TOTAL STABILITY IN ABSTRACT FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY ^{∗}

YOSHIYUKI HINO ^{†} and SATORU MURAKAMI ^{‡}

Department of Mathematics and Informatics, Chiba University 1-33 Yayoicho, Inageku, Chiba 263–8522, Japan E-mail: hino@math.s.chiba-u.ac.jp

and

Department of Applied Mathematics, Okayama University of Science 1-1 Ridai-cho, Okayama 700–0005, Japan E-mail:murakami@xmath.ous.ac.jp

1. INTRODUCTION

Recently, authors [2] have discussed some equivalent relations for ρ-uniform stabilities of a given equation and those of its limiting equations by using the skew product flow constructed by quasi-processes on a general metric space. In 1992, Murakami and Yoshizawa [6] pointed out that for functional differential equations with infinite delay on a fading memory space B = B((−∞, 0]; R^{n}) ρ-stability is a useful tool in the study of the existence of almost periodic solutions for almost periodic systems and they proved that ρ-total stability is equivalent to BC-total stability.

The purpose of this paper is to show that equivalent relations established by Murakami and Yoshizawa [6] holds even for functional differential equations with infinite delay on a fading memory space B = B((−∞, 0]; X) with a general Banach space X.

2. FADING MEMORY SPACES AND SOME DEFINITIONS

∗This paper is in final form and no version of it will be submitted for publication elsewhere.

†Partly supported in part by Grant-in-Aid for Scientific Research (C), No.12640155, Japanese Ministry of Education, Science, Sports and Culture.

‡Partly supported in part by Grant-in-Aid for Scientific Research (C), No.11640191, Japanese Ministry of Education, Science, Sports and Culture.

Let X be a Banach space with norm | · |_{X}. For any interval J ⊂ R := (−∞, ∞), we denote by BC(J; X) the space of all bounded and continuous functions mapping J into X. Clearly BC(J; X) is a Banach space with the norm | · |_{BC(J;X)} defined by |ϕ|_{BC(J;X)} = sup{|ϕ(t)|_{X} : t ∈ J}. If J = R^{−} := (−∞, 0], then we simply write BC(J; X) and | · |_{BC(J;X)} as BC and | · |_{BC}, respectively. For any function u : (−∞, a) ↦ X and t < a, we define a function u_{t} : R^{−} ↦ X by u_{t}(s) = u(t + s) for s ∈ R^{−}. Let B = B(R^{−}; X) be a real Banach space of functions mapping R^{−} into X with a norm | · |_{B}. The space B is assumed to have the following properties:

(A1) There exist a positive constant N and locally bounded functions K(·) and M(·) on R^{+} := [0, ∞) with the property that if u : (−∞, a) ↦ X is continuous on [σ, a) with u_{σ} ∈ B for some σ < a, then for all t ∈ [σ, a),

(i) u_{t} ∈ B,

(ii) u_{t} is continuous in t (w.r.t. | · |_{B}),

(iii) N|u(t)|_{X} ≤ |u_{t}|_{B} ≤ K(t − σ) sup_{σ≤s≤t} |u(s)|_{X} + M(t − σ)|u_{σ}|_{B}.

(A2) If {ϕ^{n}} is a sequence in B ∩ BC converging to a function ϕ uniformly on any compact intertval in R^{−} and sup_{n} |ϕ^{n}|_{BC} < ∞, then ϕ ∈ B and |ϕ^{n} −ϕ|_{B} → 0 as n → ∞.

It is known [3, Proposition 7.1.1] that the space B contains BC and that there is a constant ℓ > 0 such that

|ϕ|_{B} ≤ ℓ|ϕ|_{BC}, ϕ ∈ BC.  (1)

Set B_{0} = {ϕ ∈ B : ϕ(0) = 0} and define an operator S_{0}(t) : B_{0} ↦ B_{0} by

[S_{0}(t)ϕ](s) = \{ \begin{matrix}ϕ(t + s) & if t + s ≤ 0, \\ 0 & if t + s > 0 \end{matrix}

for each t ≥ 0. In virtue of (A1), one gets that the family {S_{0}(t)}_{t≥0} is a strongly continuous semigroup of bounded linear operators on B_{0}. We consider the following properties:

(A3) lim_{t→∞}|S_{0}(t)ϕ|_{B} = 0, ϕ ∈ B_{0}.

