Fixed point theorems for contractive mappings of Perov type (Q2786413)

Let \((X,d)\) be a cone metric space over a solid cone \(P\) in a Banach space \(E\) and let \(\mathcal B(E)\) be the algebra of linear and bounded operators on \(E\). Further, let \(T:X\to X\) and \(O(z)\) be the orbit of an element \(z\in X\) under the mapping \(T\).NEWLINENEWLINEThe authors prove the following two Perov-type fixed point results. NEWLINENEWLINE(1) If there exist a point \(z\in X\) with \(\overline{O(z)}\) complete and a positive operator \(A\in\mathcal B(E)\) with \(r(A)<1\) such that \(d(Tx,Ty)\preceq A(d(x,y))\) for all \(x,y=Tx\in O(z)\), then \(\{T^nz\}\) converges to some \(u\in\overline{O(z)}\) and NEWLINE\[NEWLINEd(T^nz,u)\preceq A^n(I-A)^{-1}(d(z,Tz))\eqno{(*)}NEWLINE\]NEWLINE for \(n\in\mathbb N\). If, moreover, the contraction condition holds for all \(x,y\in\overline{O(z)}\), then \(u\) is a fixed point of \(T\). NEWLINENEWLINE(2) If, in addition, \(P\) is a normal cone with normal constant \(K\) and the condition \(r(A)<1\) is replaced by \(K\| A\|<1\), then \((*)\) can be replaced by \(\| d(T^nz,u)\|\leq\frac{(K\| A\|)^n}{1-K\| A\|}\| d(z,Tz)\|\).