All Title Author
Keywords Abstract

Variational-Like Inequalities and Equilibrium Problems with Generalized Monotonicity in Banach Spaces

DOI: 10.1155/2012/648070

Full-Text   Cite this paper   Add to My Lib


We introduce the notion of relaxed (ρ-θ)-η-invariant pseudomonotone mappings, which is weaker than invariant pseudomonotone maps. Using the KKM technique, we establish the existence of solutions for variational-like inequality problems with relaxed (ρ-θ)-η-invariant pseudomonotone mappings in reflexive Banach spaces. We also introduce the concept of (ρ-θ)-pseudomonotonicity for bifunctions, and we consider some examples to show that (ρ-θ)-pseudomonotonicity generalizes both monotonicity and strong pseudomonotonicity. The existence of solution for equilibrium problem with (ρ-θ)-pseudomonotone mappings in reflexive Banach spaces are demonstrated by using the KKM technique. 1. Introduction Let be a nonempty subset of a real reflexive Banach space , and let be the dual space of . Consider the operator and the bifunction . Then the variational-like inequality problem (in short, VLIP) is to find , such that where denote the pairing between and . If we take , then (1.1) becomes to find , such that which is classical variational inequality problems (VIPs). These problems have been studied in both finite and infinite dimensional spaces by many authors [1–3]. VIP has numerous applications in optimization, nonlinear analysis, and engineering sciences. In the study of VLIP and VIP, monotonicity is the most common assumption for the operator . Recently many authors established the existence of solutions for (VIP) and VLIP under generalized monotonicity assumptions, such as quasimonotonicity, relaxed monotonicity, densely pseudomonotonicity, relaxed - -monotonicity, and relaxed - -pseudomonotonicity (see [1, 3–6] and the references therein). In 2008 [7], Behera et al. defined various concepts of generalized ( - )- -invariant monotonicities which are proper generalization of generalized invariant monotonicity introduced by Yang et al. [8]. Chen [9] defined semimonotonicity and studied semimonotone scalar variational inequalities problems in Banach spaces. Fang and Huang [3] obtained the existence of solution for VLIP using relaxed - -monotone mappings in the reflexive Banach spaces. In [1], Bai et al. extended the results of [3] with relaxed - -pseudomonotone mappings and provided the existence of solution of the variational-like inequalities problems in reflexive Banach spaces. Bai et al. [10] studied variational inequalities problems with the setting of densely relaxed -pseudomonotone operators and relaxed -quasimonotone operators, respectively. Inspired and motivated by [1, 3, 10], we introduce the concept of relaxed ( )- -invariant pseudomonotone mappings. Using


[1]  M.-R. Bai, S.-Z. Zhou, and G.-Y. Ni, “Variational-like inequalities with relaxed η-α pseudomonotone mappings in Banach spaces,” Applied Mathematics Letters, vol. 19, no. 6, pp. 547–554, 2006.
[2]  E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994.
[3]  Y. P. Fang and N. J. Huang, “Variational-like inequalities with generalized monotone mappings in Banach spaces,” Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 327–338, 2003.
[4]  B.-S. Lee and G.-M. Lee, “Variational inequalities for (η, θ)-pseudomonotone operators in nonreflexive Banach spaces,” Applied Mathematics Letters, vol. 12, no. 5, pp. 13–17, 1999.
[5]  D. T. Luc, “Existence results for densely pseudomonotone variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 254, no. 1, pp. 291–308, 2001.
[6]  N. Hadjisavvas and S. Schaible, “Quasimonotone variational inequalities in Banach spaces,” Journal of Optimization Theory and Applications, vol. 90, no. 1, pp. 95–111, 1996.
[7]  N. Behera, C. Nahak, and S. Nanda, “Generalized (ρ-θ)-η-invexity and generalized (ρ-θ)-η-invariant-monotonicity,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 8, pp. 2495–2506, 2008.
[8]  X. M. Yang, X. Q. Yang, and K. L. Teo, “Generalized invexity and generalized invariant monotonicity,” Journal of Optimization Theory and Applications, vol. 117, no. 3, pp. 607–625, 2003.
[9]  Y.-Q. Chen, “On the semi-monotone operator theory and applications,” Journal of Mathematical Analysis and Applications, vol. 231, no. 1, pp. 177–192, 1999.
[10]  M.-R. Bai, S.-Z. Zhou, and G.-Y. Ni, “On the generalized monotonicity of variational inequalities,” Computers & Mathematics with Applications, vol. 53, no. 6, pp. 910–917, 2007.
[11]  K. Fan, “Some properties of convex sets related to fixed point theorems,” Mathematische Annalen, vol. 266, no. 4, pp. 519–537, 1984.
[12]  H. Brezis, Analyse Fonctionnelle: Théorie et Applications, Collection Mathématiques Appliquées pour la Ma?trise, Masson, Paris, France, 1983.


comments powered by Disqus

Contact Us


微信:OALib Journal