In mathematics, Gronwall’s inequality (also called Grönwall’s lemma, Gronwall’s lemma or Gronwall–Bellman inequality) allows one to bound a function that is known to satisfy a certain differential or integral inequality by the solution of the corresponding differential or integral equation. The differential form was proven by Grönwall in 1919. The integral form was proven by Richard Bellman in 1943. A nonlinear generalization of the Gronwall–Bellman inequality is known as Bihari’s inequality.

First, we consider the Gronwall inequality.

**Type 1**. Bounds by integrals based on lower bound .

Let and be real-valued continuous functions defined on . If is differentiable in and satisfies the differential inequality

then is bounded by the solution of the corresponding differential equation , that is to say

for all .

*Proof*. We define , a solution of the equation , i.e.

for all . Clearly and for all . By the quotient rule, we know that

Hence the quotient is monotone decreasing in . In particular, we obtain

for all , which is Gronwall’s inequality.

**Type 2**. Bounds by integrals based on upper bound .

Let and be real-valued continuous functions defined on . If is differentiable in and satisfies the differential inequality

then is bounded by the solution of the corresponding differential equation , that is to say

for all .

*Proof*. We basically use Type 1. Simply writing

we obtain

for all .

We now consider the Bellman inequality.

**Type 1**. Bounds by integrals based on lower bound .

Let , and be real-valued functions defined on . Assume that and are continuous and that the negative part of is integrable on every closed and bounded subinterval of .

- If is non-negative and if satisfies the integral inequality
for all then

for all .

- If, in addition, the function is non-decreasing, then
for all .

- If, in addition, the function , then
for all .

*Proof*. If we differentiate the RHS of the integral inequality, we then obtain

Hence, a solution to should take the following from

Note that, upon integrating by parts, we further obtain

If we need a plus sign after “= \alpha (t)”, we simply change to , hence we obtain the RHS of the desired inequality.

If the function is non-decreasing, then , and the fundamental theorem of calculus implies the desired estimate.

**Type 2**. Bounds by integrals based on upper bound .

Let , and be real-valued functions defined on . Assume that and are continuous and that the negative part of is integrable on every closed and bounded subinterval of .

- If is non-negative and if satisfies the integral inequality
for all then

for all .

- If, in addition, the function is non-decreasing, then
for all .

- If, in addition, the function , then
for all .

*Proof*. We again use Type 1.

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