Second Order Delay Differential Equations
Theorem 5.2.1. Equation (5.1) has a nonoscillatory solution x such that
t1
[x(∞)]α−
∞
s n
i=1
pi(r) [x (gi(r))]αdr α1
ds, t≥ t1, where t1> t0 is chosen so that inft≥t1gi(t)≥ t0, 1≤ i ≤ n. Let x be an eventually positive solution of (5.1) satisfying (5.3). Then we have
(5.5) x(t) = x(∞) +
∞
t
∞
s n
i=1
pi(r) [x (gi(r))]αdr 1α
ds, t≥ t1, after integrating (5.1) twice from t to∞.
Based on these integral representations (5.4) and (5.5), we can prove the follow-ing existence theorems.
Theorem 5.2.1. Equation (5.1) has a nonoscillatory solution x such that
t→∞lim x(t)
t = constant= 0 if and only if
(5.6)
∞
pi(t)[gi(t)]αdt <∞ for all i∈ {1, . . . , n}.
Proof. (The “only if” part) Let x be a nonoscillatory solution of (5.1) satisfying limt→∞x(t)
t = c > 0. Then from (5.4) we see that
∞ n
i=1
pi(t) [x(gi(t))]αdt <∞.
This, combined with the relation limt→∞x(gi(t))
gi(t) = c, 1 ≤ i ≤ n, immediately implies that (5.6) holds. Let k > 0 be arbitrary and fixed and take T > a so large that
(5.7) T∗= min
1≤i≤n
t≥Tinf gi(t)
≥ a
and n
i=1
∞
T
pi(t)[gi(t)]αdt≤2α− 1 2α .
Consider the set X⊂ C[T∗,∞) and the mapping F : X → C[T∗,∞) defined by X =
x∈ C[T∗,∞) : k
2(t− T ) ≤ x(t) ≤ k(t − T ), t ≥ T, x(t) = 0, T∗≤ t ≤ T
and
(F x)(t) =
⎧⎪
⎪⎨
⎪⎪
⎩
t T
kα−
∞
s n
i=1
pi(r) [x(gi(r))]αdr α1
ds if t≥ T,
0 if T∗≤ t < T.
It is clear that X is a closed convex subset of the Fr´echet space C[T,∞) of continu-ous functions on [T∗,∞) with the usual metric topology and that F is well defined and continuous on X. It can be shown without difficulty that F maps X into itself and F (X) is relatively compact in C[T∗,∞). Therefore, by the Schauder–Tychonov fixed point theorem (Theorem 1.4.25), F has a fixed element x∈ X, which satisfies
x(t) =
t T
kα−
∞
s n
i=1
pi(r) [x(gi(r))]αdr 1/α
ds, t≥ T.
By differentiating this equation, we see that x is a solution of (5.1) on [T,∞) and limt→∞x(t)t = limt→∞x(t) = k.
Theorem 5.2.2. Equation (5.1) has a nonoscillatory solution x such that
tlim→∞x(t) = constant= 0 if and only if
(5.8)
∞ ∞
t
pi(s)ds 1/α
dt <∞ for all i∈ {1, . . . , n} .
Proof. Note that our assumptions imply (5.3), and then the “only if” part follows readily from (5.5). To prove the “if” part, suppose that (5.8) is satisfied. Choose T > a so that (5.7) holds and
∞
T
∞
t n
i=1
pi(s)ds 1/α
dt≤1 2. Define Y ⊂ C[T∗,∞) and G : Y → C[T∗,∞) by
Y =
y∈ C[T∗,∞) : k ≤ y(t) ≤ 2k, t ≥ T∗
, k > 0 being a fixed constant, and
(Gy)(t) =
⎧⎪
⎪⎨
⎪⎪
⎩ k +
∞
t
∞
s n
i=1
pi(r) [y(gi(r))]αdr 1α
ds if t≥ T,
(Gy)(T ) if T∗≤ t < T.
As in the proof of Theorem 5.2.1 one can verify that G maps Y into a relatively compact subset of Y , so that there exists y∈ Y such that
y(t) = k +
∞
t
∞
s n
i=1
pi(r) [y(gi(r))]αdr 1/α
ds, t≥ T.
Differentiating this equation twice, one sees that y satisfies (5.1) on [T,∞). Since y(t)→ k as t → ∞, y is a solution of (5.1) with the desired asymptotic property.
