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Zbl 1221.54048
Aage, C.T.; Salunke, J.N.
Fixed points of $(\psi, \varphi )$-weak contractions on cone metric spaces. (English)
[J] Ann. Funct. Anal. AFA 2, No. 1, 59-71, electronic only (2011). ISSN 2008-8752/e

Let $(X,d)$ be a cone metric space in the sense of [{\it L-G. Huang} and {\it X. Zhang}, J. Math. Anal. Appl. 332, No.~2, 1468--1476 (2007; Zbl 1118.54022)] over a regular cone $P$, such that $d(x,y)\in\text{int}\,P$ for $x\ne y$. Let $T:X\to X$ be a mapping satisfying $\psi(d(Tx,Ty))\preceq\psi(d(x,y))-\phi(d(x,y))$ for all $x,y\in X$, where $\psi,\phi:\text{int}\,P\cup\{0\}\to\text{int}\,P\cup\{0\}$ are continuous and increasing, satisfying: (a)~$\psi(t)=\phi(t)=0$ iff $t=0$; (b)~$t-\psi(t)\in \text{int}\,P\cup\{0\}$, $\phi(t)\in\text{int}\,P$ for $t\in \text{int}\,P$; (c)~$\psi(t_1+t_2)\preceq\psi(t_1)+\psi(t_2)$ for $t_1,t_2\in\text{int}\,P$ and (d)~either $\psi(t),\phi(t)\preceq d(x,y)$ or $d(x,y)\preceq\psi(t),\phi(t)$ for $t\in \text{int}\,P$ and $x,y\in X$. Under these assumptions, the authors prove that $T$ has a unique fixed point in~$X$. A similar result is obtained for the existence of a common fixed point for two self-mappings.
[Zoran Kadelburg (Beograd)]

MSC 2000:: *54H25 Fixed-point theorems in topological spaces
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces

Keywords: cone metric space; weak contraction; common fixed point

Citations: Zbl 1118.54022

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