Fixed point theorems for nonself single-valued almost contractions (Q2863548)

Let \(X\) be a Banach space, \(K\) a nonempty closed subset of \(X\) and \(T : K \to X\) a non-self map. Let \(x \in K\) with \(Tx \notin K\) and let \(y \in \partial K\) be the corresponding elements given by \(d(x, Tx) = d(x, y) + d(y, Tx)\), \(y \in \partial K\). If, for any such elements \(x\), we have \(d(y, Ty) < d(x, Tx)\) for at least one of the corresponding \(y \in Y\), then \(T\) is said to have property \((M)\). Suppose that \(T\) satisfies the condition \(d(Tx, Ty) \leq \delta dx, y) + Ld(y, Tx)\) for all \(x, y \in K\). The authors prove that, if \(T\) has property \((M)\) and satisfies the boundary condition \(T(\partial K) \subset K\), then \(T\) has a fixed point in \(K\).