Some fixed point theorems in Banach space
Theorem 1. Let K be a nonempty weakly compact convex subset of a Banach space X, and \(T:K\to 2^ x\) be a mapping such that the mapping \(x\to d(x,Tx)\) is lower semicontinuous. If inf \(\{\) d(x,Tx);x\(\in K\}=0\) and for all \(x,y\in K\), \(0\leq \lambda \leq 1\), \(u=\lambda x+(1-\lambda)y\) we have d(u,Tu)\(\leq \phi (\max [d(x,Tx),d(y,Ty)])\), where \(\phi:{\mathbb{R}}^+\to {\mathbb{R}}^+\) is nondecreasing, continuous from the right at 0 with \(\phi (0)=0\), \(d(x,Tx)=\inf \{\| x-y\|,y\in Tx\}\). Then T has a fixed point. From this theorem the authors deduce some well known fixed point theorem in Banach space.
weakly compact convex subset of a Banach space
scientific article; zbMATH DE number 4098120
scientific article