EXISTENCE OF POSITIVE ALMOST PERIODIC OR ERGODIC

Di↵erential and Integral Equations
Volume 22, Numbers 11-12 (2009), 1075–1096
EXISTENCE OF POSITIVE ALMOST PERIODIC OR
ERGODIC SOLUTIONS FOR SOME NEUTRAL
NONLINEAR INTEGRAL EQUATIONS
E. Ait Dads
Université Cadi Ayyad, Département de Mathématiques
Faculté des Sciences, B.P. 2390 Marrakech, Morocco
P. Cieutat
Laboratoire de Mathématiques de Versailles
Université de Versailles Saint-Quentin en Yvelines
45 Avenue des Etats-Unis, 78035 Versailles Cedex, France
L. Lhachimi
Université Cadi Ayyad, Département de Mathématiques
Faculté des Sciences, B.P. 2390 Marrakech, Morocco
(Submitted by: Jean Mawhin)
We state sufficient conditions for the existence of the positive, almost
periodic or ergodic solutions of the following neutral integral equation:
Z t
x(t) = x(t
) + (1
)
f (s, x(s))ds,
t
where 0  < 1 and f : R ⇥ R+ ! R+ is a continuous map. We also
treat the asymptotically, weakly and pseudo almost periodic solutions.
Our results do not need the monotonicity of f (t, .).
1. Introduction
As we all know, the existence of periodic solutions of functional di↵erential
equations (FDE) has great theoretical and practical significance, and is one
of the problems of great interest to scholars in the field. Since Yoshizawa
[28] presented an excellent result for the existence of periodic solutions to
FDE with bounded delay, Cooke and Huang [10], Burton and Hatvani [6]
generalized Yoshizawa’s result to FDE with infinite delay. We remark that,
in nature, there is no phenomenon which is purely periodic; this motivates
the idea of considering the almost periodic situation.
Accepted for publication: February 2009.
AMS Subject Classifications: 42A75; 44A35, 45G10.
1075