An Existence Theorem of Solutions for the System of Generalized Vector Quasi-Variational-Like Inequalities

Affiliation(s)

Department of Applied Mathematics, Faculty of Engineering & Technology, Aligarh Muslim University, Aligarh, India.

Department of Applied Mathematics & Humanities, Sardar Vallabhbhai National Institute of Technology, Surat, India.

Department of Applied Mathematics, Faculty of Engineering & Technology, Aligarh Muslim University, Aligarh, India.

Department of Applied Mathematics & Humanities, Sardar Vallabhbhai National Institute of Technology, Surat, India.

ABSTRACT

In this paper, we introduce and study the system of generalized
vector quasi-variational-like inequalities in Hausdorff topological vector spaces,
which include the system of vector quasi-variational-like inequalities, the
system of vector variational-like inequalities, the system of vector
quasi-variational inequalities, and several other systems as special cases.
Moreover, a number of C-diagonal quasiconvexity properties are proposed for
set-valued maps, which are natural generalizations of the g-diagonal
quasiconvexity for real functions. Together with an application of continuous
selection and fixed-point theorems, these conditions enable us to prove unified
existence results of solutions for the system of generalized vector
quasi-variational-like inequalities. The results of this paper can be seen as
extensions and generalizations of several known results in the
literature.

Cite this paper

S. Husain, S. Gupta and V. Mishra, "An Existence Theorem of Solutions for the System of Generalized Vector Quasi-Variational-Like Inequalities,"*American Journal of Operations Research*, Vol. 3 No. 3, 2013, pp. 329-336. doi: 10.4236/ajor.2013.33029.

S. Husain, S. Gupta and V. Mishra, "An Existence Theorem of Solutions for the System of Generalized Vector Quasi-Variational-Like Inequalities,"

References

[1] J.-W. Peng, “System of Generalized Set-Valued QuasiVariational-Like Inequalities,” Bulletin of the Australian Mathematical Society, Vol. 68, No. 3, 2003, pp. 501-515. doi:10.1017/S0004972700037904

[2] Q. H. Ansari, S. Schaible and J. C. Yao, “The System of Generalized Vector Equilibrium Problems with Applications,” Journal of Global Optimization, Vol. 22, No. 1, 2002, pp. 3-16. doi:10.1023/A:1013857924393

[3] H. H. Schaefer, “Topological Vector Space,” 2nd Edition, Graduate Texts in Mathematics, Springer, Berlin, 1999.

[4] X. P. Ding and E. Tarafdar, “Generalized VariationalLike Inequalities with Pseudomonotone Set-Valued Mappings,” Archive der Mathematik, Vol. 74, No. 4, 2000, pp. 302-313. doi:10.1007/s000130050447

[5] Q. H. Ansari and J. C. Yao, “System of Generalized Varitional Ineqalities and Their Applications,” Applicable Analysis, Vol. 76, No. 3-4, 2000, pp. 203-217. doi:10.1080/00036810008840877

[6] G. Xiao and S. Liu, “Existence of Solutions for Generalized Vector Quasi-Variational-Like Inequalities without Monotonicity,” Computers and Mathematics with Applications, Vol. 58, No. 8, 2009, pp. 1550-1557. doi:10.1016/j.camwa.2009.05.021

[7] G. Xiao, Z. Fan and X. Qi, “Existence Results for Generalized Nonlinear Vector Variational-Like Inequalities with Set-Valued Mapping,” Applied Mathematics Letter, Vol. 23, No. 1, 2010, pp. 44-47. doi:10.1016/j.aml.2009.07.023

[8] J.-W. Peng and X.-M. Yang, “Generalized Vector QuasiVariational-Like Inequalities,” Journal of Inequalities and Applications, Vol. 2006, No. 1, 2006, pp. 1-11.

[9] S. Husain and S. Gupta, “Existence of Solutions for Generalized Nonlinear Vector Quasi-Variational-Like Inequalities with Set-Valued Mappings,” Filomat, Vol. 26, No. 5, 2012, pp. 909-916. doi:10.2298/FIL1205909H

[10] S. Husain and S. Gupta, “Generalized Nonlinear Vector Quasi-Variational-Like Inequality,” International Journal of Pure and Applied Mathematics, Vol. 66, No. 2, 2011, pp. 157-169.

[11] P. Hartman and G. Stampacchia, “On Some Nonlinear Elliptic Differential Function Equations,” Acta Mathematica, Vol. 115, No. 1, 1966, pp. 271-310. doi:10.1007/BF02392210

[12] S. H. Hou, H. Yu and G. Y. Chen, “On Vector QuasiEquilibrium Problems with Set-Valued Maps,” Journal of Optimization Theory and Applications, Vol. 119, No. 3, 2003, pp. 485-498. doi:10.1023/B:JOTA.0000006686.19635.ad

[13] Q. M. Liu, L. Y. Fan and G. H. Wang, “Generalized Vector Quasi-Equilibrium Problems with Set-Valued Mappings,” Applied Mathematics Letters, Vol. 21, No. 9, 2008, pp. 946-950. doi:10.1016/j.aml.2007.10.007

[14] G. Q. Tian and J. Zhou, “Quasi-Variational Inequalities without the Concavity Assumption,” Journal of Mathematical Analysis and Applications, Vol. 172, No. 1, 1993, pp. 289-299. doi:10.1006/jmaa.1993.1025

[15] J. P. Aubin and I. Ekeland, “Applied Nonlinear Analysis,” Wiley-Interscience, New York, 1984.

