This paper presents a new class of Lur'e type Lyapunov functions for a discrete-time switched system interconnected with a switched nonlinearity satisfying a mode-dependent cone bounded condition. This function includes the mode-dependent nonlinearity, but not its integral. Such a Lyapunov function allows to obtain sufficient conditions in terms of linear matrix inequalities (LMI), for the stability analysis in two different frameworks: global stability analysis for the considered systems and local stability analysis for these systems with an additional saturating input consisting of a switched linear state feedback. In the second case, an optimization problem based on these sufficient conditions is provided to enlarge the estimation of the basin of attraction, which may be composed of non-convex and disconnected sets, because of the presence of the nonlinearities in the Lyapunov function. Some numerical examples are presented to highlight the relevance of the new Lyapunov function and of the proposed method.