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The multidirectional mean value inequalities with second order information

Published online by Cambridge University Press:  09 April 2009

Mohammed Bachir
Affiliation:
Laboratoire de Mathématiques, Université Bordeaux I, 351, cours de la Libération, 33405 Talence Cedex, France, e-mail: [email protected]
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Abstract

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We give a multidirectional mean value inequality with second order information. This result extends the classical Clarke-Ledyaev's inequality to the second order. As application, we give the uniqueness of viscosity solution of second order Hamilton-Jacobi equations in finite dimensions.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Clarke, F. H. and Ledyaev, Yu., ‘Mean value inequalities’, Proc. Amer. Math. Soc. 122 (1994), 10751083.Google Scholar
[2]Clarke, F. H. and Ledyaev, Yu., ‘Mean value inequalities in Hilbert space’, Trans. Amer. Math. Soc. 344 (1994), 307324.CrossRefGoogle Scholar
[3]Clarke, F. H. and Radulescu, M. L., ‘The multidirectional mean value theorem in Bancah space’, Canad. Math. Bull. 40 (1997), 88102.Google Scholar
[4]Crandall, M. G., Ishii, H. and Lions, P. L., ‘User's guide to viscosity solutions of second order partial differential equations’, Bull. Amer. Math. Soc. (N. S.) 27 (1992), 167.Google Scholar
[5]Crandall, M. G. and Lions, P. L., ‘Viscosity solutions of Hamilton-Jacobi equations’, Trans. Amer. Math. Soc. 277 (1983), 142.Google Scholar
[6]Deville, R., Godefroy, G. and Zizler, V., ‘A smooth variations principle with applications to Hamilton-Jacobi equations in infinite dimensions’, J. Funct. Anal. 111 (1993), 197212.Google Scholar
[7]Deville, R., Godefroy, G. and Zizler, V., Smoothness and renormings in Banach spaces, Pitman Monographs 64 (Longman, New York, 1993).Google Scholar
[8]Deville, R. and Haddad, El., ‘The subdifferential of the sum of two functions in Banach space, II. Second order case’, Bull. Austral. Math. Soc. 51 (1995), 235248.Google Scholar
[9]Deville, R. and Ivanov, M., ‘Smooth variations principles with constraints’, Arch. Math. 69 (1997), 418426.CrossRefGoogle Scholar
[10]Zhu, Q. J., ‘The Clarke-Ledyaev mean value inequality in smooth Banach spaces’, nonlinear Anal. 32 (1998), 315324.CrossRefGoogle Scholar