Georgian Mathematical Journal 1(1994), No. 3, 267-276

STURM’S THEOREM FOR EQUATIONS WITH DELAYED ARGUMENT

A.DOMOSHNITSKY

Abstract. Sturm’s type theorems on separation of zeros of solutions are proved for the second order linear differential equations with delayed argument.

1. Introduction

In this article the distribution of zeros of solutions is investigated for the following differential equation with delayed arguments

x′′(t) + ∑_{i=1}^{m} p_{i}(t)x(h_{i}(t)) = 0, t ∈ [0, +∞),  (1)

where p_{i} are locally summable nonnegative functions and h_{i} are nonnegative measurable functions for i = 1, …, m.

The classical result of Sturm is the following: if x_{1} and x_{2} are linearly independent solutions of the ordinary differential equation

x′′(t) + p(t)x(t) = 0, t ∈ [0, +∞),

then between two adjacent zeros of x_{1} there is one and only one zero of x_{2}. This article deals with the extension of the Sturm’s theorem to equation (1) with delayed argument.

The first result of this type was obtained by N.V.Azbelev [1]. Namely, if for almost all t ∈ [0, +∞) there is at most one zero of each nontrivial solution of equation (1) on the interval [h(t), t], where h(t) = min_{i=1,... ,m} h_{i}(t),

then Sturm’s theorem holds for equation (1), i.e. the interval [h(t), t] must be ”small enough”. The generalization of this result of N.V.Azbelev to the ”neutral” equation

x′′(t) − ∑_{j=1}^{n} q_{j}(t)x′′(g_{j}(t)) + ∑_{i=1}^{m} p_{i}(t)x(h_{i}(t)) = 0, t ∈ [0, +∞),

was obtained in [3]. Our approach assumes that [h_{1}(t), h_{m}(t)] is ”small enough” for almost all t ∈ [0, +∞) (note that we consider the case h_{1}(t) ≤ ⋯≤h_{m}(t)≤t in this article). Namely, if a solution has zero on [h_{1}(t), h_{m}(t)], then its derivative has no zero on this interval.

Note the close result of S.M.Labovsky [10] for equation (1) in the case m = 1 and another version of Sturm’s separation theorem proposed by Yu.I.Domshlak [4,5].

It is known [1] that the space of solutions of equation (1) is two-dimen-sional, the Wronskian

W (t) = | \begin{matrix}u(t) & v(t) \\ u′(t) & v′(t) \end{matrix} |

of a fundamental system u, v of the solution (1) can vanish, zeros of W (t) do not depend on a fundamental system, W (0) is not equal to zero. Nonvanishing of Wronskian selects the class of homogeneous equations such that each of them is equivalent to a corresponding ordinary differential equation. In this case each nontrivial solution of equation (1) can have only finite number of zeros on any finite interval, moreover, all zeros are simple. It is also known [1] that nonvanishing of the Wronskian is equivalent to the validity of Sturm’s theorem about separation of zeros.

The important part of this article concerns with estimates of the distance between adjacent zeros. These results are usually connected with Sturm’s comparison theorem. Note in this connection the following investigations [1,4-9,11,12].

We reduce the question about lower bounds for the distance between adjacent zeros and between zero of a solution and zero of its derivative to an estimation of the spectral radius of the corresponding completely continuous operator in the space of continuous functions, i.e. to the well-known problem of functional analysis.

Our interest in the lower bounds of this distance is connected with the problem of existence and uniqueness of a solution of boundary value problems. For example, if b − a is less than the distance between adjacent zeros of solutions of (1), then the boundary value problem

x′′(t) + ∑_{i=1}^{m} p_{i}(t)x(h_{i}(t)) = f(t), t ∈ [0, +∞),

x(a) = A, x(b) = B,

has for each A, B, f(t) the unique solution.

