Abstract
Some examples are given to show that some known generalized convex spaces are so abstract that some basic properties related to the convexity are lost. In order to improve the convexity structure for applications, the concepts of path-convex space, path-convex set and path-convex function are introduced. And their properties are discussed.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ben-El-Mechaiekh, H., Chebbi, S.: Abstract convexity and fixed points. Journal of Mathematical Analysis and Applications 222, 138–150 (1998)
Ding, X.P.: Maximal element theorems in product FC-spaces and generalized games. Journal of Mathematical Analysis and Applications 305, 29–42 (2005)
Ding, X.P.: Generalized KKM type theorems in FC-spaces with applications. Journal of Global Optimization 36, 581–596 (2006)
Horvath, C.D.: Contractibility and generalized convexity. Journal of Mathematical Analysis and Applications 156, 341–357 (1991)
Horvath, C.D.: Extension and selection theorems in topological spaces with a generalized convexity structure. Annales de Faculté des Sciences de Toulouse 2, 253–269 (1993)
Huang, J.: The matching theorems and coincidence theorems for generalized R-KKM mapping in topological spaces. Journal of Mathematical Analysis and Applications 312, 1374–3821 (2005)
Kindler, J., Trost, R.: Minimax theorems for interval spaces. Acta Mathematica Hungarica 54, 38–49 (1989)
Park, S.: On generalizations of the KKM principle on abstract convex spaces. Nonlinear Analysis Forum 11, 67–77 (2006)
Park, S.: Various subclasses of abstract convex spaces for the KKM theory. Proceedings of the National Institute of Mathematical Science 2, 35–47 (2007)
Park, S.: Elements of the KKM theory on abstract convex spaces. Journal of the Korean Mathematical Society 45, 1–27 (2008)
Park, S.: Equilibrium existence theorems in KKM spaces. Nonlinear Analysis, Theory, Methods and Applications 69, 4352–4364 (2008)
Park, S., Kim, H.: Admissible classes of multifunctions on generalized convex spaces. Proceedings College of Natural Science Seoul National University 18, 1–21 (1993)
Stachó, L.L.: Minimax theorems beyond topological vector spaces. Acta Scientiarum Mathematicarum 42, 157–164 (1980)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fan, X., Cheng, Y. (2011). Note on Generalized Convex Spaces. In: Li, S., Wang, X., Okazaki, Y., Kawabe, J., Murofushi, T., Guan, L. (eds) Nonlinear Mathematics for Uncertainty and its Applications. Advances in Intelligent and Soft Computing, vol 100. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22833-9_46
Download citation
DOI: https://doi.org/10.1007/978-3-642-22833-9_46
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22832-2
Online ISBN: 978-3-642-22833-9
eBook Packages: EngineeringEngineering (R0)