Discrete inequalities of the Gronwall-Bellman type in n independent variables
The author proved three elegant results for n-independent-variables discrete inequalities of the Gronwall-Bellman type, which generalize some well-known theorems of \textit{S. M. Singare} and \textit{B. G. Pachpatte} [Pac. J. Math. 82, 197-210 (1979; Zbl 0413.26009); and Indian J. Pure Appl. Math. 11, 727-736 (1980; Zbl 0453.26010)] established for the special case \(n=3\). The fundamental result in this paper is the following: Theorem 1. Let \(u(x), f(x), h(x)\) be real-valued nonnegative functions defined on \(N^ n\) (here N is the set of nonnegative integers), and let \(H(r)\in C(R_+,R_+)\) be a nondecreasing function such that \(Q(r):=\int^{r}_{r_ 0}ds/H(s)\) exists for \(r>0\) with \(r_ 0>0\) fixed, but arbitrary. If the inequality \[ u(x)\leq f(x)+\sum^{x-\hat 1}_{t=\hat 0}h(t)H(u(t)) \] holds, then for \(\hat 0\leq x\leq b\) we have \[ (*)\quad u(x)\leq Q^{-1}[Q(\bar f(x))+\sum^{x-\hat 1}_{t=\hat 0}h(t)], \] where \(Q^{-1}\) is the inverse of Q, \(\bar f(x):=\max \{f(y): \hat 0\leq y\leq x\}\), here \(x=(x_ 1,...,x_ n)\), \(y=(y_ 1,...,y_ n)\), \(\hat 0=(0,...,0)\in N^ n\), and \(b=(b_ 1,...,b_ n)\) is chosen so that the expression in the bracket of the above inequality (*) belongs to the range of Q as long as \(\hat 0\leq x\leq b\). Here the n- fold summation sign \(\sum^{x_ 1-1}_{t_ 1=0}...\sum^{x_ n- 1}_{t_ n=0}\) is denoted by \(\sum^{x-\hat 1}_{t=\hat 0}\).
n-independent-variables discrete inequalities of the Gronwall-Bellman type
scientific article; zbMATH DE number 3975267