GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 2, 1996, 101-120

ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF A SECOND ORDER NONLINEAR DIFFERENTIAL EQUATION

V. M. EVTUKHOV AND N. G. DRIK

Abstract. Asymptotic properties of proper solutions of a certain class of essentially nonlinear binomial differential equations of the second order are investigated.

Introduction

Let us consider a nonlinear differential equation of the second order

y′′ = α_{0}p(t) exp(σy)|y′|^{λ},  (0.1)

where α_{0} ∈ {−1; 1}; σ, λ ∈ ℝ, σ ≠ 0, λ ≠ 1, λ ≠ 2; p : [a, ω[→]0, +∞[ (−∞ < a < ω ≤ +∞) is a continuously differentiable function. Opposite to the well-studied Emden–Fowler equation of the type

y′′ = α_{0}p(t)|y|^{σ}|y′|^{λ} sign y,  (0.2)

the above binomial equation has nonlinearity of another type. The main results about the behavior of the solutions of (0.2) when λ = 0 are given in the monograph [1]. Asymptotic behavior of monotonic solutions of (0.2) when λ ≠ 0 is investigated in [2]–[6].

Equation of type (0.1) as well as of (0.2) are derived while describing different physical processes. In particular, the equation \frac{1}{r} \frac{d}{dr} (r \frac{dφ}{dr}) = A exp(νφ) + B exp(−νφ) from electrodynamics and the equation u′′ = u exp(αx − u)/2 from combustion theory reduce to the equation of type (0.1) with the help of some transformations [7].

In this work asymptotic representations of all proper solutions of (0.1) and their first derivatives are obtained when certain conditions on the function p are satisfied.

101

1991 Mathematics Subject Classification. 34E05.

Key words and phrases. Second-order nonlinear differential equation, proper solution, asymptotic representation.

1072-947X/96/0300-0101$09.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1996 Plenum Publishing Corporation

§ 1. Formulation of Basic Results

A real solution y of equation (0.1) is said to be proper if it is defined in the left neighborhood of ω, and for certain t_{0} from this neighborhood y′(t) ≠ 0 for t ∈ [t_{0}; ω[.

Let us introduce the auxiliary notation

Γ(t) = \frac{α_{0}σ}{λ − 2}[\frac{1}{2 − λ}p^{\frac{λ−3}{2−λ}} (t)p′(t) ∫_{γ_{0}}^{t} p^{\frac{1}{2−λ}} (s)ds − 1],

V (t) = |\frac{σ}{λ − 2} ∫_{γ_{0}}^{t} p^{\frac{1}{2−λ}} (s)ds|^{\frac{λ−2}{σ}};

γ_{0} = \begin{cases}a, if ∫_{a}^{ω} p^{\frac{1}{2−λ}} (s)ds = +∞ \\ ω if ∫_{a}^{ω} p^{\frac{1}{2−λ}} (s)ds < +∞ \end{cases} ; β_{0} = \begin{cases}−1, if lim_{t↑ω} V (t) = 0 \\ 1, if lim_{t↑ω} V (t) = +∞ \end{cases} .

When the conditions

lim_{t↑ω} Γ(t) = Γ_{0}, 0 < |Γ_{0}| < +∞  (1.1)

are fulfilled, the following statements hold.

Theorem 1.1. Let ω ≤ +∞. If Γ_{0} < 0, then each proper solution y of equation (0.1) admits one of the representations

y(t) = c + o(1), t ↑ ω for ω < +∞,  (1.2_{1})
y(t) = c_{1}t + o(1), t → +∞ for ω = +∞,  (1.2_{2})

where c ∈ ℝ, c_{1}σ ≤ 0.

If Γ_{0} > 0 and α_{0}σ > 0, then each proper solution y of (0.1) admits one of the representations (1.2_{i}) (i ∈ {1; 2}) or

y(t) = ln[|Γ_{0}|^{\frac{1}{σ}} V (t)] + o(1), t ↑ ω.  (1.3)

If Γ_{0} > 0 and α_{0}σ < 0, then each proper solution y of (0.1) either admits one of the representations (1.2_{i}) (i ∈ {1; 2}), (1.3) or there exists a sequence \{t_{k}\} ↑ ω, k → ∞, such that y(t_{k}) = \frac{1}{σ} ln[V^{σ}(t_{k})Γ(t_{k})], k = 1, 2, . . ..

