$({\phi }_1,{\phi }_2)-$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 $\left(\varphi_1, \varphi_2\right)-$convexfunction$g, $witharbitrarilysmallnorm,suchthat$f + g $attainsitsstrongminimumon$X. $ This result extends some of the well-known varitional principles as that of Ekeland [18], that of Borwein-Preiss [6] and that of Deville-Godefroy-Zizler [14, 15].