STABILITY OF SIMPLE PERIODIC SOLUTIONS OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS

YOUMIN LU & ZHOUDE SHAO

Abstract. We study the stability property of a simple periodic solution of an autonomous neutral functional differential equation (NFDE) of the form \frac{d}{dt}D(x_{t}) = f(x_{t}). A new proof based on local integral manifold theory and the implicit function theorem is given for the classical result that a simple periodic orbit of the equation above is asymptotically orbitally stable with asymptotic phase. The technique used overcomes the difficulty that the solution operator of a NFDE does not smooth as t increases.

1. Introduction

Suppose that r ≥ 0 is a given real number, and ℝ^{n} is the n-dimensional Euclidean space with norm | · |. Let C = C([−r, 0], ℝ^{n}) be the Banach space of all the continuous functions mapping the interval [−r, 0] into ℝ^{n} with norm |ϕ|_{C} = max_{−r≤θ≤0} |ϕ(θ)|, ϕ ∈ C. If

t_{0} ∈ ℝ, σ ≥ 0, and x : [t_{0} − r, t_{0} + σ] → ℝ^{n} is a continuous function, for any t ∈ [t_{0}, t_{0} + σ], x_{t} ∈ C is defined by x_{t}(θ) = x (t + θ), θ ∈ [−r, 0].

We consider an autonomous neutral functional differential equation of the form

(1.1)  \frac{d}{dt}D(x_{t}) = f(x_{t}).

Throughout this paper, the functional f is assumed to be twice continuously Fréchet differentiable on C and the linear operator D is assumed to be stable and to have an integral representation given by

D(φ) = φ(0) − ∫_{−r}^{0}[dμ(θ)]φ(θ),

where μ is an n × n matrix function on (−∞, ∞) of bounded variation which is left continuous on (−r, 0] and satisfies the following

(1). μ(θ) = μ(−r) for θ ≤ −r, and μ(θ) = μ(0) = 0 for θ ≥ 0.

(2). There is a continuous nonnegative scalar function γ such that γ(0) = 0 and

| ∫_{−δ}^{0}[dμ(θ)]φ(θ)| ≤ γ(δ)|φ|_{C}.

1991 Mathematics Subject Classification. 34C45, 34Dxx, 34K20.

Key words and phrases. Periodic solution, orbital stability, asymptotic phase, local integral manifolds, implicit function theorem.

It is noted that the representation for D given above are equivalent to saying that D is atomic at 0. For more details on stable D-operators, see Hale [8] (Sections 12.4 and 12.10) or Hale and Lunel [9] (Sections 9.2 and 9.3).

We assume that equation (1.1) has a nontrivial periodic solution p(t) with period ω and corresponding orbit W = {p_{t} : t ∈ [0, ω]} in C. We are interested in the stability property of p(t). For any ϕ ∈ C, we will let x(ϕ)(t) or x_{t}(ϕ) represent the solution of (1.1) through (0, ϕ). For completeness, we state the following definitions, which follow naturally from the corresponding definitions for ordinary and delay functional differential equations.

Definition 1.1. A periodic solution p(t) of (1.1) is said to be orbitally asymptotically stable if

(1). For any given ε > 0, there exists a δ > 0 such that, for any ϕ ∈ C with

dist(ϕ, W) < δ, dist(x_{t}(ϕ), W) < ε, for all t ≥ 0. (2). There is a δ_{0} > 0 such that, for any ϕ ∈ C with dist(ϕ, W) < δ_{0},

dist(x_{t}(ϕ), W) → 0 as t → ∞.

Definition 1.2. A periodic solution p(t) of (1.1) is said to be orbitally asymptotically stable with asymptotic phase if p(t) is orbitally asymptotically stable and the δ_{0} in Definition 1.1 can be chosen so that, for any ϕ ∈ C with dist(ϕ, W) < δ_{0}, there is a constant c = c(ϕ) such that

|x_{t+c}(ϕ) − p_{t}|_{C} → 0 as t → ∞.

For ordinary differential equations, it is proved that any simple periodic orbit is orbitally asymptotically stable with asymptotic phase. By a simple periodic orbit, we mean that the periodic orbit has 1 as a simple characteristic multiplier and all the other characteristic multipliers have norm less than 1. This result was extended to the case of a family of periodic solutions by Hale and Stokes [10], and to delay functional differential equations by Hale [5] and Stokes [13,14]. For neutral functional differential equations, Hale [6,7] studied the behavior of solutions near constant solutions. See Hale and Lunel [9] for other related results. As to the behavior of solutions near a periodic orbit of a neutral functional differential equation, one can derive the following result from Hale and Lunel [9] (Theorem 3 in Chapter 10).

