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An auxiliary problem principle for the solutions of mixed invex equilibrium problems in Banach spaces

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Abstract

In this paper, we establish some existence results on solutions for mixed invex equilibrium problems. Some auxiliary mixed invex equilibrium problems corresponding to the mixed invex equilibrium problem along with their iterative algorithm are then presented. Finally, the solvability and convergence criteria of the iterative algorithms are analyzed.

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References

  1. Ansari, Q.H., Yao, J.C.: Iterative schemes for solving mixed variational-like inequalities. J. Optim. Theory Appl. 108, 527–541 (2001)

    Article  Google Scholar 

  2. Bianchi, M., Schaible, S.: Equilibrium problems under generalized convexity and generalized monotonicity. J. Global Optim. 30, 121–134 (2004)

    Article  Google Scholar 

  3. Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud.-India 63, 123–145 (1994)

    Google Scholar 

  4. Chadli, O., Chbani, Z., Riahi, H.: Equilibrium problems with generalized monotone bifunctions and applications to variational inequalities. J. Optim. Theory Appl. 105, 299–323 (2000)

    Article  Google Scholar 

  5. Chadli, O., Pany, G., Mohapatra, R.N.: Existence and iterative approximation method for solving mixed equilibrium problem under generalized monotonicity in Banach spaces. Numer. Algebra Control Optim. 10, 75–92 (2020)

    Article  Google Scholar 

  6. Cohen, G.: Auxiliary problem principle extended to variational inequalities. J. Optim. Theory Appl. 59, 325–333 (1988)

    Article  Google Scholar 

  7. Ding, X.P., Tan, K.K.: A minimax inequality with applications to existence of equilibrium point and fixed point theorems. Colloquium Mathematicae 63, 233–247 (1992)

    Article  Google Scholar 

  8. Ding, X.P., Yao, J.C.: Existence and algorithm of solutions for Mixed quasi-variational-like inclusions in banach spaces. Comput. Math. Appl. 49, 857–869 (2005)

    Article  Google Scholar 

  9. Ekeland, I., Temam, R.: Convex analysis and variational problems. SIAM, Philadelphia (1999)

    Book  Google Scholar 

  10. Fang, Y.P., Huang, N.J.: Variational-like inequalities with generalized monotone mappings in Banach spaces. J. Optim. Theory Appl. 118, 327–338 (2003)

    Article  Google Scholar 

  11. Giannessi, F., Maugeri, A., Pardalos, P.M.: Equilibrium Problems Nonsmooth Optimization and Variational Inequality Models. Kluwer Academic Publishers, Dordrecht-Boston-London (2004)

    Book  Google Scholar 

  12. Hanson, M.A.: On Sufficiency of the Kuhn-Tucker Conditions. J. Math. Anal. Appl. 80, 545–550 (1981)

    Article  Google Scholar 

  13. Konnov, I.V.: Equilibrium Models and Variational Inequalities, Elsevier (2007)

  14. Liu, Z., Migórski, S., Zeng, B.: Existence results and optimal control for a class of quasi mixed equilibrium problems involving the \((f, g, h)\)-quasimonotonicity. Appl. Math. Optim. 79, 257–277 (2019)

    Article  Google Scholar 

  15. Liu, Z., Zeng, S.: Equilibrium problems with generalized monotone mapping and its applications. Math. Methods Appl. Sci. 39, 152–163 (2016)

    Article  Google Scholar 

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

    Article  Google Scholar 

  17. Noor, M.A.: Invex equilibrium problems. J. Math. Anal. Appl. 302, 463–475 (2005)

    Article  Google Scholar 

  18. 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)

    Article  Google Scholar 

  19. Pany, G., Pani, S.: On mixed equilibrium problems in banach spaces. Pan Am. Math. J. 26, 12–24 (2016)

    Google Scholar 

  20. Popovici, N.: A decomposition approach to vector equilibrium problems. Ann. Oper. Res. 251, 105–115 (2017)

    Article  Google Scholar 

  21. Sahu, B.K., Pani, S., Mohapatra, R.N.: Mixed invex equilibrium problems with generalized relaxed monotone and relaxed invariant pseudomonotone mappings. Math. Inequal. Appl. 23, 201–215 (2020)

    Google Scholar 

  22. Verma, R.U.: A-monotonicity and applications to nonlinear variational inclusion problems. Int. J. Stochastic Anal. 2004, 193–195 (2004)

    Article  Google Scholar 

  23. Zhou, J.X., Chen, G.: Diagonal convexity conditions for problems in convex analysis and quasi-variational inequalities. J. Math. Anal. Appl. 132, 213–225 (1988)

    Article  Google Scholar 

  24. Zhu, D.L., Marcotte, P.: Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities. SIAM J. Optim. 6, 714–726 (1996)

    Article  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referee for their valuable comments and suggestions.

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Author notes

  1. Sujeet Kumar and Sabyasachi Pani have contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, Chandbali College, Chandbali, Bhadrak, Odisha, 756133, India

    Bijaya Kumar Sahu & Sabyasachi Pani

  2. School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Jatni, Bhubaneswar, Odisha, 752050, India

    Bijaya Kumar Sahu, Sujeet Kumar & Sabyasachi Pani

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  1. Bijaya Kumar Sahu
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  2. Sujeet Kumar
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  3. Sabyasachi Pani
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Correspondence to Bijaya Kumar Sahu.

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Sahu, B.K., Kumar, S. & Pani, S. An auxiliary problem principle for the solutions of mixed invex equilibrium problems in Banach spaces. OPSEARCH 60, 1777–1792 (2023). https://doi.org/10.1007/s12597-023-00655-y

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