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 $ such that $ \mathbb{R}^2\setminus\{\phi(x):x\in\mathbb{R}\} $ is not connected, an integer $ b\ge 2 $ and $ \lambda\in (c/{b^{\frac12}},1) $, we prove the following for the generalized Weierstrass-type function $ W(x) = \sum\limits_{n = 0}^{\infty}{{\lambda}^n\phi(b^nx)} $: the box dimension of its graph is equal to $ 3+2\log_b\lambda $, where $ c $ is a constant depending on $ \phi $.