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,

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[edit]

  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.