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
$$f(x,y)\to \inf, \ \ x\in M,$$
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 $\epsilon$ -optimal points is given by
$$a_{\epsilon}(y)=\{ x\in M \,|\, f(x,y)\leq \inf_{x\in M}f(x,y) +\epsilon\},$$
where $\epsilon>0.$ Conditions for the semicontinuity and continuity of the multivalued mapping $a_{\epsilon}$ are discussed. Using gradient projection and linearization methods, we obtain theorems on the existence of continuous selections of the multivalued mapping $a_{\epsilon}.$ 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 $f'_x(x,y)$ on the set $M\times Y.$ 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.