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Coincidence and common fixed point results in partially ordered cone metric spaces and applications to integral equations. (English) Zbl 1226.54043
The main result of the paper under review is the following. Let $\left(X,⪯,d\right)$ be a partially ordered complete cone metric space over a regular cone $P$ in a Banach space $E$ in the sense of L.-G. Huang and 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 $\left(T,G\right)$ and $\left(S,G\right)$ are compatible, and $T$ and $S$ are $G$-weakly increasing. Finally, let for all $x,y\in X$ such that $Gx$ and $Gy$ are $⪯$-comparable, the following contractive condition hold: $\psi \left(d\left(Tx,Sy\right)\right){\le }_{P}\psi \left(\frac{1}{2}\left[d\left(Tx,Gx\right)+d\left(Sy,Gy\right)\right]\right)-\varphi \left(d\left(Gx,Gy\right)\right)$, where $\psi :P\to P$ and $\varphi :\text{int}P\cup \left\{{0}_{E}\right\}\to \text{int}P\cup \left\{{0}_{E}\right\}$ 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 $\left(X,⪯,d\right)$. 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:
 54H25 Fixed-point and coincidence theorems in topological spaces 54E50 Complete metric spaces 54F05 Linearly, generalized, and partial ordered topological spaces