Main Theorem: Assume that the linear operator D : C → ℝ^{n} in (1.1) is stable and p(t) is a nontrivial periodic solution of (1.1). If 1 is a simple characteristic multiplier of p(t) and all the other characteristic multipliers of p(t) have norm less than 1, then p(t) is orbitally asymptotically stable with asymptotic phase.

The result is a direct extension of the corresponding results for ordinary and delay functional differential equations. However, the techniques used in the proof of the corresponding results for ordinary and delay functional differential equations do not extend directly to the case of neutral functional differential equations. The major difficulty lies in the fact that the associated solution operator does not smooth with time t. Actually, this is the main reason that the qualitative theory of neutral functional differential equations is less satisfactory than that for ordinary and delay functional differential equations. The proof of above result in Hale and Lunel [9] uses the theory of synchronized stable and unstable manifolds. Here we provide a proof that uses local integral manifolds and the implicit function theorem, and is similar to that used in Hale [6, 7] and Stokes [13, 14]. To overcome the difficulty that the solutions are not smooth enough as t increases, we first prove that the desired property holds for smooth solutions and then use continuity argument to show that the property is actually valid for all solutions. We need a result on the differentiability of the solutions of neutral functional differential equations, which states that the set of initial functions such that the corresponding solutions of (1.1) is differentiable in time is a dense set in the phase space C, see Shao [11]. The idea used here extends to other problems in the study of the asymptotic behavior of solutions of NFDE. For example, similar technique can be used to show the stability result of an integral manifold.

This paper is organized as follows. In Section 2, we present some preliminary results related to the linear variational equation of (1.1) associated with p(t). In Section 3, we prove the existence of local stable manifolds, and Section 4 is devoted to the proof of the Main Theorem stated above.

2. Preliminary Results

Performing the change of variable y_{t} = x_{t} − p_{t} on (1.1), we have

(2.1)  \frac{d}{dt}D(y_{t}) = L(t, y_{t}) + F (t, y_{t}),

where L is the linear operator defined by L(t, ϕ) = f′_{ϕ}(p_{t})ϕ, and F is the functional defined by F (t, ϕ) = f(p_{t} +ϕ)−f(p_{t})−f′_{ϕ}(p_{t})ϕ. One can easily see that F is continuously Fréchet differentiable and satisfies

(2.2)  F (t, 0) = 0,  F′_{ϕ}(t, 0) = 0.

The compactness of W and (2.2) imply that

(2.3)  |F (t, ϕ_{1}) − F (t, ϕ_{2})| ≤ l(max{|ϕ_{1}|_{C}, |ϕ_{2}|_{C}})|ϕ_{1} − ϕ_{2}|_{C}

for all t ∈ ℝ and ϕ_{1}, ϕ_{2} in a neighborhood of 0 in C, where l : [0, ∞) → [0, ∞) is a continuous function with l(0) = 0. Because of (2.2) and (2.3), we can regard (2.1) as a perturbation of the equation

(2.4)  \frac{d}{dt}D(y_{t}) = L(t, y_{t})

near the solution y ≡ 0. Equation (2.4) is a periodic linear equation and ṗ(t) = \frac{d}{dt}p(t) is a periodic solution.

For any s ∈ ℝ, t ≥ s, let T (t, s) be the solution operator of equation (2.4), that is, T (t, s)ϕ = y_{t}(s, ϕ), where y_{t}(s, ϕ) is the solution of (2.4) through (s,ϕ), and T (s) = T (s, s + ω) for any s ∈ ℝ. Let σ(T (s)) denote the spectrum of T (s), and σ_{n}(T (s)) the set of all the normal eigenvalues of T (s). It has been shown that σ_{n}(T (s)) and the dimension of the generalized eigenspace corresponding to each element in σ_{n}(T (s)) are independent of s. If μ ∈ σ_{n}(T (s)), μ is called a characteristic multiplier of (2.4) or the periodic solution p(t), and the dimension of the generalized eigenspace of μ is defined to be the multiplicity of μ. Since ṗ(t) is a periodic solution of (2.4), 1 must be a characteristic multiplier. As mentioned earlier, if 1 has multiplicity 1, we say that the periodic orbit p (t) is simple. For more details on related definitions and results, see Hale [8], Hale and Lunel [9], and Stokes [12].

The following two lemmas provide some preliminary results on (2.4), most of which are corollaries of proven results tailored in our special case for the convenience of later reference. See Cruz and Hale [1], Hale [8](Sections 12.8 and 12.10), and Hale and Lunel [9] (Section 10.3) for proofs and more general results.

