On sufficient conditions in the classical problem of the calculus of variations

2025/12/26 by Misir Jumail ogly Mardanov, Mardanov, Misir J., T. K. Melikov +3
Computer Science · Mathematics · #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Nonlinear Differential Equations Analysis #Optimization and Control (math.OC) #Optimization and Variational Analysis

paper · doi:10.48550/arxiv.2512.21918

openalex publication_date 2025/12/26 · openalex created_date 2025/12/30 · openalex updated_date 2026/07/28

Abstract

This article is devoted to obtain new sufficient conditions for an extremum in problems of classical calculus of variations. The concept of a set of integrands is introduced. Using this concept, first and second order sufficient conditions for a weak and strong local minimum, as well as an absolute minimum were obtained. Also, this concept, in particular, allows us to define a class of variation problems for which the necessary Weierstrass condition is also sufficient condition. It is shown that the sufficient conditions for a minimum obtained here have new areas of application compared to the known sufficient conditions of the classical calculus of variations. The effectiveness of the obtained results is illustrated by examples.

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