In mathematics, Grönwall's inequality (also called Grönwall's lemma or the Grönwall–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. There are two forms of the lemma, a differential form and an integral form. For the latter there are several variants. Grönwall's inequality is an important tool to obtain various estimates in the theory of ordinary and stochastic differential equations. In particular, it provides a comparison theorem that can be used to prove uniqueness of a solution to the initial value problem; see the Picard–Lindelöf theorem. It is named for Thomas Hakon Grönwall (1877–1932). Grönwall is the Swedish spelling of his name, but he spelled his name as Gronwall in his scientific publications after emigrating to the United States. The inequality was first proven by Grönwall in 1919 (the integral form below with α and β being constants). Richard Bellman proved a slightly more general integral form in 1943. A nonlinear generalization of the Grönwall–Bellman inequality is known as Bihari–LaSalle inequality. Other variants and generalizations can be found in Pachpatte, B.G. (1998).
Differential form Let I {\displaystyle I} denote an interval of the real line of the form [ a , ∞ ) {\displaystyle [a,\infty )} or [ a , b ] {\displaystyle [a,b]} or [ a , b ) {\displaystyle [a,b)} with a < b {\displaystyle a<b} . Let β {\displaystyle \beta } and u {\displaystyle u} be real-valued continuous functions defined on I {\displaystyle I} . If u {\displaystyle u} is differentiable in the interior I ∘ {\displaystyle I^{\circ }} of I {\displaystyle I} (the interval I {\displaystyle I} without the end points a {\displaystyle a} and possibly b {\displaystyle b} ) and satisfies the differential inequality
u ′ ( t ) ≤ β ( t ) u ( t ) , t ∈ I ∘ , {\displaystyle u'(t)\leq \beta (t)\,u(t),\qquad t\in I^{\circ },}
then u {\displaystyle u} is bounded by the solution of the corresponding differential equation v ′ ( t ) = β ( t ) v ( t ) {\displaystyle v'(t)=\beta (t)\,v(t)} :
u ( t ) ≤ u ( a ) exp ( ∫ a t β ( s ) d s ) {\displaystyle u(t)\leq u(a)\exp {\biggl (}\int _{a}^{t}\beta (s)\,\mathrm {d} s{\biggr )}}
for all t ∈ I {\displaystyle t\in I} . Remark: There are no assumptions on the signs of the functions β {\displaystyle \beta } and u {\displaystyle u} .
Proof Define the function
v ( t ) = exp ( ∫ a t β ( s ) d s ) , t ∈ I . {\displaystyle v(t)=\exp {\biggl (}\int _{a}^{t}\beta (s)\,\mathrm {d} s{\biggr )},\qquad t\in I.}
Note that v {\displaystyle v} satisfies
v ′ ( t ) = β ( t ) v ( t ) , t ∈ I ∘ , {\displaystyle v'(t)=\beta (t)\,v(t),\qquad t\in I^{\circ },}
with v ( a ) = 1 {\displaystyle v(a)=1} and v ( t ) > 0 {\displaystyle v(t)>0} for all t ∈ I {\displaystyle t\in I} . By the quotient rule
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