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,
where α is a non-negative constant, then
where the function G is defined by
and G−1 is the inverse function of G and T is chosen so that
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.