A fixed point result for mappings with contractive iterate at a point in \(G\)-metric spaces (Q2853230)

Let \((X,G)\) be a complete \(G\)-metric space in the sense of \textit{Z. Mustafa} and \textit{B. Sims} [J. Nonlinear Convex Anal. 7, No. 2, 286--297 (2006; Zbl 1111.54025)] and let \(f:X\to X\) be a mapping. Suppose that there exists \(B\subset X\) such that: (a)~\(f(B)\subset B\); (b)~for some \(x_0\in B\), the limit of any sequence in \(\mathcal O(f;x_0)\) (the orbit of \(f\) at \(x_0\)), if it exists, belongs to~\(B\); (c)~for some \(q\in(0,1)\) and each \(x\in\mathcal O(f;x_0)\), there is an integer \(n(x)>1\) such that \(G(f^{n(x)}z,f^{n(x)}x,f^{n(x)}x)\leq qG(z,x,x)\) for all \(z\in B\). The authors prove that there exists a unique \(u\in B\) such that \(fu=u\) and \(f^ky_0\to u\) as \(k\to\infty\) for each \(y_0\in B\). Some consequences are derived. No example is given.