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 of the inequality
where the function satisfies the condition
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.