GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 2, 1998, 107-120

ASYMPTOTIC SOLUTIONS FOR MIXED-TYPE EQUATIONS WITH A SMALL DEVIATION

ZHANYUAN HOU AND J. S. CASSELL

Abstract. An asymptotic matrix solution is formulated for a class of mixed-type linear vector equations with a single variable deviation which is small at infinity. This matrix solution describes the asymptotic behavior of all exponentially bounded solutions. A sufficient condition is obtained for there to be no other solutions.

§ 1. Introduction

We consider the equation

\frac{d}{dt}x(t) = Ax(t − r(t)),  (1.1)

where A is an N × N constant matrix, r is measurable on [τ, ∞) with −σ ≤ r(t) ≤ ρ for some constants σ > 0 and ρ > 0, and r satisfies a suitable condition of smallness as t → ∞. Under the conditions that 0 ≤ r(t) ≤ ρ, lim_{t→∞} r(t) = 0 and either r ∈ L^{p}(τ, ∞) for some p ∈ (1, 2) with dr/dt bounded or r ∈ L(τ, ∞), Cooke [1] showed for the scalar case of (1.1) that every solution has the asymptotic form

x(t) = exp {At − A^{2}∫_{0}^{t} r(s)ds}{c + o(1)}  (1.2)

as t → ∞. Kato [2] proved that (1.2) holds also for N > 1, still with 0 ≤ r(t) ≤ ρ, if either r ∈ L(τ, ∞) or r ∈ L^{2}(τ, ∞) provided, in the latter case, r is either Lipschitzian or monotone for large t. This leaves open the case of r ∈ L^{p}(τ, ∞) with p > 2. Also, it would be desirable to have a clearer indication of the size of the o(1) term in (1.2), particularly when c = 0. If, further, we remove the restriction r ≥ 0, then another issue arises: in the retarded case all solutions are O(e^{|A|t}) as t → ∞ (see [3, §6.1]), but this

107 need not be so for the mixed-type equation. These matters are all dealt with in this present paper.

1991 Mathematics Subject Classification. 34K25.

Key words and phrases. Asymptotic solutions, small deviation.

1072-947X/98/0300-0107$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

Our results for (1.1) generalize to equations of the form

\frac{d}{dt}x(t) = A(t)x(t) + B(t)x(t − r(t)),  (1.3)

where A(t) = ∑_{i=1}^{n} a_{i}(t)M_{i} and B(t) = ∑_{i=1}^{n} b_{i}(t)M_{i}, a_{i} and b_{i} are locally integrable scalar functions and the constant matrices M_{i} all commute.

The contents of this paper are arranged as follows. We present two main theorems in §2, but leave some of the proofs to §3. In §4 sufficient conditions are given to exclude the possibility of solutions that grow faster than e^{γt} for every γ > 0. Equation (1.3) is dealt with in §5.

§ 2. Main Results

As we shall see later, (1.1) can be transformed, by a generalization of a substitution of Kato [2], to an equation of the form

\frac{d}{dt}x(t) = F (t)x(t) + G(t)\{x(t − r(t)) − x(t)\},  (2.1)

where F ∈ L(τ, ∞) and G ∈ L^{∞}(τ, ∞). We shall give a result for (2.1) before tackling (1.1).

It will be assumed that

M{t ≥ τ : |r(t)| > α} < ∞ for every α > 0,  (2.2)

where bM denotes Lebesgue measure. This condition certainly holds if r ∈ L^{p}(τ, ∞) for some p ∈ [1, ∞) or if lim_{t→∞} r(t) = 0. For s ≥ T ≥ τ, let

μ_{T} (s) = M {t ≥ T : min{t, t − r(t)} ≤ s ≤ max{t, t − r(t)}} . (2.3)

Since (2.2) implies that, for α > 0,

lim_{T →∞} M{t ≥ T : |r(t)| > α} = 0,

for any δ > 0 there exists T_{δ} ≥ τ such that M{t ≥ T_{δ} : |r(t)| ≥ δ/3} < δ/3. Then, for s ≥ T ≥ T_{δ}, μ_{T} (s) < δ. Thus if

ε_{T} = sup_{s≥T} μ_{T} (s),  (2.4)

then ε_{T} is finite and tends to zero as T → ∞.

