New Multirectional Mean Value Inequality

Mihail Hamamdjiev; Milen Ivanov

doi:10.68381/jca25078

Abstract

We establish new inequality of Clarke-Ledyaev type by direct construction.

Author information

Contact details are reproduced from the original publication and may be historical.

Mihail Hamamdjiev

Faculty of Mathematics and Informatics, Sofia University, 5 Av. James Baucher, 1164 Sofia, Bulgaria

m_hamamdjiev@abv.bg

Milen Ivanov

Faculty of Mathematics and Informatics, Sofia University, 5 Av. James Baucher, 1164 Sofia, Bulgaria

milen@fmi.uni-sofia.bg

Keywords

Multidirectional mean value theorem
subdifferential

Mathematics Subject Classification

49K27, 49J53, 49J52

References

D. Aussel, J. N. Corvellec, M. Lassonde: Nonsmooth constrained optimization and multidirectional mean value inequalities, SIAM J. Optim. 9(3) (1999) 690–706.https://doi.org/10.1137/S105262349732339X Google Scholar
F. H. Clarke, Yu. S. Ledyaev: Mean value inequalities, Proc. Amer. Math. Soc. 122(4) (1994) 1075–1083.https://doi.org/10.1090/S0002-9939-1994-1212282-4 Google Scholar
F. H. Clarke, Yu. S. Ledyaev: Mean value inequalities in Hilbert spaces, Trans. Amer. Math. Soc. 344(1) (1994) 307–324.https://doi.org/10.1090/S0002-9947-1994-1227093-8 Google Scholar
M. Ivanov, N. Zlateva: On nonconvex version of the inequality of Clarke and Ledyaev, Nonlinear Anal.: Theory, Methods Appl. 49(8) (2002) 1023–1036.https://doi.org/10.1016/S0362-546X(01)00145-6 Google Scholar
Yu. S. Ledyaev, Q. Zhu: Multidirectional mean value inequalities and weak monotonicity, J. London Math. Soc. 71(1) (2005) 187–202.https://doi.org/10.1112/S0024610704005964 Google Scholar
D. T. Luc: A strong mean value theorem and applications, Nonlinear Anal.: Theory, Methods Appl. 26(5) (1996) 915–923.https://doi.org/10.1016/0362-546X(94)00336-6 Google Scholar
J.-P. Penot: Calculus Without Derivatives, Grad. Texts Math. 266, Springer, Berlin (2012).https://doi.org/10.1007/978-1-4614-4538-8 Google Scholar
L. Thibault, N. Zlateva: Integrability of subdifferentials of directionally Lipschitz functions, Proc. Amer. Math. Soc. 133(10) (2005) 2939–2948.https://doi.org/10.1090/S0002-9939-05-07883-4 Google Scholar
Q. Zhu: Clarke-Ledyaev mean value inequalities in smooth Banach spaces, Nonlinear Anal.: Theory, Methods Appl. 32(3) (1998) 315–324.https://doi.org/10.1016/S0362-546X(97)00488-4 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2018. Rights now held by Banach Press.

[1] D. Aussel, J. N. Corvellec, M. Lassonde: Nonsmooth constrained optimization and multidirectional mean value inequalities, SIAM J. Optim. 9(3) (1999) 690–706.https://doi.org/10.1137/S105262349732339X Google Scholar

[2] F. H. Clarke, Yu. S. Ledyaev: Mean value inequalities, Proc. Amer. Math. Soc. 122(4) (1994) 1075–1083.https://doi.org/10.1090/S0002-9939-1994-1212282-4 Google Scholar

[3] F. H. Clarke, Yu. S. Ledyaev: Mean value inequalities in Hilbert spaces, Trans. Amer. Math. Soc. 344(1) (1994) 307–324.https://doi.org/10.1090/S0002-9947-1994-1227093-8 Google Scholar

[4] M. Ivanov, N. Zlateva: On nonconvex version of the inequality of Clarke and Ledyaev, Nonlinear Anal.: Theory, Methods Appl. 49(8) (2002) 1023–1036.https://doi.org/10.1016/S0362-546X(01)00145-6 Google Scholar

[5] Yu. S. Ledyaev, Q. Zhu: Multidirectional mean value inequalities and weak monotonicity, J. London Math. Soc. 71(1) (2005) 187–202.https://doi.org/10.1112/S0024610704005964 Google Scholar

[6] D. T. Luc: A strong mean value theorem and applications, Nonlinear Anal.: Theory, Methods Appl. 26(5) (1996) 915–923.https://doi.org/10.1016/0362-546X(94)00336-6 Google Scholar

[7] J.-P. Penot: Calculus Without Derivatives, Grad. Texts Math. 266, Springer, Berlin (2012).https://doi.org/10.1007/978-1-4614-4538-8 Google Scholar

[8] L. Thibault, N. Zlateva: Integrability of subdifferentials of directionally Lipschitz functions, Proc. Amer. Math. Soc. 133(10) (2005) 2939–2948.https://doi.org/10.1090/S0002-9939-05-07883-4 Google Scholar

[9] Q. Zhu: Clarke-Ledyaev mean value inequalities in smooth Banach spaces, Nonlinear Anal.: Theory, Methods Appl. 32(3) (1998) 315–324.https://doi.org/10.1016/S0362-546X(97)00488-4 Google Scholar