Convergence and divergence of the solutions of a neutral difference equation (Q410775)

Summary: We investigate the asymptotic behavior of the solutions of a neutral type difference equation of the form \(\Delta [x(n) + cx(\tau(n))] + p(n)x(\sigma(n)) = 0\), where \(\tau(n)\) is a general retarded argument, \(\sigma(n)\) is a general deviated argument (retarded or advanced), \(c \in \mathbb R, (p(n))_{n \geq 0}\) is a sequence of positive real numbers such that \(p(n) \geq p, p \in \mathbb R_+\), and \(\Delta\) denotes the forward difference operator \(\Delta x(n) = x(n + 1) - x(n)\). Also, we examine the asymptotic behavior of the solutions in case they are continuous and differentiable with respect to \(c\).