On the Existence of Almost Periodic Solutions

of Neutral Functional Differential Equations

Jianhua Xu^{1),2)}, Zhicheng Wang^{1)} and Zuxiu Zheng ^{2)}

Abstract. 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.

Key words: Almost periodic solution; Neutral functional differential equation; Liapunov function.

1991 AMS Classification: 34K15

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)  \frac{d}{dt}Dx_{t} = f(t, x_{t})

and its product systems

(1^{∗})  \{ \begin{matrix}\frac{d}{dt}Dx_{t} = f(t, x_{t}) \\ \frac{d}{dt}Dy_{t} = f(t, y_{t}) \end{matrix}

Typeset by AMS-TEX

This project was supported by NNSF of China

1). Dept. of Appl. Math., Hunan University, Changsha 410082, P. R. China

2). Dept. of Math., Anhui University, Hefei 230039, P. R. China

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 K_{0} = K_{0}(H) > 0 such that for ϕ, ψ ∈ C_{H},

|f(t, ϕ) − f(t, ψ)| ≤ K_{0}|ϕ − ψ|,

where C_{H} := {ϕ ∈ C : |ϕ| ≤ H}.

Under the above hypotheses, there is a unique solution x(t) = x(σ, ϕ)(t) of Eq.

(1) through a given intial value (σ, ϕ) ∈ R × C_{H^{∗}} (see [9]).

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

Dy_{t} = h(t), t ≥ σ

satisfies

(2)  |y_{t}| ≤ be^{−a(t−σ)}|y_{σ}| + b sup_{σ≤u≤t} |h(u)|, t ≥ σ.

Suppose that V : R^{+} × R^{n} × R^{n} → R^{+} is continuous. For any ϕ, ψ ∈ C, we define the derivative of V along the solution of (1^{∗}) by

V̇_{(1^{∗})}(t, ϕ, ψ) = lim sup_{h→0^{+}}[V (t + h, Dx_{t+h}(t, ϕ), Dy_{t+h}(t, ψ)) − V (t, Dϕ, Dψ)].

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).

Lemma 2^{[3]}. Suppose D is stable, and Eq.(1) has a solution ξ : R → R with |ξ_{t}| ≤ H < H^{∗} for t ≥ 0. If ξ is an asymptotically almost periodic function, then Eq.(1) has an almost periodic solution.

In what follows, we assume D is a stable operator, ||D|| = K. Let 0 ≤ u(s) ≤ v(s), s ≥ 0, be continuous and nondecreasing functions, u(s) → ∞ as s → ∞, v(0) = 0, and suppose that there is a continuous function α : R^{+} → R satisfying v(Kη) ≤ u(α(η)). Let β(η) be an arbitrary function of η > 0 such that β(η) > bα(η) for η > 0 ( where b > 0 is defined in inequality (2) ). Also assume α(0) = β(0) = 0. The main result of this work is as follows:

Theorem. Suppose f(t, ϕ) is almost periodic in t ∈ R uniformly for ϕ ∈ C_{H^{∗}}. If there exists a Liapunov function V : R^{+} × R^{n} × R^{n} → R^{+} such that

(i). u(|x − y|) ≤ V (t, x, y) ≤ v(|x − y|) for (t, x, y) ∈ R^{+} × R^{n} × R^{n}; (ii). |V (t, x_{1}, y_{1})−V (t, x_{2}, y_{2})| ≤ L(|x_{1}−x_{2}|+|y_{1}−y_{2}|), where L > 0, (t, x_{i}, y_{i}) ∈

R^{+} × Ω × Ω, i = 1, 2, Ω = {x ∈ R^{n} : |x| < H^{∗}};

(iii). For t ∈ R, ϕ, ψ ∈ C_{H^{∗}} with F (V (t, Dϕ, Dψ)) ≥ V (t + θ, ϕ(θ), ψ(θ)) for θ ∈ [−τ, 0], we have

V̇_{(1^{∗})}(t, ϕ, ψ) ≤ −ω(|Dϕ − Dψ|),

where F : [0, ∞) → R^{+} is continuous and nondecreasing such that F (v(Kη)) > v(β(η)), η > 0.

Moreover, Assume that Eq. (1) has a bounded solution ξ : R → R with |ξ_{t}| ≤ H < H^{∗} for t ≥ 0. Then Eq. (1) has a unique almost periodic solution p : R → R with |p(t)| ≤ H for t ∈ R, mod (p) ⊂ mod (f), and p is uniformly asymptotically stable.