2. Main Results

Let λ_{νμ} be the smallest positive characteristic number of the operator F_{νμ} : C_{[ν,μ]} → C_{[ν,μ]} (C_{[ν,μ]} is the space of continuous functions x : [ν, μ] → R) which is defined by

(F_{νμ}x)(t) = − ∫_{ν}^{μ} G_{νμ}(t, s) ∑_{i=1}^{m} p_{i}(s)x(h_{i}(s))γ(ν, h_{i}(s))ds,

where

γ(ν, h_{i}(s)) = 0 if h_{i}(s) < ν, γ(ν, h_{i}(s)) = 1 if h_{i}(s) ≥ ν,  (2)

G_{νμ}(t, s) = \begin{cases}−(μ − t)(s − ν)/(μ − ν) & for ν ≤ s ≤ t ≤ μ, \\ −(t − ν)(μ − s)/(μ − ν) & for ν ≤ t < s ≤ μ, \end{cases}

G_{νμ}(t, s) is the Green’s function of the boundary value problem

x′′(t) = f(t), t ∈ [ν, μ], x(ν) = 0, x(μ) = 0.

It is clear that the operator F_{νμ} is positive.

Theorem 1. Let

1) the functions h_{i} be nondecreasing and the inequalities h_{i}(t) ≤ h_{i+1}(t) hold for i = 1, …, m − 1 and almost all t ∈ [0, +∞);

2) the functions p_{i+1}/p_{i} be nondecreasing for i = 1, …, m − 1;

3) at least one of the following inequalities be fulfilled

ess sup_{s∈[h_{1}(t),h_{m}(t)]} ∑_{i=1}^{m} p_{i}(s)[h_{m}(t) − h_{1}(t)]^{2} < 2,  (3)

[h_{m}(t) − h_{1}(t)] ∫_{h_{1}(t)}^{h_{m}(t)} ∑_{i=1}^{m} p_{i}(s)ds ≤ 1,  (4)

for almost all t ∈ [0, +∞).

Then

a) W (t) doesn’t vanish for t ∈ [0, +∞);

b) if ν and μ are two zeros of some nontrivial solution x of equation (1), then λ_{νμ} ≤ 1;

c) there is one and only one zero of the derivative of a nontrivial solution between any two adjacent zeros of this nontrivial solution.

Examples: the condition 2) of Theorem 1 is fulfilled for the following cases:

1) if m = 2, p_{1} is nonincreasing and p_{2} is nondecreasing;

2) if p_{i}(t) = a_{i}f(t), where a_{i} = const, i = 1, …, m;

3) if p_{i}(t) = a_{i}t^{i}, i = 1, …, m;

4) if p_{i}(t) = a_{i}t + b_{i} (a_{i} > 0, b_{i} > 0), where b_{i}/a_{i} are nonincreasing for i = 1, …, m.

The condition that the functions \frac{p_{i+1}}{p_{i}} are nondecreasing for i = 1, …, m is essential, as the following example shows.

Example 1. The function

x(t) = \begin{cases}1 − t^{2}, & 0 ≤ t ≤ 2, \\ 0, 01t^{2} − 4, 04t + 5, 04, & 2 < t ≤ 210, \\ 2(t − 5239, 5)^{2}, & 210 < t \end{cases}

has a multiple zero at the point t = 5239, 5. This function is the solution of the equation

x′′(t) + p_{1}(t)x(h_{1}(t)) + x(h_{2}(t)) = 0,

where

h_{1}(t) = \begin{cases}0, & 0 ≤ t ≤ 2, \\ 0, 9, & t > 2, \end{cases} h_{2}(t) = \begin{cases}0, & 0 ≤ t ≤ 2, \\ 1, 1, & t > 2, \end{cases}

p_{1}(t) = \begin{cases}1, & 0 ≤ t ≤ 210, \\ (21 + 32/10059)/19, & t > 210. \end{cases}

It is clear that W (5239, 5) = 0.

The following fact follows from Theorem 1.

Corollary. If m = 1 and h_{1} is nondecreasing, then the assertions a),b),c) of Theorem 1 are fulfilled.

The condition that the functions h_{i} are nondecreasing is essential, as the example of N.V.Azbelev [1] shows.

