Oscillation criteria for difference equations with non-monotone arguments

This paper is concerned with the oscillatory behavior of first-order retarded [advanced] difference equation of the form Delta x(n) + p(n)x(tau(n)) = 0, n is an element of N-0 [del x(n) - q(n)x(sigma(n)) = 0, n is an element of N], where (p(n))(n >= 0) [(q(n))(n >= 1)] is a sequence of nonnegative real numbers and tau(n) [sigma(n)] is a non-monotone sequence of integers such that tau(n) <= n - 1, for n is an element of N-0 and lim(n ->infinity) tau(n) = infinity[sigma(n) >= n + 1, for n is an element of N]. Sufficient conditions, involving lim sup, which guarantee the oscillation of all solutions are established. These conditions improve all previous well-known results in the literature. Also, using algorithms on MATLAB software, examples illustrating the significance of the results are given.