On the comparison method in the study of problems of minimization of functionals

The problem of minimization of a functional U, defined on the metric space (X, ρ) (where ρ is a metric on the non-empty set X), is considered.It is assumed that the values of the functional U are bounded from below by some number γ, i.e., U(x) ≥ γ for all x ∈ X.This functional is compared with a "model" function u, which is continuous and decreasing on the interval [0, r] and such that u(0) = U(x 0 )γ for some x 0 ∈ X and u(r) = 0. Conditions which guarantee that the functional U has a minimum at some point x ∈ X, with ρ(x 0 , x) ≤ r are obtained.It is shown that the theorems on the minimum of functional that use Caristi-type conditions can be derived from the obtained statement.Applications of the obtained statement to the study of fixed points of mappings in metric spaces are also provided.