## Journal of Applied Mathematics

- J. Appl. Math.
- Volume 2012, Special Issue (2012), Article ID 194509, 38 pages.

### Well-Posedness by Perturbations of Generalized Mixed Variational Inequalities in Banach Spaces

Lu-Chuan Ceng and Ching-Feng Wen

**Full-text: Open access**

#### Abstract

We consider an extension of the notion of well-posedness by perturbations, introduced by Zolezzi (1995, 1996) for a minimization problem, to a class of generalized mixed variational inequalities in Banach spaces, which includes as a special case the class of mixed variational inequalities. We establish some metric characterizations of the well-posedness by perturbations. On the other hand, it is also proven that, under suitable conditions, the well-posedness by perturbations of a generalized mixed variational inequality is equivalent to the well-posedness by perturbations of the corresponding inclusion problem and corresponding fixed point problem. Furthermore, we derive some conditions under which the well-posedness by perturbations of a generalized mixed variational inequality is equivalent to the existence and uniqueness of its solution.

#### Article information

**Source**

J. Appl. Math., Volume 2012, Special Issue (2012), Article ID 194509, 38 pages.

**Dates**

First available in Project Euclid: 3 January 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.jam/1357180315

**Digital Object Identifier**

doi:10.1155/2012/194509

**Mathematical Reviews number (MathSciNet)**

MR2861934

**Zentralblatt MATH identifier**

1242.49014

#### Citation

Ceng, Lu-Chuan; Wen, Ching-Feng. Well-Posedness by Perturbations of Generalized Mixed Variational Inequalities in Banach Spaces. J. Appl. Math. 2012, Special Issue (2012), Article ID 194509, 38 pages. doi:10.1155/2012/194509. https://projecteuclid.org/euclid.jam/1357180315

#### References

- A. N. Tikhonov, “On the stability of the functional optimization problem,”
*USSR Computational Mathematics and Mathematical Physics*, vol. 6, no. 4, pp. 631–634, 1966. - T. Zolezzi, “Well-posedness criteria in optimization with application to the calculus of variations,”
*Nonlinear Analysis*, vol. 25, no. 5, pp. 437–453, 1995. - T. Zolezzi, “Extended well-posedness of optimization problems,”
*Journal of Optimization Theory and Applications*, vol. 91, no. 1, pp. 257–266, 1996.Zentralblatt MATH: 0873.90094

Mathematical Reviews (MathSciNet): MR1411643

Digital Object Identifier: doi:10.1007/BF02192292 - R. Lucchetti and F. Patrone, “A characterization of Tikhonov well-posedness for minimum problems, with applications to variational inequalities,”
*Numerical Functional Analysis and Optimization*, vol. 3, no. 4, pp. 461–476, 1981.Mathematical Reviews (MathSciNet): MR636739

Zentralblatt MATH: 0479.49025

Digital Object Identifier: doi:10.1080/01630568108816100 - E. Bednarczuk and J. P. Penot, “Metrically well-set minimization problems,”
*Applied Mathematics & Optimization*, vol. 26, no. 3, pp. 273–285, 1992.Zentralblatt MATH: 0762.90073

Mathematical Reviews (MathSciNet): MR1175482

Digital Object Identifier: doi:10.1007/BF01371085 - A.L. Dontchev and T. Zolezzi,
*Well-Posed Optimization Problems*, vol. 1543 of*Lecture Notes in Math*, Springer, Berlin, Germany, 1993. - X. X. Huang, “Extended and strongly extended well-posedness of set-valued optimization problems,”
*Mathematical Methods of Operations Research*, vol. 53, no. 1, pp. 101–116, 2001.Zentralblatt MATH: 1018.49019

Mathematical Reviews (MathSciNet): MR1825251

Digital Object Identifier: doi:10.1007/s001860000100 - L. C. Ceng, N. Hadjisavvas, S. Schaible, and J. C. Yao, “Well-posedness for mixed quasivariational-like inequalities,”
*Journal of Optimization Theory and Applications*, vol. 139, no. 1, pp. 109–125, 2008.Zentralblatt MATH: 1157.49033

Mathematical Reviews (MathSciNet): MR2438597

Digital Object Identifier: doi:10.1007/s10957-008-9428-9 - Y. P. Fang, N. J. Huang, and J. C. Yao, “Well-posedness of mixed variational inequalities, inclusion problems and fixed point problems,”
*Journal of Global Optimization*, vol. 41, no. 1, pp. 117–133, 2008.Zentralblatt MATH: 1149.49009

Mathematical Reviews (MathSciNet): MR2386599

Digital Object Identifier: doi:10.1007/s10898-007-9169-6 - Y. P. Fang, N. J. Huang, and J. C. Yao, “Well-posedness by perturbations of mixed variational inequalities in Banach spaces,”
*European Journal of Operational Research*, vol. 201, no. 3, pp. 682–692, 2010.Zentralblatt MATH: 1177.49018

