Existence of fixed points of nonexpansive mappings satisfying the Rothe condition

In this paper we consider the following situation: H is a Hilbert space, A is a nonempty bounded closed (not necessarily convex) subset of H, f:DHH is a nonexpansive mapping andAD. In the basic result (Theorem 1) it is shown that in this situation the nonexpansive map f has a fixed point in A, if f satisfies the Rothe condition on A:f(A)A.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 122, pp. 5–12, 1982.