A continuous vector function x : [T −ρ, ∞) → ℂ^{N} is said to be a solution of (2.1) on [T − ρ, ∞) if, on [T, ∞), it is locally absolutely continuous and satisfies (2.1) almost everywhere. A solution is said to be exponentially bounded if it is O(e^{γt}) as t → ∞ for some γ > 0. By a large solution we mean one that is not exponentially bounded. For any p ∈ [1, ∞] and any f ∈ L^{p}(T, ∞), we denote the norm of f by ∥f∥_{p} or, to mark the relevance of [T, ∞), by ∥f∥_{p}(T, ∞).

Theorem 1. Assume that F ∈ L(τ, ∞), G ∈ L^{∞}(τ, ∞), and (2.2) holds. Then, for T ≥ τ satisfying

∥F ∥_{1}(T, ∞) + ε_{T} ∥G∥_{∞}(T, ∞) < 1,  (2.5)

(2.1) has solutions x_{n} : [T − ρ, ∞) → ℂ^{N} (n = 1, 2, …, N) such that

x_{n}(t) = e_{n} + o(1)  (2.6)

as t → ∞, where e_{n} is the nth coordinate vector, and x_{n} is constant on [T − ρ, T]. Moreover, every exponentially bounded solution of (2.1) has the asymptotic representation

x(t) = X(t)c + o(e^{−βt})  (2.7)

for arbitrary β > 0, where X = (x_{1}, x_{2}, …, x_{N}) and c ∈ ℂ^{N} is a constant dependent on x.

The proof of Theorem 1 is left to §3. The following two remarks also apply to Theorem 2 given below.

Remarks. (i) Equation (2.1) may have large solutions which certainly cannot be represented by (2.7). For instance, the equation

\frac{d}{dt}x(t) = 2te^{−2t^{1/2}−t^{−1}}x(t + t^{−1/2})

meets all the requirements of Theorem 1 and it has a large solution x(t) = e^{t^{2}}.

(ii) From representation (2.7) we see that, if every solution of (2.1) is exponentially bounded, then the space of solutions on [T − ρ, ∞) can be decomposed as S_{1} ⊕ S_{2}, where S_{1} is N dimensional with a basis X, and S_{2} consists of solutions tending to zero faster than any exponential.

Extend r to all of ℝ ≡ (−∞, ∞) by putting r(t) = 0 for t < τ and, for k = 0, 1, 2, … and t ∈ ℝ, let

r_{k+1}(t) = ∫_{t}^{t−r(t)} |r_{k}(s)|ds,  (2.8)
A_{k+1}(t) = A exp { ∫_{t}^{t−r(t)} A_{k}(s)ds}  (2.9)

with r_{0}(t) = r(t) and A_{0}(t) = A.

Theorem 2. Assume that (2.2) holds and r_{M} ∈ L(τ, ∞) for some integer M ≥ 0. Then, for sufficiently large T ≥ τ, (1.1) has solutions x_{n} on [T − ρ, ∞) satisfying

x_{n}(t) = exp { ∫_{T}^{t} A_{M}(s)ds}{e_{n} + o(1)} (n = 1, 2, …, N) (2.10)

as t → ∞. Moreover, every exponentially bounded solution of (1.1) has the asymptotic representation (2.7) with X composed of N solutions (2.10).

Proof. Since each A_{k} in (2.9) can be expanded as a power series in A with scalar functions as coefficients, they all commute with A and with each other. Let

c_{0} = |A| and c_{k+1} = |A| exp {c_{k} max{ρ, σ}} (k ≥ 0). (2.11)

Then, from (2.9) and (2.11),

\begin{matrix}∥A_{0}∥_{∞} = c_{0}, \\ ∥A_{k+1}∥_{∞} ≤ |A| exp \{∥A_{k}∥_{∞} max\{ρ, σ\}\} ≤ c_{k+1} (k ≥ 0). \end{matrix} (2.12)

Since (2.11) implies c_{k} ≤ c_{k+1} (k ≥ 0) and, for any a and b in ℂ,

|e^{a} − e^{b}| ≤ e^{max{|a|,|b|}}|a − b|,

we have

|A_{k+1}(t) − A_{k}(t)| ≤ c_{k+1}| ∫_{t}^{t−r(t)}(A_{k}(s) − A_{k−1}(s))ds| (k ≥ 0),

where A_{−1} = 0. Inductively, from this and (2.8), we obtain

|A_{k+1}(t) − A_{k}(t)| ≤ c_{0}c_{1} ⋯ c_{k+1}|r_{k}(t)| (k ≥ 0).  (2.13)

By the transformation

x(t) = exp { ∫_{T}^{t} A_{M}(s)ds}y(t)  (2.14)

(cf. Kato [2, p163]), equation (1.1) can be rewritten as

\frac{d}{dt}y(t) = (A_{M+1}(t) − A_{M}(t)) y(t) + A_{M+1}(t)\{y(t − r(t)) − y(t)\}. (2.15)

Then, since A_{M+1} is bounded and (2.13), along with the condition r_{M} ∈ L(τ, ∞), implies A_{M+1} − A_{M} ∈ L(τ, ∞), the conclusion follows from Theorem 1 and (2.14).

