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We address the problem of global Lyapunov stability of discrete-time systems with known coefficients. We develop a method for reduction of dissipativity domain effectively testing if the system has a convex Lyapunov function. Our implementation is immediately applicable to differentiable systems with bounded nonlinearities, but the method proposed is more general and applicable to nondifferentiable systems with bounded right-hand sides. Our main application emphasis is on stability analysis of recurrent neural networks. We illustrate how to use our approach with examples.