Sergey Evgenevich Zhukovskiy

We study the simplest problem of the calculus of variations in the space of absolutely continuous functions. In the considered formulation, it is assumed that the Lagrangian is a lower-bounded continuous function taking values in the extended real line, which means that it may take the value +∞ . The well-posedness of the problem is shown. In the case where the Lagrangian has the special form L(y, y0 ) = α(y)β(y 0 ) , necessary conditions for a weak local minimum and for a strong local minimum are obtained in the form of the Euler equation. An important distinction of the obtained results from those known earlier is as follows. The obtained necessary conditions are applicable in the case where the function y, b at which the minimum is attained has a singularity, i.e., a point x such that L yb(x), yb 0 (x) = +∞ . The application of the obtained results is demonstrated on the example of the brachistochrone problem in the class of absolutely continuous functions. Using the necessary conditions for a minimum obtained in the paper, it is shown that, as in the classical formulation of the brachistochrone problem, the cycloid is the only function that can be the only candidate for an extremum. Since this curve has a singularity at zero, the example illustrates the applicability of the obtained results in the presence of singularities.

Рассматривается включение, порожденное непрерывным отображением, зависящим от параметра, и выпуклыми замкнутыми множествами. Получена теорема о неявной функции для включения в терминах $\lambda$-укорочений, которая справедлива без априорных предположений нормальности. Выведен ряд следствий, в том числе для систем уравнений и неравенств с параметром. Приведено сравнение полученных результатов с известными. Библиография: 16 названий.

Изучается устойчивость седловых точек. Приведена теорема об устойчивости и примеры, показывающие существенность ее предположений. Приведены некоторые обобщения теоремы устойчивости. Определены обобщенно седловые точки и изучены их свойства. Исследована их устойчивость. Библиография: 12 названий.