EXISTENCE OF PERIODIC SOLUTIONS IN TOTALLY NONLINEAR NEUTRAL DIFFERANCE EQUATIONS WITH VARIABLE DELAY

We study the existence of periodic solutions of the totally nonlinear neutral difference equation with variable delay 4x (t) = −a (t) x 3 (t + 1) + c (t) 4x (t − g (t)) + G t, x3 (t), x3 (t − g (t))
, ∀t ∈ Z. We invert the given equation to construct a fixed point mapping expressed as a sum of a large contraction and a compact map. We show that such a sum of mappings fits very nicely into the framework of Krasnoselskii-Burton’s fixed point theorem so that the existence of periodic solutions is readily concluded. The obtained results extend the work of Ardjouni and Djoudi [1]. Keywords: Fixed point, Large contraction, Periodic solutions, Totally nonlinear neutral differential equations