ON THE OSCILLATION OF IMPULSIVELY DAMPED HALFLINEAR OSCILLATORS

John R. Graef

Department of Mathematics University of Tennessee at Chattanooga Chattanooga, TN 37403, USA

János Karsai

Department of Medical Informatics University of Szeged Szeged, Korányi fasor 9, 6720, HUNGARY

Abstract. The authors consider the nonlinear impulsive system

(ϕ_{β}(x′))′ + ϕ_{β}(x) = 0 (t ≠ t_{n}), x′(t_{n} + 0) = b_{n}x′(t_{n}) where n = 1, 2 …, ϕ_{β}(u) = |u|^{β}sgn u with β > 0, and 0 ≤ b_{n} ≤ 1. They investigate the oscillatory behavior of the solutions. In the special case where b_{n} = b < 1 and t_{n} = t_{0} + n d, they give necessary and sufficient conditions for the oscillation of all solutions.

1. Introduction

Consider the system with impulsive perturbations

(ϕ_{β}(x′))′ + ϕ_{β}(x) = 0, t ≠ t_{n},  (1)
x(t_{n} + 0) = x(t_{n}), x′(t_{n} + 0) = b_{n}x′(t_{n}),

where 0 ≤ t_{1} < t_{2}, …, t_{n} < t_{n+1}, t_{n} → ∞ as n → ∞, 0 ≤ b_{n} ≤ 1 for n = 1, 2 …, and ϕ_{β}(u) = |u|^{β}sgn u with β > 0. System (1) is an impulsive analogue of the nonlinear oscillator equation

(ϕ_{β}(x′))′ + a(t)ϕ_{β}(x′) + ϕ_{β}(x) = 0  (2)

with a nonnegative damping coefficient a(t). A detailed description of this analogy can be found in the papers [6, 7, 8] by the present authors. Note that a negative b_{n} results in a beating effect, which has no continuous analogue (see the discussions of the case ϕ_{β}(u) = u in [5, 7]). Systems of the form (1) in the case where b_{n} ≥ 1 have been studied, for example, in [10].

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 14, p. 1

1991 Mathematics Subject Classification. 34D05, 34D20, 34C15.

Key words and phrases. Oscillation, second order nonlinear systems, impulses.

The research of J. Karsai is supported by Hungarian National Foundation for Scientific Research Grant no. T 029188.

It is known that if the function a is small, but large enough in some sense, then the solutions of (2) oscillate and tend to zero (for example, if \frac{1}{t} ≤ a(t) < 2 in the linear case ϕ_{β}(u) = u). This situation is called the small damping case. For large enough a(t), the solutions are monotone decreasing to zero in magnitude, and this is sometimes called the large damping case (for example, if 2 ≤ a(t) < t and ϕ_{β}(u) = u). If the system is overdamped, that is, a(t) grows too fast to infinity as t tends to infinity (for example, t^{1+ε} ≤ a(t) in the case ϕ_{β}(u) = u), then the solutions decrease in magnitude but may not tend to zero (see [2, 11, 12] and the references therein).

The problem of attractivity for system (1) and its special cases was investigated in [5, 6, 7, 8, 9]. In [7] and [9], conditions are given to ensure that every solution of system (1) is nonoscillatory, and these conditions turn out to be necessary and sufficient in case ϕ_{β}(u) = u, b_{n} = b, and t_{n} = t_{0} + n d.

In this paper, we improve the method applied in [6, 9] and formulate new conditions for the oscillation and nonoscillation of the solutions, and these will result in a necessary and sufficient condition for the system with constant impulses at equally spaced distances.

2. Preliminaries

We know that every solution of (1) can be continued to ∞, the solutions are piecewise differentiable, and ϕ_{β}(x′(t)) is piecewise continuous and continuous from the left hand side at every t > 0. The following result classifies the solutions of the system (1) as being either monotonic or oscillatory on some interval [T, ∞).

Lemma 1. ([9]) Suppose that 0 ≤ b_{n} ≤ 1, n = 1, 2, . . .. Let x(t) be a solution of (1) that is not identically zero on any interval [T, ∞), and let s_{1} and s_{2} be consecutive zeros of x′(t). Then there exists t̃∈ (s_{1}, s_{2}) such that x(t̃) = 0. Hence, solutions of (1) are either oscillatory or monotone nonincreasing in magnitude.

