Tag Archives: differential

Gronwall Inequality

There are a number of different statements of Gronwall’s inequality. In this post, we will consider only one of them, perhaps the weakest of all.

Proposition 1 Let {f(t)} be a non-negative continuous function on {\left[ a,b \right]} such that there are positive constants {C} and {K} satisfying

\displaystyle \begin{array}{rcl} f(t)\le C + K\int_{a}^{t}f(s)ds \end{array}

for all {t\in\left[ a,b \right]}. Then,

\displaystyle \begin{array}{rcl} f(t)\le Ce^{K(t-a)} \end{array}

for all {t \in \left[ a,b \right]}.

Proof: Define {U(t) = C + K\int_{a}^{t}f(s)ds}. Note that, by definition, {f(t)\le U(t)} and {U} is a strictly positive differentiable function. Also, we have {U'(t) = Kf(t)\le KU(t)}. In other words, {\frac{U'(t)}{U(t)}\le K} which means the relative rate of change of {U} is less than {K}. Hence, the growth of {U} is slower than an exponential function with relative rate of change {K}. That is {U(t) \le U(a) e^{K(t-a)}} (if you did not like this reasoning, you may integrate both sides of the previous inequality from {a} to {t}). So, we have the desired result {f(t) \le U(t) \le U(a)e^{K(t-a)}= Ce^{K(t-a)}}. \Box