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Fixed points of $\left(\psi ,\varphi \right)$-weak contractions on cone metric spaces. (English) Zbl 1221.54048

Let $\left(X,d\right)$ be a cone metric space in the sense of [L-G. Huang and X. Zhang, J. Math. Anal. Appl. 332, No. 2, 1468–1476 (2007; Zbl 1118.54022)] over a regular cone $P$, such that $d\left(x,y\right)\in \text{int}\phantom{\rule{0.166667em}{0ex}}P$ for $x\ne y$.

Let $T:X\to X$ be a mapping satisfying $\psi \left(d\left(Tx,Ty\right)\right)⪯\psi \left(d\left(x,y\right)\right)-\phi \left(d\left(x,y\right)\right)$ for all $x,y\in X$, where $\psi ,\phi :\text{int}\phantom{\rule{0.166667em}{0ex}}P\cup \left\{0\right\}\to \text{int}\phantom{\rule{0.166667em}{0ex}}P\cup \left\{0\right\}$ are continuous and increasing, satisfying: (a) $\psi \left(t\right)=\phi \left(t\right)=0$ iff $t=0$; (b) $t-\psi \left(t\right)\in \text{int}\phantom{\rule{0.166667em}{0ex}}P\cup \left\{0\right\}$, $\phi \left(t\right)\in \text{int}\phantom{\rule{0.166667em}{0ex}}P$ for $t\in \text{int}\phantom{\rule{0.166667em}{0ex}}P$; (c) $\psi \left({t}_{1}+{t}_{2}\right)⪯\psi \left({t}_{1}\right)+\psi \left({t}_{2}\right)$ for ${t}_{1},{t}_{2}\in \text{int}\phantom{\rule{0.166667em}{0ex}}P$ and (d) either $\psi \left(t\right),\phi \left(t\right)⪯d\left(x,y\right)$ or $d\left(x,y\right)⪯\psi \left(t\right),\phi \left(t\right)$ for $t\in \text{int}\phantom{\rule{0.166667em}{0ex}}P$ 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.

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 47H10 Fixed point theorems for nonlinear operators on topological linear spaces
##### Keywords:
cone metric space; weak contraction; common fixed point