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 k_{0}(ε), such that for m ≥ k ≥ k_{0},

A_{m,k}(t) = lim sup_{h→0^{+}} \frac{1}{h}|V (t + h, Dξ_{t+h}, Dξ_{t+α_{m}−α_{k}+h})
− V (t + h, Dx_{t+h}(t, ξ_{t}), Dy_{t+h}(t, ξ_{t} + α_{m} − α_{k}))| ≤ ε.

Proof. For any ε > 0, there exists k_{0}(ε) such that for m ≥ k ≥ k_{0}, we have

|f(t + α_{k}, ϕ) − f(t + α_{m}, ϕ)| ≤ \frac{ε}{2L}, (t, ϕ) ∈ R^{+} × C_{H}.

By condition (ii),

(3)

A_{m,k}(t) ≤ lim sup_{h→0^{+}} \frac{L}{h}(|Dξ_{t+h} − Dx_{t+h}(t, ξ_{t})|

+ |Dξ_{t+α_{m}−α_{k}+h} − Dy_{t+h}(t, ξ_{t+α_{m}−α_{k}})|)

= lim sup_{h→0^{+}} \frac{L}{h}|D(ξ_{t+α_{m}−α_{k}+h} − y_{t+h}(t, ξ_{t+α_{m}−α_{k}}))|.

Note that η_{s} := ξ_{s+α_{m}−α_{k}} satisfies

\{ \begin{matrix}\frac{d}{ds}Dη_{s} = f(s + α_{m} − α_{k}, η_{s}), \\ η_{t} = ξ_{t+α_{m}−α_{k}}, s ≥ t. \end{matrix}

Let B_{m,k}(t) = max_{t≤s≤t+h} |D(η_{s} − y_{s})|, where y_{s} := y_{s}(t, ξ_{t+α_{m}−α_{k}}). Thus,

B_{m,k} ≤ max_{t≤s≤t+h} ∫_{t}^{s} |f(s + α_{m} − α_{k}, η_{s}) − f(s, y_{s})|ds

= ∫_{t}^{t+h} |f(s + α_{m} − α_{k}, η_{s}) − f(s, y_{s})|ds

≤ ∫_{t}^{t+h} |f(s + α_{m} − α_{k}, η_{s}) − f(s, η_{s})|ds

+ ∫_{t}^{t+h} |f(s, η_{s}) − f(s, y_{s})|ds

≤ h\frac{ε}{2L} + K_{0} ∫_{t}^{t+h} |η_{s} − y_{s}|ds.

By (2) we have

|η_{s} − y_{s}| ≤ b sup_{t≤u≤s} |D(η_{u} − y_{u})|
≤ bB_{m,k}(t), t ≤ s ≤ t + h.

Hence

(4)  B_{m,k}(t) ≤ h\frac{ε}{2L} + K_{0}bhB_{m,k}(t).

Let h > 0 be sufficiently small such that K_{0}bh < 1/2. By (4) we get

B_{m,k} ≤ \frac{εh}{2L(1 − K_{0}bh)} ≤ \frac{ε}{L}h.

Then

|D(ξ_{t+α_{m}−α_{k}+h} − y_{t+h}(t, ξ_{t+α_{m}−α_{k}}))| ≤ \frac{ε}{L}h.

Form (3), it follows that

A_{m,k}(t) ≤ lim sup_{h→o^{+}} \frac{L}{h} \frac{εh}{L} = ε.

Lemma 4. Assume t_{0} ∈ R and |y_{t}| ≤ 2H and |Dy_{t}| ≤ α(δ)(δ > 0) for t ≥ t_{0}. Then there exists t_{1} > t_{0}, t_{1} = t_{1}(δ, t_{0}), such that |y_{t}| ≤ β(δ) for t ≥ t_{1}.

Proof. From inequality (2),

|y_{t}| ≤ be^{−a(t−t_{0})}|y_{t_{0}}| + b sup_{t_{o}≤u≤t} |Dy_{t}| ≤ 2Hbe^{−a(t−t_{0})} + bα(δ).

Choose

t_{1} > t_{0} + \frac{1}{a}ln\frac{2Hb}{β(δ) − bα(δ)},

then

|y_{t}| ≤ 2Hb\frac{β(δ) − bα(δ)}{2Hb} + bα(δ) = β(δ) for t ≥ t_{1}.

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 {z_{n}}_{n=1,2,···}, z_{0} = 2H such that

F (v(Kz_{n})) = v(β(z_{n−1})), n = 1, 2, · · · .

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)  |ξ(t + α_{k}) − ξ(t + α_{m})| < ε,

for m ≥ k ≥ l_{0} and t ∈ R^{+}. Let

γ = \frac{1}{2} inf_{kz_{N} ≤s≤2HK} ω(s) > 0.