Let R_{νμ}(t, s), Q_{νμ}(t, s) be Green’s functions of the boundary value problems

x′′(t) = f(t), t ∈ [ν, μ], x(ν) = 0, x′(μ) = 0,
x′′(t) = f(t), t ∈ [ν, μ], x′(ν) = 0, x(μ) = 0

respectively. It is clear that

R_{νμ}(t, s) = \begin{cases}ν − s & if ν ≤ s ≤ t ≤ μ, \\ ν − t & if ν ≤ t < s ≤ μ, \end{cases}

Q_{νμ}(t, s) = \begin{cases}t − μ & if ν ≤ s ≤ t ≤ μ, \\ s − μ & if ν ≤ t < s ≤ μ. \end{cases}

Define the operators R_{νμ}, Q_{νμ} : C_{[ν,μ]} → C_{[ν,μ]} by the formulas

(R_{νμ}x)(t) = ∫_{ν}^{μ} R_{νμ}(t, s) ∑_{i=1}^{m} p_{i}(s)x(h_{i}(s))γ(ν, h_{i}(s))ds,

(Q_{νμ}x)(t) = ∫_{ν}^{μ} Q_{νμ}(t, s) ∑_{i=1}^{m} p_{i}(s)x(g_{i}(s))ds,

here g_{i} (i = 1, …, m) are measurable functions such that ν ≤ g_{i}(t) ≤ μ. Let r_{νμ}, q_{νμ} be the smallest positive characteristic numbers of the operators R_{νμ}, Q_{νμ} respectively.

Theorem 2. Let the conditions 1), 2) of Theorem 1 be fulfilled, r_{h_{1}(t)h_{m}(t)} > 1 for almost all t ∈ [0, +∞), q_{h_{1}(t)h_{m}(t)} > 1 for almost all t ∈ [0, +∞) and all possible functions g_{i} such that g_{i}(s) ∈ [h_{1}(t), h_{m}(t)] for s ∈ [h_{1}(t), h_{m}(t)], i = 1, …, m. Then the assertions a),b),c) of Theorem 1 are fulfilled.

Remark. The inequalities r_{h_{1}(t)h_{m}(t)} > 1 and q_{h_{1}(t)h_{m}(t)} > 1 for t ∈ [0, +∞) guarantee that a solution of equation (1), having zero on the interval [h_{1}(t), h_{m}(t)] has no zero of its derivative on this interval.

3. Proofs

We start with some auxiliary results.

Lemma 1. Let α be a zero of the nontrivial solution x of equation (1), β be a zero of its derivative such that x(t) > 0 for t ∈ (α, β), α < β. Then there exists a set e⊂(α, β) of positive measure such that ∑_{i=1}^{m} p_{i}(t)x(h_{i}(t)) > 0 for t ∈ e.

Proof of Lemma 1. Let we have on the contrary, ∑_{i=1}^{m} p_{i}(t)x(h_{i}(t)) ≤ 0 for almost all t ∈ [α, β]. By the theorem of Lagrange there exists d ∈ (α, β) such

that x′(d) > 0. x′(β) = x′(d)+∫_{d}^{β} x′′(s)ds = x′(d)−∫_{d}^{β} ∑_{i=1}^{m} p_{i}(s)x(h_{i}(s))ds>

0, that contrast the assumption: x′(β)=0. ■

Lemma 2. Let

1) y be a nondecreasing function in the interval [a, b];

2) a ≤ h_{1}(t) ≤ h_{2}(t) ≤ ⋯ ≤ h_{m}(t) ≤ b for almost all t ∈ [c, d] ∈ [a, b], h_{i} be nondecreasing for i = 1, …, m;

3) the functions p_{i+1}/p_{i} be nondecreasing for i = 1, …, m − 1.

Then from the existence of a set e ⊂ [c_{1}, d_{1}] ⊂ [c, d) such that mes(e) > 0 and ∑_{i=1}^{m} p_{i}(t)y(h_{i}(t)) > 0 for t ∈ e, it follows that ∑_{i=1}^{m} p_{i}(t)y(h_{i}(t)) > 0 for almost all t ∈ [d_{1}, d].

Proof of Lemma 2. Let k be a number such that y(h_{i}(t)) ≥ 0 for t ∈ e, i ≥ k. By the condition we have the inequality − ∑_{i=1}^{k−1} p_{i}(t)y(h_{i}(t)) < ∑_{i=k}^{m} p_{i}(t)y(h_{i}(t)) for t ∈ e.