The space B is called a fading memory space, if it satisfies (A3) in addition to (A1) and (A2). It is known [3, Proposition 7.1.5] that the functions K(·) and M(·) in (A1) can be chosen as K(t) ≡ ℓ and M(t) ≡ (1 + (ℓ/N))∥S_{0}(t)∥. Here and hereafter, we denote by ∥ · ∥ the operator norm of linear bounded operators. Note that (A3) implies sup_{t≥0} ∥S_{0}(t)∥ < ∞ by the Banach-Steinhaus theorem. Therefore, whenever B is a fading memory space, we can assume that the functions K(·) and M(·) in (A1) satisfy K(·) ≡ K and M(·) ≡ M, constants.

We provide a typical example of fading memory spaces. Let g : R^{−} ↦ [1, ∞) be any continuous nonincreasing function such that g(0) = 1 and g(s) → ∞ as s → −∞. We set

C_{g}^{0} := C_{g}^{0}(X) = {ϕ : R^{−} ↦ X is continuous with lim_{s→−∞} |ϕ(s)|_{X}/g(s) = 0}.

Then the space C_{g}^{0} equipped with the norm

|ϕ|_{g} = sup_{s≤0} \frac{|ϕ(s)|_{X}}{g(s)} , ϕ ∈ C_{g}^{0},

is a Banach space and it satisfies (A1)–(A3). Hence the space C_{g}^{0} is a fading memory space. We note that the space C_{g}^{0} is separable whenever X is separable.

Throughout the remainder of this paper, we assume that B is a fading memory space which is separable.

We now consider the following functional differential equation

\frac{du}{dt} = Au(t) + F (t, u_{t}),  (2)

where A is the infinitesimal generator of a compact semigroup {T (t)}_{t≥0} of bounded linear operators on X and F : R^{+} × B → X is continuous. We assume the following conditions on F:

(H1) F (t, ϕ) is uniformly continuous on R^{+} × S for any compact set S in B.

(H2) For any H > 0, there is an L(H) > 0 such that |F (t, ϕ)|_{X} ≤ L(H) for all t ∈ R^{+} and ϕ ∈ B such that |ϕ|_{B} ≤ H.

For any topological spaces J and X, we denote by C(J; X) the set of all continuous functions from J into X. By virture of (H1) and (H2), it follows that for any (σ, ϕ) ∈ R × B, there exists a function u ∈ C((−∞, t_{1}); X) such that u_{σ} = ϕ and the following relation holds:

u(t) = T (t − σ)ϕ(0) + ∫_{σ}^{t} T (t − s)F (s, u_{s})ds, σ ≤ t < t_{1},

(cf. [1, Theorem 1]). Such a function u is called a (mild) solution of (2) through (σ, ϕ) defined on [σ, t_{1}) and denoted by u(t) := u(t, σ, ϕ, F).

In the above, t_{1} can be taken as t_{1} = ∞ if sup_{σ≤t<t_{1}} |u(t)|_{X} < ∞(cf. [1, Corollary 2]). In the following, we always assume the following condition, too:

(H3) Equation (2) has a bounded solution ū(t) defined on R^{+} such that ū_{0} ∈ BC and |ū_{t}|_{B} ≤ C_{1} for all t ∈ R^{+}.

By virtue of [4, Lemma 2], we see that the set \overline{\{ū(t) : t ∈ R^{+}\}} is compact in X, ū(t) is uniformly continuous on R^{+} and the set \overline{\{ū_{t} : t ∈ R^{+}\}} is compact in B.

Now we shall give the definition of BC-total stability.

Definition 1 The bounded solution ū(t) of (2) is said to be BC-totally stable (BC-TS) if for any ε > 0 there exists a δ(ε) > 0 with the property that σ ∈ R^{+}, ϕ ∈ BC with |ū_{σ} − ϕ|_{BC} < δ(ε) and h ∈ BC([σ, ∞); X) with sup_{t∈[σ,∞)} |h(t)|_{X} < δ(ε) imply |ū(t) − u(t, σ, ϕ, F + h)|_{X} < ε for t ≥ σ, where u(·, σ, ϕ, F + h) denotes the solution of

\frac{du}{dt} = Au(t) + F (t, u_{t}) + h(t), t ≥ σ,  (3)

through (σ, ϕ).