Lemma 2.1. Assume that the linear operator D : C → ℝ^{n} in (2.4) is stable. If 1 is a simple characteristic multiplier of (2.4), and all the other characteristic multipliers of (2.4) have absolute value less than 1, then for any s ∈ ℝ, there exist a one dimensional subspace P (s) of C and a closed subspace Q(s) of C such that

(1). C = P(s) ⊕ Q(s).

(2). T (s)P (s) ⊂ P (s), T (s)Q(s) ⊂ Q(s).

(3). σ(T (s) |_{P (s)}) = {1 }, σ(T (s) |_{Q(s)}) = σ(T (s)) − {1 }.

(4). For any ϕ ∈ P (s), ψ ∈ Q(s), the solution y_{t}(s, ϕ) of (2.4) through ( s,ϕ) is periodic with period ω and there exist constants M, α > 0 independent of s such that

|y_{t}(s, ϕ)|_{C} ≤ M |ϕ| for all t, s ∈ ℝ
|y_{t}(s, ψ)|_{C} ≤ M |ψ|_{C} e^{−α(t−s)} for all t ≥ s ∈ ℝ

For any s ∈ ℝ, let X(t, s) denote the matrix solution of

D(X_{t}(·, s)) = I + ∫_{s}^{t} L(τ, X_{τ}(·, s))ds

defined on the interval [s − r, ∞) with initial value

X_{s}(·, s) = X_{0},

where X_{0} is the matrix function defined by X_{0}(θ) = 0, for θ ∈ [−r, 0) and X_{0}(0) = I. Then, by the variation-of-constant formula, the solution y_{t} = y_{t}(s, ϕ) of (2.1) through (s, ϕ) can be written as

y_{t} = T (t, s)ϕ + ∫_{0}^{t} X(t, τ)F (τ, y_{τ})dτ
= T (t, s)ϕ + ∫_{0}^{t} d[K(t, τ)]F (τ, y_{τ}),

where K(t, ·) : [s, t] → C is defined by

K(t, τ) = ∫_{s}^{τ} X(t + θ, α)dα.

We will let π_{P (s)} : C → P (s), and π_{Q(s)} : C → Q(s) represent the canonical projections of C onto P (s) and Q(s), respectively.

Lemma 2.2. Under the assumptions of Lemma 2.1, the solution y_{t} of (2.1) through ( s, ϕ) can be written as y_{t} = y_{t}^{P (t)} + y_{t}^{Q(t)}, where y_{t}^{P (t)} = π_{P (t)}y_{t} ∈ P (t) and y_{t}^{Q(t)} = π_{Q(t)}y_{t} ∈ Q(t) are given by the following formulas

y_{t}^{P (t)} = T (t, s)π_{P (s)}ϕ + ∫_{0}^{t} d[π_{P (τ)}K(t, τ)]F (τ, y_{τ})
y_{t}^{Q(t)} = T (t, s)π_{Q(s)}ϕ + ∫_{0}^{t} d[π_{Q(τ)}K(t, τ)]F (τ, y_{τ}).

Moreover, we have (1). Var_{[s,t)}[π_{P (·)}K(t, ·)] ≤ M for all s, t ∈ (−∞, ∞), (2). Var_{[s,t)}[π_{Q(·)}K(t, ·)] ≤ Me^{−α(t−s)} for all t ≥ s, where M and α are constants

determined in Lemma 2.1.

3. Local Stable Manifolds

In this section, we will consider the nonlinear equation (2.1). Based on the results presented in the previous section, we can establish the existence of local stable manifolds for equation (2.1) near the solution y(t) ≡ 0.

Lemma 3.1. Assume that the assumptions of Lemma 2.2 hold. Then, for any ϕ ∈ C, if z^{∗}(ϕ) ∈ C([−r, ∞), ℝ^{n}) is a solution of the integral equation

(3.1) z_{t} = T (t, 0)ϕ + ∫_{0}^{t} d[π_{Q(s)}K(t, s)]F (s, z_{s}) − ∫_{t}^{∞} d[π_{P (s)}K(t, s)]F (s, z_{s}), , t ≥ 0

then z^{∗}(ϕ)( t) is also a solution of (2.1).

The proof of this Lemma uses standard arguments used by Hale and Lunel [9] (Lemma 1.1 in Section 10.1)) or Stokes [13] (Lemma 3.3).

