Nonlinear version of Holub's theorem and its application

Holub proved that any bounded linear operator T or -T defined on Banach space L¹(μ) satisfies Daugavet equation 1 + ∥ T ∥ = Max{∥ I + T ∥, ∥ I - T ∥}. Holub's theorem is generalized to the nonlinear case: any nonlinear Lipschitz operator f defined on Banach space l¹ satisfies 1 + L(f) = Max{L(I + f), L(I - f)}, where L(f) is the Lipschitz constant of f. The generalized Holub theorem has important applications in characterizing the invertibility of nonlinear operator.