Bihari–LaSalle inequality explained

The Bihari–LaSalle inequality was proved by the American mathematicianJoseph P. LaSalle (1916–1983) in 1949^[1] and by the Hungarian mathematicianImre 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 ^{t} _{0} f(s)w(u(s))ds, t\in[ 0 , inf ty),

where α is a non-negative constant, then

u(t)\leq G ^{-1} \left(G(\alpha) + \int ^{tf(s)} _{0} ds\right), t\in[ 0 ,T],

where the function G is defined by

G(x) = \int ^{x} _{x _{0}} (dy)/(w(y)) , x \geq 0 ,x _{0>0,}

and G⁻¹ is the inverse function of G and T is chosen so that

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

Notes and References

1. Uniqueness theorems and successive approximations . J. LaSalle . Annals of Mathematics . July 1949 . 50 . 3 . 722–730 . 10.2307/1969559. 1969559 .
2. A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations . I. Bihari . . March 1956 . 7 . 1 . 81–94 . 10.1007/BF02022967 . free . 10338.dmlcz/101943 . free .