A Variational Problem Governed by a Differential Inclusion in a Banach Space

Let $\mathcal{X} $ be a real separable reflexive Banach space. A correspondence $(=\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{t}\mathrm{i}-$ valued mapping) $\Gamma $ : $[0, T]\cross \mathcal{X}arrow \mathcal{X} $ and a function $u $ : $[0,T]\cross \mathcal{X}\cross \mathcal{X}arrow\overline{\mathrm{R}} $ are assumed to be given. A double $\mathrm{a}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{w}arrow \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{s} $ the domain and the range of a correspondence. The compact interval $[0,T] $ is endowed with the Lebesgu