New multivalued fixed point results in cone \(b\)-metric spaces (Q2812419)

Let \((X,d)\) be a complete cone \(b\)-metric space over the cone \(P\) in a Banach space \(E\), with the coefficient \(r\geq1\), and let \(CB(X)\) be the family of nonempty closed bounded subsets of \(X\). For \(p\in E\), \(a\in X\), and \(A,B\in CB(X)\), denote \(s(p)=\{q\in E: p\preceq q\}\), \(s(a,B)=\bigcup_{b\in B}s(d(a,b))\), and \(s(A,B)=(\bigcap_{a\in A}s(a,B))\cap(\bigcap_{b\in B}s(b,A))\). Suppose that a mapping \(T:X\to CB(X)\) is such that there exists \(u\in[0,1)\) such that, for all \(x,y\in X\), \((1+u)d(x,y)\in s(x,Tx)\) implies that \(ud(x,y)\in s(Tx,Ty)\). The authors prove that, under these assumptions, there exists a fixed point of \(T\) in~\(X\). An example is presented.