Bihari–LaSalle inequality
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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,
- [math]\displaystyle{ u(t)\leq \alpha+ \int_0^t f(s)\,w(u(s))\,ds,\qquad t\in[0,\infty), }[/math]
where α is a non-negative constant, then
- [math]\displaystyle{ u(t)\leq G^{-1}\left(G(\alpha)+\int_0^t\,f(s) \, ds\right),\qquad t\in[0,T], }[/math]
where the function G is defined by
- [math]\displaystyle{ G(x)=\int_{x_0}^x \frac{dy}{w(y)},\qquad x \geq 0,\,x_0\gt 0, }[/math]
and G−1 is the inverse function of G and T is chosen so that
- [math]\displaystyle{ G(\alpha)+\int_0^t\,f(s)\,ds\in \operatorname{Dom}(G^{-1}),\qquad \forall \, t \in [0,T]. }[/math]
References
- ↑ J. LaSalle (July 1949). "Uniqueness theorems and successive approximations". Annals of Mathematics 50 (3): 722–730. doi:10.2307/1969559.
- ↑ 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.
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