Some generalizations of Kannan's fixed point theorem in \(K\)-metric spaces (Q2845231)

Let \(K\) be a normal (but not necessarily solid) cone in a Banach space \((V,\|\cdot\|)\) and let \((X,d_K)\) be a complete \(K\)-metric space in the sense of \textit{P. P. Zabrejko} [Collect. Math. 48, No. 4--6, 825--859 (1997; Zbl 0892.46002)] (which is 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)]. If \(T,S:X\to X\) are such that \(d_K(T^px,S^qy)\leq Q(d_K(x,T^px))+R(d_K(y,S^qy))\) for some positive integers \(p,q\) and for all \(x,y\in X\), where \(Q\) and \(R\) are positive bounded linear operators on \(V\) such that \(\|Q\|+\|R\|<1\), the authors prove that then \(T\) and \(S\) have a unique common fixed point in~\(X\). In the case when \(T=S\), the result is proved in a more general setting of \(N\)-polygonal \(K\)-metric spaces in the sense of \textit{A. Azam, M. Arshad} and \textit{I. Beg} [Appl.\ Anal.\ Discrete Math., 3, No.\ 2, 236--241 (2009; Zbl 1274.54113)].