On an integral inequality and its discrete analogue. (English) Zbl 0845.26009

The authors give a generalization of the Gronwall type inequality proved by C. M. Dafermos [Arch. Ration. Mech. Anal. 70, 167-199 (1979; Zbl 0448.73004)]. The considered inequality is of the form \[ y^q(t)\leq c+ \int^{t_1}_0 [qLy^q(\tau, t^*)+ Ky(\tau, t^*)] d\tau,\tag{\(*\)} \] where \(t= (t_1, t_2, \dots, t_n)\), \(t^*= (t_2,\dots, t_n)\), \(y^q(t)= [y^q_1(t),\dots, y^q_m(t)]\), \(y_i\) are nonnegative and bounded on some interval \([0, t]\), \(K\), \(L\) are monotone, linear operators, \(q> 1\), \(p> 0\), \(c\) a constant vector. The authors present bounds for the \(\ell^p_m\) norm of the terms of solution \(y\) of the inequality \((*)\).
It is shown that all theorems contained in the paper of B. G. Pachpatte [J. Math. Anal. Appl. 182, No. 1, 143-157 (1994; Zbl 0806.26009)] are special cases of the inequality \((*)\). A discrete analogue of the inequality \((*)\) is considered, too. This inequality also covers inequalities of B. G. Pachpatte [Comput. Math. Appl. 28, No. 1-3, 227-241 (1994; Zbl 0809.26009)].


26D10 Inequalities involving derivatives and differential and integral operators
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