Perturbation Method in Musielak-Orlicz Sequence Spaces

We generalize an abstract variational principle in Banach spaces, introduced by Topalova \& Zlateva, 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) 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.