Remark. T ≥ τ required in Theorem 2 can be chosen in advance. From (2.5), (2.15), (2.13), and (2.12), we see that it is sufficient for T to satisfy

c_{0}c_{1} ⋯ c_{M+1} ∫_{T}^{∞} |r_{M}(s)|ds + ε_{T} c_{M+1} < 1.  (2.16)

Such a choice is possible since r_{M} ∈ L(τ, ∞) and ε_{T} → 0 as T → ∞.

Corollary. Assume that r ∈ L^{p}[τ, ∞) for some p ∈ (1, ∞). Then r_{M} ∈ L(τ, ∞) and the conclusion of Theorem 2 holds for M ≥ p.

The proof of Corollary is left to §3.

Remarks. (i) If r ∈ L(τ, ∞), then certainly we can take M = 0. If r ∈ L^{p}(τ, ∞) with p > 1 and r is either monotone or Lipschitzian (as in Kato [2]), then r_{k} ∈ L^{p/(k+1)}(τ, ∞) for each k and hence r_{M} ∈ L(τ, ∞) for M ≥ p − 1. This is a consequence of the following inequalities which can easily be shown from (2.8) by induction:

(a) |r_{k}(t)| ≤ |r(t)|^{k+1} if r ≤ 0 and r is increasing,

(b) |r_{k}(t + kρ)| ≤ |r(t)|^{k+1} if r ≥ 0 and r is decreasing,

(c) |r_{k}(t)| ≤ (μ + 1)^{k(k+1)/2}|r(t)|^{k+1} if |r(t) − r(s)| ≤ μ|t − s|.

(ii) For any particular value of p ≥ 1, we can further simplify the expression (2.10). If p = 1, then M = 0 and x_{n}(t) = e^{A(t−T)}{e_{n} + o(1)}. If p ∈ (1, 2] and M = 1, we have A_{1}(t) = Ae^{−Ar(t)} = A − A^{2}r(t) + L, where L stands for a function in L(τ, ∞), and we can replace (2.10) by

x̂_{n}(t) = exp {At − A^{2}∫_{T}^{t} r(s)ds}{e_{n} + o(1)} (n = 1, 2, …, N) (2.17)

as (x_{1}, x_{2}, …, x_{N}) = (x̂_{1}, x̂_{2}, …, x̂_{N})E for an invertible constant matrix E. Hence, in view of Remark (i), our result includes that of Kato referred to in §1. If p ∈ (1, 2] and M = 2, then

A_{2}(t) = A exp { ∫_{t}^{t−r(t)}Ae^{−Ar(s)}ds} = A − A^{2}r(t) − A^{3}∫_{t}^{t−r(t)} r(s)ds + L

and we can replace (2.10) by

x̂_{n}(t) = exp {At − A^{2}∫_{T}^{t} (r(s) + A∫_{s}^{s−r(s)} r(u)du)ds}{e_{n} + o(1)}. (2.18)

Similarly, we can simplify (2.10) for p > 2.

§ 3. Proofs of Theorem 1 and Corollary to Theorem 2

For t ≥ τ and each k ∈ {1, 2, …, N}, we consider the integral equation

x(t) = e_{k} − ∫_{t}^{∞} {F (s)x(s) + G(s) (x(s − r(s)) − x(s))} ds. (3.1)

Lemma 1. Assume that the conditions of Theorem 1 are met and that T ≥ τ satisfies (2.5). Then, for each k ∈ {1, 2, …, N}, (3.1) has a solution on [T − ρ, ∞) satisfying lim_{t→∞} x_{k}(t) = e_{k} and x_{k}(t) = x_{k}(T) for t ∈ [T − ρ, T].

