\lim\limits_{t \to \infty}   \bigl\Vert S^{(w)}_{(00)}(t)\widehat{\mathring{u}}\bigr\Vert_{Y^{(w)}}   = \lim\limits_{t \to \infty}   \bigl\Vert S^{(w)}_{(00)}(t)\widehat{\mathring{u}}\bigr\Vert_{\widehat{w}}   = 0,$$ 