For all i = 1, …, m y(h_{i}(t)) are nondecreasing since y and h_{i} are nondecreasing. Using the condition 3) we obtain for t ∈ e and r such that t + r ∈ [d_{1}, d]:

− ∑_{i=1}^{k−1} p_{i}(t + r)y(h_{i}(t + r)) = − ∑_{i=1}^{k−1}(p_{i}(t + r)/p_{i}(t))p_{i}(t)y(h_{i}(t + r)) ≤

≤ −(p_{k−1}(t + r)/p_{k−1}(t)) ∑_{i=1}^{k−1} p_{i}(t)y(h_{i}(t + r)) <

< (p_{k−1}(t + r)/p_{k−1}(t)) ∑_{i=k}^{m} p_{i}(t)y(h_{i}(t + r)) ≤

≤ ∑_{i=k}^{m} p_{i}(t + r)y(h_{i}(t + r)). ■

Lemma 3. Let [α, β] ⊂ [ν, μ]. Then

1) if λ_{νμ} > 1, then λ_{αβ} > 1; if λ_{αβ} ≤ 1, then λ_{νμ} ≤ 1;

2) if r_{νμ} > 1, then r_{αβ} > 1; if r_{αβ} ≤ 1, then r_{νμ} ≤ 1;

3) if q_{νμ} > 1 for each collection of functions such that g_{i}(t) ∈ [ν, μ], t ∈ [ν, μ], i = 1, …, m, then q_{αβ} > 1 each collection of functions such that ḡ_{i}(t) ∈ [α, β], t ∈ [α, β], i = 1, …, m;

if there exists a collection of functions such that ḡ_{i}(t) ∈ [α, β], i = 1, …, m, and q_{αβ} ≤ 1, then there exists a collection of functions g_{i}(t) ∈ [ν, μ], t ∈ [ν, μ], i = 1, …, m, such that q_{νμ} ≤ 1.

Proof of Lemma 3. Assertion 1) is proved in [1] and the proof of Assertion 2) is analogous, therefore we prove only Assertion 3).

Let us take an arbitrary collection of functions ḡ_{i}(t) ∈ [α, β], t ∈ [α, β], i = 1, …, m, and denote

g_{i}(t) = \begin{cases}ḡ_{i}(t) & for t ∈ [α, β], \\ α & for t ∈ [α, β]. \end{cases}

By condition q_{νμ} > 1 for this collection g_{i}, i = 1, …, m, the equation x = Q_{νμ}x + 1 has a positive solution v = lim_{n→∞} x_{n}, where x_{0} = 1, x_{n+1} = Q_{νμ}x_{n} + 1. Since Q_{νμ}(t, s) ≥ Q_{αβ}(t, s) for t, s ∈ [α, β], then v ≥ Q_{αβ}v + 1.

Now, by the theorem about integral inequalities [1], we have q_{αβ} > 1 for this collection ḡ_{i}, i = 1, …, m. The first part of the assertion 3) is proved.

The second part of the assertion 2) can be deduced from the first part. ■

Proof of Theorem 2. Let x be a nontrivial solution of the equation (1). Let us consider the case x(0) > 0, x′(0) ≥ 0 (the case x(0) ≥ 0, x′(0) < 0 can be considered analogously).

Denote by β_{1} the first zero of the solution’s derivative, by α_{1} the first zero of the solution x (β_{1} < α_{1} by our assumption). If β_{1} or α_{1} doesn’t exist, then the theorem is trivial.

Let us show that there exists a collection g_{i}(t), i = 1, …, m, such that q_{β_{1}α_{1}} ≤ 1. Really, x satisfies the following equation

x(t) = − ∫_{β_{j}}^{α_{k}} Q_{β_{j}α_{k}}(t, s) ∑_{i=1}^{m} p_{i}(s)x(h_{i}(s))γ(β_{j}, h_{i}(s))ds −

− ∫_{β_{j}}^{α_{k}} Q_{β_{j}α_{k}}(t, s) ∑_{i=1}^{m} p_{i}(s)x(h_{i}(s))[1 − γ(β_{j}, h_{i}(s))]ds

for t ∈ [β_{1}, α_{1}], where k = j = 1 and γ is defined by (2).

Rewrite the equality (4) in the following form

x(t) = − ∫_{β_{1}}^{α_{1}} Q_{β_{1}α_{1}}(t, s) ∑_{i=1}^{m} p_{i}(s)x(g_{i}(s))ds,

where the functions g_{i} such that g_{i}(t) ∈ [β_{1}, α_{1}].

