The Bihari–LaSalle inequality was proved by the American mathematician Joseph P. LaSalle (1916–1983) in 1949[1] and by the Hungarian mathematician Imre Bihari (1915–1998) in 1956.[2] It 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,
where α is a non-negative constant, then
![{\displaystyle u(t)\leq G^{-1}\left(G(\alpha )+\int _{0}^{t}\,f(s)\,ds\right),\qquad t\in [0,T],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c654b027dd13c27bad86505ef71a5d9e7e37b95)
where the function G is defined by
and G−1 is the inverse functionofG and T is chosen so that
![{\displaystyle G(\alpha )+\int _{0}^{t}\,f(s)\,ds\in \operatorname {Dom} (G^{-1}),\qquad \forall \,t\in [0,T].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7bb40b7fdca6ff6ee6c70ba47186bc3214f6e350)
