Abstract
In this paper, we introduce some concepts of convexity and semicontinuity for real set-valued mappings similar to those of real single-valued mappings. Then, we obtain different results on the existence of solutions of set-valued equilibrium problems generalizing in a common way several old ones for both single-valued and set-valued equilibrium problems. Applications to Browder variational inclusions, with weakened conditions on the involved set-valued operator, are given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alleche, B.: Multivalued mixed variational inequalities with locally Lipschitzian and locally cocoercive multivalued mappings. J. Math. Anal. Appl. 399, 625–637 (2013)
Alleche, B., Rădulescu, V.: Set-valued equilibrium problems with applications to Browder variational inclusions and to fixed point theory. Nonlinear Anal. Real World Appl. 28, 251–268 (2016)
Kristály, A., Varga, C.: Set-valued versions of Ky Fan’s inequality with application to variational inclusion theory. J. Math. Anal. Appl. 282, 8–20 (2003)
Lásló, S., Viorel, A.: Densely defined equilibrium problems. J. Optim. Theory Appl. 166, 52–75 (2015)
Shih, M.H., Tan, K.K.: Browder–Hartman–Stampacchia variational inequalities for multi-valued monotone operators. J. Math. Anal. Appl. 134, 431–440 (1988)
Lásló, S., Viorel, A.: Generalized monotone operators on dense sets. Numer. Funct. Anal. Optim. 36(7), 901–929 (2015)
Cambini, A., Martein, L.: Generalized Convexity and Optimization. Theory and Applications. Lecture Notes in Economics and Mathematical Systems, vol. 616. Springer, Berlin (2009)
Papageorgiou, N.S., Kyritsi-Yiallourou, S.T.H.: Handbook of Applied Analysis. Advances in Mechanics and Mathematics, vol. 19. Springer, Dordrecht (2009)
Alleche, B.: Semicontinuity of bifunctions and applications to regularization methods for equilibrium problems. Afr. Mat. 26(7), 1637–1649 (2015)
Alleche, B., Rădulescu, V.: Equilibrium problem techniques in the qualitative analysis of quasi-hemivariational inequalities. Optimization 64(9), 1855–1868 (2015)
Alleche, B., Rădulescu, V.: The Ekeland variational principle for equilibrium problems revisited and applications. Nonlinear Anal. Real World Appl. 23, 17–25 (2015)
Alleche, B., Rădulescu, V.: Solutions and approximate solutions of quasi-equilibrium problems in Banach spaces. J. Optim. Theory Appl. 170, 629–649 (2016)
Fan, K.: A generalization of Tychonoff’s fixed point theorem. Math. Ann. 142, 305–310 (1961)
Browder, F.E.: The fixed point theory of multi-valued mappings in topological vector spaces. Math. Ann. 177, 283–301 (1968)
Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Theory. Kluwer Academic, New York (2003)
Lee, C., Tan, K.K.: On Fan’s extensions of Browder’s fixed point theorems for multi-valued inward mappings. J. Aust. Math. Soc. 26(2), 169–174 (1978)
Acknowledgements
V.D. Rădulescu acknowledges the support through a grant of the Ministry of Research and Innovation, CNCS–UEFISCDI, project number PN-III-P4-ID-PCE-2016-0130, within PNCDI III.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Alleche, B., Rădulescu, V.D. Further on Set-Valued Equilibrium Problems and Applications to Browder Variational Inclusions. J Optim Theory Appl 175, 39–58 (2017). https://doi.org/10.1007/s10957-017-1169-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-017-1169-1