Bihari's inequality

Bihari's inequality, proved by Hungarian mathematician Imre Bihari (1915–1998), is the following nonlinear generalization of Grönwall's lemma.

Let u and ƒ be non-negative continuous functions defined on the half-infinite ray [0, ∞), and let w be a continuous non-decreasing function defined on [0, ∞) and w(u) > 0 on (0, ∞). If u satisfies the following integral inequality,

$u(t)\leq \alpha+ \int_0^t f(s)\,w(u(s))\,ds,\qquad t\in[0,\infty),$

where α is a non-negative constant, then

$u(t)\leq G^{-1}\left(G(\alpha)+\int_0^t\,f(s) \, ds\right),\qquad t\in[0,T],$

where the function G is defined by

$G(x)=\int_{x_0}^x \frac{dy}{w(y)},\qquad x \geq 0,\,x_0>0,$

and G⁻¹ is the inverse function of G and T is chosen so that

$G(\alpha)+\int_0^t\,f(s)\,ds\in \text{Dom}(G^{-1}),\qquad \forall \, t \in [0,T].$