Remark on globally Lipschitzian composition operators

Abstract

Let I C R be an interval, f : I x R —> R a fixed two-place function, and J’(Z) the linear space of all the functions u : I —> R. The function F : F(I) —> F{I} given by the formula (F(u))(x) := /(x,u(x)), x G I, u G F(Z), is said to be a composition operator. Let a G I be fixed. Denote by Lip(I) the Banach space of all the functions « E 7(f) with the norm (1) IhllLip(l) := lu(°)l + sup | I Xi — X2 xi,x2 e I-, In [2] it is proved that if a composition operator F mapping Lip(I) into itself is globally Lipschitzian with respect to the Lip(I)-norm, then /(x, y) = g(x)y + h(x), (x G I;y 6 R), for some g,h GLip(I) (Fragment tekstu).