This completes the proof.
It remains to discuss the existence of an unbounded nonoscillatory solution x of (5.1) with the property limt→∞|x(t)|t =∞ and of a bounded solution x of (5.1) with the property limt→∞x(t) = 0. Below we confine our attention to the case where at least one gi is retarded and show that some sufficient conditions can be derived under which (5.1) has a nonoscillatory solution that tends to zero. Our derivation is based on the following theorem which is essentially due to Philos [239].
Theorem 5.2.3. Suppose that there exists i0∈ {1, 2, . . . , n} such that (5.9) gi0(t) < t and pi0(t)≥ 0 for t≥ a.
Suppose, in addition, that there exists a positive decreasing function φ on [t0,∞) satisfying
(5.10) φ(t)≥
∞
t
∞
s n
i=1
pi(r) [φ(gi(r))]αdr 1/α
ds, t≥ t0,
where t0 is chosen so that inft≥t0gi(t) ≥ a for all 1 ≤ i ≤ n. Then (5.1) has a nonoscillatory solution tending to zero as t→ ∞.
Proof. Let Z denote the set Z =
z∈ C[t0,∞) : 0 ≤ z(t) ≤ φ(t), t ≥ t0
. With each z∈ Z we associate the function ˜z ∈ C[a, ∞) defined by (5.11) z(t) =˜
z(t) if t≥ t0,
z(t0) + [φ(t)− φ(t0)] if a≤ t < t0. Define the mapping H : Z→ C[t0,∞) by
(Hz)(t) =
∞
t
∞
s n
i=1
pi(r) [˜z(gi(r))]αdr 1/α
ds, t≥ t0.
Then it can be shown that H is a continuous mapping which sends Z into a relatively compact subset of Z. It follows therefore that there exists z∈ Z such that z = Hz, i.e.,
z(t) =
∞
t
∞
s n
i=1
pi(r) [˜z(gi(r))]αdr 1/α
ds, t≥ t0. Differentiating the above twice shows that
− [−z(t)]α
=
n
i=1
pi(t) [˜z(gi(t))]α, t≥ t0,
which, in view of (5.11), implies that z(t) is a solution of (5.1) for all sufficiently large t. That z(t) > 0 for t≥ t0 can be seen exactly as in Philos [239, page 170], and so the details are omitted. This completes the proof.
In order to apply Theorem 5.2.3 to construct decaying nonoscillatory solutions of (5.1), we distinguish the following three cases:
∞ n
i=1
pi(t)dt <∞ and
∞ ∞
t n
i=1
pi(s)ds 1/α
dt <∞, (5.12)
∞ n
i=1
pi(t)dt <∞ but
∞ ∞
t n
i=1
pi(s)ds 1/α
dt =∞, (5.13)
and ∞ n
i=1
pi(t)dt =∞.
(5.14)
The condition (5.12), which is nothing else but (5.8), always guarantees the exis-tence of a decaying nonoscillatory solution of (5.1).
Theorem 5.2.4. Suppose that (5.9) holds for some i0∈ {1, 2, . . . , n}. If (5.8) is satisfied, then (5.1) possesses a nonoscillatory solution tending to zero as t→ ∞.
Proof. Let t0 be large enough so that min1≤i≤n{inft≥t0gi(t)} ≥ max{a, 1} and
for t≥ t0. The conclusion follows from Theorem 5.2.3.
We now state existence theorems of decaying nonoscillatory solutions which are applicable to the cases (5.13) and (5.14).
Theorem 5.2.5. Suppose that (5.9) holds for some i0∈ {1, 2, . . . , n} and that
(5.16) lim sup
t→∞
where g∗(t) = min1≤i≤ngi(t). Then (5.1) possesses a nonoscillatory solution tend-ing to zero as t→ ∞.
Proof. We put
P (t) =
Since, for 1≤ i ≤ n,
for t≥ t0. By Theorem 5.2.3, (5.1) has a decaying nonoscillatory solution.
Theorem 5.2.6. Suppose that (5.9) holds for some i0 ∈ {1, 2, . . . , n}. Further,
Then (5.1) possesses a nonoscillatory solution tending to zero as t→ ∞.