Proof. Let S be the set of x in C([T − ρ, ∞), ℂ^{N}) satisfying x(t) = x(T) for t ∈ [T − ρ, T], x ∈ L^{∞}(T, ∞) and x(· − r(·)) − x ∈ L(T, ∞). Then S is a Banach space under the norm ∥x∥_{S} = max {∥x∥_{∞}, β∥x∥_{0}}, where

∥x∥_{0} = ∫_{T}^{∞} |x(t − r(t)) − x(t)|dt

and β > 0 is chosen so that βε_{T} < 1 and ∥F ∥_{1}(T, ∞) + \frac{1}{β} ∥G∥_{∞}(T, ∞) < 1. For any x ∈ S let

Ax(t) = − ∫_{t}^{∞} {F (s)x(s) + G(s) (x(s − r(s)) − x(s))} ds (t ≥ T)

and Ax(t) = Ax(T) for t ∈ [T −ρ, T]. Then Ax is continuous and bounded with

∥Ax∥_{∞} ≤ ∥F ∥_{1}∥x∥_{∞} + ∥G∥_{∞}∥x∥_{0} ≤ (∥F ∥_{1} + \frac{1}{β} ∥G∥_{∞})∥x∥_{S}.

Further,

|Ax(t−r(t))−Ax(t)| ≤ ∫_{max{β(t),T }}^{γ(t)}|F (s)x(s) + G(s) (x(s − r(s)) − x(s))| ds,

where γ(t) = max{t, t − r(t)} and β(t) = min{t, t − r(t)}, and so, with μ_{T} defined by (2.3),

∫_{T}^{∞} |Ax(t − r(t)) − Ax(t)|dt ≤
≤ ∫_{T}^{∞} |F (s)x(s) + G(s) (x(s − r(s)) − x(s))| μ_{T} (s)ds ≤
≤ ε_{T} (∥F ∥_{1}∥x∥_{∞} + ∥G∥_{∞}∥x∥_{0}) .

Thus

β∥Ax∥_{0} ≤ ∥F ∥_{1}∥x∥_{∞} + ∥G∥_{∞}∥x∥_{0}.

Hence A is a contraction on S and the conclusion follows.

Lemma 2. Assume that f : [R − σ_{1}, ∞) → ℂ^{N} satisfies

|f(t)| ≤ ω(t)∥f∥_{[t−σ_{1},t+σ_{2}]} (t ≥ R),  (3.2)

where ω : [R, ∞) → (0, 1) is decreasing, σ_{1} and σ_{2} are positive constants, and ∥f∥_{[a,b]} = sup_{a≤s≤b} |f(s)|. If

f(t) = o( exp \{ − \frac{1}{σ_{2}} ∫_{R}^{t−2σ_{2}} ln ω(s)ds\})  (3.3)

as t → ∞, then

|f(t)| ≤ exp \{\frac{1}{σ_{1}} ∫_{R}^{t} ln ω(s)ds\}∥f∥_{[R−σ_{1},R]} (t ≥ R).  (3.4)

Proof. From (3.2),

∥f∥_{[t,t+σ_{2}]} ≤ ω(t)∥f∥_{[t−σ_{1},t+2σ_{2}]}

and hence, since ω(t) < 1,

∥f∥_{[t,t+σ_{2}]} ≤ ω(t) max {∥f∥_{[t−σ_{1},t]}, ∥f∥_{[t+σ_{2},t+2σ_{2}]}} .  (3.5)

We shall show by induction that, for all t ≥ R and non-negative integer k,

∥f∥_{[t,t+σ_{2}]} ≤
≤max{ω(t)∥f∥_{[t−σ_{1},t]}, (∏_{j=0}^{k} ω(t+jσ_{2}))∥f∥_{[t+(k+1)σ_{2},t+(k+2)σ_{2}]}} (3.6)

and

∥f∥_{[t+kσ_{2},t+(k+1)σ_{2}]} ≤
≤ ω(t + kσ_{2}) max {∥f∥_{[t−σ_{1},t]}, ∥f∥_{[t+(k+1)σ_{2},t+(k+2)σ_{2}]}} .  (3.7)

For k = 0 these both reduce to (3.5). Suppose they hold for k. Then, from (3.7) with t replaced by t + σ_{2},

∥f∥_{[t+(k+1)σ_{2},t+(k+2)σ_{2}]} ≤ ω(t + (k + 1)σ_{2}) ×
× max {∥f∥_{[t+σ_{2}−σ_{1},t+σ_{2}]}, ∥f∥_{[t+(k+2)σ_{2},t+(k+3)σ_{2}]}} .  (3.8)

Further,

∥f∥_{[t+σ_{2}−σ_{1},t+σ_{2}]} ≤ max {∥f∥_{[t−σ_{1},t]}, ∥f∥_{[t,t+σ_{2}]}} ≤
≤ max {∥f∥_{[t−σ_{1},t]}, ∥f∥_{[t+(k+1)σ_{2},t+(k+2)σ_{2}]}}

by (3.6), and this together with (3.8) implies (3.7) with k replaced by k + 1. By substitution we also obtain (3.6) with k replaced by k + 1.