Define the energy function

V (x, y) = y ϕ_{β}(y) − ∫_{0}^{y} ϕ_{β}(s) ds + ∫_{0}^{x} ϕ_{β}(s) ds =: Φ_{β}(y) + F_{β}(x), (3)

where in explicit form Φ_{β}(y) = \frac{β}{β+1}|y|^{β+1} and F_{β}(x) = \frac{1}{β+1}|x|^{β+1}. Note that the functions F_{β} and Φ_{β} are both even and positive definite.

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 14, p. 2

The function V (t) = V (x(t), x′(t)) is constant along the solutions of the equation without impulses

(ϕ_{β}(x′))′ + ϕ_{β}(x) = 0.  (4)

Since the solutions of (1) are identical with those of (4) between the instants of impulses, some basic knowledge of their behavior will be useful here. It is easy to see, for example, that every nonzero solution of (4) is periodic. In addition, since we have assumed that ϕ_{β} is odd, the length of the time intervals on which x(t)x′(t) ≥ 0 or x(t)x′(t) ≤ 0 are equal. The distance between two consecutive zeros of x′(t), i.e., the half-period, will be equal for every solution due to the fact that the equation (4) is homogeneous and autonomous (see [3]). We denote the half-period by Δ_{β}. A formula for Δ_{β} can be obtained as a special case of the following lemma.

Lemma 2. Let x(t) be a solution of (4) with V (t) ≡ r > 0, and let τ_{1} < τ_{2} be such that F_{β}(τ_{1}) = δr, F_{β}(τ_{2}) = γr, 0 < τ_{2} − τ_{1} < Δ_{β}/2, and 0 ≤ δ < γ ≤ 1. Then, the time, τ_{2} − τ_{1}, elapsed by changing F_{β} from δr to γr can be expressed by the following integral:

τ_{2} − τ_{1} = ∫_{δ}^{γ} \frac{dv}{ϕ_{β}(F_{β}^{−1}(v))Φ_{β}^{−1}(1 − v)} = \frac{β^{\frac{1}{1+β}}}{1 + β} ∫_{δ}^{γ} \frac{dv}{(1 − v)^{\frac{1}{1+β}} v^{\frac{β}{1+β}}} . (5)

Although the above lemma is proved in [10], the proof itself is short and contains basic arguments concerning the solutions, so we provide it below.

Proof. Let x(t) be a solution of (4) with V (t) ≡ r, F_{β}(x(τ_{1})) = δr, and F_{β}(x(τ_{2})) = γr. From (3), we have x′(t) = Φ_{β}^{−1}(r −F_{β}(x(t))). Dividing by the right-hand side and integrating, we obtain

τ_{2} − τ_{1} = ∫_{τ_{1}}^{τ_{2}} \frac{x′(t) dt}{Φ_{β}^{−1}(r − F_{β}(x(t)))}.  (6)

Making the substitution x = x(t), τ_{2} − τ_{1} can be expressed in the form

τ_{2} − τ_{1} = ∫_{F_{β}^{−1}(δr)}^{F_{β}^{−1}(γr)} \frac{dx}{Φ_{β}^{−1}(r − F_{β}(x))} = ∫_{δ}^{γ} \frac{dv}{ϕ_{β}(F_{β}^{−1}(v))Φ_{β}^{−1}(1 − v)} (7)

where u = F_{β}(x), v = u/r, and F_{β}^{−1} and Φ_{β}^{−1} are the inverses of F_{β} and Φ_{β} on [0, ∞), respectively.

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 14, p. 3

To simplify the formulation of our results, we will use the following notation:

H_{β}(v) := \frac{1}{ϕ_{β}(F_{β}^{−1}(v))Φ_{β}^{−1}(1 − v)}.  (8)

Taking δ = 0 and γ = 1 in (5), we obtain the following expression for Δ_{β}:

Δ_{β} = \frac{2 β^{\frac{1}{1+β}} Γ(\frac{1}{1+β}) Γ(\frac{β}{1+β})}{1 + β} = \frac{2πβ^{\frac{1}{1+β}}}{(1 + β) sin \frac{π}{1+β}} ,  (9)

and in particular for the linear case β = 1, we have Δ_{1} = π .

