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MONOTONE OPERATORS AND THE PROXIMAL POINT ALGORITHM IN COMPLETE CAT(0) METRIC SPACES

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

Published online by Cambridge University Press:  23 September 2016

HADI KHATIBZADEH  and
SAJAD RANJBAR
HADI KHATIBZADEH*
Affiliation:
Department of Mathematics, University of Zanjan, Zanjan, PO Box 45195-313, Iran email [email protected]
SAJAD RANJBAR
Affiliation:
Department of Mathematics, University of Zanjan, Zanjan, PO Box 45195-313, Iran email [email protected] Department of Mathematics, College of Sciences, Higher Education Center of Eghlid, Eghlid, Iran email [email protected]
*
✉[email protected]
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Abstract

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In this paper, we generalize monotone operators, their resolvents and the proximal point algorithm to complete CAT(0) spaces. We study some properties of monotone operators and their resolvents. We show that the sequence generated by the inexact proximal point algorithm $\unicode[STIX]{x1D6E5}$-converges to a zero of the monotone operator in complete CAT(0) spaces. A strong convergence (convergence in metric) result is also presented. Finally, we consider two important special cases of monotone operators and we prove that they satisfy the range condition (see Section 4 for the definition), which guarantees the existence of the sequence generated by the proximal point algorithm.

Keywords

Hadamard spacemonotone operatorproximal point algorithm𝛥-convergencesubdifferentialnonexpansive mapping

MSC classification

Primary: 47H05: Monotone operators and generalizations
Secondary: 47J05: Equations involving nonlinear operators (general) 47J20: Variational and other types of inequalities involving nonlinear operators (general) 65K05: Mathematical programming methods
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 103 , Issue 1 , August 2017 , pp. 70 - 90
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

Footnotes

The authors are grateful to the referees for valuable comments and suggestions. The second author was supported by Higher Education Center of Eghlid in Iran.

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