For any ϕ, ψ ∈ BC, we set

ρ(ϕ, ψ) = ∑_{j=1}^{∞} 2^{−j}|ϕ − ψ|_{j}/{1 + |ϕ − ψ|_{j}},

where | · |_{j} = | · |_{[−j,0]}. Then (BC,ρ) is a metric space. Furthermore, it is clear that ρ(ϕ^{k}, ϕ) → 0 as k → ∞ if and only if ϕ^{k} → ϕ compactly on R^{−}. Let U be a closed bounded subset of X whose interior U^{i} contains the set \overline{\{ū(t) : t ∈ R\}}, where ū is the one in (H3). Whenever ϕ ∈ BC satisfies ϕ(s) ∈ U for all s ∈ R^{−}, we write as ϕ(·) ∈ U, for simply.

We shall give the definition of ρ-total stability.

Definition 2 The bounded solution ū(t) of (2) is said to be ρ-totally stable with respect to U (ρ-TS w.r.t.U) if for any ε > 0 there exists a δ(ε) > 0 with the property that σ ∈ R^{+}, ϕ(·) ∈ U with ρ(ū_{σ}, ϕ) < δ(ε) and h ∈ BC([σ, ∞); X) with sup_{t∈[σ,∞)} |h(t)|_{X} < δ(ε) imply ρ(ū_{t}, u_{t}(σ, ϕ, F + h)) < ε for t ≥ σ.

In the above, if the term ρ(u_{t}(σ, ϕ, F + h), ū_{t}) is replaced by |u(t, σ, F + h) − ū(t)|_{X}, then we have another concept of ρ-total stability, which will be referred to as the (ρ, X)- total stability.

As was shown in [6, Lemma 2], these two concepts of ρ-total stability are equivalent.

3. EQUIVALENCE OF BC-TOTAL STABILITY AND ρ-TOTAL STABILITY

In this section, we shall discuss the equivalence between the BC-total stability and the ρ-total stability, and extend a result due to Murakami and Yoshizawa [6, Theorems 1]) for X = R^{n}.

We now state our main result of this section.

Theorem The solution ū(t) of (2) is BC-TS if and only if it is ρ-TS w.r.t.U for any bounded set U in X such that U^{i} ⊃ Oū := \overline{\{ū(t) : t ∈ R\}}.

In order to prove the theorem, we need a lemma. A subset F of C(R^{+}; X) is said to be uniformly equicontinuous on R^{+}, if sup{|x(t + δ) − x(t)|_{X} : t ∈ R^{+}, x ∈ F} → 0 as δ → 0^{+}. For any set F in C(R^{+}; X) and any set S in B, we set

R(F) = {x(t) : t ∈ R^{+}, x ∈ F}
W (S, F) = {x(·) : R ↦ X : x_{0} ∈ S, x|_{R^{+}} ∈ F}

and

V (S, F) = {x_{t} : t ∈ R^{+}, x ∈ W (S, F)}.

Lemma ([5, Lemma 1]) If S is a compact subset in B and if F is a uniformly equicontinuous set in C(R^{+}; X) such that the set R(F) is relatively compact in X, then the set V (S, F) is relatively compact in B.

Proof of Theorem The “if” part is easily shown by noting that ρ(ϕ, ψ) ≤ |ϕ − ψ|_{BC} for ϕ, ψ ∈ BC. We shall establish the “only if” part. We assume that the solution ū(t) of (2) is BC-TS but not (ρ, X)-TS w.r.t.U; here U ⊂ {x ∈ X : |x|_{X} ≤ c} for some c > 0. Since the solution ū(t) of (2) is not (ρ, X)-TS w.r.t.U, there exist an ε ∈ (0, 1), sequences {τ_{m}} ⊂ R^{+}, {t_{m}}(t_{m} > τ_{m}), {ϕ^{m}} ⊂ BC with ϕ^{m}(·) ∈ U, {h_{m}} with h_{m} ∈ BC([τ_{m}, ∞)), and solutions {u(t, τ_{m}, ϕ^{m}, F + h_{m}) := û^{m}(t)} of

