## Abstract and Applied Analysis

### A New System of Multivalued Mixed Variational Inequality Problem

#### Abstract

We consider a new system of multivalued mixed variational inequality problem, which includes some known systems of variational inequalities as special cases. Under suitable conditions, the existence of solutions for the system of multivalued mixed variational inequality problem and the convergence of iterative sequences generated by the generalized $f$-projection algorithm are proved. A perturbational algorithm for solving a special case of multivalued mixed variational inequality problem is formally constructed. The results concerned with the existence of solutions and the convergence of iterative sequences generated by the perturbational algorithm are also given. Some known results are improved and generalized.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 982606, 7 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412606942

Digital Object Identifier
doi:10.1155/2014/982606

Mathematical Reviews number (MathSciNet)
MR3198279

Zentralblatt MATH identifier
07023452

#### Citation

Li, Xi; Li, Xue-song. A New System of Multivalued Mixed Variational Inequality Problem. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 982606, 7 pages. doi:10.1155/2014/982606. https://projecteuclid.org/euclid.aaa/1412606942

#### References

• C. Lescarret, “Cas d'addition des applications monotones maximales dans un espace de Hilbert,” Comptes Rendus de l'Académie des Sciences, vol. 261, pp. 1160–1163, 1965 (French).
• F. E. Browder, “On the unification of the calculus of variations and the theory of monotone nonlinear operators in Banach spaces,” Proceedings of the National Academy of Sciences of the United States of America, vol. 56, pp. 419–425, 1966.
• I. V. Konnov and E. O. Volotskaya, “Mixed variational inequalities and economic equilibrium problems,” Journal of Applied Mathematics, vol. 2, no. 6, pp. 289–314, 2002.
• S. S. Chang, H. W. Joseph Lee, and C. K. Chan, “Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces,” Applied Mathematics Letters, vol. 20, no. 3, pp. 329–334, 2007.
• N. Petrot, “A resolvent operator technique for approximate solving of generalized system mixed variational inequality and fixed point problems,” Applied Mathematics Letters, vol. 23, no. 4, pp. 440–445, 2010.
• Z. He and F. Gu, “Generalized system for relaxed cocoercive mixed variational inequalities in Hilbert spaces,” Applied Mathematics and Computation, vol. 214, no. 1, pp. 26–30, 2009.
• H. Nie, Z. Liu, K. H. Kim, and S. M. Kang, “A system of nonlinear variational inequalities involving strongly monotone and pseudocontractive mappings,” Advances in Nonlinear Variational Inequalities, vol. 6, no. 2, pp. 91–99, 2003.
• R. U. Verma, “Projection methods, algorithms, and a new system of nonlinear variational inequalities,” Computers & Mathematics with Applications, vol. 41, no. 7-8, pp. 1025–1031, 2001.
• R. U. Verma, “Generalized system for relaxed cocoercive variational inequalities and projection methods,” Journal of Optimization Theory and Applications, vol. 121, no. 1, pp. 203–210, 2004.
• N. H. Xiu and J. Z. Zhang, “Local convergence analysis of projection-type algorithms: unified approach,” Journal of Optimization Theory and Applications, vol. 115, no. 1, pp. 211–230, 2002.
• Y. Yao, Y.-C. Liou, and S. M. Kang, “Two-step projection methods for a system of variational inequality problems in Banach spaces,” Journal of Global Optimization, vol. 55, no. 4, pp. 801–811, 2013.
• Y. J. Cho and X. Qin, “Systems of generalized nonlinear variational inequalities and its projection methods,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 12, pp. 4443–4451, 2008.
• Ya. Alber, “Generalized projection operators in Banach spaces: properties and applications,” in Proceedings of the Israel Seminar on Functional Differential Equations, pp. 1–21, Ariel, Israel, 1994.
• K.-Q. Wu and N.-J. Huang, “The generalised $f$-projection operator with an application,” Bulletin of the Australian Mathematical Society, vol. 73, no. 2, pp. 307–317, 2006.
• Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 15–50, Dekker, New York, NY, USA, 1996.
• J. Fan, X. Liu, and J. Li, “Iterative schemes for approximating solutions of generalized variational inequalities in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 11, pp. 3997–4007, 2009.
• X. Li, Y. Z. Zou, and N. J. Huang, “On the stability of generalized f-projection operators with an application,” Acta Mathematica Sinica, Chinese Series, vol. 53, no. 2, pp. 375–384, 2010.
• X. Li, X. S. Li, and N. J. Huang, “A generalized f-projection algorithm for inverse mixed variational inequalities,” Optimization Letters, vol. 8, no. 3, pp. 1063–1076, 2014.
• L. S. Liu, “Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 194, no. 1, pp. 114–125, 1995.
• H.-Y. Lan, J. H. Kim, and Y. J. Cho, “On a new system of nonlinear $A$-monotone multivalued variational inclusions,” Journal of Mathematical Analysis and Applications, vol. 327, no. 1, pp. 481–493, 2007.
• R. U. Verma, “General convergence analysis for two-step projection methods and applications to variational problems,” Applied Mathematics Letters, vol. 18, no. 11, pp. 1286–1292, 2005.
• S. B. Nadler Jr., “Multi-valued contraction mappings,” Pacific Journal of Mathematics, vol. 30, pp. 475–488, 1969. \endinput