Fixed point theorems for strongly inward non-self mappings defined on non-convex domains (Q1933905)

The main result of this paper is the following. Theorem 3.2. Let \(K\) be a non-empty weakly compact subset of a uniformly convex Banach space. Let \(T_1:K\to X\) and \(T_2:K\to X\) be \(\lambda\)-strongly inward (for some \(\lambda\in (0,1]\)) and nonexpansive mappings satisfying the following conditions: {\parindent=8mm \begin{itemize}\item[(i)] \(\|T_1x-T_2y \|\leq \|x-y\|\) for all \(x,y\in K\) with \(x\neq y\); \item[(ii)] \(\|T_1x-T_2x\|\leq \max\{\|x-T_1x\|, \|x-T_2x\| \}\) for each \(x\in K\); \item[(iii)] \(T_i(\partial_{\lambda}[K, T_1])\subset K\) for \(i\in \{1,2 \}\). \end{itemize}} Then \(T_1\) and \(T_2\) have a common fixed point in \(K\).