Abstract
It is shown that if E is a separable and uniformly convex Banach space with Opial’s property and C is a nonempty bounded closed convex subset of E , then for some asymptotically regular self-mappings of C the set of fixed points is not only connected but even a retract of C . 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].
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