Box dimension of the graphs of the generalized Weierstrass-type functions

For a Lipschitz $\mathbb{Z}-$periodic function $\phi :\mathbb{R}\to {\mathbb{R}}^2$ satisfied that ${\mathbb{R}}^2\setminus \{\phi (x):x\in \mathbb{R}\}$ is not connected, an integer $b\ge 2$ and $\lambda \in (c\mathrm{/}{b}^{\frac{1}{2}},1)$, we prove the following for the generalized Weierstrass-type function $W(x)={\sum }_{n=0}^{\mathrm{\infty }}{\lambda }^n\phi (b^{n}x)$: the box dimension of its graph is equal to $3+2{\log }_b\lambda$, where $c$ is a constant depending on $\phi$.