Existence of these functions g_{i}, i = 1, …, m, follows from the next arguments. Since x′′(t) = − ∑_{i=1}^{m} p_{i}(t)x(h_{i}(t)) < 0, then x′ is not increasing. Therefore x(β_{1}) = max_{t∈[0,α_{1}]} x(t) and the set of values of function x on the in-

terval [0, β_{1}] is included in the set of values of the function x on the interval [β_{1}, α_{1}], hence the solution g_{i} of the functional equation x(h_{i}(t)) = x(g_{i}(t)), t ∈ [β_{1}, α_{1}] exists.

It is obvious that for this collection of functions g_{i}, i = 1, …, m, we have q_{α_{1}β_{1}} ≤ 1.

We show that x′(α_{1}) < 0. Indeed, by the theorem of Lagrange there exists d ∈ (β_{1}, α_{1}) such that x(α_{1}) − x(β_{1}) = x′(d)(α_{1} − β_{1}). From here x′(d) < 0 and x′(α_{1}) = x′(d) + ∫_{d}^{α_{1}} x′′(t)dt < 0.

Let β_{2} be the first zero of the solution x after α_{1}. By Lemma 1 there exists a set e ∈ [α_{1}, β_{1}] with mes(e) > 0 such that x′′(t) = − ∑_{i=1}^{m} p_{i}(t)x(h_{i}(t)) > 0 for t ∈ e. From here it follows that h_{m}(t) > α_{1} for almost all t ≥ β_{2}. Since q_{h_{1}(t)h_{m}(t)} > 1, independently of collection of functions g_{i}, i = 1, …, m we obtain by Lemma 3 that h_{1}(t) ≥ β_{1} for almost all t ≥ β_{2}.

Next, show that r_{α_{1}β_{2}} ≤ 1. Indeed, on the interval [α_{1}, β_{2}] the solution x of equation (1) satisfies the following integral equation

x(t) = − ∫_{α_{k}}^{β_{j}} R_{α_{k}β_{j}}(t, s) ∑_{i=1}^{m} p_{i}(s)x(h_{i}(s))γ(α_{k}, h_{i}(s))ds −

− ∫_{α_{k}}^{β_{j}} R_{α_{k}β_{j}}(t, s) ∑_{i=1}^{m} p_{i}(s)x(h_{i}(s))[1 − γ(α_{k}, h_{i}(s))]ds,

where k = 1, j = 2.

Taking v(t) = −x(t), we obtain the inequality v(t) ≤ (R_{α_{k}β_{j}}v)(t) for k = 1, j = 2. By the theorem about the integral inequalities (see, for example, [1]) we obtain r_{α_{1}β_{2}} ≤ 1.

Denote α_{2} the first zero of the solution x after β_{2} (if the solution x hasn’t a second zero α_{2}, then Theorem 2 is trivial).

If there exist d ∈ (β_{2}, α_{2}) such that h_{1}(t) ≥ α_{1} for almost all t ≥ d, then x(h_{i}(t)) ≤ 0 for almost all t ∈ [d, α_{2}], hence x′′(t) > 0 for almost all t ∈ [d, α_{2}].

If β_{1} ≤ h_{1}(t) < α_{1}, then by Lemma 2, with the use of the condition r_{h_{1}(t)h_{m}(t)} > 1 and Lemma 3, we can conclude that x′′(t) > 0 for t ∈ [β_{2}, d].

By the theorem of Lagrange, there exists c ∈ (β_{2}, α_{2}) such that x(α_{2}) − x(β_{2}) = x′(c)(α_{2} − β_{2}), this implies x′(c) > 0 and x′(α_{2}) = x′(c) + ∫_{c}^{α_{2}} x′′(t)dt > 0. It means that α_{2} < β_{3} (we denote β_{3} the first zero of derivative of the solution x after α_{2}). Now we show that q_{β_{2}α_{2}} ≤ 1 for some collection g_{i}, i = 1, …, m. On the interval [β_{2}, α_{2}] the solution x of equation (1) satisfies equation (4) for j = k = 2. Rewrite this equation in the following form:

x(t) = − ∫_{β_{j}}^{α_{k}} Q_{β_{j}α_{k}}(t, s) ∑_{i=1}^{m} p_{i}(s)x(g_{i}(s))γ(α_{k−1}, h_{i}(s))ds −