Theorem 1.2. Let ω < ∞. The derivative of each proper solution y of the type (1.2_{1}) of equation (0.1) satisfies one of the asymptotic representations

y′(t) = c_{0} + o(1), t ↑ ω,  (1.4)

or

y′(t) = |(1 − λ) exp(σc) ∫_{ρ}^{t} p(s)ds|^{\frac{1}{1−λ}} [ν + o(1)], t ↑ ω,  (1.5)

where ρ = ω, ν = − sign(1 − λ) if ∫_{a}^{ω} p(t)dt < iy, and ρ = a, ν = α_{0} sign(1 − λ) otherwise.

For a proper solution y of equation (0.1) admitting one of the representations (1.2_{1}), (1.4) ((1.2_{1}), (1.5)) to exist, it is necessary and sufficient that

∫_{a}^{ω} p(t)dt < ∞ (σβ_{0}(1 − λ) > 0).

Theorem 1.3. Let ω = +∞. For a proper solution of equation (0.1), y of the type (1.2_{2}), where c_{1} = 0, to exist, it is necessary and sufficient that σβ_{0}(1 − λ) > 0. The derivative of each of such solutions satisfies (1.5).

Theorem 1.4. Let ω = +∞. For arbitrary c_{1} satisfying the inequality σc_{1} < 0 and c ∈ ℝ, equation (0.1) possesses a proper solution y admitting representation (1.2_{2}). The derivative of each of such solutions is represented in the form

y′(t) = c_{1} + o(1), t → +∞.

Theorem 1.5. Let ω ≤ +∞. For a proper solution y of equation (0.1) of the type (1.3) to exist, it is necessary and sufficient that Γ_{0} > 0. The derivative of each of such solutions satisfies the relation

y′(t) = \frac{V′(t)}{V (t)}[1 + o(1)], t ↑ ω.

§ 2. Some Auxiliary Statements

Let us consider the system of differential equations

\begin{cases}u′_{1} = f_{1}(τ) + a_{11}(τ)u_{1} + a_{12}(τ)u_{2} + g_{1}(τ)X_{1}(τ, u_{1}, u_{2}) \\ u′_{2} = f_{2}(τ) + a_{21}(τ)u_{1} + a_{22}(τ)u_{2} + g_{2}(τ)X_{2}(τ, u_{1}, u_{2}) \end{cases} , (2.1)

where the functions f_{1}, g_{1} : [T, +∞[→ ℝ (i = 1, 2), a_{ij} : [T, +∞[→ ℝ (i, j = 1, 2) are continuous and the functions X_{i} : Ω → ℝ (i = 1, 2) are continuous in r, u_{1}, u_{2} in the domain

Ω = [T, +∞[×D, D = {(u_{1}, u_{2}) : |u_{1}| ≤ δ, |u_{2}| ≤ δ, δ > 0}. (2.2)

Introduce the following notation: a_{i}(τ, t) = exp ∫_{t}^{τ} a_{ii}(s)ds (i = 1, 2);

A_{2}(τ) = | ∫_{α_{2}}^{τ} |a_{21}(t)|a_{2}(τ, t)dt|; A_{1}(τ) = | ∫_{α_{1}}^{τ} |a_{12}(t)|A_{2}(t)a_{1}(τ, t)dt|;

F_{2}(τ) = | ∫_{β_{2}}^{τ} |f_{2}(t)|a_{2}(τ, t)dt|; G_{2}(τ) = | ∫_{γ_{2}}^{τ} |g_{2}(t)|a_{2}(τ, t)dt|;

F_{1}(τ) = | ∫_{β_{1}}^{τ} |f_{1}(t)|a_{1}(τ, t)dt| + | ∫_{β_{12}}^{τ} |a_{12}(t)|F_{2}(t)a_{1}(τ, t)dt|;

