Generalized Nadler \(G\)-contraction in cone metric spaces over Banach algebras endowed with a graph (Q2811559)

Let \((X,d)\) be a cone metric space over a solid cone \(P\) in a Banach algebra \(B\), let \(A\) be a collection of nonempty subsets of~\(X\), and let \(H:A\times A\to B\) be a \(H\)-cone metric in the sense of \textit{D. Wardowski} [Appl. Math. Lett. 24, No. 3, 275--278 (2011; Zbl 1206.54067)]. Suppose further that \(G=(V_G,E_G)\) is a graph such that its set of vertices \(V_G=X\) and the set of edges \(E_G\) contains all loops. The author calls a mapping \(T:X\to A\) a generalized Nadler \(G\)-contraction if (i)~there exists \(a\in P\) such that \(\rho(a)<1\), and \(H(Tx,Ty)\preceq ad(x,y)\), for all \(x,y\in X\); (ii)~if \(u\in Tx\), \(v\in Ty\) are such that \(d(u,v)\preceq d(x,y)\), then \((u,v)\in E_G\). He obtains sufficient conditions for the existence of fixed point for such mappings. The result is illustrated by an example.NEWLINENEWLINEEditor's remark: Meanwhile, an erratum has appeared in [ibid. 10, No. 1, 165--166 (2019; Zbl 1442.54050)].