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Zbl 0839.26013
Pachpatte, B.G.
On certain nonlinear integral inequalities and their discrete analogues. (English)
[J] Facta Univ., Ser. Math. Inf. 8, 21-34 (1993). ISSN 0352-9665

The author proves some generalizations of the result obtained by {\it 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 $$u(t)\le a(t)+ b(t) \int^t_0 r_1(t_1) \int^{t_1}_0 r_2(t_2)\cdots \int^{t_{n- 1}}_0 F(t_n, u(t_n))dt_n\cdots dt_2 dt_1,\tag{$*$}$$ where the function $F$ satisfies the condition $$0\le F(t, v)- F(t, w)\le K(t, w)(v- w)\quad\text{for}\quad t\in R_+,\ v\ge w\ge 0.$$ 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.
[J.Popenda (Poznań)]

MSC 2000:: *26D10 Inequalities involving derivatives, diff. and integral operators
26D15 Inequalities for sums, series and integrals of real functions

Keywords: Gronwall type inequalities; discrete analogues

Citations: Zbl 0636.45004

Cited in: Zbl 0901.26006

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