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Solution of a class of equilibrium problems and variational inequalities in FC spaces

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Abstract

In this paper, we study a class of mixed variational-like inequalities with respect to generalized weakly relaxed \(\eta {-}\alpha \) monotone mappings, involving nonlinear bifunctions, in finitely continuous topological space, in short FC space. Existence of the solution to the problem is established relaxing convexity and linearity condition by using generalized RKKM theorem. We have proposed a proximal iterative scheme using auxiliary principle technique. Solvability of the auxiliary variational inequality problem is established. Finally convergence of the iterates to the exact solution is proved. Some results with application to equilibrium problem are also discussed.

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References

  • Bai, M. R., Zhou, S. Z., & Ni, G. Y. (2006). Variational-like inequalities with relaxed pseudomonotone mappings in Banach spaces. Applied Mathematics Letters, 19(6), 547–554.

    Article  Google Scholar 

  • Ben-El-Mechaiekh, H., Chebbi, S., Florenzano, M., & Llinares, J. V. (1998). Abstract convexity and fixed points. Journal of Mathematical Analysis and Applications, 222(1), 138–150.

    Article  Google Scholar 

  • Chadli, O., Ansari, Q. H., & Al-Homidan, S. (2017). Existence of solutions and algorithms for bilevel vector equilibrium problems: An auxiliary principle technique. Journal of Optimization Theory and Applications, 172(3), 726–758.

    Article  Google Scholar 

  • Chaipunya, P., & Kumam, P. (2013). Topological aspects of circular metric spaces and some observations on the KKM property towards quasi-equilibrium problems. Journal of Inequalities and Applications, 2013(1), 1–9.

    Article  Google Scholar 

  • Chaipunya, P., & Kumam, P. (2015). Nonself KKM maps and corresponding theorems in Hadamard manifolds. Applied General Topology, 16(1), 37–44.

    Article  Google Scholar 

  • Charles, D. (1987). Some results on multivalued mappings and inequalities without convexity. In Nonlinear and convex analysis: Proceedings in honor of Ky Fan (Vol. 107, p. 99). CRC Press.

  • Chen, Q., & Luo, J. (2013). Relaxed \(\eta \)–\(\alpha \) quasimonotone and application to the generalized variational-like inequality problem. Journal of Inequalities and Applications, 2013(1), 488–495.

    Article  Google Scholar 

  • Chen, Y. Q. (1999). On the semi-monotone operator theory and applications. Journal of Mathematical Analysis and Applications, 231(1), 177–192.

    Article  Google Scholar 

  • Deng, L., & Xia, X. (2003). Generalized R-KKM theorems in topological space and their applications. Journal of mathematical analysis and applications, 285(2), 679–690.

    Article  Google Scholar 

  • Ding, X. (2011). New systems of generalized quasi-variational inclusions in FC-spaces and applications. Acta Mathematica Scientia, 31(3), 1142–1154.

    Article  Google Scholar 

  • Ding, X., & Tan, K. (1992). A minimax inequality with applications to existence of equilibrium point and fixed point theorems. Colloquium Mathematicum, 63, 233–247.

    Article  Google Scholar 

  • Ding, X. P. (1998). General algorithm for nonlinear variational-like inequalities in reflexive Banach spaces. Indian Journal of Pure and Applied Mathematics, 29(2), 109–120.

    Google Scholar 

  • Ding, X. P. (2006). Generalized KKM type theorems in FC-spaces with applications (I). Journal of Global Optimization, 36(4), 581–596.

    Article  Google Scholar 

  • Ding, X. P. (2007). Generalized KKM type theorems in FC-spaces with applications (II). Journal of Global Optimization, 38(3), 367–385.

    Article  Google Scholar 

  • Ding, X. P. (2012). Equilibrium existence theorems for multi-leader-follower generalized multiobjective games in FC-spaces. Journal of Global Optimization, 53(3), 381–390.

    Article  Google Scholar 

  • Ding, X. P., & Ding, T. M. (2007). KKM type theorems and generalized vector equilibrium problems in noncompact FC-spaces. Journal of Mathematical Analysis and Applications, 331(2), 1230–1245.

    Article  Google Scholar 

  • Fang, Y., & Huang, N. (2003). Variational-like inequalities with generalized monotone mappings in Banach spaces. Journal of Optimization Theory and Applications, 118(2), 327–338.

    Article  Google Scholar 

  • Farajzadeh, A., & Zafarani, J. (2010). Equilibrium problems and variational inequalities in topological vector spaces. Optimization, 59(4), 485–499.

    Article  Google Scholar 

  • Farajzadeh, A., Plubtieng, S., Ungchittrakool, K., & Kumtaeng, D. (2015). Generalized mixed equilibrium problems with generalized \(\alpha \)-\(\eta \)-monotone bifunction in topological vector spaces. Applied Mathematics and Computation, 265, 313–319.

    Article  Google Scholar 

  • Heinemann, C., & Sturm, K. (2016). Shape optimization for a class of semilinear variational inequalities with applications to damage models. SIAM Journal on Mathematical Analysis, 48(5), 3579–3617.

    Article  Google Scholar 

  • Horvath, C. D. (1991). Contractibility and generalized convexity. Journal of Mathematical Analysis and Applications, 156(2), 341–357.

    Article  Google Scholar 

  • Horvath, C. D. (1993). Extension and selection theorems in topological spaces with a generalized convexity structure. Annales de la Faculté des sciences de Toulouse: Mathématiques, 2, 253–269.

    Article  Google Scholar 

  • Huang, J. (2005). The matching theorems and coincidence theorems for generalized R-KKM mapping in topological spaces. Journal of Mathematical Analysis and Applications, 312(1), 374–382.

