## Taiwanese Journal of Mathematics

### NOTE ON F-IMPLICIT GENERALIZED VECTOR VARIATIONAL INEQUALITIES

#### Abstract

In this paper, we deal with weak and strong solutions to $F$-implicit generalized vector variational inequalities and $F$-implicit generalized (weak) vector variational inequalities. Several results of the existence for the weak solutions and strong solutions to both problems are derived.

#### Article information

Source
Taiwanese J. Math., Volume 14, Number 2 (2010), 707-718.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500405815

Digital Object Identifier
doi:10.11650/twjm/1500405815

Mathematical Reviews number (MathSciNet)
MR2655795

Zentralblatt MATH identifier
1250.90086

#### Citation

Lin, Yen-Cherng; Wong, Mu-Ming. NOTE ON F-IMPLICIT GENERALIZED VECTOR VARIATIONAL INEQUALITIES. Taiwanese J. Math. 14 (2010), no. 2, 707--718. doi:10.11650/twjm/1500405815. https://projecteuclid.org/euclid.twjm/1500405815

#### References

• \item[1.] K. Fan, A generalization of Tychonoff's fixed point theorem, Mathematische Annalen, 142 (1961), 305-310.
• \item[2.] F. Ferro, A minimax theorem for vector-valued functions, J. Opti. Theory Appl., 60 (1989), 19-31.
• \item[3.] L. C. Zeng, Y. C. Lin and J. C. Yao, On weak and strong solutions of $F$-implicit generalized variational inequalities with applications, Appl. Math. Let., 19 (2006), 684-689.
• \item[4.] L. J. Lin and Z. T. Yu, On some equilibrium problems for multimaps, J. Comput. Appl. Math., 129 (2001), 171-183.
• \item[5.] Y. C. Lin, On $F$-implicit generalized vector variational inequalities, J. Opti. Theory Appl., (2009), to appear.
• \item[6.] J. Li and N.J. Huang, Vector $F$-implicit complementarity problems in Banach spaces, Appl. Math. Let., 19 (2006), 464-471.
• \item[7.] L. C. Zeng, S. Y. Wu and J. C. Yao, Generalized KKM Theorem with Applications to Generalized Minimax Inequalities and Generalized Equilibrium Problems, Taiwanese J. Math., 10 (2006), 1497-1514.
• \item[8.] L. C. Zeng and J. C. Yao, Existence of Solutions of Generalized Vector Variational Inequalities in Reflexive Banach Spaces, J. Global Opti., 36 (2006), 483-497.
• \item[9.] L. C. Zeng and J. C. Yao, Strong Convergence Theorem by an Extragradient Method for Fixed Point problems and Variational Inequality Problems, Taiwanese J. Math., 10 (2006), 1293-1303.
• \item[10.] S. Schaible, J. C. Yao and L. C. Zeng, A Proximal Method for Pseudomonotone Type Variational-Like Inequalities, Taiwanese J. Math., 10 (2006), 497-513.
• \item[11.] L. C. Zeng, L. J. Lin and J. C. Yao, Auxiliary Problem Method for Mixed Variational-Like Inequalities, Taiwanese J. Math., 10 (2006), 515-529.
• \item[12.] L. C. Ceng, C. Lee and J. C. Yao, Strong Weak Convergence Theorems of Implicit Hybrid Steepest-Descent Methods for Variational Inequalities, Taiwanese J. Math., 12 (2008), 227-244.
• \item[13.] L. C. Ceng and J. C. Yao, Approximate proximal methods in vector optimization, Eur. J. Oper. Res., 183 (2007), 1-19. \item[14.] L. C. Ceng and J. C. Yao, Approximate Proximal Algorithms for Generalized Variational Inequalities with Pseudomonotone Multifunctions, J. Comput. Appl. Math., 213 (2008), 423-438.
• \item[15.] B. T. Kien, N. C. Wong and J. C. Yao, Generalized Vector Variational Inequalities with Star-Pseudomonotone and Discontinuous Operators, Nonlinear Anal. Ser. A: Theory, Methods Appl., 68 (2008), 2859-2871.
• \item[16.] Yakov I. Alber and Jen-Chih Yao, On the projection dynamical systems in Banach spaces, Taiwanese J. Math., 11 (2007), 819-848.
• \item[17.] K. Kimura and J. C. Yao, Sensitivity analysis of solution mappings of parametric generalized quasi vector equilibrium problems, Taiwanese J. Math., 12 (2008), 2233-2268.
• \item[18.] L. C. Ceng, S. Schaible and J. C. Yao, Existence of Solutions for Generalized Vector Variational-like Inequalities, J. Opti. Theory Appl., 137 (2008), 121-133.
• \item[19.] L. C. Ceng, P. Cubiotti and J. C. Yao, An Implicit Iterative Scheme for Monotone Variational Inequalities and Fixed Point Problems, Nonlinear Anal. Ser. A: Theory, Methods Appl., 69 (2008), 2445-2457.
• \item[20.] L. C. Ceng, G. Mastroeni and J. C. Yao, Existence of Solutions and Variational Principles for Generalized Vector Systems, J. Opti. Theory and Appl., 137 (2008), 485-496.
• \item[21.] L. C. Ceng and J. C. Yao, Relaxed Viscosity Approximation Methods for Fixed Point Problems and Variational Inequality Problems, Nonlinear Anal. Ser. A: Theory, Methods Appl., 69 (2008), 3299-3309.
• \item[21.] L. C. Ceng and J. C. Yao, Mixed Projection Methods for Systems of Variational Inequalities, J. Global Opti., 41 (2008), 465-478.
• \item[22.] L. C. Ceng and J. C. Yao, Well-posedness of generalized mixed variational inequalities, inclusion problems and fixed point problems, Nonlinear Anal. Ser. A: Theory, Methods Appl., 69 (2008), 4585-4603.
• \item[23.] B. T. Kien, M. M. Wong, N. C. Wong and J. C. Yao, Degree theory for generalized variational inequalities and applications, Eur. J. Oper. Res., 192 (2009), 730-736.
• \item[24.] L. C. Ceng, G. Y. Chen, X. X. Huang and J. C. Yao, Existence Theorems for Generalized Vector Variational Inequalities with Pseudomonotonicity and Their Applications, Taiwanese J. Math., 12 (2008), 151-172. \item[25.] L. C. Zeng and J. C. Yao, A Hybrid Extragradient Method for General Variational Inequalities, Math. Methods Oper. Res., 69 (2009), 141-158.
• \item[26.] B. T. Kien, M. M. Wong, N. C. Wong and J. C. Yao, Solution existence of variational inequalities with pseudomonotone operators in the sense of Brézis, J. Opti. Theory Appl., 140 (2009), 249-263. \item[27.] L. C. Ceng, Q. H. Ansari and J. C. Yao, Mann type steepest-descent and modified hybrid steepest-descent methods for variational inequalities in Banach spaces, Numerical Funct. Anal. Opti., 29(9-10) (2008), 987-1033. \item[28.] J. W. Peng and J. C. Yao, A new hybrid-extragradient method for generalized mixed equilibrium problems and fixed point problems and variational inequality problems, Taiwanese J. Math., 12 (2008), 1401-1433.
• \item[29.] J. W. Peng and J. C. Yao, Some new iterative algorithms for generalized mixed equilibrium problems with strict pseudo-contractions and monotone mappings, Taiwanese J. Math, (2009), to appear.