\frac{du}{dt} = Au(t) + F (t, u_{t}) + h_{m}(t)

such that

ρ(ϕ^{m}, ū_{τ_{m}}) < 1/m and |h_{m}|_{[τ_{m},∞)} < 1/m  (4)

and that

|û^{m}(t_{m}) − ū(t_{m})|_{X} = ε and |û^{m}(t) − ū(t)|_{X} < ε on [τ_{m}, t_{m})  (5)

for m ∈ N, where N denotes the set of all positive integers. For each m ∈ N and r ∈ R^{+}, we define ϕ^{m,r} ∈ BC by

ϕ^{m,r}(θ) = \{ \begin{matrix}ϕ^{m}(θ) & if − r ≤ θ ≤ 0, \\ ϕ^{m}(−r) + ū(τ_{m} + θ) − ū(τ_{m} − r) & if θ < −r. \end{matrix}

We note that

sup{|ϕ^{m,r} − ϕ^{m}|_{B} : m ∈ N} → 0 as r → ∞.  (6)

Indeed, if (6) is false, then there exist an ε > 0 and sequences {m_{k}} ⊂ N and {r_{k}}, r_{k} → ∞ as k → ∞, such that |ϕ^{m_{k},r_{k}} − ϕ^{m_{k}}|_{B} ≥ ε for k = 1, 2, · · ·. Put ψ^{k} := ϕ^{m_{k},r_{k}} − ϕ^{m_{k}}. Clearly, {ψ^{k}} is a sequence in BC which converges to zero function compactly on R^{−} and sup_{k} |ψ^{k}|_{BC} < ∞. Then Axiom (A2) yield that |ψ^{k}|_{B} → 0 as k → ∞, a contradiction.

Next we shall show that the set {ϕ^{m}, ϕ^{m,r} : m ∈ N, r ∈ R^{+}} is relatively compact in B. Since the set \overline{\{ū_{t} : t ∈ R^{+}\}} is compact in B as noted in the preceding section, (4) and Axiom (A2) yield that any sequence {ϕ^{m_{j}}}_{j=1}^{∞}(m_{j} ∈ N) has a convergent subsequence in B. Therefore, it sufficies to show that any sequence {ϕ^{m_{j},r_{j}}}_{j=1}^{∞}(m_{j} ∈ N, r_{j} ∈ R^{+}) has a convergent subsequence in B. We assert that the sequence of functions {ϕ^{m_{j},r_{j}}(θ)}_{j=1}^{∞} contains a subsequence which is equicontinuous on any compact set in R^{−}. If this is the case, then the sequence {ϕ^{m_{j},r_{j}}}_{j=1}^{∞} would have a convergent subsequence in B by Ascoli’s theorem and Axiom (A2), as required. Now, notice that the sequence of functions {ū(τ_{m_{j}} + θ)} is equicontinuous on any compact set in R^{−}. Then the assertion obviously holds true when the sequence {m_{j}} is bounded. Taking a subsequence if necessary, it is thus sufficient to consider the case m_{j} → ∞ as j → ∞. In this case, it follows from (4) that ϕ^{m_{j}}(θ) − ū(τ_{m_{j}} + θ) =: w^{j}(θ) → 0 uniformly on any compact set in R^{−}. Consequently, {w^{j}(θ)} is equicontinuous on any compact set in R^{−}, and so is {ϕ^{m_{j}}(θ)}. Therefore the assertion immediately follows from this observation.

Now, for any m ∈ N, set u^{m}(t) = û^{m}(t + τ_{m}) if t ≤ t_{m} − τ_{m} and u^{m}(t) = u^{m}(t − τ_{m}) if t > t_{m} − τ_{m}. Moreover, set u^{m,r}(t) = ϕ^{m,r}(t) if t ∈ R^{−} and u^{m,r}(t) = u^{m}(t) if t ∈ R^{+}. In what follows, we shall show that {u^{m}(t)} is a family of uniformly equicontinuous functions on R^{+}. To do this, we first prove that

inf_{m}(t_{m} − τ_{m}) > 0.  (7)

