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Distance in cone metric spaces and common fixed point theorems. (English) Zbl 1230.54048
Let $(X,d)$ be a cone metric space over a normal cone $P$ in a real Banach space $E$ 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)]. Let $q:X\times X\to E$ satisfy the following properties: (q1) $\theta\preceq q(x,y)$ for all $x,y\in X$; (q2) $q(x,z)\preceq q(x,y)+q(y,z)$ for all $x,y,z\in X$; (q3) if $\{y_n\}$ is a sequence in $(X,d)$ converging to $y\in X$ and for some $x\in X$ and $u=u_x\in P$, $q(x,y_n)\preceq u$ for each $n\geq1$, then $q(x,y)\preceq u$; (q4) for each $c\in E$ with $\theta\ll c$, there exists $e\in E$ with $\theta\ll e$, such that $q(z,x)\ll e$ and $q(z,y)\ll e$ imply $d(x,y)\ll c$. The function $q$ is called a $c$-distance in $(X,d)$ (this notion is a cone metric version of the notion of $\omega$-distance of {\it O. Kada, T. Suzuki} and {\it W. Takahashi} [Math. Japon. 44, No. 2, 381--391 (1996; Zbl 0897.54029)]). The authors prove the following common fixed point result in terms of a $c$-distance. Let $a_i\in(0,1)$, $i=1,2,3,4$ be constants with $a_1+2a_2+a_3+a_4<1$ and let $f,g:X\to X$ be two mappings satisfying the condition $q(fx,fy)\preceq a_1q(gx,gy)+a_2q(gx,fy)+a_3q(gx,fx)+a_4q(gy,fy)$ for all $x,y\in X$. Suppose that $gX\subset fX$ and $gX$ is a complete subset of $X$. If $f$ and $g$ satisfy $\inf\{\|q(fx,y)\|+\|q(gx,fy)\|+\|q(gx,fx)\|:x\in X\}>0$ for all $y\in X$ with $y\ne fy$ or $y\ne gy$, then $f$ and $g$ have a common fixed point in $X$. No compatibility assumptions have to be used.

54H25Fixed-point and coincidence theorems in topological spaces
Full Text: DOI
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[18] Kada, O.; Suzuki, T.; Takahashi, W.: Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. japon. 44, 381-391 (1996) · Zbl 0897.54029
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