G_{1}(τ) = | ∫_{γ_{1}}^{τ} |g_{1}(t)|a_{1}(τ, t)dt| + | ∫_{γ_{12}}^{τ} |a_{12}(t)|G_{2}(t)a_{1}(τ, t)dt|,

where each of the limits of integration α_{i}, β_{i}, γ_{i} (i = 1, 2), β_{12}, γ_{12} is equal either to T or to +∞ and is chosen in a special way: in every integral defining the functions F_{i}, A_{i}, G_{i} (i = 1, 2) and having the form

I(μ, τ) = ∫_{μ}^{τ} |b(t)| exp ∫_{t}^{τ} a(s) ds dt,  (2.3)

we put μ = +∞ if the integral I(T, +∞) converges, and μ = T otherwise.

Theorem 2.1. Let the functions X_{i} (i = 1, 2) have bounded partial derivatives with respect to the variables u_{1}, u_{2} in the domain Ω and let X_{i}(τ, 0, 0) ≡ 0 (i = 1, 2) for τ ∈ [T; +∞[. If

lim_{τ→+∞} F_{i}(τ) = lim_{τ→+∞} G_{i}(τ) = 0, lim_{τ→+∞} A_{i}(τ) = A_{i}^{o} < 1 (i = 1, 2),

then (2.1) possesses at least one real solution (u_{1}(τ), u_{2}(τ)) tending to zero as τ → +∞.

Theorem 2.2. Let X_{i}(τ, 0, 0) ≡ 0 (i = 1, 2) for τ ≥ T, and let the functions \frac{∂X_{i}(τ,u_{1},u_{2})}{∂u_{k}} (i, k = 1, 2) tend to zero as |u_{1}| + |u_{2}| → 0 uniformly with respect to τ ∈ [T, +∞[. If

lim_{τ→+∞} F_{i}(τ) = 0, lim_{τ→+∞} A_{i}(τ) = A_{i}^{o} < 1, lim_{τ→+∞} G_{i}(τ) = const (i = 1, 2),

then (2.1) possesses at least one real solution (u_{1}(τ), u_{2}(τ)) tending to zero as τ → +∞.

Theorems 2.1 and 2.2 immediately follow from the results of Kostin’s work [8].

We will use also the following statements dealing with limit properties of integrals of the type (2.3) ([2], [8]).

Lemma 2.1. Let a function a : [T, +∞[→ ℝ be continuous and represented in the form a(t) = a_{0}(t) + α(t), where a_{0} : [T, +∞[→ ℝ is a continuous function of constant sign (in particular, it can be a_{0}(t) ≡ 0) in a certain neighborhood of +∞; α : [T, +∞[→ ℝ is such that ∫_{T}^{+∞} α(t)dt converges. If b : [T, +∞[→ ℝ is continuous and ∫_{T}^{+∞} |b(t)|dt < ∞, then lim_{τ→+∞} I(μ, τ) = 0, where μ is chosen as stated above.

Lemma 2.2. Let the function a satisfy the conditions of Lemma 2.1. If | ∫_{T}^{+∞} a_{0}(t)dt| = ∞ and the function b : [T; +∞[→ ℝ is continuous and satisfies the asymptotic correlation |b(t)| = a_{0}(t)[q + o(1)], t → +∞ with q ∈ ℝ, then lim_{τ→+∞} I(τ, μ) = 0, where μ is chosen as stated above.

§ 3. Investigation of an Auxiliary Equation

Let us consider a second-order nonlinear differential equation

(\frac{ξ′(τ)}{ξ(τ)} + β_{0})′ + β_{0}S(τ)(\frac{ξ′(τ)}{ξ(τ)} + β_{0}) = α_{0}ξ^{σ}(τ)|\frac{ξ′(τ)}{ξ(τ)} + β_{0}|^{λ}, (3.1)

where α_{0}, β_{0} ∈ {−1, 1}; λ, σ ∈ ℝ, σ ≠ 0, λ ≠ 1, λ ≠ 2, and the function S : [b, +∞[→ ℝ is continuous and satisfies

lim_{τ→+∞} S(τ) = S_{0}, 0 < |S_{0}| < ∞.  (3.2)

A real solution ξ of equation (3.1) will be said to be proper if it is defined in a certain neighborhood of +∞, and for some τ_{0} from this neighborhood it satisfies the inequalities ξ(τ) > 0, ξ′(τ) + β_{0}ξ(τ) ≠ 0 for τ ≥ τ_{0}.

