e., \(m(x_1,\dots, x_n)\leq m(y_1,\dots, y_n)\) holds for \(x_i\leq y_i\) \((i= 1,2,\dots, n)\); \(\varphi\in C^1(R_+, R_+)\), \(\phi'(u)> 0\) when \(u> 0\) and \(\phi'(u)\) is nondecreasing; \(W\in C(R_+, R_+)\) is nondecreasing and \(W(u)> 0\) for \(u> 0\). The a priori bound on solutions to \((*)\) is given in the form NEWLINE\[NEWLINEu(x)\leq G^{-1} \{G\lfloor\xi_g(x) (\phi^{-1}(m(x))+ S\{x; h(s)\})\rfloor+ \xi_g(x) S\{x; f(s)\}\},\;0\leq x\leq\alpha,NEWLINE\]NEWLINE where \(G(\theta)= \int^\theta_{\theta_0} {ds\over W(s)}\), \(\theta\geq\theta_0> 0\), \(G^{-1}\), \(\phi^{-1}\) are inverse functions of \(G\) and \(\phi\), respectively, and \(\xi_g(x)\) is a known function depending on the function \(g\), and the positive vector \(\alpha\) is chosen so that the quantity in the first brace of \((*)\) is in the range of \(G\). Some applications to certain 3-variable nonlinear difference equation are also indicated.