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On certain nonlinear integral inequalities and their discrete analogues. (English) Zbl 0839.26013

The author proves some generalizations of the result obtained by S. S. Dragomir [The Gronwall type lemmas and applications (1987; Zbl 0636.45004)]devoted to the Gronwall type inequalities. For example, he gives a bound for the solution u of the inequality

u(t)a(t)+b(t) 0 t r 1 (t 1 ) 0 t 1 r 2 (t 2 ) 0 t n-1 F(t n ,u(t n ))dt n dt 2 dt 1 ,(*)

where the function F satisfies the condition

0F(t,v)-F(t,w)K(t,w)(v-w)fortR + ,vw0·

All the functions are assumed to be continuous and nonnegative. Some special cases of (*) for functions of two and several independent variables as well as their discrete analogues are also considered.

26D10Inequalities involving derivatives, differential and integral operators
26D15Inequalities for sums, series and integrals of real functions