Theorem 3.1. Each proper solution ξ of equation (3.1) either has no limit as τ → +∞, and then there exists a sequence {τ_{k}}_{k=1}^{∞} converging to +∞ with ξ^{σ}(τ_{k}) = α_{0}S(τ_{k}), k = 1, 2, … or it possesses one of the properties

lim_{τ→+∞} ξ(τ) = ξ_{0}, 0 < ξ_{0} < +∞;  (3.3)
lim_{τ→+∞} ξ′(τ)/ξ(τ) = −β_{0};  (3.4)
lim_{τ→+∞} ξ′(τ)/ξ(τ) = ±∞.  (3.5)

Proof. Assume that a proper solution ξ of equation (3.1) has no limit as τ → +∞. Then there exists a sequence {s_{k}}_{k=1}^{∞} of extremum points of this solution converging to +∞. Taking into account that ξ′(s_{k}) = 0, k = 1, 2, …, equation (3.1) implies

ξ′′(s_{k}) = ξ(s_{k})[ξ^{σ}(s_{k}) − β_{0}S(s_{k})], k = 1, 2, ….  (3.6)

Owing to the continuity of the functions S(τ) and ξ^{σ}(τ), if their graphs have no common points, then the right-hand side of equality (3.6) has the same sign when k = 1, 2, . . .. But this is impossible because it means that the solution ξ has only maximums or only minimums.

Let now ξ be a proper solution of (3.1), and let lim_{τ→+∞} ξ(τ) (finite or infinite) exist.To prove the theorem it suffices to show that if this limit is equal to zero or +∞, then the solution ξ has one of the properties (3.4) and (3.5). Assume that

lim_{τ→+∞} ξ^{σ}(τ) = 0  (3.7)

and consider the function U_{c}(τ) = −β_{0}cS(τ) + α_{0}|c|^{λ}ξ^{σ}(τ) with c ≠ 0. According to (3.2) and (3.7), the function U_{c} retains the sign in a certain interval [τ_{c}, +∞[⊂ [τ_{0}, +∞[, i.e.,

U_{c}(τ) > 0 or U_{c}(τ) < 0 when τ ≥ τ_{c}.  (3.8)

If the function u(τ) = β_{0} + ξ′(τ)/ξ(τ) has no limit as τ → +∞, then there exists a constant c ≠ 0 such that for any T ≥ τ_{c} there is T_{1} ≥ T such that u(T_{1}) = c. In view of (3.1) this contradicts (3.8). It means that lim_{τ→+∞} u(τ) (finite or infinite) exists. Suppose now that

lim_{τ→+∞} u(τ) = u_{0}.  (3.9)

Then taking into account (3.2) and (3.7), it follows from (3.1) that lim_{τ→+∞} u′(τ) = −β_{0}S_{0} ≠ 0, but this contradicts (3.9). Hence each proper solution ξ of (3.1) satisfying (3.7) possesses one of the properties (3.4) and (3.5).

In the case where the solution ξ instead of (3.7) satisfies the condition lim_{τ→+∞} ξ^{σ}(τ) = ∞, the proof of the theorem is analogous.

Corollary 3.1. If one of the inequalities α_{0}S_{0} < 0 or α_{0}σ > 0 is fulfilled, then each proper solution ξ of (3.1) possesses one of the properties (3.3)– (3.5).

