Fixed point theorem on two complete cone metric spaces
Let \((X,d_1)\) and \((Y,d_2)\) be two cone metric spaces in the sense of [\textit{L.-G. Huang} and \textit{X. Zhang}, J. Math. Anal. Appl. 332, No. 2, 1468--1476 (2007; Zbl 1118.54022)] and let \(T:X\to Y\) and \(S:Y\to X\) be two mappings such that, for \(x\in X\) and \(y\in Y\), there exist \[ u\in\{d_1(x,Sy),d_2(y,Tx),d_2(y,TSy)\} \] and \[ v\in\{d_2(y,Tx),d_1(x,Sy),d_1(x,STx)\} \] satisfying \[ d_2(Tx,TSy)\leq\lambda u \] and \[ d_1(Sy,STx)\leq\lambda v, \] where \(0<\lambda<1\). The authors prove that \(ST\) and \(TS\) have unique fixed points in \(X\) and \(Y\), respectively. A similar assertion is proved when four mappings are given. An example is provided to demonstrate the use of these results.
scientific article; zbMATH DE number 6107672
scientific article