Abstract
In this paper, we study the existence of solutions of both equilibrium problems and mixed equilibrium problems on Hadamard manifolds. Under relaxed \(\alpha \)-pseudomonotonicity assumption on the underlying bifunction we prove that the solution set of the equilibrium problem is nonempty. We also provide the existence of solution of mixed equilibrium problems with relaxed \(\alpha \)-monotonicity. The results presented in this paper generalize and improve some known results given in literature, see for example (Colao et al. in J Math Anal Appl 388:61–77, 2012; Jana and Nahak in Rend Circ Mat Palermo(2) 65(1):97–109, 2016; Mahato and Nahak in OPSEARCH, 2013. https://doi.org/10.1007/s12597-013-0142-5).
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References
Bianchi, M., Schaible, S.: Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory Appl. 90, 31–43 (1996)
Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Student 63, 123–145 (1994)
Colao, V., López, G., Marino, G., Martín-Márquez, V.: Equilibrium problems in Hadamard manifolds. J. Math. Anal. Appl. 388, 61–77 (2012)
Cruz Neto, J.X., Ferreira, O.P., Lucambio Pérez, L.R., Németh, S.Z.: Convex- and Monotone-Transformable Mathematical Programming Problems and a Proximal-Like Point Method. J. Global Optim. 35, 53–69 (2006)
Ferreira, O.P., Oliveira, P.R.: Proximal point algorithm on Riemannian manifolds. Optimization 51, 257–270 (2002)
Jana, S., Nahak, C.: Mixed equilibrium problems on Hadamard manifolds. Rend. Circ. Mat. Palermo(2) 65(1), 97–109 (2016)
Jost, J.: Nonpositive Curvature: Geometric and Analytic Aspects. Birkhauser Verlag, Basel (1997)
Li, C., López, G., Martín-Márquez, V.: Monotone vector fields and the proximal point algorithm on Hadamard manifolds. J. Lond. Math. Soc. 79, 663–683 (2009)
Li, S.L., Li, C., Liou, Y.C., Yao, J.C.: Existence of solutions for variational inequalities on Riemannian manifolds. Nonlinear Anal. 71, 5695–5706 (2009)
Li, X.B., Huang, N.J.: Generalized vector quasi-equilibrium problems on Hadamard manifolds. Optim. Lett. https://doi.org/10.1007/s11590-013-0703-9
Luc, D.T.: Existence results for densely pseudomonotone variational inequalities. J. Math. Anal. Appl. 254, 309–320 (2001)
Németh, S.Z.: Variational inequalities on Hadamard manifolds. Nonlinear Anal. 52, 1491–1498 (2003)
Németh, S.Z.: Geodesic monotone vector fields. Lobachevskii J. Math. 5, 13–28 (1999)
Mahato, N.K., Nahak, C.: Equilibrium problems with generalized relaxed monotonicities in Banach spaces. OPSEARCH. https://doi.org/10.1007/s12597-013-0142-5
Rapcsák, T.: Geodesic convexity in nonlinear optimization. J. Optim. Theory Appl. 69, 169–183 (1991)
Reich, S., Sabach, S.: Three strong convergence theorems regarding iterative methods for solving equilibrium problems in reflexive Banach spaces. Contempor. Math. 68, 225–240 (2012)
Sakai, T.: Riemannian Geometry, Translations of Mathematical Monographs, vol. 149. American Mathematical Society, Providence (1996)
Tang, G.J., Zhou, L.W., Huang, N.J.: The proximal point algorithm for pseudomonotone variational inequalities on Hadamard manifolds. Optim. Lett. (2012). https://doi.org/10.1007/s11590-012-0459-7
Udriste, C.: Convex Functions and Optimization Methods on Riemannian Manifolds, vol. 297. Kluwer Academic (1994)
Wang, J.H., López, G., Martín-Márquez, V., Li, C.: Monotone and accretive vector fields on Riemannian manifolds. J. Optim. Theory Appl. 146, 691–708 (2010)
Yao, Y., Noor, M.A., Zainab, S., Liou, Y.C.: Mixed equilibrium problems and optimization problems. J. Math. Anal. Appl. 354, 319–329 (2009)
Zhou, L.W., Huang, N.J.: Generalized KKM theorems on Hadamard manifolds with applications. http://www.paper.edu.cn/index.php/default/releasepaper/content/200906-669 (2009)
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The author wishes to thank the referees for their valuable comments and suggestions to improve the presentation of this paper.
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Jana, S. Equilibrium problems under relaxed α-monotonicity on Hadamard manifolds. Rend. Circ. Mat. Palermo, II. Ser 71, 361–372 (2022). https://doi.org/10.1007/s12215-021-00595-w
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DOI: https://doi.org/10.1007/s12215-021-00595-w