On the systems of discrete inequalities of Gronwall type (Q1329303)

The basic result of the paper embodied in Theorem 1 yields a discrete version of the next very special system of Volterra integral inequalities \[ u_ i(x) \leq f_ i(x) + g_ i(x)W \left[ \int^ x_{x_ 0} B \bigl( t,u_ 1(t), \dots, u_ s(t) \bigr) dt \right],\;i = 1,2, \dots, s, \tag{I} \] where all the functions involved are real-valued nonnegative continuous functions satisfying some additional conditions. The same functions \(B\) and \(W\) are involved in every inequality of system (I). The following assumptions on \(B\) are required: (1) \(0 \leq B(t,y_ 1, \dots,y_ s) \leq B(t,x_ 1, \dots, x_ s)\) for \(t \in [0,\infty)\), \(0 \leq y_ i \leq x_ i\); and (2) there exists a positive function \(A:[1,\infty) \to (0, \infty)\) which is continuous, nondecreasing, and such that \(B(t,ay_ 1, \dots, ay_ s) \leq A(a) B(t,y_ 1, \dots, y_ s)\) holds for \(t \in [0, \infty)\), \(0 \leq y_ i\), \(i = 1, \dots,s\), and \(a \geq 1\). Two slightly generalized systems of the above-mentioned discrete system are also obtained. A few misprints are in the paper. For example, ``\(f_ i:N\to S\)'' in the condition (i) should be \(f_ i:N \to R_ 0\), and ``\(0<y_ i<x_ i\)'' should be \(0 \leq y_ i \leq x_ i\).