Gronwall lemma

2020 Mathematics Subject Classification: Primary: 34A40 [MSN][ZBL]

The Gronwall lemma is a fundamental estimate for (nonnegative) functions on one real variable satisfying a certain differential inequality. The lemma is extensively used in several areas of mathematics where evolution problems are studied (e.g. partial and ordinary differential equations, continuous dynamical systems) to bound quantities which depend on time.

The most elementary version of the inequality is stated in the following

Theorem 1 Let $\phi: [0, T]\to \mathbb R$ be a nonnegative differentiable function for which there exists a constant $C$ such that \begin{equation}\label{e:diff_ineq} \phi' (t) \leq C \phi (t) \qquad \mbox{for all } t\in [0,T]\, . \end{equation} Then \[ \phi (t) \leq e^{C t} \phi (0) \qquad \mbox{for all } t\in [0, T]\, . \]

A more general version of this theorem assumes the inequality $\phi' (t) \leq C (t) \phi (t)$ (where $C$ is a nonnegative summable function): the conclusion is then the bound \[ \phi (t) \leq \phi (0)\, {\rm exp}\, \left(\int_0^t C (\tau)\, d\tau\right)\, . \] The assumption of differentiability can be severely relaxed and this is of great importance since often the function $\phi$ is not known to be differentiable or it is known to be differentiable only in a weak sense. In many situations $\phi$ is for instance

• only absolutely continuous, in which case \eqref{e:diff_ineq} is assumed to hold for almost every $t$;
• only a function of bounded variation, in which case \eqref{e:diff_ineq} is assumed to hold almost everywhere and, in addition, the singular part of the distributional derivative of $\phi$ is assumed to be a nonpositive measure (see Function of bounded variation for the relevant definitions).

In order to avoid the issue of the differentiability of $\phi$ the differential inequality is often stated in an integral form. Thus a rather general and popular version of Gronwall's lemma is the following

Theorem 2 Assume $\phi: [0,T]\to \mathbb R$ is a bounded nonnegative measurable function, $C: [0,T]\to \mathbb R$ is a nonnegative integrable function and $B\geq 0$ is a constant with the property that \begin{equation}\label{e:int_diff} \phi (t) \leq B + \int_0^t C (\tau) \phi (\tau)\, d\tau \qquad \mbox{for all } t\in [0,T]\, . \end{equation} Then \begin{equation}\label{e:conclusion} \phi (t) \leq B\, {\rm exp}\, \left(\int_0^t C (\tau)\, d\tau\right) \qquad \mbox{for all } t\in [0,T]\, . \end{equation}

The inequality can be further generalized if $B$ in \eqref{e:int_diff} is also allowed to depend on time. More precisely we have the following theorem, which is often called Bellman-Gronwall inequality.

Theorem 3 Assume $\phi, B: [0,T]\to \mathbb R$ are bounded nonnegative measurable function and $C: [0,T]\to \mathbb R$ is a nonnegative integrable function with the property that \begin{equation}\label{e:int_diff_2} \phi (t) \leq B (t) + \int_0^t C (\tau) \phi (\tau)\, d\tau \qquad \mbox{for all } t\in [0,T]\, . \end{equation} Then \begin{equation}\label{e:conclusion2} \phi (t) \leq B (t) + \int_0^t B (s)\, C (s)\, {\rm exp} \left(\int_s^t C (\tau)\, d\tau\right)\, ds\qquad \mbox{for all } t\in [0,T]\, . \end{equation}

Note that, when $B (t)$ is constant, \eqref{e:conclusion2} coincides with \eqref{e:conclusion}.

Remark 4 Observe also that all the theorems are sharp. If in fact we assume that the equality sign holds in any of the assumptions, then the equality sign holds also in the corresponding conclusions. For instance, the equality in \eqref{e:conclusion2} yields just the well-known formula for solutions of the linear ordinary differential equation $\phi'(t) = B'(t) + C \phi (t)$ with initial condition $\phi (0)= B (0)$.

References

[Am]	H. Amann, "Ordinary differential equations. An introduction to nonlinear analysis." de Gruyter Studies in Mathematics, 13. Walter de Gruyter & Co., Berlin, 1990.
[Gr]	T. H. Gronwall, "Note on the derivatives with respect to a parameter of the solutions of a system of differential equations", Ann. of Math. 20 (2): 292-296 (1919).
[Ha]	P. Hartman, "Ordinary differential equations" , Birkhäuser (1982).
[Pet]	I.G. Petrovskii, "Ordinary differential equations" , Prentice-Hall (1966) (Translated from Russian).