Grönwall's inequality

Grönwall's inequality bounds a nonnegative function satisfying an integral or differential inequality by an exponential expression determined by its initial value and coefficient.

Integral inequality: An inequality relating an integral of a function to another quantity or function. The integral form of Grönwall's inequality starts from a bound involving accumulated values of the unknown function.
Picard–Lindelöf theorem: A theorem guaranteeing local existence and uniqueness for ordinary differential equations with a Lipschitz continuous right-hand side. Grönwall's inequality supplies the uniqueness estimate in its usual proof.
Absolute continuity: A regularity property ensuring that a function's total change over disjoint intervals is controlled by their total length. It lets an almost-everywhere derivative bound recover an integral estimate.
Thomas Hakon Grönwall: A Swedish mathematician known for proving an influential integral inequality in 1919. His result gave the inequality its familiar name.
Differential inequality: An inequality that constrains a function through its derivative rather than requiring an exact differential equation. The differential form applies when a derivative bound controls the function's rate of growth.
Continuous dependence on initial conditions: A property stating that nearby initial data produce nearby solutions of a differential equation. The inequality bounds the separation between solutions as their initial data vary.
Fundamental theorem of calculus: The theorem connecting integration of a derivative with the change in a function's values. It links differential inequalities to the integral estimates used by Grönwall's inequality.
Richard Bellman: An American mathematician whose work in dynamic programming and differential equations extended Grönwall's inequality. His generalization popularized the variable-coefficient form.
Exponential function: A function whose rate of change is proportional to its value. The inequality's exponential bound reflects how a proportional growth limit accumulates over time.
Numerical stability: The tendency of a numerical method to control the propagation of perturbations or errors. Its exponential estimate can bound how discretization errors accumulate over time.

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