$(varphi_1, varphi_2)$-variational principle

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 $left(varphi_1, varphi_2right)$-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].