Mem. Differential Equations Math. Phys. 13(1997), 150–153

V. M. Evtukhov and E. V. Shebanina

ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF n-th ORDER DIFFERENTIAL EQUATIONS

(Reported on December 15, 1997)

Consider the differential equation

y^{(n)} = ∑_{i=1}^{m} α_{i}p_{i}(t)|y|^{σ_{0i}}|y′|^{σ_{1i}} … |y^{(l−1)}|^{σ_{l−1i}} sign y,  (1)

where n ≥ 2, α_{i} ∈ {−1, 1} (i = 1, …, m), l ∈ {1, …, n}; σ_{0i}, …, σ_{l−1i} are real con-

stants such that ∑_{k=0}^{l−1} σ_{ki} ≠ 1 for any i ∈ {1, …, m}, ∑_{k=0}^{l−1} σ_{ki} ≠ ∑_{k=0}^{l−1} σ_{kj} for i ≠ j, and

p_{i} : [a, ω[⟶]0, +∞ [(i = 1, …, m) are continuous functions with −∞ < a < ω ≤ +∞.

We call a solution of the equation (1) defined on some interval [t_{0}, +∞[⊂ [a, +∞[ a P_{ω}–solution if it satisfies the following three conditions:

a) y^{(n)}(t) ≠ 0;

b) for any k ∈ {0, …, n − 1}, either lim_{t↑ω} y^{(k)}(t) = 0 or lim_{t↑ω} y^{(k)}(t) = ±∞;

c) the limit

lim_{t↑ω} \frac{[ y^{(n−1)}(t) ]^{2}}{y^{(n)}(t) y^{(n−2)}(t)} = λ_{n−1}^{0}

exists and is finite or infinite.

Some asymptotic properties of solutions of the equations (1) satisfying the conditions a), b) can be obtained according to results from the monograph of I. T. Kiguradze and T. A. Chanturia [1] and in the scientific works of A. V. Kostin [2]. Exact asymptotic formulas for all types of P_{ω}– solutions are established only in the case m = 1 (see [3], [4]).

In this paper, the question of asymptotics of all P_{ω}–solutions of the equation (1) with λ_{n−1}^{0} ≠ 1 is considered.

We introduce the auxiliary notation:

M = {1, 2, …, m} , J = {(i, j) ∈ M × M : i ≠ j};

γ_{i} = 1 − ∑_{k=0}^{l−1} σ_{ki} , μ_{ri} = (n − r) + ∑_{k=0}^{l−1}(r − 1 − k)σ_{ki} ,

i = 1, …, m, r = 1, …, n;

a_{k} = (n − k)λ_{n−1}^{0} − (n − k − 1), k = 1, …, n − 1 ;

π_{ω}(t) = \{ \begin{matrix}t & for ω = +∞ \\ t − ω & for ω < +∞ \end{matrix} , I_{ri}(t) = ∫_{A_{ri}}^{t} p_{i}(τ)|π_{ω}(τ)|^{μ_{ri}}dτ,

1991 Mathematics Subject Classification. 34B05. Key words and phrases. High order differential equation of Emden–Fowler type,

asymptotic representations of proper and singular solutions.

A_{ri} = \{ \begin{matrix}a & for ∫_{a}^{ω} p_{i}(τ)|π_{ω}(τ)|^{μ_{ri}}dτ = +∞ \\ ω & for ∫_{a}^{ω} p_{i}(τ)|π_{ω}(τ)|^{μ_{ri}}dτ < +∞ \end{matrix} , i = 1, … m , r = 1, …, n.

Further, we investigate the case where for each fixed pair (i, j) ∈ J the following two conditions are fulfilled:

1) the functions

φ_{rs}(t) = p_{s}(t) [\frac{p_{i}(t)}{p_{j}(t)}]^{\frac{γs}{γj −γi}} |π_{ω}(t)|^{μ_{rs}+ \frac{(μri−μrj)γs}{γj −γi}} (s = 1, …, m)

are asymptotically comparable if t ↑ ω;

2) the limit lim_{t↑ω} \frac{h_{rij}(t)}{φ_{rg}(t)} = h_{g}^{0} exists and is finite or infinite for the function

h_{rij}(t) = \frac{1}{γ_{i} − γ_{j}} [\frac{p′_{i}(t)}{p_{i}(t)} − \frac{p′_{j}(t)}{p_{j}(t)} + \frac{μ_{ri} − μ_{rj}}{π_{ω}(t)} ] ,

where r ∈ \{1, …, n\} and φ_{rg} (g ∈ M) are such that lim_{t↑ω} \frac{φ_{ri}(t)}{φ_{rg}(t)} = Φ_{rig}^{0} = const for any

i ∈ M.