Lemma 3.2. Under the assumptions of Lemma 2.2, for any given constant β, 0 < β <

α, there exists a constant η_{0} > 0 such that, for any ϕ ∈ U_{Q}(η_{0})\begin{matrix}def \\ = \end{matrix}\{ϕ ∈ Q(0) : |ϕ|_{C} ≤ η_{0}\}, there exists a solution z^{∗}(ϕ) ∈ C([−r, ∞), ℝ^{n}) of (3.1) satisfying

|z^{∗}(ϕ)(t)| ≤ ρ(η_{0})e^{−βt}, t ≥ 0,

where ρ : [0, ∞) → [0, ∞) is a continuous function satisfying ρ(0) = 0. Furthermore, H : U_{Q}(η_{0}) → C defined by

H(ϕ)(·) = ∫_{0}^{∞} d[π_{P (s)}K(·, s)]F (s, z_{s}^{∗}(ϕ))

is continuously Fréchet differentiable and satisfies

H′_{ϕ}(0) = 0 : Q(0) → C.

Proof. For any δ > 0, we define

Z(δ) = {z ∈ C([−r, ∞), ℝ^{n}) : |z(t)| ≤ δe^{−βt}, t ≥ 0, |z_{0}| ≤ δ}
Z̄(δ) = {z ∈ C([−r, ∞), ℝ^{n}) : |z̄(t)| ≤ δ, t ≥ −r}.

Then Z(δ) and Z̄(δ) are bounded and convex in the Banach space C([−r, ∞), ℝ^{n}) with the uniform topology. Moreover, there exists a one-to-one correspondence h: Z(δ) → Z̄(δ), defined by, for any z ∈ Z(δ),

h(z) = z̄ ∈ Z̄(δ) : z̄(t) = z(t)e^{βt}, t ≥ 0; z̄_{0} = z_{0}.

For any ϕ ∈ C, let A(ϕ, ·) : Z(δ) → C([−r, ∞), ℝ^{n}) be the operator defined by the righthand side of (3.1), that is, for each z ∈ Z(δ) and ϕ ∈ C, A(ϕ, z) ∈ C([−r, ∞), ℝ^{n}) is the function defined by

A(ϕ, z)(t) = the righthand side of (3.1) for t ≥ 0

and let B(ϕ, ·) : Z̄(δ) → C([−r, ∞), ℝ^{n}) be the operator defined as B = h ∘ A(ϕ, ·) ∘ h^{−1}, i.e.,

B(ϕ, z̄)(t) = e^{βt}A(ϕ, z)(t), for t ≥ 0
B(ϕ, z̄)_{0} = A(ϕ, z)_{0},

where z = h^{−1}(z̄).

Similarly as in the proof presented in Stokes [13,14] (see Lemma 3.4 in Stokes [13], for example), using (2.2), (2.3), and Lemma 2.2, we can prove that there exist constants η_{0} > 0 and δ_{0} > 0 such that B(ϕ, ·) : Z̄(δ_{0}) → Z̄(δ_{0}) is a uniform contraction operator with respect to ϕ ∈ U_{Q}(η_{0}). This implies that B(ϕ, ·) : Z̄(δ_{0}) → Z̄(δ_{0}) has a unique fixed point z̄^{∗}(ϕ) ∈ Z̄(δ_{0}). Therefore, z^{∗}(ϕ) = h^{−1}(z̄^{∗}(ϕ)) ∈ Z(δ_{0}) is a unique fixed point of A(ϕ, ·) : Z(δ_{0}) → Z(δ_{0}). Hence z^{∗}(ϕ) is a solution of (3.1) and (3.2). Also, from (2.2), it is clear that z^{∗}(0) = 0.

The existence of function ρ with required properties can be obtained from estimating the right-hand side of (3.1) directly using (2.2) and (2.3). Furthermore, we can choose δ_{0} = ρ(η_{0}).

In order to show that H is continuously Fréchet differentiable, we first show that z̄^{∗}(ϕ) is continuously Fréchet differentiable on U_{Q}(η_{0}). By Theorem 3.2 in Chapter 0 of Hale [4], we only need to show that B(ϕ, z̄) is continuously Fréchet differentiable with respect to both ϕ and z̄ on U_{Q}(η_{0}) × Z̄(ρ(η_{0})). It is quite easy to see that B′_{ϕ}(ϕ, z̄) exists and is continuous. To show that B′ ; z̄(ϕ, z̄) exists on U_{Q}(η_{0})×Z̄(ρ(η_{0})), we first show that B′ ; z̄(ϕ, z̄) is the linear operator from C([−r, ∞), ℝ^{n}) to itself defined by

B′ ; z̄(ϕ, z̄)ȳ(t) = e^{βt}[∫_{0}^{t} d[π_{Q(s)}K(t, s)]F′_{ϕ}(s, z_{s})y_{s}

− ∫_{t}^{∞} d[π_{P (s)}K(t, s)]F′_{ϕ}(s, z_{s})y_{s}, t ≥ 0 (3.2)

