(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 infinite dimensions, j. funct. anal. 111 (1993) 197–212].