Now, from (3.3), for any ε > 0,

|f(t)| ≤ ε exp \{ − \frac{1}{σ_{2}} ∫_{R}^{t−2σ_{2}} ln ω(s)ds\}

for all sufficiently large t. Hence, for any t ≥ R and large enough k,

ω(t + σ_{2})ω(t + 2σ_{2}) ⋯ ω(t + kσ_{2})∥f∥_{[t+(k+1)σ_{2},t+(k+2)σ_{2}]} ≤

≤ ε exp \{ ∑_{j=1}^{k} ln ω(t + jσ_{2}) − \frac{1}{σ_{2}} ∫_{R}^{t+kσ_{2}} ln ω(s)ds\} ≤

≤ ε exp \{ − \frac{1}{σ_{2}} ∫_{R}^{t} ln ω(s)ds\},

and so, by letting k → ∞ in (3.6), we have

∥f∥_{[t,t+σ_{2}]} ≤ ω(t)∥f∥_{[t−σ_{1},t]}.

From this it follows that ∥f∥_{[t,t+σ_{1}]} ≤ ω(t)∥f∥_{[t−σ_{1},t]} for all t ≥ R and hence that, for R + mσ_{1} ≤ t < R + (m + 1)σ_{1}, where m is an integer,

|f(t)|≤ ∏_{j=0}^{m} ω(t−jσ_{1})∥f∥_{[R−σ_{1},R]} ≤ exp \{\frac{1}{σ_{1}} ∫_{R}^{t} ln ω(s)ds\}∥f∥_{[R−σ_{1},R]}.

Remark. There are functions that satisfy (3.2) but not (3.3). For instance,

f(t) = exp \{ − \frac{1}{σ_{2}} ∫_{R}^{t+δ} ln ω(s)ds\} (t ≥ R)

with f(t) = f(R) for t ∈ [R − σ_{1}, R], is such a function for any δ ≥ 0.

With the help of Lemmas 1 and 2 and an adaptation of the method used in Driver [4], we now complete the proof of Lemma 2.

Proof of Theorem 1. Since each solution of (3.1) is also a solution of (2.1), by Lemma 1 (2.1) has solutions x_{n} (n = 1, 2, …, N) with asymptotic behavior (2.6) and X = (x_{1}, x_{2}, …, x_{N}) is a matrix solution of (2.1). With r_{1} given by (2.8), since (2.2) implies lim_{t→∞} r_{1}(t) = 0, there is T_{1} ≥ T + 2ρ such that X^{−1} is bounded and ξ < 1 on [T_{1} − 3ρ, ∞), where

ξ(t) = (∥X∥_{∞}∥X^{−1}∥_{∞}∥G∥_{∞})^{2} sup_{s≥t} |r_{1}(s)|.  (3.9)

The substitution of x = Xy into (2.1) leads to

\frac{d}{dt}y(t) = X^{−1}(t)G(t)X(t − r(t))\{y(t − r(t)) − y(t)\} (t ≥ T_{1} − 2ρ). (3.10)

Integration of (3.10) from t to t − r(t) produces

z(t) = ∫_{t}^{t−r(t)} X^{−1}(s)G(s)X(s − r(s))z(s)ds (t ≥ T_{1} − ρ),

where z(t) = y(t − r(t)) − y(t). Thus z(t) satisfies

z(t) = ∫_{t}^{t−r(t)} X^{−1}(s)G(s)X(s − r(s)) ×
× ∫_{s}^{s−r(s)} X^{−1}(u)G(u)X(u − r(u))z(u)du ds  (3.11)

for t ≥ T_{1}. By (3.9) and (3.11),

|z(t)| ≤ ξ(t)∥z∥_{[t−2ρ,t+2σ]} (t ≥ T_{1}).  (3.12)

Now suppose that x(t) = O(e^{γt}) as t → ∞ for some γ > 0. Then certainly

x(t) = o( exp \{ − \frac{1}{2σ} ∫_{T_{1}}^{t−5σ} ln ξ(s)ds\})  (3.13)

as t → ∞, and, by the boundedness of X^{−1}, (3.13) also holds if x is replaced by y. Hence, from the definition of z,

z(t) = o( exp \{ − \frac{1}{2σ} ∫_{T_{1}}^{t−4σ} ln ξ(s)ds\}).