First, we prove that there is a T_{1} > 0 such that

(6)  V (t) := V (t, Dξ_{t}, Dξ_{t+α_{m}−α_{k}}) ≤ v(kz_{1}),

for t ≥ T_{1} + v(2HK)/γ and m ≥ k ≥ k_{0}(γ). From

u(|D(ξ_{t} − ξ_{t+α_{m}−α_{k}})|) ≤ V (t) ≤ v(|D(ξ_{t} − ξ_{t+α_{m}−α_{k}})|)
≤ v(2HK) ≤ u(α(2H)),

we deduce

|D(ξ_{t} − ξ_{t+α_{m}−α_{k}})| ≤ α(2H), t ≥ 0.

Applying Lemma 4, there is a T_{1} ≥ 0 such that

(7)  |ξ_{t} − ξ_{t+α_{m}−α_{k}}| ≤ β(2H), t ≥ T_{1}.

We now consider the following two cases:

Case 1. V (t) > v(Kz_{1}) for T_{1} ≤ t ≤ T_{1} + v(2HK)/γ. In this case we have

F (V (t)) ≥ F (v(Kz_{1})) = v(β(2H))
≥ v(|ξ_{t} − ξ_{t+α_{m}−α_{k}}|)
≥ V (t + θ, ξ(t + θ), ξ(t + α_{m} − α_{k} + θ)), −τ ≤ θ ≤ 0,

which yields

V̇_{(1^{∗})}(t) ≤ −ω(|D(ξ_{t} − ξ_{t+α_{m}−α_{k}})|).

Since

v(|D(ξ_{t} − ξ_{t+α_{m}−α_{k}})|) ≥ V (t) > v(Kz_{1}),

we obtain

|D(ξ_{t} − ξ_{t+α_{m}−α_{k}})| ≥ Kz_{1} ≥ Kz_{N}.

Moreover,

|D(ξ_{t} − ξ_{t+α_{m}−α_{k}})| ≤ 2KH.

Then

V̇_{(1^{∗})}(t) ≤ −2γ.

Applying Lemma 3 with m ≥ k ≥ k_{0}(γ_{0}), we obtain that

V′(t) = lim sup_{h→0^{+}} \frac{1}{h}[V (t + h) − V (t)]

≤ lim sup_{h→0^{+}} \frac{1}{h}[V (t + h) − V (t + h, Dx_{t+h}(t, ξ_{t}), Dy_{t+h}(t, ξ_{t+α_{m}−α_{k}})]

+ lim sup_{h→0^{+}} \frac{1}{h}[V (t + h, Dx_{t+h}(t, ξ_{t}), Dy_{t+h}(t, ξ_{t+α_{m}−α_{k}})) − V (t)]

≤ γ − 2γ = −γ for T_{1} ≤ t ≤ T_{1} + \frac{v(2HK)}{γ}.

Thus,

V (t) ≤ V (T_{1}) − γ(t − T_{1}) for T_{1} ≤ t ≤ T_{1} + \frac{v(2HK)}{γ},

which yields

V (T_{1} + \frac{v(2HK)}{γ}) ≤ v(2HK) − γ(T_{1} + \frac{v(2HK)}{γ} − T_{1}) = 0.

This contradicts V (t) > v(Kz_{1}).

EJQTDE, 1998 No. 4, p.6

Case 2. There is a t_{1} ∈ [T_{1}, T_{1} + v(2HK)/γ] such that V (t_{1}) ≤ v(Kz_{1}). In this case, we can suppose that there is t_{2} ≥ t_{1} such that V (t_{2}) = v(Kz_{1}). Then,

F (V (t_{2})) = F (v(Kz_{1})) = v(β(2H))
≥ v(|ξ_{t_{2}} − ξ_{t_{2}+α_{m}−α_{k}}|)
≥ V (t_{2} + θ, ξ(t_{2} + θ), ξ(t_{2} + α_{m} − α_{k} + θ)),

where −τ ≤ θ ≤ 0. Thus, condition(iii) implies

V̇_{(1^{∗})}(t_{2}) ≤ −ω(|Dξ_{t_{2}} − ξ_{t_{2}+α_{m}−α_{k}})|).

An argument similar to Case 1 leads to

V′(t_{2}) ≤ −γ < 0 for m ≥ k ≥ k_{0}(γ).

Consequently, in both Case 1 and Case 2, (6) turns to be true.

