A Continuous Method for Finding a Generalized Fixed Point of a Nonexpansive Mapping on a Set in a Hilbert Space

We introduce the concept of a generalized fixed point of a nonexpansive operator on aconvex closed set in a Hilbert space. To find this point, we construct a regularizing algorithm inthe form of the Cauchy problem for a first-order differential equation and establish sufficientconditions for the strong convergence of the resulting approximations to a normal generalized fixedpoint under approximate specification of the nonexpansive operator and the convex closed set onwhich the desired generalized fixed point of the operator is located. Examples of parametricfunctions are given that ensure the convergence of the approximations in the norm of the Hilbertspace to a normal generalized fixed point of the operator on the convex closed set in this space.