Mem. Differential Equations Math. Phys. 6(1995), 116–118

R. Koplatadze

ON ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS

(Reported on October 2 and 9, 1995)

Consider the equation

u^{(n)}(t) + ∑_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)}u(s)d_{s}r_{i}(s, t) = 0,  (1)

where n ≥ 2, m ∈ N, τ_{i}; σ_{i} ∈ C(R_{+}; R_{+}), τ_{i}(t) ≤ σ_{i}(t) ≤ t for t ∈ R_{+}, lim_{t→+∞} τ_{i}(t) = +∞, (i = 1, …, m), the functions r_{i}(s, t) are measurable, and r_{i}(·, t) is nondecreasing (i = 1, …, m).

Let t_{0} ∈ R_{+}. A function u : [t_{0}, +∞[→ R is called a proper solution of the equation (1) if it is absolutely continuous along with its derivatives up to the n − 1-th order inclusively,

sup {|u(s)| : s ∈ [t, +∞[} > 0 for t ≥ t_{0},

and there exists \overline{u} ∈ C(R_{+}; R) such that \overline{u}(t) ≡ u(t) for t ∈ [t_{0}, +∞[ and

\overline{u}^{(n)}(t) + ∑_{i=1}^{m} ∫_{τ_{i}(t)}^{σ_{i}(t)}\overline{u}(s)d_{s}r_{i}(s, t) = 0

almost everywhere on [t_{0}, +∞[. A proper solution u : [t_{0}, +∞[→ R is said to be oscillatory if it has a sequence of zeroes tending to +∞. Otherwise the solution is said to be nonoscillatory.

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

Theorem. Let for some i_{0} ∈ {1, …, m} there exist a nondecreasong δ ∈ C(R_{+}; R_{+}) such that τ_{i_{0}}(t) ≤ δ(t) ≤ σ_{i_{0}}(t) for t ∈ R_{+},

\underline{lim}_{t→+∞} ∫_{δ(t)}^{t} ∫_{τ_{i0} (s)}^{δ(s)} ξ^{n−1}d_{ξ}r_{i_{0}}(ξ, s)ds > 0,

vrai sup {t ∫_{τ_{i0} (t)}^{δ(t)} s^{n−1}d_{s}r_{i_{0}}(s, t) : t ∈ R_{+}} < +∞

1991 Mathematics Subject Classification. 34K15.

Key words and phrases. Functional differential equation, proper solution, equation with property A. and

\overline{lim}_{t→+∞} \frac{ln t}{ln τ_{i}(t)} < +∞ (i = 1, …, m).  (2)

Then the condition

inf \{ \underline{lim}_{t→+∞} ln^{λ} t ∫_{t}^{+∞}∑_{i=1}^{m} ∫_{τ_{i}(s)}^{σ_{i}(s)}ξ^{n−1} ×

× ln^{−λ} ξ d_{ξ}r_{i}(ξ, s)ds : λ ∈]0, k]} > (n − 1)! for all k ∈ N  (3)

is sufficient for the equation (1) to have the property A.

Corollary 1. Let c_{i} ∈]0, +∞[, α_{i}; \overline{α}_{i} ∈]0, 1], and α_{i} < \overline{α}_{i} (i = 1, …, m). Then for the equation

u^{(n)}(t) + ∑_{i=1}^{m} \frac{c_{i}}{t} ∫_{t^{αi}}^{t^{\overline{α}i}} \frac{u(s)}{s^{n} ln^{2} s} ds = 0

to have the property A, it is sufficient that

inf \{\frac{1}{λ(λ + 1)} ∑_{i=1}^{m} c_{i}(α_{i}^{−λ−1} − \overline{α}_{i}^{−λ−1}) : λ ∈]0, +∞[\} > (n − 1)!.

Corollary 2. Let c_{i} ∈]0, +∞[, α_{i} ∈]0, 1], and α_{i_{0}} < 1 for some i_{0} ∈ {1, …, m}. Then for the equation

u^{(n)}(t) + \frac{1}{t ln t} ∑_{i=1}^{m} \frac{c_{i}}{t^{α_{i}(n−1)}} u(t^{α_{i}})ds = 0

to have the property A, it is sufficient that

inf \{\frac{1}{λ} ∑_{i=1}^{m} c_{i}α_{i}^{−λ} : λ ∈]0, +∞[\} > (n − 1)!.

In the case where the condition

\overline{lim}_{t→+∞} \frac{t}{τ_{i}(t)} < +∞ (i = 1, …, m)

holds instead of (2), analogous questions are considered in [1].

Remark. Note that the inequality (2) cannot be replaced by

inf \{ \underline{lim}_{t→+∞} ln^{λ} t ∫_{t}^{+∞}∑_{i=1}^{m} ∫_{τ_{i}(s)}^{σ_{i}(s)}ξ^{n−1} ×

× ln^{−λ} ξ d_{ξ}r_{i}(ξ, s)ds : λ ∈]0, k]} > (n − 1)! − ε

for any whatever small ε.

References

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

Author’s address:

R. Koplatadze I. Vekua Institute of Applied Mathematics Tbilisi State University 2, University St., Tbilisi 380043 Ceorgia