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This article advances the understanding and development of the Grönwall integral inequalities. The first part examines the ability of the standard Grönwall integral inequalities to extend themselves. In such an extended setting, we also analyze the degree of importance of a usual non-negativity assumption. The second part establishes a new theorem on flexible variants of the Grönwall integral inequalities. This flexibility is characterized by five adaptable functions. Complementary results provide more tractable bounds under certain technical assumptions. Detailed proofs of all results ensure that the article remains self-contained.

Grönwall integral inequalities, integration, primitive, integral inequalities, exponential function, logarithmic function.