Mem. Differential Equations Math. Phys. 15(1998), 141–144

R. Koplatadze

COMPARISON THEOREMS FOR DEVIATED DIFFERENTIAL EQUATIONS WITH PROPERTY A

(Reported on March 23, 30, 1998)

1. Short Survey of Known Results

Consider the equations

u^{(n)}(t) + p(t)u(τ(t)) = 0,  (1.1)
v^{(n)}(t) + q(t)v(σ(t)) = 0,  (1.2)

where p, q ∈ L_{loc}(R_{+}; R_{+}) and τ, σ ∈ C(R_{+}; R+), lim_{t→+∞} τ(t) = lim_{t→+∞} σ(t) = +∞ and σ, τ are nondecreasing.

Comparison theorems play an important role in studying oscillatory properties of differential equations. Sturm [1] was the first who proved a comparison theorem on distribution of zeros for second order linear ordinary differential equations. On the basis of this theorem Kneser [2] obtained effective and somehow optimal criteria for oscillation of all solutions.

For higher order equations, comparison theorems are formulated in terms of the so called Properties A and B, analogues of oscillation of all solutions. In the present note we deal with the case of Property A. Analogous results for Property B will be reported in the nearest future.

Definition 1.1. We say that the equation (1.1) has Property A if any of its proper solutions is oscillatory when n is even and either is oscillatory or satisfies |u^{(i)}(t)| ↓ 0 as t ↑ +∞ (i = 1, …, n − 1), when n is odd.

In [3] V. Kondrat’ev first proved a comparison theorem for Property A.

Theorem 1.1 (V. Kondrat’ev). Let τ(t) ≡ σ(t) ≡ t for t ∈ R_{+}, p(t) ≥ q(t) for t ∈ R_{+} and (1.2) have Property A. Then (1.1) also has Property A.

Basing on this theorem, he obtained an effective condition for (1.1) to have Property A. Theorem 1.2 (V. Kondrat’ev). Let τ(t) ≡ t for t ∈ R_{+}, p(t) ≥ \frac{M_{n}+ε}{t^{n}} for t ≥ 1,

where ε > 0 and M_{n} = max { − x(x − 1) ⋯ (x − n + 1) : x ∈ [n − 2, n − 1]}. Then (1.1) has Property A.

In this theorem one can not take ε = 0, so it is optimal in a sense.

Generalizing Kondrat’ev’s results, T. Chanturia [4] obtained integral conditions.

Theorem 1.3 (T. Chanturia). Let τ(t) ≡ σ(t) ≡ t,

∫_{t}^{+∞} s^{n−2}p(s) ds ≥ ∫_{t}^{+∞} s^{n−2}q(s) ds for t ∈ R_{+},

1991 Mathematics Subject Classification. 34K15.

Key words and phrases. Advanced differential equation, property A, comparison theorem. and (1.2) have Property A. Then (1.1) also has Property A.

Theorem 1.4 (T. Chanturia). Let τ(t) ≡ t and

lim inf_{t→+∞} t ∫_{t}^{+∞} s^{n−2}p(s) ds > M_{n},

with M_{n} defined in Theorem 1.2. Then (1.1) has Property A.

2. Comparison Theorems

Below we give integral comparison theorems of two types for (1.1) and (1.2). Theorems of the first type (Theorems 2.1 and 2.2) generalize T. Chanturia’s Theorem 1.3 to deviated differential equations. As to theorems of the second type (Theorems 2.3 and 2.4), as far as we know, they have not been considered earlier even in the case of ordinary differential equations. All the four theorems of this section generalize Kondrat’ev’s Theorem 1.1.

Theorem 2.1. Let σ(t) ≤ τ(t) ≤ t, for sufficiently large t_{0}

∫_{t}^{+∞} τ^{n−2}(s)p(s) ds ≥ ∫_{t}^{+∞} σ^{n−2}(s)q(s) ds for t ≥ t_{0},  (2.1)

and (1.2) have Property A. Then (1.1) also has Property A.

Theorem 2.2. Let τ(t) ≥ max{t, σ(t)} for t ∈ R_{+}. Let for sufficiently large t_{0}

∫_{t}^{+∞} s^{n−2}p(s) ds ≥ ∫_{t}^{+∞} s^{n−2}q(s) ds for t ≥ t_{0},

be satisfied if n is even, and (2.1),

∫_{t}^{+∞} s^{n−3}τ(s)p(s) ds ≥ ∫_{t}^{+∞} s^{n−3}σ(s)q(s) ds for t ≥ t_{0} .

and

∫_{0}^{+∞} s^{n−1}p(s) ds = +∞.  (2.2)

be satisfied if n is odd. Let, moreover, (1.2) have Property A. Then (1.1) also has Property A.

Theorem 2.3. Let τ(t) ≤ max{t, σ(t)} for t ∈ R_{+} and for sufficiently large t_{0} there exist t_{1} = t_{1}(t_{0}) ≥ t_{0} such that

∫_{t_{0}}^{t} sτ^{n−1}(s)p(s) ds ≥ ∫_{t_{0}}^{t} sσ^{n−1}(s)q(s) ds for t ≥ t_{1} .  (2.3)

Let, moreover, (1.2) have Property A. Then (1.1) also has Property A.

Theorem 2.4. Let t ≤ τ(t) ≤ σ(t) for t ∈ R_{+} and for sufficiently large t_{0} there exists t_{1} = t_{1}(t_{0}) such that

∫_{t_{0}}^{t} s^{n−1}τ(s)p(s) ds ≥ ∫_{t_{0}}^{t} s^{n−1}σ(s)q(s) ds for t ≥ t_{1},

if n is even, and (2.2), (2.3) and

∫_{t_{0}}^{t} s^{n−2}τ^{2}(s)p(s) ds ≥ ∫_{t_{0}}^{t} s^{n−2}σ^{2}(s)q(s) ds for t ≥ t_{1},

be fulfilled if n is odd. Let, moreover, (1.2) have Property A. Then (1.1) also has Property A.

