Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Mean value inequalities in Hilbert space

Authors: F. H. Clarke and Yu. S. Ledyaev
Journal: Trans. Amer. Math. Soc. 344 (1994), 307-324
MSC: Primary 49J52; Secondary 26A24, 47H99, 47N10, 49L25
MathSciNet review: 1227093
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We establish a new mean value theorem applicable to lower semi-continuous functions on Hilbert space. A novel feature of the result is its "multidirectionality": it compares the value of a function at a point to its values on a set. We then discuss some refinements and consequences of the theorem, including applications to calculus, flow invariance, and generalized solutions to partial differential equations.

Résumé. On établit un nouveau théorème de la valeur moyenne qui s'applique aux fonctions semicontinues inférieurement sur un espace de Hilbert. On déduit plusieurs conséquences du résultat portant, par exemple, sur les fonctions monotones et sur les solutions généralisées des équations aux dérivées partielles.

References [Enhancements On Off] (What's this?)

  • [1] J. M. Borwein and D. Preiss, A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. Amer. Math. Soc. 303 (1987), 517-527. MR 902782 (88k:49013)
  • [2] F. H. Clarke, Methods of dynamic and nonsmooth optimization, CBMS-NSF Regional Conf. Ser. in Appl. Math., vol. 57, SIAM, Philadelphia, Pa., 1989. MR 1085948 (91j:49001)
  • [3] F. H. Clarke and Yu. S. Ledyaev, Mean value inequalities, Proc. Amer. Math. Soc. (to appear). MR 1212282 (95b:26030)
  • [4] F. H. Clarke, R. J. Stern and P. R. Wolenski, Subgradient criteria for monotonicity, the Lipschitz condition, and convexity, Canad. J. Math. 45 (1993), 1167-1183. MR 1247540 (94j:49018)
  • [5] M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27 (1992), 1-67. MR 1118699 (92j:35050)
  • [6] H. G. Guseinov, A. I. Subbotin and V. N. Ushakov, Derivatives for multivalued mappings with applications to game-theoretical problems of control, Problems Control Inform. Theory 14 (1985), 155-167. MR 806060 (86k:90159)
  • [7] P. D. Loewen, Optimal control via nonsmooth analysis, CRM Proc. & Lecture Notes, vol. 2, Amer. Math. Soc., Providence, R.I., 1993. MR 1232864 (94h:49003)
  • [8] A. I. Subbotin, A generalization of the basic equation of the theory of differential games, Soviet Math. Dokl. 22 (1980), 358-362.
  • [9] -, On a property of a subdifferential, Mat. Sb. 1982 (1991), 1315-1330. (Russian) MR 1133572 (92i:49033)
  • [10] -, Continuous and discontinuous solutions of boundary value problems for first-order partial differential equations, Dokl. Akad. Nauk SSSR 323 (1992), no. 2. (Russian)
  • [11] -, Viable characteristics of Hamilton-Jacobi equations, preprint.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 49J52, 26A24, 47H99, 47N10, 49L25

Retrieve articles in all journals with MSC: 49J52, 26A24, 47H99, 47N10, 49L25

Additional Information

Keywords: Mean value theorem, nonsmooth analysis, flow invariance, monotone functions, generalized solutions of partial differential equations
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society