EXISTENCE, UNIQUENESS AND POSITIVITY OF SOLUTIONS FOR A NEUTRAL NONLINEAR PERIODIC DYNAMIC EQUATION ON A TIME SCALE

ABDELOUAHEB  ARDJOUNI; AHCENE  DJOUDI

EXISTENCE, UNIQUENESS AND POSITIVITY OF SOLUTIONS FOR A NEUTRAL NONLINEAR PERIODIC DYNAMIC EQUATION ON A TIME SCALE

Authors

ABDELOUAHEB ARDJOUNI Department of Mathematics and Informatics, Souk Ahras University, Souk Ahras, 41000, Algeria
AHCENE DJOUDI Department of Mathematics, Annaba University, Annaba, 23000, Algeria

Keywords:

Fixed point theory, Nonlinear neutral dynamic equation, Periodic solutions, Positivity, Time scales

Abstract

Let T be a periodic time scale. We use Krasnoselskii's fixed point theorem, to show new results on the existence and positivity of solutions for the nonlinear periodic dynamic equation with variable delay of the form
$x^{\triangle}(t) =-a(t)x(t)+(Q(t,x(g(t))))^{\triangle}+G(t,x(t),x(g(t)))$,
$x(t+T) =x(t)$.
Also, by transforming the problem to an integral equation we are able, using the contraction mapping principle, to show that the periodic solution is unique.