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Zbl 1226.54043
Aydi, H.; Nashine, H.K.; Samet, B.; Yazidi, H.
Coincidence and common fixed point results in partially ordered cone metric spaces and applications to integral equations.
(English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 17, 6814-6825 (2011). ISSN 0362-546X

The main result of the paper under review is the following. Let $(X,\preceq,d)$ be a partially ordered complete cone metric space over a regular cone $P$ in a 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 $T,S,G:X\to X$ be continuous mappings such that $TX\subset GX$, $SX\subset GX$, the pairs $(T,G)$ and $(S,G)$ are compatible, and $T$ and $S$ are $G$-weakly increasing. Finally, let for all $x,y\in X$ such that $Gx$ and $Gy$ are $\preceq$-comparable, the following contractive condition hold: $\psi(d(Tx,Sy))\leq_P \psi(\frac12[d(Tx,Gx)+d(Sy,Gy)])-\varphi(d(Gx,Gy))$, where $\psi:P\to P$ and $\varphi:\text {int}P\cup\{0_E\}\to \text {int}P\cup\{0_E\}$ satisfy certain conditions. Then $T$, $S$ and $G$ have a coincidence point in $u\in X$, that is, $Tu=Su=Gu$ holds. A version of this result is given using so-called regularity of the space $(X,\preceq,d)$. The existence and uniqueness of a common fixed point is obtained under additional assumptions. Finally, as an application, a theorem on existence of a common solution for a pair of integral equations is obtained.
MSC 2000:
*54H25 Fixed-point theorems in topological spaces
54E50 Complete metric spaces
54F05 Ordered topological spaces

Keywords: partially ordered cone metric space; coincidence point; common fixed point, weakly increasing mapping; compatible mappings.

Citations: Zbl 1118.54022

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