Bihari's inequality

Bihari's inequality, proved by Hungarian mathematician Imre Bihari (1915–1998), is the following nonlinear generalization of Grönwall's lemma.[1]

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

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

References

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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