Fixed point theorems for some discontinuous operators (Q800615)

Let K be a closed subset of a Banach space X and \(T:K\to K\) be a not necessarily continuous map satisfying the condition \(\| Tx-Ty\|\leq a\| x-y\| +b\{\| x-Tx\| +\| y-Ty\|\}\) for all x,\(y\in K\), where \(0\leq a,b<1\). These operators are refered as belonging to the class \(D(a,b).\) The existence and uniqueness of fixed points for such operators is established. This generalizes results of Kirk, Goebel, Shimi, Hardy and Rogers et al., who assume X to be uniformly convex, T continuous and \(a+2b\leq 1\). Some applications to differential and integral operators are also considered.