− ∫_{β_{j}}^{α_{k}} Q_{β_{j}α_{k}}(t, s) ∑_{i=1}^{m} p_{i}(s)x(h_{i}(s))[1 − γ(α_{k−1}, h_{i}(s))]ds

where k = j = 2, g_{i}(t) ∈ [β_{2}, α_{2}] such that γ(α_{1}, h_{i}(t)) = x(g_{i}(t)) for t ∈ [β_{2}, α_{2}]. The collection g_{i}, i = 1, …, m exists since x′′(t) ≥ 0 for t ∈ [β_{2}, α_{2}]. For, since x′ isn’t decreasing on the interval [β_{2}, α_{2}], therefore |x(β_{2})| = max_{α_{1}≤t≤α_{2}} |x(t)| and the set of values of the function x on the interval

[α_{1}, β_{2}] is included in the set of values of x on the interval [β_{2}, α_{2}]. It follows from [1] that q_{β_{2}α_{2}} ≤ 1. The inequality λ_{α_{1}α_{2}} ≤ 1 is proved analogously to r_{α_{1}β_{2}} ≤ 1. If α_{m} is the last zero of the nontrivial solution, then repeating the argu-

ments for j, k = 3, 4, 5, …, m, we obtain the proof of the theorem. If the solution x has an infinite number of zeros, then the sequence α_{k} of zeros is unbounded. Indeed, we have proved that λ_{α_{k}α_{k+1}} ≤ 1, this implies that [1] (α_{k+1} − α_{k}) ∫_{α_{k}}^{α_{k+1}} ∑_{i=1}^{m} p_{i}(t)dt > 4 and, consequently, the increasing sequence α_{k} cannot be bounded from above.

It is clear that all zeros of the solution x belong to this sequence α_{k}. In this case the repetition of our arguments completes the proof of the assertions a) and b) of Theorem 2.

The assertion c) follows from the following argument. For each j we have proved that sign x(t) = sign ∑_{i=1}^{m} p_{i}(t)x(h_{i}(t)) for t ∈ (β_{j}, α_{j}), this implies x′(t) = x′(β_{j}) − ∫_{β_{j}}^{t} ∑_{i=1}^{m} p_{i}(s)x(h_{i}(s))ds ≠ 0 for t ∈ (β_{j}, α_{j}). ■

Proof of Theorem 1. Theorem 1 can be obtained as a corollary of Theorem 2.

Indeed, from the theorem about the integral inequalities [1] we have the following. If there exists a continuous positive function v such that v(t) > Q_{νμ}v)(t) (v(t) > (R_{νμ}v)(t)) for t ∈ (ν, μ), then q_{νμ} > 1 (r_{νμ} > 1). Substituting v = 1, we obtain that the condition (4) guarantees the inequalities r_{h_{1}(t)h_{m}(t)} > 1, q_{h_{1}(t)h_{m}(t)} > 1.

If there exists a positive function v such that

φ(t) = v′′(t) + ∑_{i=1}^{m} p_{i}(t)v(h_{i}(t))γ(v, h_{i}(t)) ≤ 0, ∫_{ν}^{μ} φ(t)dt < 0,

v′(ν) = 0, v(μ) = 0 (v(ν) = 0, v′(μ) = 0),

then q_{νμ} > 1 (r_{νμ} > 1) [2]. Substituting v(t) = (t − 2ν + μ)(μ − t) (v(t) = (t − ν)(2μ − ν − t)), we conclude that inequality (3) implies the inequality q_{h_{1}(t)h_{m}(t)} > 1 (r_{h_{1}(t)h_{m}(t)} > 1). ■

Acknowledgement.

The research was supported in part by grant from the Ministry of Science and the ”MA-AGARA”-special project for absorption of new immigrants, in the Department of Mathematics, Technion.

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(Received 04.01.1993)

Author’s address:

Technion–Israel Institute of Technology

Department of Mathematics

Technion City, 32000 Haifa

Israel