Now, let x(t) be a solution of (1). The jump in the energy along x(t) at t_{n} is given by

V (t_{n+1}) − V (t_{n}) = V (t_{n} + 0) − V (t_{n})  (10)

= Φ_{β}(x′(t_{n} + 0)) + F_{β}(x(t_{n} + 0)) − Φ_{β}(x′(t_{n})) − F_{β}(x(t_{n}))

= b_{n}^{β+1}Φ_{β}(x′(t_{n})) − Φ_{β}(x′(t_{n})) = Φ_{β}(x′(t_{n}))(b_{n}^{β+1} − 1).

Denoting r_{n} = V (t_{n−1} + 0) = V (t_{n} − 0) and F_{β}(x(t_{n})) = σ_{n}r_{n}, and calculating F_{β}(x(t_{n})) in terms of r_{n+1} = V (t_{n} + 0), we obtain

r_{n+1} = F_{β}(x(t_{n})) + b_{n}^{β+1}Φ_{β}(x′(t_{n} − 0))
= F_{β}(x(t_{n})) + b_{n}^{β+1}(r_{n} − F_{β}(x(t_{n})))
= b_{n}^{β+1}r_{n} + F_{β}(x(t_{n}))(1 − b_{n}^{β+1})
= b_{n}^{β+1}r_{n} + (1 − b_{n}^{β+1})r_{n}σ_{n}
= r_{n} [b_{n}^{β+1} + (1 − b_{n}^{β+1})σ_{n}] .

Hence,

F_{β}(x(t_{n})) = \frac{σ_{n}}{b_{n}^{β+1} + (1 − b_{n}^{β+1})σ_{n}} r_{n+1}.

In order to simplify the notation, we let

Θ(u, b) := \frac{u}{b^{β+1}(1 − u) + u}.  (11)

The function Θ measures the jump in the quantity F_{β}(x(t))/V (t). It is clear that Θ(0, b) = 0, 0 < u < Θ(u, b) for 0 < u < 1 and b ≤ 1, and that Θ is monotone increasing with respect to u and decreasing with respect to b.

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 14, p. 4

3. Oscillation and Nonoscillation Results

Our main nonoscillation theorem is as follows.

Theorem 3. Assume there exist a constant N > 0 and a sequence {γ_{n}} with 0 < γ_{n+1} ≤ Θ(γ_{n}, b_{n}) < 1 such that

t_{n+1} − t_{n} ≤ ∫_{γ_{n+1}}^{Θ(γ_{n},b_{n})} H_{β}(v) dv  (12)

holds for every n > N. Then every solution of (1) is nonoscillatory and

\frac{F (x(t_{n}))}{V (t_{n} − 0)} ≥ γ_{n}.

Proof. Let x(t) be a nontrivial solution of (1). Clearly, the trajectory of x(t) cannot remain in either the first or the third quadrant, i.e., x(t)x′(t) ≥ 0 cannot hold on any half-ray [T, ∞). We will show that x(t)x′(t) < 0 for t ≥ T for some T > 0.

Now V (t) = F_{β}(x(t))+Φ_{β}(x′(t)) is constant on each interval (t_{n−1}, t_{n}), and we denote this value by r_{n}. Define σ_{n} by F_{β}(x(t_{n})) = σ_{n}r_{n}. If we can show that

0 < γ_{n}r_{n} ≤ σ_{n}r_{n}  (13)

holds for sufficiently large n, this would imply the nonoscillation of x(t).

Letting s_{0} be a zero of x′(t) with t_{n−1} ≤ s_{0} < t_{n}, we can assume that x(s_{0}) > 0; otherwise, we can consider −x(t) and use the symmetry of the equation. Now (5) and (12) imply

t_{n} − s_{0} = ∫_{σ_{n}}^{1} H_{β}(v) dv ≤ t_{n} − t_{n−1}

≤ ∫_{γ_{n}}^{Θ(γ_{n−1},b_{n−1})} H_{β}(v) dv < ∫_{γ_{n}}^{1} H_{β}(v) dv,

which, by the monotonicity of the integral on the right-hand-side of the above inequality, implies (13) holds. From the definition of the function Θ, we have

F_{β}(x(t_{n})) = Θ(σ_{n}, b_{n})r_{n+1}.

