Variational Principles in Gauge Spaces

Abstract

In Part 1, we show that the 1993 Cârjă–Ursescu Ordering Principle [An. Şt. Univ. “A. I. Cuza” Iasi (Mat.) 39, 367–396 (1993)] is equivalent with the Dependent Choice Principle (DC); and as such, equivalent with Ekeland’s Variational Principle (EVP) [J. Math. Anal. Appl. 47, 324–353 (1974)]. This conclusion is valid for all intermediary principles; hence, in particular, for the Brezis–Browder’s (BB) [Adv. Math. 21, 355–364 (1976)]. In Part 2, it is established that the vectorial Zhu–Li Variational Principle in Fang spaces is in the logical segment between (BB) and (EVP); hence, it is equivalent with both (BB) and (EVP). In particular, the conclusion is applicable to Hamel’s Variational Principle (HVP); moreover, a proof of (HVP) ⇔ (EVP) is provided, by means of a direct approach that avoids (DC). Finally, in Part 3, we show that the gauge Brezis–Browder Principle in Turinici [Bull. Acad. Pol. Sci. (Math.) 30, 161–166 (1982)] is obtainable from (DC) and implies (EVP); hence, it is equivalent with both (DC) and (EVP). This is also true for the gauge variational principle deductible from it, including the one in Bae, Cho, and Kim [Bull. Korean Math. Soc. 48, 1023–1032 (2011)].