A class of Lyapunov functions is proposed for discrete-time linear systems interconnected with a cone bounded nonlinearity. Using these functions, we propose sufficient conditions for the global stability analysis, in terms of linear matrix inequalities (LMI), only taking the bounded sector condition into account. Unlike frameworks based on the Lure-type function, the additional assumptions about the derivative or discrete variation of the nonlinearity are not necessary. Hence, a wider range of cone bounded nonlinearities can be covered. We also show that there is a link between global stability LMI conditions based on this new Lyapunov function and a transfer function of an auxiliary system being strictly positive real. In addition, the novel function is considered in the local stability analysis problem of discrete-time Lur'e systems subject to a saturating feedback. A convex optimization problem based on sufficient LMI conditions is formulated to maximize an estimate of the basin of attraction. Another specificity of this new Lyapunov function is the fact that the estimate is composed of disconnected sets. Numerical examples reveal the effectiveness of this new Lyapunov function in providing a less conservative estimate with respect to the quadratic function. (C) 2012 Elsevier Ltd. All rights reserved.