On some systems of two discrete inequalities of Gronwall type (Q1359596)

In the present paper, some linear and nonlinear systems of two discrete inequalities of Gronwall-type are discussed. A priori bound on solutions to these systems are established. The typical result embodied in the Theorem 1 is concerned with the system \[ u_i(x)\leq a_i(t)+ p_i(t) R_1(t)+ q_i(t) R_2(t)\quad (i=1,2) \] with \[ R_i(t)= \sum^{t-1}_{s=0} \sum^{s-1}_{t_1=0} f_1(t_1) \sum^{t_1-1}_{t_2= 0} f_2(t_2)\cdots\sum^{t_{n-1}- 1}_{t_n=0} f_n(t_n)u(t_n), \] where \(t=0,1,2,\dots\); \(a_i(t)\), \(p_i(t)\), \(q_i(t)\) \((i=1,2)\) are positive and non-decreasing, and \(f_j(t)\) \((j=1,2,\dots, n)\) are real-valued nonnegative functions, and the empty sum is assumed to be zero. We note that, according to the proof (see page 555), the bound on solutions to the above system given by (4) and (5) can be easily improved by replacing \(E(t,p_1(t)+ p_2(t)+ q_1(t)+ q_2(t))\) with \(\max\{E(t,p_1(t)+ p_2(t)),E(t,q_1(t)+ q_2(t))\}\). Some application examples are also indicated. Several misprints are in the paper.