By Lemma 2, z satisfies

|z(t)| ≤ exp \{\frac{1}{2ρ} ∫_{T_{1}}^{t} ln ξ(s)ds\}∥z∥_{[T_{1}−2ρ,T_{1}]} (t ≥ T_{1}). (3.14)

For any δ > 0 and t ≥ T_{1}, (3.10) and (3.14) imply that

|y(t + δ) − y(t)| ≤ γ_{0} ∫_{t}^{t+δ} exp \{\frac{1}{2ρ} ∫_{T_{1}}^{s} ln ξ(θ)dθ\}ds

for some constant γ_{0}. Since ξ decreases to zero as t → ∞ and ξ(t) < 1, we have

|y(t + δ) − y(t)| ≤ γ_{0}( exp \{\frac{1}{2ρ} ∫_{T_{1}}^{t} ln ξ(θ)dθ\}) ×

× ∫_{t}^{t+δ} exp {(s − t)(2ρ)^{−1} ln ξ(T_{1})} ds ≤
≤ γ_{1} exp \{\frac{1}{2ρ} ∫_{T_{1}}^{t} ln ξ(θ)dθ\} (t ≥ T_{1}),  (3.15)

where γ_{1} is a constant. Further, for any β > 0,

exp \{\frac{1}{2ρ} ∫_{T_{1}}^{t} ln ξ(θ)dθ\} = o(e^{−βt})  (3.16)

as t → ∞. Then, by (3.15), (3.16) and the Cauchy convergence principle, there is c ∈ ℂ^{N} such that lim_{t→∞} y(t) = c. Letting δ → ∞ in (3.15), we obtain |y(t) − c| = o(e^{−βt}) as t → ∞. Thus

x(t) = X(t)(y(t) − c) + X(t)c = X(t)c + o(e^{−βt}) ,

i.e., every exponentially bounded solution of (2.1) has form (2.7).

Remark. Any solution of (2.1) that is not of form (2.7) must fail to satisfy (3.13). This, by (3.9), relates the separation of large solutions of (2.1) from those in (2.7) to the rate at which r_{1} tends to zero.

Lemma 3. If r ∈ L^{p}(τ, ∞) for some p ∈ [1, ∞), then r_{n} ∈ L^{β_{n}}(τ, ∞), where

\frac{1}{β_{n}} = \frac{n + 1}{p} − \frac{n}{p^{2}} + \frac{n − 1}{p^{3}} − ⋯ + (−1)^{n} \frac{1}{p^{n+1}} .

Proof. We show by induction that

|r_{n}(t)| ≤ |r(t)|^{η_{n}}q_{n}(t),  (3.17)

where

η_{n} = 1 − \frac{1}{p} + \frac{1}{p^{2}} − ⋯ + (−1)^{n} \frac{1}{p^{n}} ,

for some function q_{n} satisfying

∥q_{n}∥_{[t−ρ,t+σ]} ∈ L^{β_{n−1}}(τ, ∞).  (3.18)

We take q_{0}(t) = 1 for t ≥ τ and β_{−1} = ∞, and take q_{n}(t) = 0 for t < τ and all n. The conclusion of the lemma follows from (3.17) with (3.18) since

\frac{η_{n}}{p} + \frac{1}{β_{n−1}} = \frac{1}{β_{n}} .  (3.19)

Suppose then that (3.17) with (3.18) holds for a particular value of n. Then

|r_{n+1}(t)| ≤ ∥q_{n}∥_{[t−ρ,t+σ]}| ∫_{t}^{t−r(t)} |r(s)|^{η_{n}}ds| ≤

≤ ∥q_{n}∥_{[t−ρ,t+σ]}|r(t)|^{1−η_{n}/p}| ∫_{t}^{t−r(t)} |r(s)|^{p}ds|^{η_{n}/p}

by Hölder’s inequality. Thus we have

|r_{n+1}(t)| ≤ |r(t)|^{η_{n+1}}q_{n+1}(t)

with

q_{n+1}(t) = ∥q_{n}∥_{[t−ρ,t+σ]}| ∫_{t}^{t−r(t)} |r(s)|^{p}ds|^{η_{n}/p} (t ≥ τ).

