Some fixed point theorems in logarithmic convex structures

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