Mem. Differential Equations Math. Phys. 16(1998), 148–149

G. Kvinikadze

ON MONOTONE SOLUTIONS OF LINEAR ADVANCED EQUATIONS

(Reported on September 28, 1998)

In the present note we consider the equation

u^{(n)}(t) = p(t)u(τ(t)),  (1)

where p : R_{+} → R_{+} is locally integrable and τ : R_{+} → R_{+} is measurable with τ(t) ≥ t for t ≥ 0.

A solution u of (1) is said to be proper if it is defined in a neighbourhood [t_{0}, +∞[ of +∞ and

sup{|u(s)| : s ≥ t} > 0 for t ≥ t_{0}.

We will treat the problem of existence of a proper solution u : [t_{0}, +∞[ → R satisfying

u^{(i)}(t) > 0 for t ≥ t_{0} (i = 0, …, n − 1).  (2)

The problem of this kind is specific for advanced equations. It does not arise for ordinary and delay equations since in that case every solution of (1) with positive initial conditions is continuable up to infinity and satisfies (2).

The following two cases will be considered:

(1)  \overline{lim}_{t→+∞}(τ(t) − t) = Δ < +∞ with Δ ≥ 0;  (3)
(2)  \overline{lim}_{t→+∞} \frac{τ(t)}{t} = α < +∞ with α ≥ 1.  (4)

R. Koplatadze in [1] has established sufficient conditions for (1) not to have a solution satisfying (2). From Theorems 11.4 and 11.8 of [1] there follow the following results, respectively.

1) if τ ≡ t + Δ with Δ ∈ R_{+} and for any λ ∈]0, +∞[

lim inf_{t→+∞} e^{−λt} ∫_{0}^{t} e^{λs}p(s) ds > λ^{n−1}e^{−λΔ},

then the equation (1) has no solution satisfying (2).

2) if τ ≡ αt with α ≥ 1 and for any λ ∈]n − 1, +∞[

lim inf_{t→+∞} t^{n−λ−1} ∫_{0}^{t} s^{λ}p(s) ds < α^{−λ} ∏_{i=0}^{n−2}(λ − i),

then the equation (1) has no solution satisfying (2).

Theorems 1 and 2 below invert in a sense these results.

1991 Mathematics Subject Classification. 34C11. Key words and phrases. Linear advanced equations, monotone solution, proper

solution.

Theorem 1. Let (3) be fulfilled and there exist λ_{0} ∈]0, +∞[ such that

lim sup_{t→+∞} e^{−λ_{0}t} ∫_{0}^{t} e^{λ_{0}s}p(s) ds < λ_{0}^{n−1}e^{−λ_{0}Δ}.

Then the equation (1) has a solution satisfying (2).

Theorem 2. Let (4) be fulfilled and there exist λ_{0} ∈]n − 1, +∞[ such that

lim sup_{t→+∞} t^{n−λ_{0}−1} ∫_{0}^{t} s^{λ_{0}}p(s) ds < α^{−λ_{0}} ∏_{i=0}^{n−2}(λ_{0} − i).

Then the equation (1) has a solution satisfying (2).

The following below comparison theorem is a basic tool in establishing Theorems 1 and 2.

Along with (1), consider the equation

v^{(n)} + q(t)v(σ(t)) = 0,  (5)

where q : R_{+} → R_{+} is locally integrable and σ : R_{+} → R_{+} is measurable with σ(t) ≥ t for t ∈ R_{+}.

It will be assumed that

τ(t) ≤ σ(t) for t ∈ R_{+}.  (6)

For the equation (5) the problem on existence of a solution v satisfying

v^{(i)}(t) > 0 for t ≥ t_{0}  (7)

will be likewise considered.

Theorem 3. Let (6) be fulfilled and the equation (5) have a proper solution v : [t_{0}, +∞[ → R satisfying (7). Let, moreover, there exist a continuous function φ : [t_{0}, +∞[ →]0, +∞[ such that v/φ is nonincreasing and

∫_{t_{0}}^{t} φ(σ(s))p(s) ds ≤ ∫_{t_{0}}^{t} φ(σ(s))q(s) ds for t ≥ t_{0}.

Then the equation (1) has a solution u : [t_{0}, +∞[ → R satisfying (2).

Comparing, according to this theorem, the equation (6) with the equations

v^{(n)} + q_{1}v(t + Δ) = 0 and v^{(n)} + \frac{q_{2}}{t^{n}} v(αt) = 0,

where q_{1} and q_{2} are appropriately chosen, we derive Theorems 1 and 2, respectively.

Acknowledgement

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

References

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

Author’s address:

A. Razmadze Mathematical Institute

Georgian Academy of Sciences

1, M. Aleksidze St., Tbilisi 380093

Georgia