On the Existence of Almost Periodic Solutions of Neutral Functional Differential Equations

This paper discuss the existence of almost periodic solutions of neutral functional differential equations. Using a Liapunov function and the Razumikhin’s technique, we obtain the existence, uniqueness and stability of almost periodic solutions.

In the theory of functional differential equations, the existence, uniqueness and stability of almost periodic solutions is an important subject.Hale [1], Yoshizawa [2] and Yuan [3,4] et al, have provided some existence results for certain kind of retarded functional differential equations by means of Liapunov functions.The focus of our present work is to establish the existence of almost periodic solutions of neutral functional differential equations by using the Razumikhin-type argument.The problem of uniqueness and stability of the solution is also addressed.As a corollary to our results, the corresponding theorem of Yuan [4] is included and the proof in [4] is also simplified.
Consider the following almost periodic neutral functional differential equation (1) and its product systems where D : C → R n is linear, autononous and atomic at zero(see Hale [9]), C := C([−τ, 0], R n ), f : R × C → R n is continous and local Lipschitzian with respect to φ ∈ C. Namely, for any H > 0, there is where C H := {φ ∈ C : |φ| ≤ H}.
In addition, we always suppose that f : R × C H * → R n is almost periodic in t uniformly for φ ∈ C H * (see [8]).
Definition.Let C D = {φ ∈ C : Dφ = 0}.D is said to be stable if the zero solution of the homogeneous difference equation Dy t = 0, t ≥ 0, y 0 = ψ ∈ C D is uniformly asymptotically stable.
It is shown (see [9]) that when D is linear autonomous and atomic at zero, D is stable if and only if D is uniformly stable.Namely, there are two constant a, b > 0 such that for any h ∈ C(R + , R n ), the solutions of the equation For any φ, ψ ∈ C, we define the derivative of V along the solution of (1 * ) by Similar to the proof in [8, p.207], we can obtain Lemma 1. Suppose p : R → R is the unique almost periodic solution of (1) with p t ∈ C H for t ∈ R. Then mod (p) ⊂ mod (f ).
We first prove the following two lemmas.Lemma 3. Assume all conditions of the Theorem are satisfied.If a sequence {α n } is given so that f (t + α n , φ) coverges uniformly on R + × C H , then for any ε > 0, there is a positive integer Proof.For any ε > 0, there exists k 0 (ε) such that for m ≥ k ≥ k 0 , we have By (2) we have Let h > 0 be sufficiently small such that K 0 bh < 1/2.By (4) we get Proof.From inequality (2), EJQTDE, 1998 No. 4, p.4 Choose This complete the proof of Lemma 4.
Proof of the Theorem.Let S = Cl{ξ t : t ≥ 0}.It is easy to see that S is a compact set in C(see, for example, [4]).Let α = {α n }, a n → ∞ as n → ∞, be a given sequence.Since f (t, φ) is almost periodic in t uniformly for φ ∈ C H * , there exists a subsequence {α n } ⊂ α such that lim n→∞ f (t + α n , φ) exists uniformly on R × S. Also we can suppose that {α n } is increasing.
From the condition F (v(Kη)) > v(β(η)), η > 0, we know that there exists a sequence Obviously, z n is decreasing and tends to zero as n → ∞ .For any given ε > 0, we may assume ε < β(2H), and select a N such that β(z N ) < ε.In the following, we prove that there exists l 0 = l 0 (ε) such that ( 5) First, we prove that there is a T 1 > 0 such that (6) Applying Lemma 4, there is a T 1 ≥ 0 such that (7) EJQTDE, 1998 No. 4, p.5 We now consider the following two cases: In this case we have Applying Lemma 3 with m ≥ k ≥ k 0 (γ 0 ), we obtain that Thus, which yields This contradicts V (t) > v(Kz 1 ).
By the same reasoning as above, we obtain that if then there exists T j+1 > T j + v(2HK)/γ such that Finally, V (t) ≤ v(Kz N ) for all t ≥ T N +1 .
Thus, we have Applying Lemma 4, there is a T * > T N +1 such that Then, for all t ≥ T * , m ≥ k ≥ k 0 .We can select l 0 ≥ k 0 such that a l 0 ≥ T * .Therefore, (8) implies Thus, ξ(t) is an asymptotially almost periodic solution of Eq.( 1).Applying Lemma 2, Eq.( 1) has an almost periodic solution p with p(t) ∈ C H for t ∈ R.
Similarly to the proof above, we can obtain that p is quasi-uniformly asymptotically stable.At last, we prove that p is uniformly stable.For any ε ≥ 0 and t 0 ∈ R, let δ 1 > 0 so that β(δ 1 ) < ε.Denote We will prove that when |φ − p t 0 | < δ, we have where x(t) := x(t 0 , φ)(t).Suppose that there is a t 1 > t 0 , such that Then, Therefore, |D(x t − p t )| ≤ α(δ 1 ).From inequality (2), we have Consequently, Then, from condition (iii), we have Using the almost periodicity, we obtain p(t) = p(t) for all t ∈ R.This implies that Eq. ( 1) has only one almost periodic solution in C H . And, from Lemma 2, we have mod(p) ⊂ mod(f ), completing the proof.We conclude the paper with an example to illustrate the theorem.
Example.Consider the following equation e−1 η, and ψ(t) = e −2t + p(t) is a bounded solution of Eq. (9).Then it is easy to see that the conditions of the Theorem are satisfied, thus, Eq.( 9) has a unique almost periodic solution x(t) = p(t), which is uniformly asymptotically stable.