A class of switched non-quadratic Lyapunov functions is considered in this article. The function is associated with discrete-time switched systems subject to mode-dependent cone-bounded nonlinearities and saturation actuator. These Lyapunov functions depend on the switched nonlinearities and on the active mode, instead of the time-invariant Lur'e-type function. The only assumption considered here is the mode-dependent sector condition, without constraint related to the slope of the nonlinearities. The stability analysis and control design problems lead to linear matrix inequality conditions. Optimisation problems are provided to cope with both issues in order to maximise the size of the basin of attraction estimate, which may be composed of disconnected sets. Numerical examples illustrate via comparisons that we improve the results of the literature.