= ∫_{0}^{∞} d[π_{Q(s)}K(t, s)]F′_{ϕ}(s, z_{s})y_{s}, −r ≤ t ≤ 0,

where y = h^{−1}(ȳ), z = h^{−1}(z̄). In fact, if we let

w(ϕ, z̄, ȳ, t) = B(ϕ, z̄ + ȳ)(t) − B(ϕ, z̄)(t) − B′_{ϕ}(ϕ, z̄)ȳ(t),

we have, for all t ≥ 0,

|w(ϕ, z̄, ȳ, t)|
≤ e^{βt}[| ∫_{0}^{t} d[π_{Q(s)}K(t, s)](F (s, z_{s} + y_{s}) − F (s, z_{s}) − F′_{ϕ}(s, z_{s})y_{s})|
+| ∫_{t}^{∞} d[π_{P (s)}K(t, s)](F (s, z_{s} + y_{s}) − F (s, z_{s}) − F′_{ϕ}(s, z_{s})y_{s})|]
≤ e^{βt}[∫_{0}^{t} Me^{−α(t−s)}∥f′_{ϕ}(p_{s} + z_{s} + τy_{s}) − f′_{ϕ}(p_{s} + z_{s})∥|y_{s}|_{C}
+ ∫_{t}^{∞} M ∥ f′_{ϕ}(p_{s} + z_{s} + τy_{s}) − f′_{ϕ}(p_{s} + z_{s}) ∥ |y_{s}|_{C} ].

Since for any fixed z ∈ Z(ρ(η_{0})), {p_{s} + z_{s} : s ∈ [0, ∞)} is a compact set in C, by the continuity of f′_{ϕ}, for any given ϵ > 0, we can choose η > 0 so small that

∥ f′_{ϕ}(p_{s} + z_{s} + ψ) − f′_{ϕ}(p_{s} + z_{s}) ∥< ϵ, 0 ≤ s < ∞

provided that |ψ|_{C} < η. Therefore, using the fact that |y_{s}|_{C} = sup_{s−r≤θ≤s} |y(s + θ)| =

sup_{s−r≤θ≤s} e^{−β(s+θ)}|ȳ(s + θ)| ≤ e^{βr} ∥ ȳ ∥, we have

(3.3)  |w(ϕ, z̄, ȳ, t)| ≤ ϵMe^{βr}(\frac{1}{α − β} + \frac{1}{β}) ∥ ȳ ∥

provided that ∥ ȳ ∥< ηe^{−βr}, where ∥ ȳ ∥= sup{|ȳ(t)|, −r ≤ t < ∞}.

Similarly, we can show that (3.3) holds for −r ≤ t ≤ 0. This says that B′ ; z̄(ϕ, z̄) exists and is given by (3.2). To show the continuity of Bz̄(ϕ, z̄), we note that f′′_{ϕϕ} is assumed to be continuous and W is a compact set in C. Thus we may assume that f′′_{ϕϕ} is bounded on the set {p_{s} + z_{s} : for all s ∈ [−r, ∞), z ∈ Z(ρ(η_{0}))} (choose a smaller η_{0} if necessary). From this fact, the continuity of B′ ; z̄(ϕ, z̄) on U_{Q}(η_{0}) × Z̄(ρ(η_{0})) can be proved by using the continuity of f′′_{ϕϕ}. Details are omitted. Therefore, we have proved that z̄^{∗}(ϕ), and thus z^{∗}(ϕ), is continuously Fréchet differentiable on U_{Q}(η_{0}). Moreover, one has, for any ψ ∈ C([−r, ∞), ℝ^{n}),

z_{ϕ}^{∗}(ϕ)ψ(t) = e^{−βt}z̄_{ϕ}^{∗}(ϕ)ψ(t), t ≥ 0.