By the same reasoning as above, we obtain that if

V (t) ≤ v(Kz_{j}) (j = 1, 2, ..., N − 1) for all t ≥ T_{j} + \frac{v(2HK)}{γ},

then there exists T_{j+1} > T_{j} + v(2HK)/γ such that

V (t) ≤ v(Kz_{j+1}) for all t ≥ T_{j+1} + \frac{v(2HK)}{γ}.

Finally,

V (t) ≤ v(Kz_{N}) for all t ≥ T_{N+1}.

Thus, we have

u(|D(ξ_{t} − ξ_{t+α_{m}−α_{k}})|) ≤ V (t) ≤ v(Kz_{N}) ≤ u(α(z_{N})).

Therefore,

|D(ξ_{t} − ξ_{t+α_{m}−α_{k}})| ≤ α(z_{N}).

Applying Lemma 4, there is a T^{∗} > T_{N+1} such that

|ξ_{t} − ξ_{t+α_{m}−α−k}| ≤ β(z_{N}) < ε for t ≥ T^{∗}.

Then,

(8)  |ξ(t) − ξ(t + α_{m} − α_{k})| ≤ ε,

for all t ≥ T^{∗}, m ≥ k ≥ k_{0}. We can select l_{0} ≥ k_{0} such that a_{l_{0}} ≥ T^{∗}. Therefore,

(8) implies

|ξ(t + α_{k}) − ξ(t + α_{m})| ≤ ε, t ∈ R^{+}m ≥ k ≥ l_{0}.

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

δ := \frac{1}{b}(β(δ_{1}) − bα(δ_{1})) > 0.

We will prove that when |ϕ − p_{t_{0}}| < δ, we have

V (t_{1}, Dx_{t}, Dp_{t}) ≤ v(Kδ_{1}), t ≥ t_{0},

where x(t) := x(t_{0}, ϕ)(t). Suppose that there is a t_{1} > t_{0}, such that

V (t, Dx_{t_{1}}, Dp_{t_{1}}) = v(Kδ_{1})

and

V (t, Dx_{t}, Dp_{t}) ≤ v(Kδ_{1}) for t_{0} ≤ t ≤ t_{1}.

Then,

u(|D(x_{t} − p_{t})|) ≤ V (t, Dx_{t}, Dp_{t}) ≤ v(Kδ_{1}) ≤ u(α(δ_{1})), t_{0} ≤ t ≤ t_{1}.

Therefore, |D(x_{t} − p_{t})| ≤ α(δ_{1}). From inequality (2), we have

|x_{t_{1}} − p_{t_{1}}| ≤ be^{−a(t_{1}−t_{0})}|ϕ − p_{t_{0}}| + b sup_{t_{0}≤u≤t_{1}} |D(x_{u} − p_{u})|
≤ bδ + bα(δ_{1}) = β(δ_{1}).

Consequently,

F (V (t_{1}, Dx_{t_{1}}, Dp_{t_{1}})) = F (v(Kδ_{1})) > v(β(δ_{1})) ≥ v(|x_{t_{1}} − p_{t_{1}}|)
≥ V (t_{1} + θ, x(t_{1} + θ), p(t_{1} + θ)), −τ ≤ θ ≤ 0.

Then, from condition (iii), we have

V′(t_{1}, x_{t_{1}}, p_{t_{1}}) ≤ −ω(|D(x_{t_{1}} − p_{t_{1}})|) ≤ 0.

Thus,

V (t, Dx_{t}, Dp_{t})|) ≤ v(Kδ_{1}), t ≥ t_{0},

which yields

u(|D(x_{t} − p_{t})|) ≤ v(Kδ_{1}) ≤ u(α(δ_{1})).

That is,

|D(x_{t} − p_{t})| ≤ α(δ_{1}), t ≥ t_{0}.

It follows from (2) that

|x_{t} − p_{t}| ≤ β(δ_{1}) < ε,

and this implies that p is uniformly stable. Since p is asymptotically stable, it follows that for any almost periodic solution p̄(t) of Eq. (1), |p̄(t)| < H for t ∈ R, we have

|p(t) − p̄(t)| → 0 as t → ∞.

Using the almost periodicity, we obtain p(t) = p̄(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

(9)  \frac{d}{dt}[x(t) − e^{−1}x(t − r)] = −x(t) + (p(t) − e^{−1}p(t − r))′ + p(t),

where r = \frac{1}{2}(1 − ln2), p : R → R is an almost periodic function such that p′ is uniformly continous on R.

Let V (x, y) = (x − y)^{2}, u(s) = v(s) = s^{2}, F (s) = A^{2}s, where A > \frac{1}{(1−e^{−1})}, α(η) = (1 + e^{−1})η, β(η) = \frac{e+1}{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.

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