We generalize an abstract variational principle in Banach spaces, introduced by Topalova & Zlateva [3], by showing that the set $\mathbb{P}_{0}$ of perturbations for which a perturbed lower semi-continuous function $f$ is WPMC (Well Posed Modulus Compact) (equivalently, well posed in generalized sense in [1] and [2]) not only contains a dense $G_{\delta }$ subset, but is also a complement to a $\sigma $ -porous subset in a specifically defined positive cone. Moreover, if the space is a Musielak-Orlicz sequence space satisfying $\ell _{\Phi }\cong h_{\Phi }$ , then the notion WPMC is replaced by the stronger notion of Tikhonov well posedness, which is proved to be equivalent to the single-valuedness and upper semi-continuity of the multivalued mapping assigning a parameter to the solution set. We give several applications. The first one is that the Musielak-Orlicz sequence spaces have the Radon-Nikodym property and, therefore, are dentable by proving the validity of Stegall’s variational principle. As a consequence we obtain that the duals of Musielak-Orlicz sequence spaces are $w^{*}$ -Asplund. We establish also a sufficient condition for Musielak-Orlicz and Nakano sequence spaces to be Asplund spaces. The next applications are for determining the type of the smoothness of certain Musielak-Orlicz, Nakano, and weighted Orlicz sequence spaces. We illustrate by an example that it is possible to consider an Orlicz function without the $\Delta _{2}$ condition, by a particular choice of the weighted sequence $\{w_{n}\}_{n=1}^{\infty }$ to get $\ell _{M}(w)\cong h_{M}(w)$ and to be able to apply the main result.