3. Effective Criteria for Property A

Using the above comparison theorems and some results of [5] (see section 7 of Chapter 2), we can obtain new effective criteria for (1.1) to have Property A.

Let l ∈ {1, …, n − 1}, α ∈]0, +∞[. Denote

M_{l}(α) = max { − α^{−x}x(x − 1) ⋯ (x − n + 1) : l − 1 ≤ x ≤ l}.

Theorem 3.1. Let α ∈]0, 1], αt ≤ τ(t) ≤ t for t ∈ R_{+} and

lim inf_{t→+∞} t ∫_{t}^{+∞} τ^{n−2}(s)p(s) ds > α^{n−2}M_{n−1}(α).

Then (1.1) has Property A.

Theorem 3.2. Let α ∈ [1, +∞[, τ(t) ≥ αt for t ∈ R_{+}. Let

lim inf_{t→+∞} t ∫_{t}^{+∞} s^{n−2}p(s) ds > M_{1}(α)

be fulfilled if n is even, and (2.2) and

lim inf_{t→+∞} t ∫_{t}^{+∞} s^{n−3}τ(s)p(s) ds > αM_{2}(α),

lim inf_{t→+∞} t ∫_{t}^{+∞} τ^{n−2}(s)p(s) ds > α^{n−2}M_{n−1}(α)

be fulfilled if n is odd. Then (1.1) has Property A.

Theorem 3.3. let α ∈]0, 1], τ(t) ≤ αt for t ∈ R_{+} and

lim inf_{t→+∞} t^{−1} ∫_{0}^{t} sτ^{n−2}(s)p(s) ds > α^{n−1}M_{n−1}(α).

Then (1.1) has Property A.

Theorem 3.4. Let α ∈ [1 + ∞[, t ≤ τ(t) ≤ αt. Let

lim inf_{t→+∞} t^{−1} ∫_{0}^{t} s^{n−1}τ(s)p(s) ds > αM_{1}(α)

be fulfilled if n is even, and

lim inf_{t→+∞} t^{−1} ∫_{0}^{t} sτ^{n−1}(s)p(s) ds > α^{n−1}M_{n−1}(α),

lim inf_{t→+∞} t^{−1} ∫_{0}^{t} s^{n−2}τ^{2}(s)p(s) ds > α^{2}M_{2}(α)

be fulfilled. Then (1.1) has Property A.

Remark. In Theorems 3.1.–3.4 strict inequalities cannot be replaced by nonstrict ones. Moreover, none of the hypotheses of Theorems 3.2–3.4 can be omited without affecting their validity.

Now we consider the case where the condition τ(t) ≥ αt (τ(t) ≤ αt) is violated.

Theorem 3.5. Let σ(t)≤τ(t)≤t for t∈R_{+}, \frac{σ(t)}{t}↓0 as t↑+∞, and the conditions (2.2),

∫ ^{+∞} \frac{dt}{tσ(t)} < +∞, lim inf_{t→+∞} φ(t) ∫_{t}^{+∞} τ^{n−2}(s)p(s) ds > 0

be fulfilled, where φ(t) = ( ∫_{t}^{+∞} \frac{ds}{sσ(s)})^{−1}. Then (1.1) has Property A.

Corollary 3.1. Let α ∈]0, 1[, t^{α} ≤ τ(t) ≤ t for t ∈ R_{+} and

lim inf_{t→+∞} t^{α} ∫_{t}^{+∞} τ^{n−2}(s)p(s) ds > 0.

Then (1.1) has Property A.

Theorem 3.6. Let t ≤ τ(t) ≤ σ(t) for t ∈ R_{+}, \frac{σ(t)}{t} ↑ +∞ as t ↑ +∞, and (2.2) and

lim inf_{t→+∞} φ(t) ∫_{0}^{t} s^{n−1}τ(s)p(s) ds > 0

be fulfilled, where φ(t) = ( ∫_{1}^{t} \frac{σ(s)ds}{s})^{−1}. Then (1.1) has Property A.

Corollary 3.2. Let α ∈]1, +∞[, t ≤ τ(t) ≤ t^{α} for t ∈ R_{+} and

lim inf_{t→+∞} t^{−α} ∫_{0}^{t} s^{n−1}τ(s)p(s) ds > 0.

Then (1.1) has Property A.

Acknowledgement

This work was supported by Grant No. 1.6/1998 of the Georgian Academy of Sciences.

References

1. C. Sturm, Sur les équations différentielles linéaires de second ordre. J. Math. Pures et Appl. 1(1836), 106–186.

2. A. Kneser, Untersuchungen über die reelen Nulstellen der Integrale linearer Differentialgleichungen. Math. Ann. 42(1893), No. 3, 178–212.

3. V.A. Kondrat’ev, On Oscillation of solutions of the equation y^{(n)} + p(x)y = 0. (Russian) Trudy Moskov. Mat. Obshch. 10(1961), 419–436.

4. I. Kiguradze, T. Chanturia, Asymptotic properties of solutions of nonautonomous ordinary differential equations. Kluwer Academic Publishers, Dordrecht-Bos-ton-London, 1993.

5. R. Koplatadze, On Oscillatory properties of solutions of functional defferential equations. Mem. Differential Equations Math. Phys. 3(1994), 1–177.

Authors address:

A. Razmadze Mathematical Institute

Georgian Academy of Sciences

1, Aleksidze St., Tbilisi 380093

Georgia