If we define the function h_{n}(u) implicitly by

t_{n+1} − t_{n} = ∫_{h_{n}(u)}^{u} H_{β}(v) dv,

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 14, p. 5 a differentiation shows that h_{n}(u) is monotone decreasing. Clearly,

F_{β}(x(t_{n+1})) = h_{n}(Θ(σ_{n}, b_{n}))r_{n+1}.

From the monotonicity of Θ, we obtain

t_{n+1} − t_{n} = ∫_{σ_{n+1}}^{Θ(σ_{n},b_{n})} H_{β}(v) dv

= ∫_{h_{n}(Θ(γ_{n},b_{n}))}^{Θ(γ_{n},b_{n})} H_{β}(v) dv ≤ ∫_{γ_{n+1}}^{Θ(γ_{n},b_{n})} H_{β}(v) dv.

Finally, the monotonicity of h_{n} implies σ_{n+1} ≥ γ_{n+1}, and this proves that x(t) is nonoscillatory.

Theorem 3 can be applied to particular problems by finding appropriate sequences \{γ_{n}\}. Let 0 < γ < \frac{1}{2} be given such that Θ(γ, b_{n}) ≥ 1 − γ and

t_{n+1} − t_{n} ≤ ∫_{γ}^{1−γ} H_{β}(v) dv, n = 0, 1, 2, ...;

then (12) holds ([9]).

Note that any choice 0 < γ_{n} = γ < 1 results in a nonoscillation criterion, but to formulate sharp conditions, we need some further investigation.

Next, we find a monotone nonincreasing sequence {γ_{n}} for which (12) holds. In this case, the integral in (12) is estimated from below by

∫_{γ_{n}}^{Θ(γ_{n},b_{n})} H_{β}(v) dv.

Applying the usual methods, we obtain that this integral takes its maximum at

γ̄_{n} = \frac{b_{n}^{β}(b_{n} − 1)}{b_{n}^{β+1} − 1}.  (14)

Note that γ̄_{n} is monotone increasing with respect to b_{n}. If the sequence {b_{n}} is nonincreasing, then γ̄_{n} is also nonincreasing. Hence, in this case γ_{n} := γ̄_{n} gives a better estimate in (12). In general, a nonincreasing {γ_{n}} can be defined by γ_{n} := min_{i=1,...,n} γ̄_{i}. In particular, if b_{n} ≤ b < 1, then we can apply

γ_{n} := γ̄ = \frac{b^{β}(b − 1)}{b^{β+1} − 1}.  (15)

On the other hand, numerical simulations show that for the case lim_{n→∞} b_{n} = 1, the sharpest criterion can be obtained by choosing

γ_{n} = lim_{b→1} \frac{b^{β}(b − 1)}{b^{β+1} − 1} = \frac{1}{1 + β} .

Summarizing the above arguments, we can formulate the following corollary.

Corollary 4. Let the sequence γ_{n} be defined as follows:

If the sequence {b_{n}} is nonincreasing, let γ_{n} := γ̄_{n}.

If b_{n} ≤ b < 1, let γ_{n} := γ̄.

If lim_{n→∞} b_{n} = 1, let γ_{n} := \frac{1}{1+β}.

Otherwise, let γ_{n} := γ ∈ (0, 1). If (12) holds, then every solution of (1) is nonoscillatory.

Next, we prove our main oscillation result.

Theorem 5. Assume that there exist a constant N > 0 and a sequence {λ_{n}} with 0 < λ_{n} ≤ 1 and ∑_{n=1}^{∞} λ_{n} = ∞ such that for every γ ∈ [λ_{n}, 1],

t_{n+1} − t_{n} ≥ ∫_{γ−λ_{n}}^{Θ(γ,b_{n})} H_{β}(v) dv  (16)

holds for every n > N. Then every solution of (1) is oscillatory.