Assume that (7) is false. By taking a subsequence if necessary, we may assume that lim_{m→∞}(t_{m} − τ_{m}) = 0. If m ≥ 3 and 0 ≤ t ≤ min{t_{m} − τ_{m}, 1}, then

|u^{m}(t) − ū(t + τ_{m})|_{X} = |T (t)ϕ^{m}(0) + ∫_{0}^{t} T (t − s){F (s + τ_{m}, u_{s}^{m}) + h_{m}(s + τ_{m})}ds
−T (t)ū(τ_{m}) − ∫_{0}^{t} T (t − s)F (s + τ_{m}, ū_{s+τ_{m}})ds|_{X}
≤ C_{2}{|ϕ^{m} − ū_{τ_{m}}|_{1} + ∫_{0}^{t}(2L(H) + 1)ds}
≤ C_{2}{2/(m − 2) + t(2L(H) + 1)},

where H = sup{|ū_{s}|_{B}, |u_{s}^{m}|_{B} : 0 ≤ s ≤ t_{m} − τ_{m}, m ∈ N} and C_{2} = sup_{0≤s≤1} ||T (s)||. Then (5) yields that

ε ≤ C_{2}{2/(m − 2) + (t_{m} − τ_{m})(2L(H) + 1)} → 0

as m → ∞, a contradiction. We next prove that the set O := {u^{m}(t) : t ∈ R^{+}, m ∈ N} is relatively compact in X. To do this, for each η such that 0 < η < inf_{m}(t_{m} − τ_{m}) we consider the sets O_{η} = {u^{m}(t) : t ≥ η, m ∈ N} and Õ_{η} = {u^{m}(t) : 0 ≤ t ≤ η, m ∈ N}. Then α(O) = max{α(O_{η}), α(Õ_{η})}, where α(·) is the Kuratowski’s measure of noncompactness of sets in X. For the details of the properties of α(·), see [5; Section 1.4]. Let 0 < ν < min{1, η}. If η ≤ t ≤ t_{m} − τ_{m}, then

u^{m}(t) = T (t)ϕ^{m}(0) + ∫_{0}^{t} T (t − s){F (s + τ_{m}, u_{s}^{m}) + h_{m}(s + τ_{m})}ds
= T (ν)[T (t − ν)u^{m}(0) + ∫_{0}^{t−ν} T (t − ν − s){F (s + τ_{m}, u_{s}^{m}) + h(s + τ_{m})}ds]
+ ∫_{t−ν}^{t} T (t − s){F (s + τ_{m}, u_{s}^{m}) + h(s + τ_{m})}ds
= T (ν)u^{m}(t − ν) + ∫_{t−ν}^{t} T (t − s){F (s + τ_{m}, u_{s}^{m}) + h(s + τ_{m})}ds.

Since the set T (ν){u^{m}(t − ν) : t ≥ η, m ∈ N} is relatively compact in X because of the compactness of the semigroup {T (t)}_{t≥0}, it follows that

α(O_{η}) ≤ C_{2}{L(H) + 1}ν.

Letting ν → 0 in the above, we get α(O_{η}) = 0 for all η such that 0 < η < inf_{m}(t_{m} − τ_{m}). Observe that the set {T (t)ϕ^{m}(0) : 0 ≤ t ≤ η, m ∈ N} is relatively compact in X. Then

α(O) = α(Õ_{η})
= α({T (t)ϕ^{m}(0) + ∫_{0}^{t} T (t − s){F (s + τ_{m}, u_{s}^{m}) + h(s + τ_{m})}ds : 0 ≤ t ≤ η, m ∈ N})
= α({∫_{0}^{t} T (t − s){F (s + τ_{m}, u_{s}^{m}) + h(s + τ_{m})}ds : 0 ≤ t ≤ η, m ∈ N})
= C_{2}(L(H) + 1)η

for all η such that 0 < η < inf_{m}(t_{m} − τ_{m}), which shows α(O) = 0; consequently O must be relatively compact in X.

Now, in order to establish the uniform equicontinuity of the family {u^{m}(t)} on R^{+}, let σ ≤ s ≤ t ≤ s + 1 and t ≤ t_{m} − τ_{m}. Then

|u^{m}(t) − u^{m}(s)|_{X} ≤ |T (t − s)u^{m}(s) − u^{m}(s)|_{X} + | ∫_{s}^{t} T (t − τ){F (τ + τ_{m}, u_{τ}^{m})
+h(τ + τ_{m})}dτ|_{X}
≤ sup{|T (t − s)z − z|_{X} : z ∈ O} + C_{2}{L(H) + 1}|t − s|.

