On the existence of continuous selections of a multivalued mapping related to the problem of minimizing a functional

The article considers a parametric problem of the form f(x , y)→ inf, x∈M , where M is a convex closed subset of a Hilbert or uniformly convex space X , y is a parameter belonging to a topological space Y . For this problem, the set of ϵ -optimal points is given by a ϵy = x ∈ M | f ( x , y )≤ infx ∈ M fx , y + ϵ , where ϵ>0 . Conditions for the semicontinuity and continuity of the multivalued mapping a ϵ are discussed. Using gradient projection and linearization methods, we obtain theorems on the existence of continuous selections of the multivalued mapping a ϵ . One of the main assumptions of these theorems is the convexity of the functional f(x,y) with respect to the variable x on the set M and continuity of the derivative fx ' ( x,y ) on the set M×Y . Examples that confirm the significance of the assumptions made are given, as well as examples illustrating the application of the obtained statements to optimization problems.

strictly convex functions, projection operator, fixed points of a mapping, multivalued mapping, continuous selections, set of ϵ-optimal points