Abstract
One of the main questions that arise in the investigation of extremal problems is the existence of solutions. The general approach to establishing solvability of extremal problems are typically in the form of sufficient conditions. In some cases, the verification of these conditions is quite complex. In this article, we consider the existence of solutions of the problem f(x) {r_arrow}inf, x {element_of}{Omega} defined by the nonempty closed set {Omega} in the n-dimensional Euclidean space R{sup n} and a continuous function f on {Omega}. We assume that for some set {Omega}{prime} {improper_subset} R{sup n} and some finite-valued continuous function {psi}(x) {le} 0 and the problem f(x){r_arrow}inf x {element_of} {Omega}{prime} has a finite value f{sub *} {equivalent_to} inf/x {element_of} {Omega} and a nonempty solution set X{sub *} {equivalent_to} (x {element_of} {Omega}{prime}{vert_bar}f(x) = f{sub *}{prime}).
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