Cyclic contractions and fixed point theorems (Q2867606)

Let \(K_1\) and \(K_2\) be closed subsets of a Banach space \(X\). Suppose that an operator \(T\colon K_1\cup K_2\to K_1\cup K_2\) with \(T(K_1)\subset K_2\) and \(T(K_2)\subset K_1\) satisfies \(\| Tx-Ty\| \leq a\| x-y\| +b[\| x-Tx\| +\| y-Ty\| ]\) for some \(a,b\in[0,1]\) and all \(x\in K_1\), \(y\in K_2\). The authors prove that a sequence \((x_n)\) in \(K_1\cup K_2\) converges to the unique fixed point of~\(T\) if and only if \(\lim_{n\to\infty}(x_n-Tx_n)=0\). In particular, if \(0\leq a+2b<1\), then \(T\) has a unique fixed point \(p\in K_1\cap K_2\) and \(\| Tx-p\| <\| x-p\| \) for all \(x\in K_1\cup K_2\).