Assume X is a rotund Banach space with Eisenfeld-Lakshmikantham measure of nonconvexity \(\nu \) . Let \(Y\subset X\) be nonvoid and bounded, although not necessarily convex. Then, every isometric self-map \(f:Y\rightarrow Y\) for which \(\lim _{n\rightarrow \infty }\nu (f^n(Y))=0\) has a fixed point, under either one of two additional requirements: (a) Y is weakly compact; (b) X is reflexive, Y is closed and \(\lim _{n\rightarrow \infty }\nu (\widehat{Y}_n)=0\) , where, for each \(n\in \mathbb {N}\) , \(\widehat{Y}_n\) consists of all those \(z\in Y\) satisfying \(\rho (Y)\le \rho (z)\le \rho (Y)+n^{-1}\) and \(\rho (Y)\) denotes the Chebyshev radius of Y. More precisely, if (b) holds then Y has a unique Chebyshev center c, which is fixed by any such isometry f. Thus, previous results of Lim et al. (2003) and of Gordon (2020) are generalized by weakening the hypotheses on X (just rotundity and reflexivity instead of uniform convexity) and/or dropping the condition that Y be convex.
