Institute of Mathematics

Some fixed point theorems in logarithmic convex structures

Alireza Moazzen, Yoel-Je Cho, Choonkil Park, Madjid Eshaghi Gordji

Received November 18, 2014.  First published October 17, 2016.

Abstract:  In this paper, we introduce the concept of a logarithmic convex structure. Let $X$ be a set and $D\colon X\times X\rightarrow[1,\infty)$ a function satisfying the following conditions: \item{(i)} For all $x,y\in X$, $D(x,y)\geq1$ and $D(x,y)=1$ if and only if $x=y$. \item{(ii)} For all $x,y\in X$, $D(x,y)=D(y,x)$. \item{(iii)} For all $x,y,z\in X$, $D(x,y)\leq D(x,z)D(z,y)$. \item{(iv)} For all $x,y,z\in X$, $z\neq x,y$ and $\lambda\in(0,1)$, \begin{gather} D(z,W(x,y,\lambda))\leq D^\lambda(x,z)D^{1-\lambda}(y,z),\nonumber
D(x,y)= D(x,W(x,y,\lambda))D(y,W(x,y,\lambda)),\nonumber\end{gather} where $W X\times X\times[0,1]\rightarrow X$ is a continuous mapping. We name this the logarithmic convex structure. In this work we prove some fixed point theorems in the logarithmic convex structure.
Keywords:  fixed point; logarithmic convex structure; convex metric space
Classification MSC:  47H09, 47H10, 54H25
DOI:  10.21136/MB.2017.0074-14

References:
[1] S. S. Chang, Y. J. Cho, S. M. Kang: Nonlinear Operator Theory in Probabilistic Metric Spaces. Nova Science Publishers Huntington (2001). MR 2018691 | Zbl 1080.47054
[2] L. B. Ćirić: On some discontinuous fixed point mappings in convex metric spaces. Czech. Math. J. 43 (1993), 319-326. MR 1211753 | Zbl 0814.47065
[3] M. D. Guay, K. L. Singh, J. H. M. Whitfieled: Fixed point theorems for nonexpansive mappings in convex metric spaces. Nonlinear Analysis and Applications. Proc. Int. Conf. at Memorial University of Newfoundland, 1981 S. P. Singh at al. Lect. Notes Pure Appl. Math. 80, Marcel Dekker, New York (1982), 179-189. MR 0689554 | Zbl 0501.54030
[4] H. V. Machado: A characterization of convex subsets of normed spaces. Kōdai Math. Semin. Rep. 25 (1973), 307-320. DOI 10.2996/kmj/1138846819 | MR 0326359 | Zbl 0271.54021
[5] T. Shimizu, W. Takahashi: Fixed point theorems in certain convex metric spaces. Math. Jap. 37 (1992), 855-859. MR 1186552 | Zbl 0764.47030
[6] W. Takahashi: A convexity in metric spaces and nonexpansive mapping I. Kōdai Math. Semin. Rep. 22 (1970), 142-149. DOI 10.2996/kmj/1138846111 | MR 0267565 | Zbl 0268.54048
[7] L. A. Talman: Fixed points for condensing multifunctions in metric spaces with convex structure. Kōdai Math. Semin. Rep. 29 (1977), 62-70. DOI 10.2996/kmj/1138833572 | MR 0463985 | Zbl 0423.54039

Affiliations:   Alireza Moazzen, Department of Mathematics, Kosar University of Bojnord, Farabi street 41, Bojnord, Iran, e-mail: ar.moazzen@yahoo.com, ar.moazzen@kub.ac.ir; Yeol-Je Cho, Department of Mathematics Education and the Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, and Jaeil Poongkyung Chae Appartment 105-402, Gwaza-Dong, Jinju City 660-701, Korea, e-mail: yjcho@gnu.ac.kr; Choonkil Park, Research Institute for Natural Sciences, Hanyang University, Seoul 04763, and Dosandaero 96Gil 39, 104dong 901ho, GangNam Gu, Seoul 06070, Korea, e-mail: baak@hanyang.ac.kr; Madjid Eshaghi Gordji, Department of Mathematics, Semnan University, Imam Khomaini Street 50, P. O. Box 35195-363, Semnan, Iran, e-mail: meshaghi@semnan.ac.ir

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