Relations Between Two Sets of Functions Defined by the Two Interrelated One-Side Lipschitz Conditions

Abstract

In the theoretical study of numerical solution of stiff ODEs, it usually assumes that the right-hand function $f(y)$ satisfy one-side Lipschitz condition $$ ≤ v' ||y-z||^2,f: \Omega \subseteq C^m → C^m,$$ or another related one-side Lipschitz condition $$[F(Y)-F(Z),Y-Z]_D ≤ v'' ||Y-Z||^2_D, F:\Omega^s \subseteq C^{ms} → C^{ms},$$ this paper demonstrates that the difference of the two sets of all functions satisfying the above two conditions respectively is at most that $v'-v''$ only is constant independent of stiffness of function $f$.