Theorem 1. Let the conditions 1), 2) be true for each fixed pair (i, j) ∈ J and for r = n. Then any positive P_{ω}–solution of the equation (1) with λ_{n−1}^{0} ∉ \{0, \frac{1}{2} , …, \frac{n−2}{n−1} , 1, ±∞\} assumes either the asymptotic representations

\begin{matrix}y^{(k−1)}(t) ∼ b_{0k}[π_{ω}(t)]^{n−k}y^{(n−1)}(t) , k = 1, …, n − 1, \\ y^{(n−1)}(t) ∼ c_{ij} [\frac{p_{i}(t)}{p_{j}(t)} |π_{ω}(t)|^{μ_{ni}−μ_{nj}}]^{\frac{1}{γi−γj}} , (i, j) ∈ J; \end{matrix}  (2)

or the representations

\begin{matrix}y^{(k−1)}(t) ∼ b_{0k}[π_{ω}(t)]^{n−k}y^{(n−1)}(t) , k = 1, …, n − 1, \\ y^{(n−1)}(t) ∼ α_{i}|γ_{i}d_{oi}I_{ni}(t)|^{\frac{1}{γi}} sign[γ_{i}I_{ni}(t)] , i ∈ M \end{matrix}  (3)

as t ↑ ω, where

c_{ij} ≠ 0, b_{0k} = \frac{[λ_{n−1}^{0} − 1]^{n−k}}{∏_{i=k}^{n−1} a_{i}} , d_{0i} = |λ_{n−1}^{0} − 1|^{μ_{ni}} ∏_{k=0}^{l−1} | ∏_{j=k+1}^{n−1} a_{j} |^{−σ_{ki}} .

Theorem 2. Let the conditions 1), 2) be true for each fixed pair (i, j) ∈ J and for r = n. Then any positive P_{ω}–solution of the equation (1) with λ_{n−1}^{0} = ±∞ assumes either the asymptotic representations (2) or the representations (3), in which the constants b_{0k}, d_{0i} are replaced by b_{k}^{0}, d_{i}^{0}, where

b_{k}^{0} = lim_{λ_{n−1}^{0}→∞} b_{0k} = \frac{1}{(n − k)!} , d_{i}^{0} = lim_{λ_{n−1}^{0}→∞} d_{0i} = ∏_{k=0}^{l−1}[(n − 1 − k)!]^{−σ_{ki}}.

Theorem 3. Let r ∈ {1, …, n − 1}, l ≤ r and the conditions 1), 2) be true for each fixed pair (i, j) ∈ J. Then any positive P_{ω}–solution of the equation (1) with

λ_{n−1}^{0} = \frac{n − r − 1}{n − r} assumes either the asymptotic representations

\begin{matrix}y^{(k−1)}(t) ∼ \frac{[π_{ω}(t)]^{r−k}}{(r − k)!} y^{(r−1)}(t) , k = 1, …, r − 1, \\ y^{(r−1)}(t) ∼ c_{ij} [\frac{p_{i}(t)}{p_{j}(t)} |π_{ω}(t)|^{μ_{ri}−μ_{rj}}]^{\frac{1}{γi−γj}} , (i, j) ∈ J, \\ y^{(r)}(t) = φ_{rg}(t)y^{(r−1)}(t) [h_{g}^{0} + 0(1)] , \\ y^{(k)}(t) ∼ (−1)^{k−r} \frac{(k − r)!}{[π_{ω}(t)]^{k−r}} y^{(r)}(t) , k = r + 1, …, n , \end{matrix}  (4)

or the representations

\begin{matrix}y^{(k−1)}(t) ∼ \frac{[π_{ω}(t)]^{r−k}}{(r − k)!} y^{(r−1)}(t) , k = 1, …, r − 1, \\ y^{(r−1)}(t) ∼ α_{i}γ_{i}|γ_{i}d_{ri}I_{ri}(t)|^{\frac{1}{γi}} sign I_{ri} , i ∈ M, \\ y^{(r)}(t) ∼ α_{i}d_{ri}p_{i}(t)|π_{ω}(t)|^{μ_{ri}}|y^{(r−1)}(t)|^{1−γ_{i}}, \\ y^{(k)}(t) ∼ (−1)^{k−r} \frac{(k − r)!}{[π_{ω}(t)]^{k−r}} y^{(r)}(t) , k = r + 1, …, n , \end{matrix}  (5)

as t ↑ ω, where

c_{ij} ≠ 0, d_{ri} = \frac{∏_{k=o}^{l−1}[(r − k − 1)!]^{−σ_{ki}}}{(n − r)!} .