Mathematical Reviews (MathSciNet): MR2552486

Digital Object Identifier: doi:10.1016/j.ejor.2009.04.001 - L. C. Ceng and J. C. Yao, “Well-posedness of generalized mixed variational inequalities, inclusion problems and fixed-point problems,”
*Nonlinear Analysis, Theory, Methods and Applications*, vol. 69, no. 12, pp. 4585–4603, 2008. - M. B. Lignola and J. Morgan, “Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution,”
*Journal of Global Optimization*, vol. 16, no. 1, pp. 57–67, 2000.Zentralblatt MATH: 0960.90079

Mathematical Reviews (MathSciNet): MR1770522

Digital Object Identifier: doi:10.1023/A:1008370910807 - I. Del Prete, M. B. Lignola, and J. Morgan, “New concepts of well-posedness for optimization problems with variational inequality constraints,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 4, no. 1, article 5, 2003. - M. B. Lignola and J. Morgan, “Approximating solutions and well-posedness for variational inequalities and Nash equilibria,” in
*Decision and Control in Management Science*, pp. 367–378, Kluwer Academic, 2002. - E. Cavazzuti and J. Morgan, “Well-posed saddle point problems,” in
*Optimization, Theory and Algorithms*, J. B. Hirriart-Urruty, W. Oettli, and J. Stoer, Eds., pp. 61–76, Marcel Dekker, New York, NY, USA, 1983. - J. Morgan, “Approximations and well-posedness in multicriteria games,”
*Annals of Operations Research*, vol. 137, no. 1, pp. 257–268, 2005.Zentralblatt MATH: 1138.91407

Mathematical Reviews (MathSciNet): MR2166443

Digital Object Identifier: doi:10.1007/s10479-005-2260-9 - R. Lucchetti and J. Revalski, Eds.,
*Recent Developments in Well-Posed Variational Problems*, Kluwer Academic, Dodrecht, The Netherlands, 1995.Mathematical Reviews (MathSciNet): MR1351737 - M. Margiocco, F. Patrone, and L. Pusillo, “On the Tikhonov well-posedness of concave games and Cournot oligopoly games,”
*Journal of Optimization Theory and Applications*, vol. 112, no. 2, pp. 361–379, 2002.Zentralblatt MATH: 1011.91004

Mathematical Reviews (MathSciNet): MR1883076

Digital Object Identifier: doi:10.1023/A:1013658023971 - Y. P. Fang, R. Hu, and N. J. Huang, “Well-posedness for equilibrium problems and for optimization problems with equilibrium constraints,”
*Computers and Mathematics with Applications*, vol. 55, no. 1, pp. 89–100, 2008. - B. Lemaire, C. Ould Ahmed Salem, and J. P. Revalski, “Well-posedness by perturbations of variational problems,”
*Journal of Optimization Theory and Applications*, vol. 115, no. 2, pp. 345–368, 2002.Zentralblatt MATH: 1047.90067

Mathematical Reviews (MathSciNet): MR1950699

Digital Object Identifier: doi:10.1023/A:1020840322436 - B. Lemaire, “Well-posedness, conditioning, and regularization of minimization, inclusion, and fixed point problems,”
*Pliska Studia Mathematica Bulgarica*, vol. 12, pp. 71–84, 1998.Mathematical Reviews (MathSciNet): MR1686513 - H. Yang and J. Yu, “Unified approaches to well-posedness with some applications,”
*Journal of Global Optimization*, vol. 31, no. 3, pp. 371–381, 2005.Zentralblatt MATH: 1080.49021

Mathematical Reviews (MathSciNet): MR2165723

Digital Object Identifier: doi:10.1007/s10898-004-4275-1 - S. Schaible, J. C. Yao, and L. C. Zeng, “Iterative method for set-valued mixed quasi-variational inequalities in a Banach space,”
*Journal of Optimization Theory and Applications*, vol. 129, no. 3, pp. 425–436, 2006.Zentralblatt MATH: 1123.49006

Mathematical Reviews (MathSciNet): MR2281149

Digital Object Identifier: doi:10.1007/s10957-006-9077-9 - L. C. Zeng and J. C. Yao, “Existence of solutions of generalized vector variational inequalities in reflexive banach spaces,”
*Journal of Global Optimization*, vol. 36, no. 4, pp. 483–497, 2006.Zentralblatt MATH: 1115.49005

Mathematical Reviews (MathSciNet): MR2269293

Digital Object Identifier: doi:10.1007/s10898-005-5509-6 - L.C. Zeng, “Perturbed proximal point algorithm for generalized nonlinear set-valued mixed quasi-variational inclusions,”
*Acta Mathematica Sinica*, vol. 47, no. 1, pp. 11–18, 2004. - H. Brezis,
*Operateurs Maximaux Monotone et Semigroups de Contractions dans les Es-Paces de Hilbert*, North-Holland, Amsterdam, The Netherlands, 1973.Zentralblatt MATH: 0325.35033 - S. B. Nadler Jr., “Multi-valued contraction mappings,”
*Pacific Journal of Mathematics*, vol. 30, pp. 475–488, 1969.Mathematical Reviews (MathSciNet): MR254828

