Abstract
Recently, Du (J Nonlinear Anal 72:2259–2261, 2010) by using a nonlinear scalarization function, in the setting of locally convex topological vector spaces, could transfer a cone metric space to a usual metric space. Simultaneously, Amini-Harandi and Fakhar (Com Math Appl 59:3529–3534, 2010) by using a notion of base for the cone , in the setting of Banach spaces, could do the same. In this note we will see that two methods coincide and moreover they are valid for topological vector spaces and it is not necessary that we only consider the cones which have a compact base. Finally, it is worth noting that the nature of this note is similar to Caglar and Ercan (Order-unit-metric spaces, arXiv:1305.6070 [math.FA], 2013).
Similar content being viewed by others
References
Amini-Harandi, A., Fakhar, M.: Fixed point theory incone metric spaces obtained via the scalarization method. Com. Math. Appl. 59, 3529–3534 (2010)
Caglar, M., Ercan, Z.: Order-unit-metric spaces, arXiv:1305.6070 [math.FA] (2013)
Chen, G.Y., Yang, X.Q., Yu, H.: A nonlinear scalarization function and generalized quasi-vector equilibrium. J. Glob. Optim. 32, 451–466 (2005)
Deimling, K.: Nonlinear functional analysis. Springer, Berlin (1988)
Du, W.-S.: A note on cone metric fixed point theory and its equivalence. Nonlinear Anal. 72, 2259–2261 (2010)
Gerstewitz (Tammer), Chr.: Nichtkonvexe Dualitt in der Vektoroptimierung, Wiss. Zeitschr. TH Leuna-Merseburg. 25, 357–364 (1983)
Gerstewitz ( Tammer), Chr., Weinder, P.:Nonconvex separation theorems and some applications in vector optimization, J. Optim. Theory Appl. 67(2), 297–320 (1990)
Huang, L.G., Zhang, X.: Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. 32, 1468–1476 (2007)
Kantorovich, L.V.: The majorant principle and Newton’s method. Dokl. Akad. Nauk SSSR (NS) 76, 17–20 (1951)
Rudin, W.: Functional Analysis. McGraw-Hill company, USA (1972)
Vandergraft, J.S.: Newton’s method for convex operators in partially ordered spaces. SIAM J. Numer. Anal. 4, 406–432 (1967)
Zabreiko, P.P.: \(K-\)metric and \(K-\)normed spaces: survey. Collect. Math. 48(4–6), 825–859 (1997)
Acknowledgments
The author is very thankful to referee for valuable comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Farajzadeh, A.P. On the scalarization method in cone metric spaces. Positivity 18, 703–708 (2014). https://doi.org/10.1007/s11117-013-0271-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11117-013-0271-3