Geodesic
Some fixed point theorems in logarithmic convex structures
Mathematica Bohemica, Tome 142 (2017) no. 1, pp. 1-7
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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.
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.
DOI : 10.21136/MB.2017.0074-14
Classification : 47H09, 47H10, 54H25
Keywords: fixed point; logarithmic convex structure; convex metric space 
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Moazzen, Alireza; Cho, Yoel-Je; Park, Choonkil; Eshaghi Gordji, Madjid. Some fixed point theorems in logarithmic convex structures. Mathematica Bohemica, Tome 142 (2017) no. 1, pp. 1-7. doi: 10.21136/MB.2017.0074-14

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