Zentralblatt MATH: 0187.45002

Digital Object Identifier: doi:10.2140/pjm.1969.30.475

Project Euclid: euclid.pjm/1102978504 - K. Kuratowski,
*Topology*, vol. 1-2, Academic Press, New York, NY, USA, 1968.Mathematical Reviews (MathSciNet): MR259836 - E. Zeidler,
*Nonlinear Functional Analysis and Its Applications II: Monotone Operators*, Springer, Berlin, Germany, 1985. - S. Adly, E. Ernst, and M. Thera, “Well-positioned closed convex sets and well-positioned closed convex functions,”
*Journal of Global Optimization*, vol. 29, no. 4, pp. 337–351, 2004.Zentralblatt MATH: 1072.52005

Mathematical Reviews (MathSciNet): MR2104402

Digital Object Identifier: doi:10.1023/B:JOGO.0000047907.66385.5d - Y. R. He, X. Z. Mao, and M. Zhou, “Strict feasibility of variational inequalities in reflexive Banach spaces,”
*Acta Mathematica Sinica*, vol. 23, no. 3, pp. 563–570, 2007.Zentralblatt MATH: 1126.47049

Mathematical Reviews (MathSciNet): MR2292702

Digital Object Identifier: doi:10.1007/s10114-005-0918-5 - A. Brondsted and R. T. Rockafellar, “On the subdifferentiability of convex functions,”
*Proceedings of the American Mathematical Society*, vol. 16, no. 4, pp. 605–611, 1965. - I. Cioranescu,
*Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems*, Kluwer, 1990. - Z. B. Xu and G. F. Roach, “On the uniform continuity of metric projections in Banach spaces,”
*Approximation Theory and its Applications*, vol. 8, no. 3, pp. 11–20, 1992.

### More like this

- Well-Posedness for a Class of Strongly Mixed Variational-Hemivariational
Inequalities with Perturbations

Ceng, Lu-Chuan, Wong, Ngai-Ching, and Yao, Jen-Chih, Journal of Applied Mathematics, 2012 - Well-Posedness by Perturbations for Variational-Hemivariational
Inequalities

Lv, Shu, Xiao, Yi-bin, Liu, Zhi-bin, and Li, Xue-song, Journal of Applied Mathematics, 2012 - Metric Characterizations of
α
-Well-Posedness for a System ofMixed Quasivariational-Like Inequalities in
Banach Spaces

Ceng, L. C. and Lin, Y. C., Journal of Applied Mathematics, 2012

- Well-Posedness for a Class of Strongly Mixed Variational-Hemivariational
Inequalities with Perturbations

Ceng, Lu-Chuan, Wong, Ngai-Ching, and Yao, Jen-Chih, Journal of Applied Mathematics, 2012 - Well-Posedness by Perturbations for Variational-Hemivariational
Inequalities

Lv, Shu, Xiao, Yi-bin, Liu, Zhi-bin, and Li, Xue-song, Journal of Applied Mathematics, 2012 - Metric Characterizations of
α
-Well-Posedness for a System ofMixed Quasivariational-Like Inequalities in
Banach Spaces

Ceng, L. C. and Lin, Y. C., Journal of Applied Mathematics, 2012 - A New System of Generalized Mixed Quasivariational Inclusions with Relaxed Cocoercive Operators and Applications

Wan, Zhongping, Chen, Jia-Wei, Sun, Hai, and Yuan, Liuyang, Journal of Applied Mathematics, 2011 - STORNG LEVITIN-POLYAK WELL-POSEDNESS FOR GENERALIZED QUASI-VARIATIONAL INCLUSION PROBLEMS WITH APPLICATIONS

Wang, San-Hua, Huang, Nan-Jing, and Wong, Mu-Ming, Taiwanese Journal of Mathematics, 2012 - Well-posedness of Hemivariational Inequalities and Inclusion Problems

Xiao, Yi-bin, Huang, Nan-jing, and Wong, Mu-Ming, Taiwanese Journal of Mathematics, 2011 - THE EXISTENCE OF SOLUTIONS AND WELL-POSEDNESS FOR BILEVEL MIXED EQUILIBRIUM PROBLEMS IN BANACH SPACES

Chen, Jia-wei, Cho, Yeol Je, and Wan, Zhongping, Taiwanese Journal of Mathematics, 2013 - WELL-POSEDNESS FOR VECTOR VARIATIONAL INEQUALITY AND CONSTRAINED VECTOR OPTIMIZATION

Fang, Ya-Ping and Huang, Nan-Jing, Taiwanese Journal of Mathematics, 2007 - A New General Iterative Method for Solution of a New General System of Variational Inclusions for Nonexpansive Semigroups in Banach Spaces

Sunthrayuth, Pongsakorn and Kumam, Poom, Journal of Applied Mathematics, 2011 - α
-Well-Posedness for Mixed Quasi Variational-Like Inequality Problems

Peng, Jian-Wen and Tang, Jing, Abstract and Applied Analysis, 2011