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MONOTONE AND PSEUDO-MONOTONE EQUILIBRIUM PROBLEMS IN HADAMARD SPACES

Part of: Existence theories Equations and inequalities involving nonlinear operators 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: [email protected]
*
e-mail: [email protected]
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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

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