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MONOTONE AND PSEUDO-MONOTONE EQUILIBRIUM PROBLEMS IN HADAMARD SPACES

Part of: Equations and inequalities involving nonlinear operators Existence theories Nonlinear operators and their properties

Published online by Cambridge University Press:  11 March 2019

HADI KHATIBZADEH Open the ORCID record for HADI KHATIBZADEH [Opens in a new window]  and
VAHID MOHEBBI Open the ORCID record for VAHID MOHEBBI [Opens in a new window]
HADI KHATIBZADEH*
Affiliation:
Department of Mathematics, University of Zanjan, PO Box 45195-313, Zanjan, Iran
VAHID MOHEBBI
Affiliation:
Department of Mathematics, University of Zanjan, PO Box 45195-313, Zanjan, Iran e-mail: mohebbi@znu.ac.ir
*
e-mail: hkhatibzadeh@znu.ac.ir
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Abstract

As a continuation of previous work of the first author with Ranjbar [‘A variational inequality in complete CAT(0) spaces’, J. Fixed Point Theory Appl. 17 (2015), 557–574] on a special form of variational inequalities in Hadamard spaces, in this paper we study equilibrium problems in Hadamard spaces, which extend variational inequalities and many other problems in nonlinear analysis. In this paper, first we study the existence of solutions of equilibrium problems associated with pseudo-monotone bifunctions with suitable conditions on the bifunctions in Hadamard spaces. Then, to approximate an equilibrium point, we consider the proximal point algorithm for pseudo-monotone bifunctions. We prove existence of the sequence generated by the algorithm in several cases in Hadamard spaces. Next, we introduce the resolvent of a bifunction in Hadamard spaces. We prove convergence of the resolvent to an equilibrium point. We also prove $\triangle$-convergence of the sequence generated by the proximal point algorithm to an equilibrium point of the pseudo-monotone bifunction and also the strong convergence under additional assumptions on the bifunction. Finally, we study a regularization of Halpern type and prove the strong convergence of the generated sequence to an equilibrium point without any additional assumption on the pseudo-monotone bifunction. Some examples in fixed point theory and convex minimization are also presented.

Keywords

Hadamard spaceequilibrium problemHalpern regularizationproximal point algorithmpseudo-monotone bifunctionstrong convergence△-convergence

MSC classification

Primary: 47H05: Monotone operators and generalizations 47J20: Variational and other types of inequalities involving nonlinear operators (general) 47J25: Iterative procedures 49J40: Variational methods including variational inequalities
Secondary: 90C25: Convex programming 65K10: Optimization and variational techniques
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 110 , Issue 2 , April 2021 , pp. 220 - 242
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Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by G. Willis

