A pair of non-self mappings in cone metric spaces (Q2853294)

Let \((X,\preceq,d)\) be a cone metric space 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)] over a cone which is not necessarily normal. Let \(K\) be a closed nonempty subset of \(X\) such that for all \(x\in K\), \(y\notin K\) there exists \(z\in\partial K\) satisfying \(d(x,z)+d(z,y)=d(x,y)\). Let \(f,g:K\to X\) be two (non-self) mappings such that \(d(fx,fy)\preceq Cu(x,y)+C'(d(fx,gy)+d(fy,gx))\) for some real constants \(C,C'\) and some \(u(x,y)\in\{d(fx,gx),d(fy,gy)\}\); also suppose that \(f,g\) satisfy the following conditions: (i)~\(\partial K\subset gK\), \(fK\cap K\subset gK\); (ii)~\(gx\in\partial K\implies fx\in K\); (iii)~\(gK\) is closed in~\(X\). The author proves that under these conditions, there exists a point of coincidence \(p\) of \(f\) and \(g\) in \(K\). In particular, if the pair \((f,g)\) is coincidentally commuting, then \(p\) is a unique common fixed point of \(f\) and~\(g\).