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  • Author: George E. Chatzarakis x
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In this paper, we investigate the asymptotic behavior of the solutions of a neutral type difference equation of the form

where τj (n), j = 1, . . . , w are general retarded arguments, σ(n) is a general deviated argument, [###] is a sequence of positive real numbers such that p(n) ≥ p, p ∈ R+, and Δ denotes the forward difference operator Δx(n) = x(n + 1) − x(n).


Sufficient oscillation conditions involving lim sup and lim inf for first-order differential equations with non-monotone deviating arguments and nonnegative coefficients are obtained. The results are based on the iterative application of the Grönwall inequality. Examples, numerically solved in MATLAB, are also given to illustrate the applicability and strength of the obtained conditions over known ones.


In this paper, we study the oscillatory behavior of solutions of the fractional difference equation of the form


where Δα denotes the Riemann-Liouville fractional difference operator of order α, 0 < α ≤ 1, ℕt0+1−α={t 0+1−αt 0+2−α…}, t 0 > 0 and γ > 0 is a quotient of odd positive integers. We establish some oscillatory criteria for the above equation, using the Riccati transformation and Hardy type inequalities. Examples are provided to illustrate the theoretical results.