Geometrical coefficients and the structure of the fixed-point set of asymptotically regular mappings in Banach spaces

It is shown that if EE is a separable and uniformly convex Banach space with Opial’s property and CC is a nonempty bounded closed convex subset of EE, then for some asymptotically regular self-mappings of CC the set of fixed points is not only connected but even a retract of CC. Our results qualitatively complement, in the case of a uniformly convex Banach space, a corresponding result presented in [T. Domínguez, M.A. Japón, G. López, Metric fixed point results concerning measures of noncompactness mappings, in: W.A. Kirk, B. Sims (Eds.), Handbook of Metric Fixed Point Theory, Kluwer Acad. Publishers, Dordrecht, 2001, pp. 239–268].