$(\varphi_1, \varphi_2)$-variational principle

Maaden, Abdelhakim; Abdelkader, Stouti

doi:10.22075/ijnaa.2017.1664.1439

$(\varphi_1, \varphi_2)$-variational principle

Document Type : Research Paper

Authors

Abdelhakim Maaden

Stouti Abdelkader

Universit'e Sultan Moulay Slimane, Facult'e des Sciences et Techniques, Laboratoire de Math'ematiques et Applications, B.P. 523, Beni-Mellal 23000, Marocco

10.22075/ijnaa.2017.1664.1439

Abstract

In this paper we prove that if $X $ is a Banach space, then for every lower semi-continuous bounded below function $f, $ there exists a $\left(\varphi_1, \varphi_2\right)$-convex function $g, $ with arbitrarily small norm, such that $f + g $ attains its strong minimum on $X. $ This result extends some of the well-known varitional principles as that of Ekeland [On the variational principle, J. Math. Anal. Appl. 47 (1974) 323-353], that of Borwein-Preiss [A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. Amer. Math. Soc. 303 (1987) 517-527] and that of Deville-Godefroy-Zizler [Un principe variationel utilisant des fonctions bosses, C. R. Acad. Sci. (Paris). Ser.I 312 (1991) 281--286] and [A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions, J. Funct. Anal. 111 (1993) 197-212].

Keywords

$(\varphi_1, \varphi_2)$-convex function

$(\varphi_1, \varphi_2)$-variational principle

Ekeland's variational principle

smooth variational principle