From this, one can show that, for any given ϵ > 0, we can choose η > 0 so that

|z_{s,ϕ}^{∗}(ϕ + ψ) − z_{s,ϕ}^{∗}(ϕ) − z_{z,ϕ}^{∗}(ϕ)ψ| ≤ e^{−βs}ϵ

for all s ∈ [0, ∞) and ψ ∈ C with |ψ|_{C} ≤ η. This implies that

|F (s, z_{s}^{∗}(ϕ + ψ)) − F (s, z_{s}^{∗}(ϕ)) − F′_{ϕ}(s, z_{s}^{∗}(ϕ))z_{s,ϕ}^{∗′}(ϕ)ψ|
≤ |f(p_{s} + z_{s}^{∗}(ϕ + ψ)) − f(p_{s} + z_{s}^{∗}(ϕ)) − f′_{ϕ}(p_{s} + z_{s}^{∗}(ϕ))z_{s,ϕ}^{∗′}(ϕ)ψ|
+|f′_{ϕ}(p_{s})[z_{s,ϕ}^{∗}(ϕ + ψ) − z_{s,ϕ}^{∗}(ϕ) − z_{z,ϕ}^{∗}(ϕ)ψ]|
≤ |f′_{ϕ}(p_{s} + z_{s}^{∗}(ϕ))(z_{s,ϕ}^{∗}(ϕ)ψ + e^{−βs}ϵ]) − f′_{ϕ}(p_{s} + z_{s}^{∗}(ϕ))z_{s,ϕ}^{∗′}(ϕ)ψ|
+||f′_{ϕ}(p_{s})||e^{−βs}ϵ.
≤ (||f′_{ϕ}(p_{s} + z_{s}^{∗}(ϕ))|| + ||f′_{ϕ}(p_{s})||)e^{−βs}ϵ.

Therefore one has the following

|F (s, z_{s}^{∗}(ϕ + ψ)) − F (s, z_{s}^{∗}(ϕ)) − F′_{ϕ}(s, z_{s}^{∗}(ϕ))z_{s,ϕ}^{∗′}(ϕ)ψ| ≤ Ne^{−βs}ϵ,

where N is constant independent of s. Now by the definition of H, we can easily show that H is Fréchet differentiable on U_{Q}(η_{0}) and

H′_{ϕ}(ϕ)ψ = ∫_{0}^{∞} d[π_{P (s)}K(t, s)]F′_{ϕ}(s, z_{s}^{∗}(ϕ))z_{s,ϕ}^{∗′}(ϕ)ψ.

The continuity of H′_{ϕ} can be proved by standard argument. By (2.2), one can show that H′_{ϕ}(0) = 0 : Q(0) → C.

Corollary 3.3. Under the assumptions in Lemma 3.2, if we let R = I − H(·) : U_{Q}(η_{0}) → C, then we have

(1). R : U_{Q}(η_{0}) → R(U_{Q}(η_{0})) is a homeomorphism.

(2). R is continuously Fréchet differentiable and R′_{ϕ}(0) = I : Q(0) → C; (3). If z is a solution of (3.1) with ϕ ∈ U_{Q}(η_{0}), then

|z_{t}| ≤ ρ(η_{0})e^{−βt,} t ≥ 0.

This is a direct consequence of the implicit function theorem and Lemma 3.2.

Corollary 3.4. Let π = π_{Q(0)} : C → Q(0) be the canonical projection, and H_{1,} R_{1} : U_{Q}(η_{0}) → Q(0) be defined by H_{1}(ϕ) = πH(ϕ), and R_{1}(ϕ) = ϕ − H_{1}(ϕ). Then there exist open neighborhoods of 0 in Q(0), W_{1} and W_{2}, such that R_{1} : W_{1} → W_{2} is a diffeomorphism. Moreover, R_{1}(0) = 0.

Proof. Since H : U_{Q}(η_{0}) → C is continuously Fréchet differentiable, H′_{ϕ} (0) = 0 : Q(0) → C, and π : C → Q(0) is a bounded linear operator, we have H_{1} = πH : U_{Q}(η_{0}) → Q(0) is continuously Fréchet differentiable and H′_{1,ϕ}(0) = 0 : Q(0) → C. Therefore R_{1} :

U_{Q}(η_{0}) → Q(0) is continuously Fréchet differentiable and R′_{1,ϕ}(0) = I : Q(0) → Q(0). The conclusion of the corollary then follows from the inverse function theorem.

4. Proof of the Main Theorem

In order to prove our main theorem, we will need the following two Lemmas. The first is a slight extension of the implicit function theorem stated in most introductory functional analysis book, see for example, Dieudonné [3]. The proof is almost identical to that of the standard implicit function theorem and will be omitted. The second concerns the differentiability of solutions of equation (1.1) with respect to time t.

Lemma 4.1. Let E_{1} be a metric space with metric d, E_{2} and E_{3} be Banach spaces with norms | · |_{E_{2}} and | · |_{E_{3}}, Ω ⊂ E_{1} × E_{2} be an open set, and (x_{0}, y_{0}) ∈ Ω. Assume that G : Ω → E_{3} satisfies

(1). G(x, y) is continuous in a neighborhood of (x_{0}, y_{0}) contained in Ω, and G′_{y}(x, y)

is continuous at (x_{0}, y_{0}).

(2). G(x_{0}, y_{0}) = 0 .

(3). G′_{y}(x_{0}, y_{0}) : E_{2} → E_{3} has bounded inverse.

