Some fixed point theorems in logarithmic convex structures

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

Mathematica Bohemica, Tome 142 (2017) no. 1, pp. 1-7.

Résumé

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)\geq 1$ 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 $W\colon X\times X\times [0,1]\rightarrow X$ 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.

Détail

DOI : 10.21136/MB.2017.0074-14

Classification : 47H09, 47H10, 54H25
Mots-clés : fixed point; logarithmic convex structure; convex metric space