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In this paper, we prove several new explicit estimations for the solutions of some classes of nonlinear dynamic inequalities of Gronwall–Bellman–Pachpatte type on time scales. Our results formulate some integral and discrete inequalities discussed in the literature as special cases and extend some known dynamic inequalities on time scales. The inequalities given here can be used in the analysis of the qualitative properties of certain classes of dynamic equations on time scales. Some examples are presented to demonstrate the applications of our results.

SOME GRONWALL{BELLMAN TYPE INEQUALITIES ON TIME SCALES FOR VOLTERRA-FREDHOLM DYNAMIC INTEGRAL EQUATIONS