Then

∥q_{n+1}∥_{[t−ρ,t+σ]} ≤ ∥q_{n}∥_{[t−2ρ,t+2σ]}| ∫_{t−2ρ}^{t+2σ} |r(s)|^{p}ds|^{η_{n}/p}

and there is in L^{β_{n}}(τ, ∞) by (3.19).

Proof of Corollary to Theorem 2. By Lemma 3, r_{n} ∈ L^{β_{n}}(τ, ∞). We have

\frac{1}{β_{n}} = \frac{(n + 1)p^{n+2} + (n + 2)p^{n+1} + (−1)^{n}}{p^{n+1}(p + 1)^{2}} ,

so β_{n} ≤ 1 if n ≥ p. Thus r_{M} ∈ L(τ, ∞) for M ≥ p.

§ 4. Elimination of Large Solutions

In this section we give a condition under which (1.1) or (2.1) cannot have any large solutions.

Lemma 4. Let {t_{n}} be a real sequence satisfying t_{n} → ∞ as n → ∞ and let ω : [R, ∞) → (0, 1) be decreasing. If f : [R − σ_{1}, ∞) → ℂ^{N} satisfies

|f(t)| ≤ ω(t)∥f∥_{[t−σ_{1},t_{n}]}  (4.1)

for each t_{n} ≥ R and all t ∈ [R, t_{n}], where σ_{1} > 0 is a constant, then inequality (3.4) holds.

Proof. For each t_{n} ≥ R and any t ∈ [R, t_{n}], the conditions imply that

|f(t)| ≤ ∥f∥_{[t,t_{n}]} ≤ ω(t)∥f∥_{[t−σ_{1},t_{n}]}.

Hence ∥f∥_{[t−σ_{1},t_{n}]} = ∥f∥_{[t−σ_{1},t]} and

|f(t)| ≤ ω(t)∥f∥_{[t−σ_{1},t]}.  (4.2)

Since lim_{n→∞} t_{n} = ∞, (4.2) actually holds for all t ≥ R. The conclusion follows as in the last part of the proof of Lemma 2.

Definition. A point t_{0} ≥ τ is said to be a zero-advanced point (ZAP) of (1.1), (1.3), or (2.1) if t − r(t) ≤ t_{0} for all t ∈ [τ, t_{0}].

Note that in the retarded case, r(t) ≥ 0, every point t_{0} ≥ τ is a ZAP. As a simple illustration of the case when r changes sign, consider (1.1) with r(t) = t^{−α} sin βt, where α, β > 0. Clearly, the points t = (2n + 1)π/β are ZAPs for all sufficiently large integer n, and the corollary of Theorem 4, below, will apply.

Theorem 3. Assume that (2.1) has a sequence {t_{n}} of ZAPs satisfying lim_{n→∞} t_{n} = ∞. Then, under the conditions of Theorem 1, every solution of (2.1) has the asymptotic representation (2.7).

Proof. We refer to the proof of Theorem 1. Take any t_{k} ≥ T_{1}. Since t − r(t) ≤ t_{k} for all t ∈ [τ, t_{k}], from (3.11) we derive

|z(t)| ≤ ξ(t)∥z∥_{[t−2ρ,t_{k}]} (T_{1} ≤ t ≤ t_{k})  (4.3)

instead of (3.12). Then Lemma 4 implies (3.14). Therefore, every solution of (2.1) on [T − ρ, ∞) has representation (2.7).

Theorem 4. Assume that (1.1) has a sequence {t_{n}} of ZAPs satisfying lim_{n→∞} t_{n} = ∞ and that the conditions of Theorem 2 are met. Then every solution of (1.1) has the asymptotic representation (2.7) with X composed of N solutions (2.10).

Proof. By the proof of Theorem 2, (1.1) is transformed to (2.15). Then the conclusion follows from Theorem 3.

Corollary. Assume that r ∈ L^{p}(τ, ∞) for some p ∈ (1, ∞) and that (1.1) has a sequence {t_{n}} of ZAPs satisfying lim_{n→∞} t_{n} = ∞. Then, for M ≥ p, r_{M} ∈ L(τ, ∞) and (1.1) has N solutions of form (2.10). Moreover, every solution on [T − ρ, ∞) can be represented by (2.7) with X composed of N solutions (2.10).