Then there exist constants τ > 0, and r > 0 such that equation G(x, y) = 0 has a unique solution y = g(x) on {x : d(x, x_{0}) < r} satisfying y_{0} = g(x_{0}) and | g(x) − y_{0}|_{E_{2}} < τ.

Lemma 4.2. Consider equation (1.1) and let S_{0} ⊂ C be defined as

S_{0} = {ϕ : ϕ̇ ∈ C, D(ϕ̇) = D_{−}ϕ(0) − ∫_{−r}^{0}[dμ(θ)]ϕ̇(θ) = f(ϕ)},

where D_{−}ϕ(0) represents the left derivative of ϕ at θ = 0. Then S_{0} is dense in C and, for any ϕ ∈ S_{0}, the solution x_{t}(ϕ) of (1.1) through (0, ϕ) is continuously differentiable with respect to t on its interval of existence.

This Lemma is a corollary of the more general result proved in Shao [11].

Proof of The Main Theorem. We first show that there exists a δ > 0 such that, if ϕ ∈ S_{0}, |ϕ − p_{0}|_{c} ≤ δ, there exists a c_{0} = c_{0}(ϕ) > 0 such that

(4.1)  x_{c_{0}}(ϕ) ∈ p_{0} + R(U_{Q}(η_{0})),

where R and U_{Q}(η_{0}) are as in Corollary 3.3.

By the continuity of solutions, |x_{t}(ϕ) − p_{t}|_{C} can be made arbitrary small over any given finite interval if we choose ϕ sufficiently close to p_{0}. Since p_{t} is periodic, |p_{t} − p_{0}|_{C} can be made as small as we want if we choose t sufficiently close to ω. Hence, there exist constants δ_{1} > 0 and ς > 0 such that if ϕ ∈ C, |ϕ − p_{0}|_{C} < δ_{1}, t ∈ (ω − ς, ω + ς), the solution x_{t}(ϕ) of (1.1) through (0, ϕ) satisfies

(4.2)  π(x_{t}(ϕ) − p_{0}) ∈ W_{2},

where π and W_{2} are defined in Corollary 3.4. Let R and R_{1} be defined as in Corollary 3.3 and 3.4. By (4.2) and Corollary 3.4, for any ϕ ∈ C, |ϕ − p_{0}|_{C} < δ_{1}, t ∈ (ω − ς, ω + ς), there exists ψ^{t}(ϕ) ∈ W_{1} such that

ψ^{t}(ϕ) = R_{1}^{−1}π(x_{t}(ϕ) − p_{0}),

where W_{1} is defined in Corollary 3.4. This leads to

π(x_{t}(ϕ) − p_{0}) = R_{1}ψ^{t}(ϕ) = πRψ^{t}(ϕ), t ∈ (ω − ς, ω + ς).

To show (4.1), we need only to show that there exists a δ : 0 < δ < δ_{1} such that if ϕ ∈ S_{0,} |ϕ − p_{0}|_{C} ≤ δ, there exists a c_{0} = c_{0}(ϕ) > 0 such that

(I − π)(x_{c_{0}}(ϕ) − p_{0}) = (I − π)R(ψ^{t}(ϕ)).

Let us define the function on G on {ϕ ∈ C, |ϕ − p_{0}|_{C} < δ_{1}} × (ω − ς, ω + ς) by

G(ϕ, t) = (I − π)(x_{t}(ϕ) − p_{0}) − (I − π)R(ψ^{t}(ϕ)).

It is obvious that G(ϕ, t) is continuous with respect to t ∈ (ω−ς, ω+ς), ϕ ∈ C, |ϕ−p_{0}|_{C} < δ_{1}. If we further restrict that ϕ ∈ S_{0}, we also have that x_{t}(ϕ) is continuously differentiable with respect to t ∈ (ω − ς, ω + ς). Since π and I − π are both bounded linear operators, by the properties of R and R_{1,} G′_{t}(ϕ, t) is continuous for t ∈ (ω − ς, ω + ς), ϕ ∈ S_{0}, |ϕ − p_{0}|_{C} < δ_{1}. By the periodicity of p_{t}, we have ψ^{ω} (p_{0)} = 0, and thus

G(p_{0}, ω) = (I − π)(p_{ω} − p_{0}) − (I − π)R(ψ^{ω} (p_{0})) = 0.

Further, G′_{t}(ϕ, t)|_{(p_{0},ω)} has a bounded inverse. In fact we have

\frac{∂}{∂t}(x_{t}(ϕ) − p_{0})|_{(p_{0},ω)} = ẋ_{ω}(p_{0}) = ṗ_{0} ∈ P (0).

