Abstract
Variational relations problems include in a general approach various problems like variational inequalities, equilibrium problems, optimization problems and variational or differential inclusions. In this paper, we study the existence of solutions for a general variational problem, using classical fixed point results for generalized contractions, both for set-valued and single-valued mappings. We particularize then the result for a class of multivalued equilibrium problems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Mansour, M.A., Riahi, H.: Sensitivity analysis for abstract equilibrium problems. J. Math. Anal. Appl. 306, 684–691 (2005)
Agarwal, R.P., Balaj, M., O’Regan, D.: Variational relation problems in a general setting. J. Fixed Point Theory Appl. doi:10.1007/s11784-016-0285-0
Anh, L.Q., Khanh, P.Q.: Uniqueness and Hölder continuity of the solution to multivalued equilibrium problems in metric spaces. J. Glob. Optim. 37, 449–465 (2007)
Aussel, D., Cotrina, J.: Quasimonotone quasivariational inequalities: existence results and applications. J. Optim. Theorey Appl. 158, 637–652 (2013)
Balaj, M., Lin, L.J.: Generalized variational relation problems with applications. J. Optim. Theory Appl. 148, 1–13 (2011)
Balaj, M., Luc, D.T.: On mixed variational relation problems. Comput. Math. Appl. 60, 2712–2722 (2010)
Ceng, L.C., Hadjisavvas, N., Schaible, S., Yao, J.C.: Well-posedness for mixed quasivariational-like inequalities. J. Optim. Theory Appl. 139, 109–125 (2008)
Granas, A., Dugundji, J.: Fixed point theory. Springer Monographs in Mathematics, Springer, New York (2003)
Khanh, P.Q., Luc, D.T.: Stability of solutions in parametric variational relation problems. Set Valued Var. Anal. 16(78), 1015–1035 (2008)
Fan, K.: A generalization of Tychonoff’s fixed point theorem. Math. Ann. 142, 305–310 (1961)
Latif, A., Luc, D.T.: Variational relation problems: existence of solutions and fixed points of contraction mappings. Fixed Point Theory Appl. 315, 1–10 (2013) (Article Id)
Lin, L.J., Ansari, Q.H.: Systems of quasi-variational relations with applications. Nonlinear Anal. 72, 1210–1220 (2010)
Luc, D.T.: An abstract problem in variational analysis. J. Optim. Theory Appl. 138, 65–76 (2008)
Matkowski, J.: Integrable solutions of functional equations. Dissertationes Math. 127, 1–63 (1972)
Matkowski, J.: Fixed point theorems for mappings with a contractive iterate at a point. Proc. Am. Mat. Soc. 62, 344–348 (1977)
Reich, S.: Fixed point of contractive functions. Boll. Un. Mat. Ital. 5, 26–42 (1972)
Rus, I.A., Petruşel, A., Petruşel, G.: Fixed point theory. Cluj University Press, Cluj-Napoca (2008)
Wȩgrzyk, R.: Fixed point theorems for multifunctions and their applications to functional equations. Dissertationes Math. 201, 1–28 (1982)
Acknowledgements
The author would like to thank Professor Mircea Balaj and the reviewers for valuable suggestions which helped to improve the paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Inoan, D. Variational relations problems via fixed points of contraction mappings. J. Fixed Point Theory Appl. 19, 1571–1580 (2017). https://doi.org/10.1007/s11784-016-0393-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11784-016-0393-x