Bihari–LaSalle inequality

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,

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

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

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

References

1. J. LaSalle (July 1949). "Uniqueness theorems and successive approximations". Annals of Mathematics. 50 (3): 722–730. doi:10.2307/1969559.
2. I. Bihari (March 1956). "A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations". Acta Mathematica Hungarica. 7 (1): 81–94. doi:10.1007/BF02022967.