EXISTENCE RESULTS FOR HEMIVARIATIONAL INEQUALITIES INVOLVING RELAXED η − α MONOTONE MAPPINGS

TitleEXISTENCE RESULTS FOR HEMIVARIATIONAL INEQUALITIES INVOLVING RELAXED η − α MONOTONE MAPPINGS
Publication TypeJournal Article
Year of Publication2009
AuthorsCOSTEA, NICUSOR, RADULESCU, VICENTIU
Volume13
Issue3
Start Page293
Pagination11
Date Published2009
ISSN1083-2564
AMS47H04, 47H05, 47J20, 58E35
Abstract

We establish some existence results for hemivariational inequalities with relaxed η − α monotone mappings on bounded, closed and convex subsets in reflexive Banach spaces. Our proofs rely essentially on a fixed point theorem for set valued mappings which is due to Tarafdar [20]. We also give a sufficient condition for the existence of solutions in the case of unbounded subsets.

URLhttp://www.acadsol.eu/en/articles/13/3/2.pdf
Refereed DesignationRefereed
Full Text

REFERENCES
[1] Y. Q. Chen, On the semimonotone operator theory and applications, J. Math. Anal. Appl. 231 (1999), 177–192.
[2] F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, 1983.
[3] Y. P. Fang and N. J. Huang, Variational-like inequalities with generalized monotone mappings
in Banach Spaces, J. Optim. Theory Appl. 118 (2003), 327–338.
[4] G. Fichera, Problemi electrostatici con vincoli unilaterali: il problema de Signorini con ambigue
condizioni al contorno, Mem. Acad. Naz. Lincei 7 (1964), 91–140.
[5] G. J. Hartman and G. Stampacchia, On some nonlinear elliptic differential equations, Acta Math. 112 (1966), 271–310.
[6] S. Karamardian and S. Schaible, Seven kinds of monotone maps, J. Optim. Theory Appl. 66 (1990), 37–46.
[7] S. Karamardian, S. Schaible and J. P. Crouzeix, Characterizations of generalized monotone
maps, J. Optim. Theory Appl. 76 (1993), 399–413.
[8] J. L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math. 20 (1967), 493–519.
[9] D. Motreanu and P. D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of
the Solutions of Hemivariational Inequalities and Applications, Kluwer Academic Publishers,
Nonconvex Optimization and its Applications, vol. 29, Boston/Dordrecht/London, 1999.
[10] D. Motreanu and V. R˘adulescu, Existence results for inequality problems with lack of convexity,
Numer. Funct. Anal. Optimiz. 21 (2000), 869–884.
[11] D. Motreanu and V. R˘adulescu, Variational and Non-variational Methods in Nonlinear Analysis
and Boundary Value Problems, Kluwer Academic Publishers, Boston/Dordrecht/London, 2003.
[12] Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities
and Applications, Marcel Dekker, New York, 1995.
[13] P. D. Panagiotopoulos, Hemivariational Inequalities: Applications to Mechanics and Engineering,
Springer-Verlag, New York/Boston/Berlin, 1993.
[14] P. D. Panagiotopoulos, Noconvex energy functions. Hemivariational inequalities and substationarity
principles, Acta Mechanica 42 (1983), 160–183.
[15] P. D. Panagiotopoulos, Hemivariational inequalities and their applications, in: Topics in Nonsmooth
Mechanics, (Ed:) J. J. Moreau, P. D. Panagiotopoulos and G. Strang, Birkh¨auserVerlag, Basel, (1988).
[16] P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex
Energy Functionals, Birkh¨auser-Verlag, Basel/Boston, 1985.
[17] P. D. Panagiotopoulos, M. Fundo and V. R˘adulescu, Existence theorems of HartmanStampacchia
type for hemivariational inequalities and applications, J. Global Optim. 15 (1999), 41–54.
[18] V. R˘adulescu, Analyse de Quelques Probl`emes aux Limites Elliptiques Non Lin´eaires, Habilitation
`a Diriger des Recherches, Universit´e Pierre et Marie Curie (Paris VI), 2003.
[19] V. R˘adulescu, Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations: Monotonicity,
Analytic, and Variational Methods, Contemporary Mathematics and Its Applications,
vol. 6, Hindawi Publ. Corp., 2008.
[20] E. Tarafdar, A fixed point theorem equivalent to the Fan-Knaster-Kuratowski-Mazurkiewicz
theorem, J. Math. Anal. Appl. 128 (1987), 352–363.
[21] R. U. Verma, On generalized variational inequalities involving relaxed Lipschitz and relaxed
monotone operators, J. Math. Anal. Appl. 213 (1997), 387–392.