Let f : C [right arrow] [Y.sup.*] be a Lipschitz mapping. Then f is [[omega].sup.*]-Gateaux differentiable off a Aronszajn null set of N(C).

Let f : C [right arrow] [Y.sup.*] be a Lipschitz mapping. Then f is said to be [[omega].sup.*]-Gateaux differentiable at x [member of] N(C) if there exists a bounded linear operator [T.sub.x] : C [right arrow] [Y.sup.*] such that for every h [member of] [C.sub.x] and every y [member of] Y the following limit exists:

It is a well-known fact that every Lipschitz mapping L : A [right arrow] B between pointed metric spaces A, B, such that L(0) = 0 extends uniquely to a linear mapping

Therefore we get by computation similar to those done in (12) and (13) that [[phi].sub.n] is a Lipschitz mapping with constant K = uc(E) + 2bc(E), which concludes the induction.

If F is a real-valued locally

Lipschitz mapping on an open set U of a Banach space X, and x [member of] U, then

in the case of [alpha] = 1 and f(x) is a

Lipschitz mapping. If F is a Cantor set, we have [H.sup.ln 2/ln 3](F [intersection] (x, [x.sub.0])) = [(x - [x.sub.0]).sup.ln2]/[ln3] wither [alpha] = ln 2/ln 3.

Suppose that [V.sub.0] is a

Lipschitz mapping from [S.sub.1](E) into [S.sub.1](F) with the constant K = 1, that is, ||[V.sub.0](x) - [V.sub.0](y)|| [less than or equal to] ||x - y|| for any x, y [member of] [S.sub.1](E).

That is, for each y [member of] [partial derivative][W.sub.r,j] we can find a hyperplane [P.sub.r,j,y] through y and a

Lipschitz mapping [F.sub.r,j,y] : [P.sub.r,j,y] - [P.sup.[perpendicular to].sub.r,j,y] such that

In this article we prove a new uniform boundedness principle for monotone, positively homogeneous, subadditive, and

Lipschitz mappings defined on a suitable cone of functions (Theorem 2).

Xu, "Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings," Fixed Point Theory and Applications, vol.

Radenovic, "Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality," Fixed Point Theory and Applications, vol.

Radenovie, "Common fixed point theorems of generalized

Lipschitz mappings in cone b-metric spaces over Banach algebras and applications," Journal of Nonlinear Sciences and Applications.