Proof. Let x(t) be a nontrivial solution of (1). It will suffice to show that x(t)x′(t) < 0 cannot hold on any interval [T, ∞), so suppose x(t) > 0 and x′(t) < 0 for t ∈ [t_{N}, ∞). Again, we let σ_{n} be defined by F_{β}(x(t_{n})) = σ_{n}r_{n}, where r_{n} = V (t_{n} − 0). It follows that σ_{n} > λ_{n} since, in the opposite case, (16) yields

t_{n+1} − t_{n} ≥ ∫_{0}^{Θ(σ_{n},b_{n})} H_{β}(v) dv,

and hence x(t_{n+1}) ≤ 0, which contradicts to the positivity of x(t).

Now assuming σ_{n} > λ_{n}, n = N, ..., it follows again from (16) that 0 < σ_{n+1} ≤ σ_{n} − λ_{n}. Hence,

σ_{n+1} ≤ σ_{N} − ∑_{i=N}^{n} λ_{n},

and since the right-hand-side tends to negative infinity as n tends to infinity, this contradicts the positivity of F_{β}(x(t)).

If the sequence {b_{n}} is bounded away from zero, the following corollary holds.

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 14, p. 7 Corollary 6. Assume that 0 < b ≤ b_{n} ≤ 1 and there exist a constant N > 0 such that the sequence {μ_{n}} defined by

μ_{n} := (t_{n+1} − t_{n}) − ∫ Θ(γ̄,b) ; γ̄ H_{β}(v) dv ≥ 0  (17)

satisfies ∑_{n=N}^{∞} μ_{n} = ∞, where γ̄ is defined by (15). Then every solution of (1) is oscillatory. In particular, if t_{n+1} − t_{n} ≥ d > 0 and

d > ∫ Θ(γ̄,b) ; γ̄ H_{β}(v) dv  (18)

holds for every n > N, then every solution of (1) is oscillatory.

Proof. We will find a sequence λ_{n} that satisfies (16). The integral on the right-hand-side of (16) can be estimated from above by replacing b_{n} with b ≤ b_{n}. For a given λ_{n}, let γ′_{n} ∈ [0, 1] be the place where the value

max_{γ∈[0,1−λ_{n}]} (∫_{γ}^{Θ(γ+λ_{n},b)} H_{β}(v) dv)

is attained. Let us define λ_{n} implicitly by the relation

∫_{γ′_{n}−λ_{n}}^{Θ(γ′_{n},b)} H_{β}(v) dv = min(μ_{n}, \frac{1}{n}) + ∫ Θ(γ̄,b) ; γ̄ H_{β}(v) dv.  (19)

By continuity arguments, we obtain that lim_{n→∞} γ′_{n} = γ̄ and lim_{n→∞} λ_{n} = 0. Hence, for sufficiently large n we have that γ′_{n} ∈ [ε, 1 − ε] for some ε ∈ (0, 1/2). Since the integral on the left-hand-side of (19) is Lipschitzian with respect to λ_{n} uniformly in γ′_{n} ∈ [ε, 1−ε], ∑_{N}^{∞} μ_{n} = ∞ implies ∑_{N}^{∞} λ_{n} = ∞, and this proves the statement.

Without assuming b_{n} ≥ b > 0, we can state the following corollary.

Corollary 7. Assume that there exist a constant N > 0 such that the sequence {μ_{n}} defined by

μ_{n} := (t_{n+1} − t_{n}) − ∫ Θ(γ̄_{n},b_{n}) ; γ̄_{n} H_{β}(v) dv ≥ 0  (20)

satisfies ∑_{n=N}^{∞} μ_{n}^{1+β} = ∞ for β ≥ 1 and ∑_{n=N}^{∞}(μ_{n})^{(1+β)/β} = ∞ for 0 < β ≤ 1, respectively, where γ̄_{n} is defined by (14). Then every solution of (1) is oscillatory.

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 14, p. 8 Proof. Similar to the proof of Corollary 6, we will find a sequence λ_{n} that satisfies (16). The integral on the right-hand-side of (16) can be estimated from above by

∫ Θ(γ̄_{n},b_{n}) ; γ̄_{n} H_{β}(v) dv + sup_{γ∈[λ_{n},1]} ∫_{γ−λ_{n}}^{γ} H_{β}(v) dv.

