Variational-like inequalities for weakly relaxed \(\eta\)-\(\alpha\) pseudomonotone set-valued mappings in Banach space. (English) Zbl 1355.49007
Summary: We introduce and study variational-like inequalities for generalized pseudomonotone set-valued mappings in Banach spaces. By using KKM technique, we obtain the existence of solutions for variational-like inequalities for generalized pseudomonotone set-valued mappings in reflexive Banach spaces. The results presented in this paper are generalizations and improvements of the several well-known results in the literature.
MSC:
| 49J40 | Variational inequalities |
| 49J53 | Set-valued and variational analysis |
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
| 47H04 | Set-valued operators |
| 47H05 | Monotone operators and generalizations |
References:
| [1] | Bai, M.-R.; Zhou, S.-Z.; Ni, G.-Y., Variational-like inequalities with relaxed \(\eta - \alpha\) pseudomonotone mappings in Banach spaces, Applied Mathematics Letters, 19, 6, 547-554 (2006) · Zbl 1144.47047 · doi:10.1016/j.aml.2005.07.010 |
| [2] | Ceng, L. C.; Yao, J. C., On generalized variational-like inequalities with generalized monotone multivalued mappings, Applied Mathematics Letters, 22, 3, 428-434 (2009) · Zbl 1158.49010 · doi:10.1016/j.aml.2008.06.011 |
| [3] | Cottle, R. W.; Yao, J. C., Pseudo-monotone complementarity problems in Hilbert space, Journal of Optimization Theory and Applications, 75, 2, 281-295 (1992) · Zbl 0795.90071 · doi:10.1007/BF00941468 |
| [4] | Fang, Y. P.; Huang, N. J., Variational-like inequalities with generalized monotone mappings in Banach spaces, Journal of Optimization Theory and Applications, 118, 2, 327-338 (2003) · Zbl 1041.49006 · doi:10.1023/A:1025499305742 |
| [5] | Goeleven, D.; Motreanu, D., Eigenvalue and dynamic problems for variational and hemivariational inequalities, Communications on Applied Nonlinear Analysis, 3, 4, 1-21 (1996) · Zbl 0911.49009 |
| [6] | Hadjisavvas, N.; Schaible, S., Quasimonotone variational inequalities in Banach spaces, Journal of Optimization Theory and Applications, 90, 1, 95-111 (1996) · Zbl 0904.49005 · doi:10.1007/BF02192248 |
| [7] | Hartman, P.; Stampacchia, G., On some non-linear elliptic differential-functional equations, Acta Mathematica, 115, 271-310 (1966) · Zbl 0142.38102 · doi:10.1007/BF02392210 |
| [8] | Siddiqi, A. H.; Ansari, Q. H.; Kazmi, K. R., On nonlinear variational inequalities, Indian Journal of Pure and Applied Mathematics, 25, 9, 969-973 (1994) · Zbl 0817.49012 |
| [9] | Verma, R. U., On generalized variational inequalities involving relaxed Lipschitz and relaxed monotone operators, Journal of Mathematical Analysis and Applications, 213, 1, 387-392 (1997) · Zbl 0902.49009 · doi:10.1006/jmaa.1997.5556 |
| [10] | Verma, R. U., Nonlinear variational inequalities on convex subsets of Banach spaces, Applied Mathematics Letters, 10, 4, 25-27 (1997) · Zbl 0905.47055 · doi:10.1016/s0893-9659(97)00054-2 |
| [11] | Verma, R. U., On monotone nonlinear variational inequality problems, Commentationes Mathematicae Universitatis Carolinae, 39, 1, 91-98 (1998) · Zbl 0937.47060 |
| [12] | Lee, B. S.; Lee, B. D., Weakly relaxed \(α\)-semi pseudomonotone setvalued variational-like inequalities, Journal of the Korea Society of Mathematical Education, Series B: Pure and Applied Mathematics, 11, 3, 231-241 (2004) · Zbl 1142.49302 |
| [13] | Kang, M. A.; Huang, N. J.; Lee, B. S., Generalized pseudomonotone setvalued variational-like inequalities, Indian Journal of Mathematics, 45, 3, 251-264 (2003) · Zbl 1083.49007 |
| [14] | Sintunavarat, W., Mixed equilibrium problems with weakly relaxed \(α\)-monotone bifunction in banach spaces, Journal of Function Spaces and Applications, 2013 (2013) · Zbl 1455.47018 · doi:10.1155/2013/374268 |
| [15] | Kutbi, M. A.; Sintunavarat, W., On the solution existence of variational-like inequalities problem for weakly relaxed \(\eta - \alpha\) monotone mapping, Abstract and Applied Analysis, 2013 (2013) · Zbl 1291.49007 · doi:10.1155/2013/207845 |
| [16] | Knaster, B.; Kuratowski, C.; Mazurkiewicz, S., Ein Beweis des Fixpunktsatzes fur n-dimensionale Simplexe, Fundamenta Mathematicae, 14, 1, 132-137 (1929) · JFM 55.0972.01 |
| [17] | Yao, J. C., Existence of generalized variational inequalities, Operations Research Letters, 15, 1, 35-40 (1994) · Zbl 0874.49012 · doi:10.1016/0167-6377(94)90011-6 |
| [18] | Yang, X. Q.; Chen, G. Y., A class of nonconvex functions and pre-variational inequalities, Journal of Mathematical Analysis and Applications, 169, 2, 359-373 (1992) · Zbl 0779.90067 · doi:10.1016/0022-247X(92)90084-Q |
| [19] | Fan, K., A generalization of Tychonoff’s fixed point theorem, Mathematische Annalen, 142, 305-310 (1961) · Zbl 0093.36701 · doi:10.1007/bf01353421 |
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