Fixed point theorems for multifunctions with stochastic domain (Q1184884)

The authors prove the following result: Theorem: Let \(K\) be a closed, bounded and convex subset of a Banach space \(X\). Let \(T: K\to K\) satisfy the conditions: (i) \(T^ 2=I\), the identity mapping, (ii) \(\| Tx-Ty\|\leq a\| x-y\|+b[\| x-Tx\|+\| y- Ty\|]\) for all \(x\), \(y\) in \(K\), where \(0\leq a+4b<2\). Then \(T\) has at least one fixed point \(x_ 0\) in \(K\). Further, if \(a<1\), then \(x_ 0\) is unique. Some other related results on common fixed points of two operators are also given.