Azagra Rueda, DanielFerrera Cuesta, JuanLópez-Mesas Colomina, Fernando2023-06-202023-06-202003-07-01[1] D. Azagra, Smooth negligibility and subdifferential ca1culus in Banach spaces, with applications, Ph.D. dissertation, Universidad Complutense de Madrid, October 1997. [2] D. Azagra, R. Deville, Subdifferential Rolle's and mean value inequality theorems, Bull. Austral. Math. Soc. 56 (1997) 319-329. [3] D. Azagra, J. Gómez, J.A. Jaramillo, Rolle's theorem and negligibility of points in infinite-dimensional Banach spaces, J. Math. Anal. Appl. 213 (1997) 487--495. [4] D. Azagra, M. Jiménez-Sevilla, The failure of Rolle's theorem in infinite-dimensiOl1al Banach spaces, J. Funct. Anal. 182 (2001) 207-226. [5] J. Bes, J. Ferrera, On a multidimensional version of Rolle's theorem, Publ. Dto. Análisis Mat. UCM 1 (1996/1997) 21-27. [6] F.H. Clark, Yu.S. Ledyaev, R.J. Stem, P.R. Wolensk.i, Nonsmooth Analysis and Control Theory, in: Graduate Texts in Mathematics, Vol. 178, Springer, 1998. [7] R. Deville, A mean value theorem for nondifferentiable mappings in Banach spaces, Serdica Math. J. 21 (1995) 59-66. [8] R. Deville, G. Godefroy, V. Zizler, Smoothness and Renorrnings in Banam Spaces, in: Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 64, 1993. [9] 1. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. (N.S.) 1 (1979) 443-474. [10] 1. Ferrer, Rolle's, theorem fail in [2, Amer. Math. Monthly 103 (1996) 161-165. [11] G. Godefroy, Sorne remarks on subdifferential calculus, Rev. Mat. Complut. 11 (1998) 269-279. [12] S.A. Shkarin, On Rolle's theorem in infinite-dimensional Banach espaces, transl. from Mat. Zametki 51 (1992) 128-136.0022-247X10.1016/S0022-247X(03)00267-1https://hdl.handle.net/20.500.14352/49793We establish approximate Rolle's theorems for the proximal subgradient and for the generalized gradient. We also show that an exact Rolle's theorem for the generalized gradient is completely false in all infinite-dimensional Banach spaces (even when they do not possess smooth bump functions).engjournal articlehttp://www.sciencedirect.com/science/journal/0022247Xopen access517.98Rolle theoremProximinal subspaceGeneralized gradientAnálisis matemático1202 Análisis y Análisis Funcional