By O. Arino, M.L. Hbid, E. Ait Dads

ISBN-10: 1402036469

ISBN-13: 9781402036460

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1987) Topics in analysis. Pub. Math. Univ. Aut. Barcelona 39, 29-84. , Murakami, S. and T. Naito (1991) Functional Diﬀerential Equations with Inﬁnite Delay. Lect. Notes Math. 1473 Springer-Verlag. V. S. Shin (2002) Almost Periodic Solutions of Diﬀerential Equations in Banach Spaces. Taylor and Francis, London. S. (1962) The existence of periodic solutions of f (x) = −αf (x− 1)[1 + f (x)]. J. Math. Anal. Appl. 5, 435-450. Kolmanovski, V. and A. Myshkis (1999) Introduction to the Theory and Applications of Functional Diﬀerential Equations.

Deﬁnition 6 : (Asymptotically smooth) A dynamical system T (t), t ∈ + , on a Banach space X is said to be asymptotically smooth if, for any bounded set B in X for which T (t)B ⊂ B for t ∈ + , there is a compact set J in X such that lim dist(T (t)B, J) = 0. t∈G+ ,t→∞ This deﬁnition stated in a diﬀerent but equivalent way is due to Hale, LaSalle and Slemrod (1973). It is very important to note that a dynamical system can be asymptotically smooth and there can be a bounded set for which the positive orbit is unbounded; for example, the dynamical system deﬁned by the ODE x = x.

By the Gronwall’s lemma, one has |x(t, ϕ1 ) − x(t, ϕ2 )| ≤ |ϕ1 − ϕ2 | exp (kt) We shall ﬁrst prove the following lemma, which will be used subsequently. Lemma 3 [22] Let f ∈ C (J × Cρ , Rn ). For t ∈ J and φ ∈ Cρ , we put G(t, r) = max f (t, φ) . Then, if x(t, t0 , φ0 ) is any solution of dx (t) = f (t, xt ) dt with φ0 as an initial value at t = t0 . Then, we have : xt (t0 , φ0 ) − φ0 ≤ r∗ (t, t0 , 0) on the common interval of existence of x(t, t0 , φ0 ) and r∗ (t, t0 , 0). Theorem 7 [22] Let f ∈ C (J × Cρ , Rn ) and for t ∈ J, ϕ, φ ∈ C1 f (t, ϕ) − f (t, φ) ≤ g(t, ϕ − φ ) where g ∈ C (J × [0, 2ρ] , R+ ).