[16] C. H. Su and V. M. Sehgal, “Some Fixed Point Theorems for Condensing Multifunctions in Locally Convex Spaces,” Proceedings of the American Mathematical Society, Vol. 50, No. 1, 1975, pp. 150-154. doi:10.1090/S0002-9939-1975-0380530-7

[17] X. Q. Yang, “Generalized Convex Functions and Vector Variational Inequalities,” Journal of Optimization Theory and Applications, Vol. 79, No. 3, 1993, pp. 563-580. doi:10.1007/BF00940559

[18] Q. H. Ansari and J. C. Yao, “A Fixed Point Theorem and Its Applications to the System of varItional Ineqalities,” Bulletin of the Australian Mathematical Society, Vol. 59, No. 3, 1999, pp. 433-442. doi:10.1017/S0004972700033116

[1] J.-W. Peng, “System of Generalized Set-Valued QuasiVariational-Like Inequalities,” Bulletin of the Australian Mathematical Society, Vol. 68, No. 3, 2003, pp. 501-515. doi:10.1017/S0004972700037904

[2] Q. H. Ansari, S. Schaible and J. C. Yao, “The System of Generalized Vector Equilibrium Problems with Applications,” Journal of Global Optimization, Vol. 22, No. 1, 2002, pp. 3-16. doi:10.1023/A:1013857924393

[3] H. H. Schaefer, “Topological Vector Space,” 2nd Edition, Graduate Texts in Mathematics, Springer, Berlin, 1999.

[4] X. P. Ding and E. Tarafdar, “Generalized VariationalLike Inequalities with Pseudomonotone Set-Valued Mappings,” Archive der Mathematik, Vol. 74, No. 4, 2000, pp. 302-313. doi:10.1007/s000130050447

[5] Q. H. Ansari and J. C. Yao, “System of Generalized Varitional Ineqalities and Their Applications,” Applicable Analysis, Vol. 76, No. 3-4, 2000, pp. 203-217. doi:10.1080/00036810008840877

[6] G. Xiao and S. Liu, “Existence of Solutions for Generalized Vector Quasi-Variational-Like Inequalities without Monotonicity,” Computers and Mathematics with Applications, Vol. 58, No. 8, 2009, pp. 1550-1557. doi:10.1016/j.camwa.2009.05.021

[7] G. Xiao, Z. Fan and X. Qi, “Existence Results for Generalized Nonlinear Vector Variational-Like Inequalities with Set-Valued Mapping,” Applied Mathematics Letter, Vol. 23, No. 1, 2010, pp. 44-47. doi:10.1016/j.aml.2009.07.023

[8] J.-W. Peng and X.-M. Yang, “Generalized Vector QuasiVariational-Like Inequalities,” Journal of Inequalities and Applications, Vol. 2006, No. 1, 2006, pp. 1-11.

[9] S. Husain and S. Gupta, “Existence of Solutions for Generalized Nonlinear Vector Quasi-Variational-Like Inequalities with Set-Valued Mappings,” Filomat, Vol. 26, No. 5, 2012, pp. 909-916. doi:10.2298/FIL1205909H

[10] S. Husain and S. Gupta, “Generalized Nonlinear Vector Quasi-Variational-Like Inequality,” International Journal of Pure and Applied Mathematics, Vol. 66, No. 2, 2011, pp. 157-169.

[11] P. Hartman and G. Stampacchia, “On Some Nonlinear Elliptic Differential Function Equations,” Acta Mathematica, Vol. 115, No. 1, 1966, pp. 271-310. doi:10.1007/BF02392210

[12] S. H. Hou, H. Yu and G. Y. Chen, “On Vector QuasiEquilibrium Problems with Set-Valued Maps,” Journal of Optimization Theory and Applications, Vol. 119, No. 3, 2003, pp. 485-498. doi:10.1023/B:JOTA.0000006686.19635.ad

[13] Q. M. Liu, L. Y. Fan and G. H. Wang, “Generalized Vector Quasi-Equilibrium Problems with Set-Valued Mappings,” Applied Mathematics Letters, Vol. 21, No. 9, 2008, pp. 946-950. doi:10.1016/j.aml.2007.10.007

[14] G. Q. Tian and J. Zhou, “Quasi-Variational Inequalities without the Concavity Assumption,” Journal of Mathematical Analysis and Applications, Vol. 172, No. 1, 1993, pp. 289-299. doi:10.1006/jmaa.1993.1025

[15] J. P. Aubin and I. Ekeland, “Applied Nonlinear Analysis,” Wiley-Interscience, New York, 1984.

[16] C. H. Su and V. M. Sehgal, “Some Fixed Point Theorems for Condensing Multifunctions in Locally Convex Spaces,” Proceedings of the American Mathematical Society, Vol. 50, No. 1, 1975, pp. 150-154. doi:10.1090/S0002-9939-1975-0380530-7

[17] X. Q. Yang, “Generalized Convex Functions and Vector Variational Inequalities,” Journal of Optimization Theory and Applications, Vol. 79, No. 3, 1993, pp. 563-580. doi:10.1007/BF00940559

[18] Q. H. Ansari and J. C. Yao, “A Fixed Point Theorem and Its Applications to the System of varItional Ineqalities,” Bulletin of the Australian Mathematical Society, Vol. 59, No. 3, 1999, pp. 433-442. doi:10.1017/S0004972700033116