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Lipschitz stability of nonlinear systems of differential equations. (English) Zbl 0595.34054
The authors define a new notion of stability - uniform Lipschitz stability (ULS) for nonlinear systems of differential equations (1) x ˙=f(t,x), where fC[J×R n ,R n ] and f x =f/x exists and is continuous on J×R n , J=[t 0 ,), t 0 0, f(t,0)=0, and x(t,t 0 ,x 0 ) is the solution of (1) with x(t 0 ,t 0 ,x 0 )=x 0 . Definition. The zero solution of (1) is said to be (ULS) if there exists M>0 and δ>0 such that |x(t,t 0 ,x 0 )|M|x 0 |, whenever |x 0 |δ and tt 0 0. Several criteria for the (ULS) are obtained. It is shown by examples that the (ULS) coincides with the uniform stability in the linear case, which means that the (ULS) is a nonlinear phenomenon. The relationship between (ULS) and various types of stability notions is illustrated by a diagram.

34D10Stability perturbations of ODE