The Bihari–LaSalle inequality was proved by the American mathematician Joseph P. LaSalle (1916–1983) in 1949 and by the Hungarian mathematician Imre Bihari (1915–1998) in 1956. It is the following nonlinear generalization of Grönwall's lemma. Let ''u'' and ''ƒ'' be non-negative

continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...

s defined on the half-infinite ray , ∞), 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 (mathematics), 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 (mathematics), constant, then :

u(t)\leq G^\left(G(\alpha)+\int_0^t\,f(s) \, ds\right),\qquad t\in,T

where the function ''G'' is defined by :

G(x)=\int_^x \frac,\qquad x \geq 0,\,x_0>0,

and ''G''⁻¹ is the

inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...

of ''G'' and ''T'' is chosen so that :

G(\alpha)+\int_0^t\,f(s)\,ds\in \operatorname(G^),\qquad \forall \, t \in,T

References

{{DEFAULTSORT:Bihari-LaSalle inequality Differential equations Inequalities (mathematics)