Proof. If α_{0}S_{0} < 0, then the validity of the statement is obvious. Let α_{0}σ > 0, and let ξ be a proper solution of (3.1) for which the limit does not exist as τ → +∞. Then, according to Theorem 3.1, there exists a sequence {τ_{k}}_{k=1}^{∞} tending to +∞ as k → +∞ such that ξ^{σ}(τ_{k}) = α_{0}S(τ_{k}), k = 1, 2, . . .. Because of (3.2) it is easy to see that there will be at least one point of the local maximum m_{1} or of the local minimum m_{2} of the function ξ^{σ}(τ) at which the inequality ξ^{σ}(m_{1}) > α_{0}S(m_{1}) or ξ^{σ}(m_{2}) < α_{0}S(m_{2}) is respectively fulfilled. From (3.1) we have

[ξ^{σ}(τ)]′′|_{τ=m_{i}} = α_{0}σξ^{σ}(m_{i})[ξ^{σ}(m_{i}) − α_{0}S(m_{i})], i ∈ {1, 2}. (3.10)

Because α_{0}σ > 0, it follows from (3.10) that [ξ^{σ}(τ)]′′|_{τ=m_{1}} > 0 or [ξ^{σ}(τ)]′′|_{τ=m_{2}} < 0. The obtained contradiction completes the proof of the Corollary.

Thus, if a proper solution ξ of (3.1) is such that lim_{τ→+∞} ξ(τ) (finite or infinite) exists, then it possesses one of properties (3.3)–(3.5), and vice versa. Corollary 3.1 shows the conditions under which the limit exists for each proper solution ξ of (3.1). Using conditions (3.3)–(3.5), these solutions can be divided into three groups. Therefore further investigation will be performed for each group separately.

3.1. On Proper Solutions of Equation (3.1) Which Have Finite Different from Zero Limit as τ → +∞.

Theorem 3.2. For equation (3.1) to have a proper solution ξ with property (3.3), it is necessary and sufficient that

α_{0}S_{0} > 0 and ξ_{0} = |S_{0}|^{\frac{1}{σ}} .  (3.11)

Moreover, each of such solutions admits the representation

ξ′(τ) + β_{0}ξ(τ) = β_{0}ξ_{0} + o(1), τ → +∞.  (3.12)

Proof. Let ξ be a proper solution of (3.1) with property (3.3). Since for every fixed value c which is different from the solutions of the equation α_{0}|c|^{λ}ξ_{0}^{σ} − β_{0}cS_{0} = 0, the function U_{c}(τ) = −β_{0}cS(τ) + α_{0}|c|^{λ}ξ^{σ}(τ) (c ∈ ℝ) retains the sign in a certain interval [c, +∞[⊂ [τ_{0}, +∞[, arguing as in proof of Theorem 3.1, it is not difficult to show that lim_{τ→+∞} ξ′(τ)/ξ(τ) (finite or infinite) exists. Then, according to (3.3), lim_{τ→+∞} ξ′(τ) also exists and equals zero. Passing to the limit as τ → +∞ in (3.1) in which ξ is the solution in question, we obtain S_{0} = α_{0}ξ_{0}^{σ} which proves (3.11).

Finally, because lim_{τ→+∞} ξ′(τ)/ξ(τ) = 0, the equality (3.12) is true due to (3.3).

Assume now that (3.11) holds. We shall prove that the equation (3.1) has at least one solution ξ satisfying the conditions (3.3) and (3.12).

Applying to equation (3.1) the transformation

ξ(τ) = β_{0} + u_{1}(τ), ξ′(τ) + β_{0}ξ(τ) = ξ_{0}β_{0} + u_{1}(τ)h + u_{2}(τ), (3.13)

where h is a constant which will be defined later on, we obtain the system

\begin{cases}u′_{1} = (h − β_{0})u_{1} + u_{2} \\ u′_{2} = −ξ_{0}[S(τ) − S_{0}] + a_{21}(τ)u_{1} + a_{22}(τ)u_{2} + X(u_{1}, u_{2}) \end{cases} , (3.14)

in which

a_{21}(τ) = −h^{2} + hβ_{0}[2 − S(τ) + λS_{0}] + S_{0}(σ + 1 − λ) − 1,
a_{22}(τ) = β_{0} − h − β_{0}[S(τ) − λS_{0}],

X(u_{1}, u_{2}) = (ξ_{0}β_{0} + hu_{1} + u_{2})^{2}(ξ_{0} + u_{1})^{−1} − [ξ_{0} + (2β_{0}h − 1)u_{1} +

+2β_{0}u_{2}] + α_{0}(|ξ_{0}β_{0} + hu_{1} + u_{2}|^{λ}|ξ_{0} + u_{1}|^{σ+1−λ} −
−ξ_{0}^{σ}[ξ_{0} + (σ + 1 − λ + β_{0}hλ)u_{1} + β_{0}λu_{2}]).

