Evgeny Zhukovskiy, Evgenii Burlakov, On the comparison method in the study of problems of minimization of functionals, Vol. 2026 (2026), No. 14, pp. 1-9

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DOI: 10.23952/cot.2026.14

Received October 8, 2024; Accepted February 14, 2025; Published online February 2, 2026

Abstract. The problem of minimization of a functional U, defined on the metric space $(X,\rho)$ (where $\rho$ 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 $\gamma,$ i.e., $U(x) \geq \gamma$ for all $x \in X$ . This functional is compared with a “model” function u, which is continuous and decreasing on the interval $[0,r]$ and such that $u(0) = U(x_0) - \gamma$ for some $x_0 \in X$ and $u(r) = 0$ . Conditions which guarantee that the functional U has a minimum at some point $x \in X,$ with $\rho(x_0,x) \leq r$ 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.

How to Cite this Article:
E. Zhukovskiy, E. Burlakov, On the comparison method in the study of problems of minimization of functionals, Commun. Optim. Theory 2026 (2026) 14.