Since the set O is relatively compact in X, T (τ)z is uniformly continuous in τ ∈ [0, 1] uniformly for z ∈ O. This leads to sup{|u^{m}(t)−u^{m}(s)|_{X} : 0 ≤ s ≤ t ≤ s+1, m ∈ N} → 0 as |t − s| → 0, which proves the uniform equicontinuity of {u^{m}} on R^{+}.

Since {ϕ^{m}, ϕ^{m,r} : m ∈ N, r ∈ R^{+}} is relatively compact in B, |u_{t}^{m}|_{B} ≤ K{1 + |ū|_{[0,∞)}} + M|ϕ^{m}|_{B} ≤ K{1 + |ū|_{[0,∞)}} + Mℓc by (1) and (A1-iii), and the family {u^{m}(t)} is uniformly equicontinuous on R^{+}, it follows from Lemma that the set W := \overline{\{u_{t}^{m,r}, u_{t}^{m} : m ∈ N, t ∈ R^{+}, r ∈ R^{+}\}} is compact in B. Hence F (t, ϕ) is uniformly continuous on R^{+} × W. Define a continuous function q_{m,r} on R^{+} by q_{m,r}(t) = F (t + τ_{m}, u_{t}^{m}) − F (t + τ_{m}, u_{t}^{m,r}) if 0 ≤ t ≤ t_{m} − τ_{m}, and q_{m,r}(t) = q_{m,r}(t_{m} − τ_{m}) if t > t_{m} − τ_{m}. Since |u_{t}^{m,r} − u_{t}^{m}|_{B} ≤ M|ϕ^{m,r} − ϕ^{m}|_{B} (t ∈ R^{+}, m ∈ N) by (A1-iii), it follows from (6) that sup{|u_{t}^{m,r} − u_{t}^{m}|_{B} : t ∈ R^{+}, m ∈ N} → 0 as r → ∞; hence one can choose an r = r(ε) ∈ N in such a way that

sup{|q_{m,r}(t)|_{X} : m ∈ N, t ∈ R^{+}} < δ(ε/2)/2,

where δ(·) is the one for BC-TS of the solution ū(t) of (2). Moreover, for this r, select an m ∈ N such that m > 2^{r}(1+δ(ε/2))/δ(ε/2). Then 2^{−r}|ϕ^{m} −ū_{t_{m}}|_{r}/[1+|ϕ^{m} −ū_{t_{m}}|_{r}] ≤ ρ(ϕ^{m}, ū_{t_{m}}) < 2^{−r}δ(ε/2)/[1 + δ(ε/2)] by (4), which implies that

|ϕ^{m} − ū_{t_{m}}|_{r} < δ(ε/2) or |ϕ^{m,r} − ū_{t_{m}}|_{BC} < δ(ε/2).

The function u^{m,r} satisfies u_{0}^{m,r} = ϕ^{m,r} and

u^{m,r}(t) = u^{m}(t)
= T (t)ϕ^{m,r}(0) + ∫_{0}^{t} T (t − s){F (s + τ_{m}, u_{s}^{m}) + h_{m}(s + τ_{m})}ds
= T (t)ϕ^{m}(0) + ∫_{0}^{t} T (t − s){F (s + τ_{m}, u_{s}^{m,r}) + q_{m,r}(s) + h_{m}(s + τ_{m})}ds

for t ∈ [0, t_{m} − τ_{m}). Since ū^{m}(t) = ū(t + τ_{m}) is a BC-TS solution of

\frac{du}{dt} = Au(t) + F (t + τ_{m}, u_{t})

with the same δ(·) as the one for ū(t), from the fact that sup_{t≥0} |q_{m,r}(t)+h_{m}(τ_{m} +t)|_{X} < δ(ε/2)/2 + 1/m < δ(ε/2) it follows that |u^{m,r}(t) − ū(t + τ_{m})|_{X} < ε/2 on [0, t_{m} − τ_{m}). In particular, we have |u^{m,r}(t_{m} − τ_{m}) − ū(t_{m})|_{X} < ε or |û^{m}(t_{m}) − ū(t_{m})|_{X} < ε, which contradicts (5).

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