Proof. Put
We see that
φ (gi(t))≤ exp
α + 1 α
φ(t), t≥ t0, 1≤ i ≤ n, and hence that
∞
for t≥ t0. Consequently, we obtain
for t≥ t0, where (5.18) has been used. This establishes the existence of a strictly decreasing positive function satisfying (5.10), and so the proof is complete via Theorem 5.2.3.
Example 5.2.7. Consider the equation (5.19) Theo-rem 5.2.1 and TheoTheo-rem 5.2.2, (5.19) has nonoscillatory solutions x1 and x2
such that limt→∞x1(t)
t = constant = 0 and limt→∞x2(t) = constant = 0 regardless of the values of θ > 0.
(ii) Let λ = 3. An easy computation shows that (5.16) is satisfied for (5.19) if 1 < θ < exp
From Theorem 5.2.5 it follows that, for such a θ, (5.19) possesses a nonoscil-latory solution tending to zero as t→ ∞.
(iii) Let 1 < λ < 3. Then (5.18) is satisfied for (5.19) since Pt0= 1 and
Therefore there exists a decaying nonoscillatory solution of (5.19) by Theo-rem 5.2.6.
5.3. Classification Schemes for Iterative Equations
In this section we are concerned with the general class of second order nonlinear differential equations
with the conditions ∞
0 ds/[r(s)]1/σ =∞ and ∞
0 ds/[r(s)]1/σ <∞, respectively.
We give a classification scheme for eventually positive solutions of this equation in terms of their asymptotic magnitude, and provide necessary and/or sufficient conditions for the existence of solutions.
Let T ∈ R+= [0,∞). Define T−1= inf{Δ(t, x) : t ≥ T, x ∈ R}.
Definition 5.3.1. The function x is called a solution of the differential equation (5.20) in the interval [T,∞), if x(t) is defined for t ≥ T−1, is twice differentiable, and satisfies (5.20) for t≥ T .
Definition 5.3.2. The solution x of (5.20) is called regular, if it is defined on some interval [Tx,∞) and sup{|x(t)| : t ≥ T } > 0 for t ≥ Tx.
Throughout this section, we assume that the following conditions hold:
(H1) r∈ C(R+,R+) and r(t) > 0, t∈ R+. (H2) f ∈ C(R+× R2,R+).
(H3) There exists T ∈ R+ such that uf (t, u, v) > 0 for t ≥ T , uv > 0, and f (t, u, v) is nondecreasing in u and v for each fixed t≥ T .
(H4) Δ∈ C(R+× R, R).
(H5) There exist a function Δ∗ ∈ C(R+,R) and T ∈ R+ such that limt→∞Δ∗(t) =∞ and Δ∗(t)≤ Δ(t, x) for t ≥ T , x ∈ R.
(H6) There exist a function Δ∗ ∈ C(R+,R) and T ∈ R+ such that Δ∗(t) is nondecreasing for t≥ T and Δ(t, x) ≤ Δ∗(t)≤ t for t ≥ T , x ∈ R.
(H7) σ is a quotient of odd integers.
For the sake of convenience, we will employ the following notation:
R(t) =
∞
t
ds
[r(s)]1/σ, R(T, t) =
t
T
ds
[r(s)]1/σ, R0=
∞
0
ds [r(s)]1/σ. Lemma 5.3.3. Suppose x is an eventually positive solution of (5.20). Then x(t) is of constant sign eventually.
Proof. Assume that there exists t0≥ 0 such that x(t) > 0 for t ≥ t0. It follows from (H6) that there exists t1 ≥ t0 such that x(Δ(t, x(t))) > 0 for t ≥ t1. From (H4) and (5.20) we conclude that (r(x)σ)(t) < 0 for t ≥ t1. If x(t) is not eventually positive, then there exists t2≥ t1such that x(t2)≤ 0. Therefore, r(t2)[x(t2)]σ ≤ 0.
Integrating (5.20) from t2 to t provides r(t) [x(t)]σ− r(t2) [x(t2)]σ+
t t2
f
s, x(s), x(Δ(s, x(s)))
ds = 0.
Thus
r(t) [x(t)]σ≤ −
t t2
f
s, x(s), x(Δ(s, x(s)))
ds < 0 for t≥ t2. This shows that x(t) < 0 for t≥ t2. The proof is complete.
As a consequence, an eventually positive solution x of (5.20) either satisfies x(t) > 0 and x(t) > 0 for all large t, or x(t) > 0 and x(t) < 0 for all large t.
Lemma 5.3.4. Suppose that