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Keywords:
fixed point; logarithmic convex structure; convex metric space
Summary:
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.
References:
 Chang, S. S., Cho, Y. J., Kang, S. M.: Nonlinear Operator Theory in Probabilistic Metric Spaces. Nova Science Publishers Huntington (2001). MR 2018691 | Zbl 1080.47054
 'Cirić, L. B.: On some discontinuous fixed point mappings in convex metric spaces. Czech. Math. J. 43 (1993), 319-326. MR 1211753 | Zbl 0814.47065
 Guay, M. D., Singh, K. L., Whitfieled, J. H. M.: 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
 Machado, H. V.: 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
 Shimizu, T., Takahashi, W.: Fixed point theorems in certain convex metric spaces. Math. Jap. 37 (1992), 855-859. MR 1186552 | Zbl 0764.47030
 Takahashi, W.: 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
 Talman, L. A.: 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