§ 5. Asymptotic Representation for Solutions of (1.3)

As a consequence of Theorems 1 and 3, some results for equation (1.3) can be easily obtained. Extend A and B to ℝ by letting A(t) = B(t) = 0 for t < τ and, for t ∈ ℝ, let

B_{k+1}(t) = B(t) exp { ∫_{t}^{t−r(t)} (B_{k}(s) + A(s)) ds} (k ≥ 0) (5.1)

with B_{0}(t) = B(t).

Theorem 5. Let A and B have the forms stated at the end of §1. Assume that A and B are bounded, (2.2) holds, and r_{M} ∈ L(τ, ∞). Then, for sufficiently large T ≥ τ, (1.3) has N solutions x_{n} on [T − ρ, ∞) satisfying

x_{n}(t) = exp { ∫_{T}^{t} (B_{M}(s) + A(s)) ds}{e_{n} + o(1)} (1 ≤ n ≤ N) (5.2)

as t → ∞, where B_{M} is determined by (5.1). Moreover, every exponentially bounded solution of (1.3) has the asymptotic representation (2.7) for arbitrary β > 0, where X is now composed of x_{n} in (5.2). If, in addition, there is a sequence {t_{n}} of ZAPs of (1.3) satisfying lim_{n→∞} t_{n} = ∞, then every solution on [T − ρ, ∞) has form (2.7).

The proof of Theorem 5 is similar to that of Theorems 2 and 4.

When A is not bounded, the conclusion of Theorem 5 may not be true. However, it is possible to determine the asymptotic behavior of solutions of (1.3) under other conditions. Let Ã(t) = ∫_{t}^{t−r(t)} A(s)ds.

Corollary. Assume that B and Ã are bounded, (2.2) holds, and r_{M} ∈ L(τ, ∞). Then, for sufficiently large T ≥ τ, (1.3) has N solutions x_{n} on [T − ρ, ∞) satisfying

x_{n}(t) = exp { ∫_{T}^{t} (B_{M+1}(s) + A(s)) ds}{e_{n} + o(1)} (1 ≤ n ≤ N) (5.3)

as t → ∞, where B_{M+1} is given by (5.1). Moreover, if x is a solution of (1.3) of order exp {∫_{T}^{t} A(s)ds + λt} for some λ > 0, then x has the asymptotic form

x(t) = X(t)c + o( exp { ∫_{T}^{t} A(s)ds − βt})  (5.4)

as r → ∞, where X consists of N solutions (5.3), c ∈ ℂ^{N} depends on x and β > 0 is arbitrary. Further, if there is a sequence {t_{n}} of ZAPs of (1.3) satisfying lim_{n→∞} t_{n} = ∞, then every solution on [T −ρ, ∞) has form (5.4).

Proof. Let

x(t) = exp { ∫_{T}^{t} (B(s) + A(s)) ds}y(t).

Then (1.3) is changed to

\frac{d}{dt}y(t) = −B(t)y(t) + B_{1}(t)y(t − r(t)).  (5.5)

The conclusion follows by applying Theorem 5 to (5.5).

Remarks. (i) The condition r_{M} ∈ L(τ, ∞) in Theorem 5 and its corollary will be met for some M if r ∈ L^{p}(τ, ∞) for some p ∈ [1, ∞); M can be chosen as in the corollary of Theorem 2 or the remark following that corollary.

(ii) We note that, if Λ is a diagonal matrix whose elements satisfy a suitable dichotomy condition and if R and S are small at infinity, then the equation

\frac{d}{dt}x(t) = \{Λ(t) + R(t)\}x(t) + S(t)x(t − r(t))  (5.6)

can be treated by the methods of Cassell and Hou [5].

References

1. K. L. Cooke, Functional differential equations close to differential equations. Bull. Amer. Math. Soc. 72(1966), 285–288.

2. J. Kato, On the existence of a solution approaching zero for functional differential equations. Proc. U.S.A.-Japan Seminar Diff. Funct. Eqns., 153–169, Benjamin, New York, 1967.

3. J. K. Hale and S. M. V. Lunel, Introduction to functional differential equations. Appl. Math. Sci. 99, Springer-Verlag, New York, 1993.

4. R. D. Driver, Linear differential systems with small delays. J. Differential Equations 21(1976), 148–166.

5. J. S. Cassell and Z. Hou, Asymptotically diagonal linear differential equations with retardation. J. London Math. Soc. (2) 47(1993), 473–483.

(Received 2.08.1996; revised 12.05.1997)

Authors’ address: Department of C.I.S. & M. London Guildhall University 100 Minories, Tower Hill London EC3N 1JY United Kingdom