Nonlinear version of Holub’s theorem and its application

Holub proved that any bounded linear operator T or -T defined on Banach space L 1(μ) satisfies Daugavet equation1+‖T‖=Max{‖I+T‖, ‖I-T‖}.Holub's theorem is generalized to the nonlinear case: any nonlinear Lipschitz operator f defined on Banach space l 1 satisfies1+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.