It can be shown that the second term is not greater than ∫_{0}^{λ_{n}} H_{β}(v) dv for β ≥ 1 and is not greater than ∫_{1−λ_{n}}^{1} H_{β}(v) dv for 0 < β ≤ 1. Consider the case β ≥ 1. Let us define λ_{n} implicitly by

∫_{0}^{λ_{n}}H_{β}(v) dv = μ_{n}.  (21)

Since

∫_{0}^{λ} H_{β}(v) dv = \frac{β^{\frac{1}{β+1}} B_{λ}(\frac{1}{β+1}, \frac{β}{β+1})}{β + 1} = O(λ^{\frac{1}{1+β}}), λ → 0,

where B_{z}(a, b) = ∫_{0}^{z} t^{a−1} (1 − t)^{b−1} dt is the incomplete Beta function, the conditions ∑_{N}^{∞} μ_{n}^{1+β} = ∞ and ∑_{N}^{∞} λ_{n} = ∞ are equivalent, and this part is proved.

For the case 0 ≤ β ≤ 1, we have only to observe that

∫_{1−λ}^{1} H_{β}(v) dv = ∫_{0}^{λ} H_{1/β}(v) dv = O(λ^{\frac{β}{1+β}}), λ → 0.

In the special case b_{n} = b and t_{n+1} − t_{n} = d for n = 1, 2, ..., combining Corollaries 4 and 6 gives a necessary and sufficient condition for nonoscillation.

Theorem 8. Assume that b_{n} = b and t_{n+1} − t_{n} = d for n = 1, 2, .... Every solution of (1) is nonoscillatory if and only if

d ≤ ∫ Θ(γ̄,b) ; γ̄ H_{β}(v) dv.  (22)

4. The linear case

Now, let us apply the results in the previous section to the linear case (β = 1)

x′′ + x = 0, t ≠ t_{n},  (23)
x(t_{n} + 0) = x(t_{n}), x′(t_{n} + 0) = b_{n}x′(t_{n}),

We have

∫_{u}^{Θ(u,b)} H_{β}(v) dv = − arcsin(\sqrt{u}) + arcsin (\sqrt{\frac{u}{b^{2}(1 − u) + u}}) .

Hence,

sin (∫_{u}^{Θ(u,b)} H_{β}(v) dv) = (1 − b) \sqrt{\frac{(1 − u) u}{b^{2} (1 − u) + u}}.  (24)

Applying Corollary 4 we obtain the following statement.

Corollary 9. Every solution of system (23) is nonoscillatory if there exists a number 0 < γ < 1 such that

sin(t_{n+1} − t_{n}) ≤ (1 − b_{n}) \sqrt{\frac{(1 − γ) γ}{b_{n}^{2} (1 − γ) + γ}} .  (25)

In particular, every solution of system (23) is nonoscillatory in the following cases:

a) b_{n} ≤ b < 1 and

sin(t_{n+1} − t_{n}) ≤ \frac{1 − b}{1 + b}.

b) b_{n+1} ≤ b_{n} < 1 and

sin(t_{n+1} − t_{n}) ≤ \frac{1 − b_{n}}{1 + b_{n}} .

c) lim_{n→∞} b_{n} = 1 and

sin(t_{n+1} − t_{n}) ≤ (1 − b_{n}) \sqrt{\frac{1}{2 + 2 b_{n}^{2}}} .

For oscillation, the following corollaries are consequences of Corollaries 6 and 7 respectively.

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 14, p. 10 Corollary 10. Assume that 0 < b ≤ b_{n} ≤ 1, and let

μ_{n} := t_{n+1} − t_{n} − arcsin \frac{1 − b}{1 + b} ≥ 0.

If ∑_{n=1}^{∞} μ_{n} = ∞, then every solution of (23) is oscillatory. In particular, if t_{n+1} − t_{n} > d > 0 and

sin d > \frac{1 − b}{1 + b},

then every solution of (23) is oscillatory.

Corollary 11. Let

μ_{n} := t_{n+1} − t_{n} − arcsin \frac{1 − b_{n}}{1 + b_{n}} ≥ 0.

If ∑_{n=1}^{∞} μ_{n}^{2} = ∞, then every solution of (23) is oscillatory.

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