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Well-posedness for the split equilibrium problem

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Abstract

We extend the concept of well-posedness to the split equilibrium problem and establish Furi–Vignoli-type characterizations for the well-posedness. We prove that the well-posedness of the split equilibrium problem is equivalent to the existence and uniqueness of its solution under certain assumptions on the bifunctions involved. We also characterize the generalized well-posedness of the split equilibrium problem via the Kuratowski measure of noncompactness. We illustrate our theoretical results by several examples.

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References

  1. Tykhonov, A.N.: On the stability of the functional optimization problem. USSR J. Comput. Math. Math. Phys. 6, 631–634 (1966)

    Google Scholar 

  2. Furi, M., Vignoli, A.: About well-posed optimization problems for functionals in metric spaces. J. Optim. Theory Appl. 5(3), 225–229 (1970)

    Article  MathSciNet  Google Scholar 

  3. Lucchetti, R., Patrone, F.: A characterization of Tyhonov well-posedness for minimum problems, with applications to variational inequalities. Numer. Funct. Anal. Optim. 3(4), 461–476 (1981)

    Article  MathSciNet  Google Scholar 

  4. Lucchetti, R., Patrone, F.: Some properties of well-posed variational inequalities governed by linear operators. Numer. Funct. Anal. Optim. 5(3), 349-361 (1982/83)

  5. Lignola, M.B., Morgan, J.: Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution. J. Glob. Optim. 16(1), 57–67 (2000)

    Article  MathSciNet  Google Scholar 

  6. Zolezzi, T.: Well-posedness criteria in optimization with application to the calculus of variations. Nonlinear Anal. TMA 25, 437–453 (1995)

    Article  MathSciNet  Google Scholar 

  7. Zolezzi, T.: Extended well-posedness of optimization problems. J. Optim. Theory Appl. 91, 257–266 (1996)

    Article  MathSciNet  Google Scholar 

  8. Lignola, M.B.: Well-posedness and L-well-posedness for quasivariational inequalities. J. Optim. Theory Appl. 128(1), 119–138 (2006)

    Article  MathSciNet  Google Scholar 

  9. Fang, Y.P., Huang, N.J., Yao, J.C.: Well-posedness of mixed variational inequalities, inclusion problems and fixed point problems. J. Glob. Optim. 41, 117–133 (2008)

    Article  MathSciNet  Google Scholar 

  10. Fang, Y.P., Huang, N.J., Yao, J.C.: Well-posedness by perturbations of mixed variational inequalities in Banach spaces. Eur. J. Oper. Res. 201(3), 682–692 (2010)

    Article  MathSciNet  Google Scholar 

  11. Huang, X.X., Yang, X.Q., Zhu, D.L.: Levitin-Polyak well-posedness of variational inequality problems with functional constraints. J. Glob. Optim. 44(2), 159–174 (2009)

    Article  MathSciNet  Google Scholar 

  12. Lalitha, C.S., Bhatia, G.: Levitin-Polyak well-posedness for parametric quasivariational inequality problem of the Minty type. Positivity 16(3), 527–541 (2012)

    Article  MathSciNet  Google Scholar 

  13. Li, X.B., Xia, F.Q.: Hadamard well-posedness of a general mixed variational inequality in Banach space. J. Glob. Optim. 64(6), 1617–1629 (2013)

    Article  MathSciNet  Google Scholar 

  14. Wong, M.M.: Well-posedness of general mixed implicit quasi-variational inequalities, inclusion problems and fixed point problems. J. Nonlinear Convex Appl. 14(2), 389–414 (2013)

    MathSciNet  Google Scholar 

  15. Xiao, Y.B., Huang, N.J.: Well-posedness for a class of variational-hemivariational inequalities with perturbations. J. Optim. Theory Appl. 151(1), 33–51 (2011)

    Article  MathSciNet  Google Scholar 

  16. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I. Springer, New York (2003)

    Google Scholar 

  17. Stampacchia, G.: Formes bilineaires coercitives sur les ensembles convexes. Comp. Rend. Hebd. séances l’Acad. Sci. 258, 4413–4416 (1964)

    MathSciNet  Google Scholar 

  18. Noor, M.D.: Well-posed variational inequalities. J. Appl. Math. Comput. 11(1–2), 165–172 (2003)

    Article  MathSciNet  Google Scholar 

  19. Dong, Q.L., Lu, Y.Y., Yang, J., He, S.: Approximately solving multi-valued variational inequalities by using a projection and contraction algorithm. Numer. Algorithms 76(3), 799–812 (2017)

    Article  MathSciNet  Google Scholar 

  20. Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 59(2), 301–323 (2012)

    Article  MathSciNet  Google Scholar 

  21. Ansari, Q.H., Nimana, N., Petrot, N.: Split hierarchical variational inequality problems and related problems. Fixed Point Theory Appl. 2014, 208 (2014)

    Article  MathSciNet  Google Scholar 

  22. Hu, R., Fang, Y.P.: Characterizations of Levitin-Polyak well-posedness by perturbations for the split variational inequality problem. Optimization 65(9), 1717–1732 (2016)

    Article  MathSciNet  Google Scholar 

  23. Bianchia, M., Kassay, G., Pini, R.: Well-posed equilibrium problems. Nonlinear Anal. 72, 460–468 (2010)

    Article  MathSciNet  Google Scholar 

  24. Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Student 63, 123–145 (1994)

    MathSciNet  Google Scholar 

  25. Eslamizadeh, L., Naraghirad, E.: Existence of solutions of set-valued equilibrium problems in topological vector spaces with applications. Optim. Lett. 14, 65–83 (2020)

    Article  MathSciNet  Google Scholar 

  26. Alleche, B., Radulescu, V.D.: Further on set-valued equilibrium problemsin the pseudo-monotone case and applicationsto Browder variational inclusions. Optim. Lett. 12, 1789–1810 (2018)

    Article  MathSciNet  Google Scholar 

  27. Bianchi, M., Kassay, G., Pini, R.: Ekland’s principle for vector equilibrium problem. Nonlinear Anal. 66, 1454–1464 (2007)

    Article  MathSciNet  Google Scholar 

  28. Kuratowski, K.: Topology, vol. I. Academic Press, New York, II (1968)

    Google Scholar 

  29. Klein, E., Thompson, A.C.: Theory of Correspondences. Wiley, New York (1984)

    Google Scholar 

  30. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)

    Book  Google Scholar 

  31. Rakocevik, V.: Measure of noncompactness and some applications. FILOMAT 12(2), 87–120 (1998)

    MathSciNet  Google Scholar 

  32. Horvath, C.: Measure of Non-compactness and Multivalued Mappings in Complete Metric Topological Vector Spaces. J. Math. Anal. Appl. 108, 403–408 (1985)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are thankful to the editor and to two anonymous referees for their useful comments and helpful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, India Institute of Technology Madras, Chennai, 600036, India

    Soumitra Dey & V. Vetrivel

  2. Department of Mathematics, The Technion–Israel Institute of Technology, 3200003, Haifa, Israel

    Soumitra Dey

  3. Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, 310018, China

    Hong-Kun Xu

  4. College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, China

    Hong-Kun Xu

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  1. Soumitra Dey
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  2. V. Vetrivel
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  3. Hong-Kun Xu
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Dey, S., Vetrivel, V. & Xu, HK. Well-posedness for the split equilibrium problem. Optim Lett 18, 977–989 (2024). https://doi.org/10.1007/s11590-023-02034-4

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