Define D by [S_{0}(λ − 1)/2]^{2} and consider two cases: D ≥ 0 and D < 0. 1^{0}. Let D ≥ 0. In this case we choose a constant h so that h^{2} − hβ_{0}[2 +

S_{0}(λ − 1)] − S_{0}(σ + 1 − λ) + 1 = 0. Note that now

h − β_{0}[1 + S_{0}(λ − 1)] ≠ 0, h − β_{0} ≠ 0.  (3.15)

Consider the system (3.14) in the domain Ω (see (2.2), where T = b, 0 < δ < xi_{0}(|h| + 1)). Partial derivatives \frac{∂X(u_{1},u_{2})}{∂u_{i}} (i = 1, 2) tend to zero as |u_{1}| + |u_{2}| → 0 and X(0, 0) = 0. The functions A_{i}, F_{i}, G_{i} (i = 1, 2) defined for (3.14) in §2 are of the form

A_{2}(τ) = | ∫_{α_{2}}^{τ} a_{21}(t) exp ∫_{t}^{τ} a_{22}(s)ds dt|;

A_{1}(τ) = | ∫_{α_{1}}^{τ} A_{2}(t) exp ∫_{t}^{τ} (h − β_{0})ds dt|;

F_{2}(τ) = | ∫_{β_{2}}^{τ} ξ_{0}|S(t) − S_{0}| exp ∫_{t}^{τ} a_{22}(s)ds dt|;

F_{1}(τ) = | ∫_{β_{1}}^{τ} F_{2}(t) exp ∫_{t}^{τ} (h − β_{0})ds dt|;

G_{2}(τ)=|∫_{γ_{2}}^{τ} exp∫_{t}^{τ} a_{22}(s)ds dt|; G_{1}(τ)=|∫_{γ_{12}}^{τ} G_{2}(t)exp∫_{t}^{τ} (h − β_{0})ds dt|.

Using Lemma 2.2 and taking into account (3.2), (3.11), and (3.15), we can easily verify that lim_{τ→+∞} A_{i}(τ) = lim_{τ→+∞} F_{i}(τ) = 0 (i = 1, 2), lim_{τ→+∞} G_{2}(τ) = |h − β_{0}| lim_{τ→+∞} G_{1}(τ) = 1/|h − β_{0}[1 + S_{0}(1 − λ)]|.

Thus system (3.14) satisfies the conditions of Theorem 2.2; hence it has at least one real solution (u_{1}(τ), u_{2}(τ)) tending to zero as τ → +∞. Because of the transformation (3.13), this implies that there exists a solution ξ of (3.1) satisfying conditions (3.3), (3.12).

2^{0}. Let D < 0. We use the following notation: q = \sqrt{−D}, p = β_{0}S_{0}(λ − 1)/2,

M(τ)=(\begin{matrix}cos(qτ) & sin(qτ) \\ (p+β_{0}) cos(qτ)−q sin(qτ) & q cos(qτ)+(p+β_{0}) sin(qτ) \end{matrix}), (3.16)
(\begin{matrix}δ_{11}(τ) & δ_{12}(τ) \\ δ_{21}(τ) & δ_{22}(τ) \end{matrix}) = M^{−1}(τ) (\begin{matrix}0 & 0 \\ 0 & −β_{0}[S(τ) − S_{0}] \end{matrix}) .  (3.17)