References

Ahmadi Kakavandi, B., ‘Weak topologies in complete CAT(0) spaces’, Proc. Amer. Math. Soc. 141 (2013), 1029–1039.CrossRefGoogle Scholar
Ahmadi Kakavandi, B. and Amini, M., ‘Duality and subdifferential for convex function on CAT(0) metric spaces’, Nonlinear Anal. 73 (2010), 3450–3455.CrossRefGoogle Scholar
Bacak, M., ‘The proximal point algorithm in metric spaces’, Israel J. Math. 194 (2013), 689–701.CrossRefGoogle Scholar
Bacak, M., Convex Analysis Optimization in Hadamard Spaces, De Gruyter Series in Nonlinear Analysis and Applications, 22 (De Gruyter, Berlin, 2014).CrossRefGoogle Scholar
Bacak, M. and Reich, S., ‘The asymptotic behavior of a class of nonlinear semigroups in Hadamard spaces’, J. Fixed Point Theory Appl. 16 (2014), 189–202.CrossRefGoogle Scholar
Bento, G. C., Ferreira, O. P. and Oliveira, P. R., ‘Local convergence of the proximal point method for a special class of nonconvex functions on Hadamard manifolds’, Nonlinear Anal. 73 (2010), 564–572.CrossRefGoogle Scholar
Berg, I. D. and Nikolaev, I. G., ‘On a distance between directions in an Alexandrov space of curvature ≤K’, Michigan Math. J. 45 (1998), 275–289.Google Scholar
Berg, I. D. and Nikolaev, I. G., ‘Quasilinearization and curvature of Alexandrov spaces’, Geom. Dedicata 133 (2008), 195–218.CrossRefGoogle Scholar
Bianchi, M. and Schaible, S., ‘Generalized monotone bifunctions and equilibrium problems’, J. Optim. Theory Appl. 90 (1996), 31–43.CrossRefGoogle Scholar
Bridson, M. R. and Haefliger, A., Metric Spaces of Non-positive Curvature (Springer, Berlin, 1999).CrossRefGoogle Scholar
Burago, D., Burago, Y. and Ivanov, S., A Course in Metric Geometry, Graduate Studies in Mathematics, 33 (American Mathematical Society, Providence, RI, 2001).CrossRefGoogle Scholar
Chadli, O., Chbani, Z. and Riahi, H., ‘Equilibrium problems with generalized monotone bifunctions and applications to variational inequalities’, J. Optim. Theory Appl. 105 (2000), 299–323.Google Scholar
Chaoha, P. and Phon-on, A., ‘A note on fixed point sets in CAT(0) spaces’, J. Math. Anal. Appl. 320 (2006), 983–987.CrossRefGoogle Scholar
Cholamjiak, P., ‘The modified proximal point algorithm in CAT(0) spaces’, Optim. Lett. 9 (2015), 1401–1410.CrossRefGoogle Scholar
Colao, V., López, G., Marino, G. and Martín-Márquez, V., ‘Equilibrium problems in Hadamard manifolds’, J. Math. Anal. Appl. 388 (2012), 61–77.CrossRefGoogle Scholar
Combetes, P. L. and Hirstoaga, S. A., ‘Equilibrium programming in Hilbert spaces’, J. Nonlinear Convex Anal. 6 (2005), 117–136.Google Scholar
Dehghan, H. and Rooin, J., ‘A characterization of metric projection in Hadamard spaces with applications’. arXiv:1311.4174.Google Scholar
Dhompongsa, S. and Panyanak, B., ‘On △-convergence theorems in CAT(0) spaces’, Comput. Math. Appl. 56 (2008), 2572–2579.CrossRefGoogle Scholar
Hadjisavvas, N. and Khatibzadeh, H., ‘Maximal monotonicity of bifunctions’, Optimization 59 (2010), 147–160.CrossRefGoogle Scholar
Hadjisavvas, N., Schaible, S. and Wong, N. C., ‘Pseudomonotone operators: a survey of the theory and its applications’, J. Optim. Theory Appl. 152 (2012), 1–20.CrossRefGoogle Scholar
Iusem, A. N., Kassay, G. and Sosa, W., ‘On certain conditions for the existence of solutions of equilibrium problems’, Math. Program. Ser. B 116 (2009), 259–273.CrossRefGoogle Scholar
Iusem, A. N. and Sosa, W., ‘New existence results for equilibrium problems’, Nonlinear Anal. 52 (2003), 621–635.CrossRefGoogle Scholar
Iusem, A. N. and Sosa, W., ‘On the proximal point method for equilibrium problems in Hilbert spaces’, Optimization 59 (2010), 1259–1274.CrossRefGoogle Scholar
Khatibzadeh, H., Mohebbi, V. and Alizadeh, M. H., ‘On the cyclic pseudomonotonicity and the proximal point algorithm’, Numer. Algebra Control Optim. 8 (2018), 441–449.CrossRefGoogle Scholar
Khatibzadeh, H. and Ranjbar, S., ‘A variational inequality in complete CAT(0) spaces’, J. Fixed Point Theory Appl. 17 (2015), 557–574.CrossRefGoogle Scholar
Kirk, W. A., ‘Geodesic geometry and fixed point theory. II’, in: Int. Conf. Fixed Point Theory and Applications (Yokohama Publishers, Yokohama, Japan, 2004), 113–142.Google Scholar
Kirk, W. A. and Panyanak, B., ‘A concept of convergence in geodesic spaces’, Nonlinear Anal. 68 (2008), 3689–3696.CrossRefGoogle Scholar
Li, C., Lopez, G. and Martin-Marquez, V., ‘Monotone vector fields and the proximal point algorithm on Hadamard manifolds’, J. Lond. Math. Soc. 79 (2009), 663–683.CrossRefGoogle Scholar
Li, C., Lopez, G., Martin-Marquez, V. and Wang, J. H., ‘Resolvent of set valued monotone vector fields in Hadamard manifolds’, Set-Valued Anal. 19 (2011), 361–383.CrossRefGoogle Scholar
Mansour, M. A., Chbani, Z. and Riahi, H., ‘Recession bifunction and solvability of noncoercive equilibrium problems’, Commun. Appl. Anal. 7 (2003), 369–377.Google Scholar
Noor, M. A. and Noor, K. I., ‘Some algorithms for equilibrium problems on Hadamard manifolds’, J. Inequal. Appl. (2012), 2012:230, 8 pp.CrossRefGoogle Scholar
Papadopoulos, A., Metric Spaces, Convexity and Nonpositive Curvature, IRMA Lectures in Mathematics and Theoretical Physics, 6 (European Mathematical Society (EMS), Zürich, 2005).Google Scholar
Papa Quiroz, E. A. and Oliveira, P. R., ‘Proximal point method for minimizing quasiconvex locally Lipschitz functions on Hadamard manifolds’, Nonlinear Anal. 75 (2012), 5924–5932.CrossRefGoogle Scholar
Saejung, S. and Yotkaew, P., ‘Approximation of zeros of inverse strongly monotone operators in Banach spaces’, Nonlinear Anal. 75 (2012), 742–750.CrossRefGoogle Scholar
Tang, G. and Xiao, Y., ‘A note on the proximal point algorithm for pseudomonotone variational inequalities on Hadamard manifolds’, Adv. Nonlinear Var. Inequal. 18 (2015), 58–69.Google Scholar
Tang, G., Zhou, L. and Huang, N., ‘The proximal point algorithm for pseudomonotone variational inequalities on Hadamard manifolds’, Optim. Lett. 7(4) (2013), 779–790.CrossRefGoogle Scholar
Wang, J. H., Lopez, G., Martin-Marquez, V. and Li, C., ‘Monotone and accretive vector fields on Riemannian manifolds’, J. Optim. Theory Appl. 146 (2010), 691–708.CrossRefGoogle Scholar