In addition, we have obtained necessary and sufficient conditions for the existence of all types of P_{ω}–solutions of the equation (1) defined by Theorems 1 − 3.

We formulate Theorems 4 and 5 for the case where λ_{n−1}^{0} = \frac{n − r − 1}{n − r}.

Theorem 4. Let the hypotheses of Theorem 3 be true. Then for the existence of posi-

tive P_{ω}–solutions of the type (5) with λ_{n−1}^{0} = \frac{n − r − 1}{n − r}, it is necessary and sufficient

that

α_{i}γ_{i}I_{ri}(t)[π_{ω}(t)]^{r−1} > 0 for t ∈ (a, ω),

lim_{t↑ω} \frac{p_{s}(t)|π_{ω}(t)|^{μ_{rs}−μ_{ri}}}{p_{i}(t)} · \begin{matrix}[I_{rs}(t)] & \frac{1−γs}{γs} \\ [I_{ri}(t)] & \frac{1−γi}{γi} \end{matrix} = 0 for any s ≠ i,

lim_{t↑ω} \frac{π_{ω}(t) G′_{ri}(t)}{G_{ri}(t)} = r + 1 − n ,

where

G_{ri}(t) = ∫_{Q_{ri}}^{t} p_{i}(τ)|π_{ω}(τ)|^{μ_{ri}−n+r+1}|I_{ri}(τ)|^{\frac{1−γi}{γi}} dτ

and Q_{ri} are defined by analogy with A_{ri}.

Theorem 5. Let the hypotheses of Theorem 3 be true. Then for the existence of

positive P_{ω}– solutions of the equation (1) of the type (4) with λ_{n−1}^{0} = \frac{n − r − 1}{n − r}, it is

necessary that

∫_{a}^{ω} φ_{rg}(t) dt = ∞ , h_{g}^{0} = const, c_{ij}[π_{ω}(t)]^{r−1} > 0 ,  (6)

−h_{g}^{0}c_{ij} sign[−π_{ω}(t)]^{n−r} + ∑_{s=1}^{m} α_{s}d_{rs}Φ_{rsg}^{0}|c_{ij}|^{1−γ_{s}} = 0 ,  (7)

lim_{t↑ω} \frac{π_{ω}(t)I′_{rij}(t)}{I_{rij}(t)} = r + 1 − n ,  (8)

where

I_{rij}(t) = ∫_{A_{rij}}^{t} φ_{rg}(τ)|π_{ω}(τ)|^{r+1−n+ \frac{μri−μrj}{γi−γj}} [\frac{p_{i}(τ)}{p_{j}(τ)}]^{\frac{1}{γi−γj}} dτ

and A_{rij} are defined by analogy with A_{ri}.

Suppose that along with (6) − (8), the conditions

h_{g}^{0} ≠ 0 , ∑_{k=1}^{m} α_{k}d_{rk}Φ_{rkg}^{0}|c_{ij}|^{−γ_{k}} [σ_{0k} + ⋯ + σ_{l−1k} − 1] ≠ 0

are fulfilled. Then the equation (1) has positive P_{ω}-solutions of the type (4).

References

1. I. T. Kiguradze and T. A. Chanturia, Asymptotic properties of solutions of nonautonomous ordinary differential equations. (Russian) Nauka, Moscow, 1990.

2. A. V. Kostin, Asymptotics of proper solutions of nonlinear ordinary differential equations. (Russian) Differentsial’nye Uravneniya 23 (1987), No. 3, 524–526.

3. V. M. Evtukhov, Asymptotic representation of monotonic solutions of a nonlinear n–th order differential equation of Emden–Fowler type. (Russian) Dokl. Akad. Nauk. 324 (1992), No. 2, 258–260.

4. V. M. Evtukhov, On a class of monotone solutions of n–th order non–linear differential equation of Emden–Fowler type. (Russian) Soobshch. Akad. Nauk Gruzii 145 (1992), No. 2, 269–273.

Authors’ address:

Faculty of Mechanics and Mathematics

I. Mechnikov Odessa State University

2, Petra Velikogo St., Odessa 270057

Ukraine