Mathematica Bohemica, Vol. 142, No. 1, pp. 1-7, 2017


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


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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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