    Article  Google Scholar 

  • Huang, N. (1997). On the generalized implicit quasivariational inequalities. Journal of Mathematical Analysis and Applications, 216(1), 197–210.

    Article  Google Scholar 

  • Huang, Nj, & Deng, Cx. (2001). Auxiliary principle and iterative algorithms for generalized set-valued strongly nonlinear mixed variational-like inequalities. Journal of Mathematical Analysis and Applications, 256(2), 345–359.

    Article  Google Scholar 

  • Kristály, A. (2014). Nash-type equilibria on riemannian manifolds: A variational approach. Journal de Mathématiques Pures et Appliquées, 101(5), 660–688.

    Article  Google Scholar 

  • Kutbi, M.A., & Sintunavarat, W. (2013). On the solution existence of variational-like inequalities problems for weakly relaxed monotone mapping. In Abstract and applied analysis (p. 8), 2013 (ID 207845).

  • Liu, Z., & Zeng, S. (2016). Equilibrium problems with generalized monotone mapping and its applications. Mathematical Methods in the Applied Sciences, 39(1), 152–163.

    Article  Google Scholar 

  • Mahato, N. K., & Nahak, C. (2014). Equilibrium problems with generalized relaxed monotonicities in Banach spaces. Opsearch, 51(2), 257–269.

    Article  Google Scholar 

  • Neto, J. D. C., Ferreira, O. P., Pérez, L. L., & Németh, S. Z. (2006). Convex-and monotone-transformable mathematical programming problems and a proximal-like point method. Journal of Global Optimization, 35(1), 53–69.

    Article  Google Scholar 

  • Noor, M. A., & Noor, K. I. (2012). General equilibrium bifunction variational inequalities. Computers & Mathematics with Applications, 64(11), 3522–3526.

    Article  Google Scholar 

  • Pany, G., & Pani, S. (2015). Nonlinear mixed variational-like inequality with respect to weakly relaxed \(\eta \)–\(\alpha \) monotone mapping in banach spaces. In Mathematical analysis and its applications: Roorkee (Vol. 143, pp. 185–196), India, Dec 2014.

  • Park, S., & Kim, H. (1993). Admissible classes of multifunctions on generalized convex spaces. In Proceedings of College of Natural Sciences, Seoul National University (Vol. 18, pp. 1–21).

  • Park, S., & Kim, H. (1997). Foundations of the KKM theory on generalized convex spaces. Journal of Mathematical Analysis and Applications, 209(2), 551–571.

    Article  Google Scholar 

  • Salahuddin, Verma R. (2016). Extended systems of nonlinear vector quasi variational inclusions and extended systems of nonlinear vector quasi optimization problems in locally FC-spaces. Journal of Communications on Applied Nonlinear Analysis, 23(1), 71–88.

    Google Scholar 

  • Solodov, M. V. (2003). Merit functions and error bounds for generalized variational inequalities. Journal of Mathematical Analysis and Applications, 287(2), 405–414.

    Article  Google Scholar 

  • Tian, G. (1993). Generalized quasi-variational-like inequality problem. Mathematics of Operations Research, 18(3), 752–764.

    Article  Google Scholar 

  • Tremolieres, R., Lions, J. L., & Glowinski, R. (2011). Numerical analysis of variational inequalities. North Holland: Elsevier.

    Google Scholar 

  • Verma, R. U. (1999). Some results on R-KKM mappings and R-KKM selections and their applications. Journal of Mathematical Analysis and Applications, 232(2), 428–433.

    Article  Google Scholar 

  • Wang, J. (2012). The existence of solutions for general variational inequality and applications in FC spaces. Journal of Inequalities and Applications, 2012(1), 1–8.

    Article  Google Scholar 

  • Xie-ping, D. (2005). Himmelberg type fixed point theorem in locally FC-spaces. Journal of Sichuan Normal University (Natural Science), 2, 034.

    Google Scholar 

  • Xie-ping, D. (2006). Generalizations of Himmelberg type fixed point theorems in locally FC spaces. Journal of Sichuan Normal University (Natural Science), 1, 001.

    Google Scholar 

  • Xieping, D. (2010). Mathematical programs with system of generalized vector quasi-equilibrium constraints in FC-spaces. Acta Mathematica Scientia, 30(4), 1257–1268.

    Article  Google Scholar 

  • Yao, J. C. (1991). The generalized quasi-variational inequality problem with applications. Journal of Mathematical Analysis and Applications, 158(1), 139–160.

    Article  Google Scholar 

Download references

Acknowledgements

We are grateful to the anonymous referees for their valuable comments which helped us in improving the paper. We are also grateful to the editors for considering our paper for the special issue. The first author wishes to thank Department of Science and Technology (DST-INSPIRE, Fellowship Code No. IF110762) for the grant of research fellowship and to IIT Bhubaneswar for providing the research facilities.

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Authors and Affiliations

  1. School of Basic Science, IIT, Bhubaneswar, 751007, India

    Gayatri Pany & Sabyasachi Pani

  2. University of Central Florida, Orlando, FL, 32816, USA

    Ram N. Mohapatra

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  1. Gayatri Pany
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  2. Ram N. Mohapatra
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  3. Sabyasachi Pani
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Correspondence to Ram N. Mohapatra.

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Pany, G., Mohapatra, R.N. & Pani, S. Solution of a class of equilibrium problems and variational inequalities in FC spaces. Ann Oper Res 269, 565–582 (2018). https://doi.org/10.1007/s10479-017-2506-3

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