(𝜑1; 𝜑2)-variational principle

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 (𝜑1; 𝜑2)-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 innite dimensions, J. Funct. Anal. 111 (1993) 197-212].