This implies that

\frac{∂}{∂t}(I − π)(x_{t}(ϕ) − p_{0})|_{(p_{0},ω)} = ṗ_{0} and \frac{∂}{∂t}π(x_{t}(ϕ) − p_{0})|_{(p_{0},ω)} = 0.

Therefore, \frac{∂}{∂t}ψ^{t}(ϕ)|_{(p_{0},ω)} = 0. By the definition of G, we have G′_{t}(ϕ, t)|_{(p_{0},ω)} = ṗ_{0}. Since ṗ_{0} is a basis of P (0), we know that G′_{t}(ϕ, t)|_{(p_{0},ω)} is a linear homeomorphism from ℝ to ℝ. Now by Lemma 4.1, there exists δ : 0 < δ < δ_{1} such that for any ϕ ∈ S_{0}, |ϕ − p_{0}| < δ, there exists an c_{0}(ϕ) > 0 such that

G(ϕ, c_{0}(ϕ)) = 0.

This proves that (4.1) holds provided that ϕ ∈ S_{0}.

Since S_{0} is dense in {ϕ ∈ C : |ϕ|_{C} < δ} , for any ϕ ∈ C with |ϕ − p_{0}|_{C} < δ, we can find a sequence {ϕ_{n}} ⊂ S_{0} satisfying that |ϕ_{n} − p_{0}|_{C} < δ and ϕ_{n} → ϕ as n → ∞. Let c_{n} = c_{0}(ϕ_{n}) be such that G(ϕ_{n}, c_{n}) = 0. Since {c_{n}} is bounded, we may assume that c_{n} → c_{0} as n → ∞. By the continuity of G, we know that

0 = lim_{n→∞} G(ϕ_{n}, c_{n}) = G(ϕ, c_{0}),

i.e., (4.1) hold for ϕ ∈ C, |ϕ|_{C} < δ.

Corollary 3.3 says that the solution z(t) of (3.1), which is identical to the solution of (2.1) through (0, x_{c_{0}}(ϕ) − p_{0}), satisfies that

|z_{t}|_{C} ≤ ρ(η_{0})e^{−βt} t ≥ 0,

Therefore we have

|x_{t+c_{0}(ϕ)} − p_{t}|_{C} ≤ ρ(η_{0})e^{−βt} t ≥ 0.

This completes the proof of the main theorem.

References

[1] Cruz, M.A. and Hale, J.K., Stability of functional differential equations of neutral type. J. Differential Equations 7 (1970), 334-355.

[2] Cruz, M.A. and Hale, J.K., Exponential estimates and saddle point properties neutral functional differential equations, J. Math. Anal. Appl. 34 (1971), 267-285.

[3] Dieudonné, J., Foundation of Modern Analysis, Academic Press, 1969.

[4] Hale, J.K., Ordinary Differential Equations, Wiley, 1969.

[5] Hale, J.K., Solutions near simple periodic orbits of functional differential equations, J. Differential Equations 7 (1970), 126-183

[6] Hale, J.K., Critical cases for neutral functional differential equations, J. Differential Equations 10 (1971), 59-82

[7] Hale, J.K., Behavior near constant solutions of functional differential equations, J. Differential Equations 15 (1974), 278-294

[8] Hale, J.K., Theory of Functional differential Equations, Applied Math. Sci., Vol. 3, Springer-Verlag, New York, 1977.

[9] Hale, J.K. and Lunel S.V., Introduction to Functional Differential Equations, Applied Math. Sci., Vol. 99, Springer-Verlag, New York, 1993.

[10] Hale, J.K. and Stokes, A. P. Behavior of solutions near integral manifolds, Arch. Rat. Mech. Anal. 2 (1960), 133-170

[11] Shao, Z., Differentiability of solutions of neutral functional differential equations, Chinese Annals of Mathematics 8A (1987), 394-402. (English Summary, Chinese Annals of Mathematics, Ser. B 8 (1987), 389)

[12] Stokes,A.P., A Floquet theory for functional differential equations, Proc. Nat. Acad. of Sci. U.S.A. 48 (1962), 1330-1334.

[13] Stokes, A.P. On the stability of a limit cycle of an autonomous functional differential equation, in Contri. Differential Equations 3 (1963), 121-139.

[14] Stokes, A.P., On the stability of integral manifolds of functional differential equations, J. Differential Equations 9 (1971), 405-419.

Youmin Lu, Department of Mathematics, Bloomsburg University, Bloomsburg, PA 17815, U.S.A

E-mail address: ylu@bloomu.edu

Zhoude Shao, Department of Mathematics, Millersville University, Millersville, PA 17551, U.S.A

E-mail address: zshao@millersville.edu