Putting h = 0 in (3.14) and applying the transformation

(\begin{matrix}u_{1}(τ) \\ u_{2}(τ) \end{matrix}) = M(τ) (\begin{matrix}z_{1}(τ) \\ z_{2}(τ) \end{matrix}) ,  (3.18)

we obtain a system

\begin{cases}z′_{1} = f_{1}(τ) + [p + δ_{11}(τ)]z_{1} + δ_{12}(τ)z_{2} + X_{1}(τ, z_{1}, z_{2}) \\ z′_{2} = f_{2}(τ) + δ_{21}(τ)z_{1} + [p + δ_{22}(τ)]z_{2} + X_{2}(τ, z_{1}, z_{2}) \end{cases} , (3.19)

in which f_{1}(τ) = \frac{ξ_{0}}{q} [S(τ) − S_{0}] sin(qτ), f_{2}(τ) = −\frac{ξ_{0}}{q} [S(τ) − S_{0}] cos(qτ), X_{1}(τ, z_{1}, z_{2}) = −\frac{1}{q} sin(qτ)X(u_{1}, u_{2}), X_{2}(τ, z_{1}, z_{2}) = \frac{1}{q} cos(qτ)X(u_{1}, u_{2}).

Partial derivatives \frac{∂X_{i}(τ,z_{1},z_{2})}{∂z_{k}} (i, k = 1, 2) tend to zero as |z_{1}| + |z_{2}| → 0 uniformly with respect to τ ∈ [b, +∞[, and X_{i}(τ, 0, 0) ≡ 0 (i = 1, 2) on [b, +∞[.

Consider system (3.19) in the domain Ω (see (2.2), where T = b, 0 < δ < \frac{ξ_{0}}{2} min\{1, 1/(|p + β_{0}| + q)\}). The functions a_{i}, A_{i}, F_{i}, G_{i} (i = 1, 2) defined for the system (3.19) in §2 are of the form

a_{i}(τ, t) = exp ∫_{t}^{τ} [δ_{ii}(s) + p]ds (i = 1, 2);

A_{2}(τ) = | ∫_{α_{2}}^{τ} |δ_{21}(t)|a_{2}(τ, t)dt|; A_{1}(τ) = | ∫_{α_{1}}^{τ} |δ_{12}(t)|A_{2}(t)a_{1}(τ, t)dt|;

F_{2}(τ) = | ∫_{β_{2}}^{τ} |f_{2}(t)|a_{1}(τ, t)dt|; G_{2}(τ) = | ∫_{γ_{2}}^{τ} a_{2}(τ, t)dt|;

F_{1}(τ) = | ∫_{β_{1}}^{τ} |f_{1}(t)|a_{1}(τ, t)dt| + | ∫_{β_{12}}^{τ} |δ_{12}(t)|F_{2}(t)a_{1}(τ, t)dt|;

G_{1}(τ) = | ∫_{γ_{1}}^{τ} a_{1}(τ, t)dt| + | ∫_{γ_{12}}^{τ} |δ_{12}(t)|G_{2}(t)a_{1}(τ, t)dt|,

It follows from (3.2), (3.16), and (3.17) that lim_{τ→+∞} δ_{ik}(τ) = 0 (i, k = 1, 2), hence ∫_{b}^{+∞}[p + δ_{ii}(τ)]dτ = +∞ (i = 1, 2). Then using Lemma 2.2 it is easy to make sure that lim_{τ→+∞} A_{i}(τ) = lim_{τ→+∞} F_{i}(τ) = 0; lim_{τ→+∞} G_{i}(τ) = \frac{1}{|p|} (i = 1, 2).

Thus system (3.19) satisfies all the conditions of Theorem 2.2. Therefore it has at least one real solution (u_{1}(τ), u_{2}(τ)) tending to zero as τ → ∞. Because of transformations (3.13) and (3.18) this implies that there exists a solution ξ of (3.1) satisfying (3.3) and (3.12).

3.2. On the Proper Solutions of Equation (3.1) with the Property (3.4.). We use the following notation:

H(τ) = ∫_{b}^{τ} S(t)dt; θ(τ) exp(−δβ_{0}τ + (1 − λ)H(τ)).

Theorem 3.3. Each proper solution ξ of equation (3.1) with property (3.4) admits the asymptotic representation

ξ(τ) = c exp(−β_{0}τ)[1 + o(1)], τ → +∞,  (3.20)

where c > 0, and its derivative satisfies one of the equalities

\frac{ξ′(τ)}{ξ(τ)} + β_{0} = c_{0} exp(−H(τ))[1 + o(1)], τ → +∞,  (3.21)

or

\frac{ξ′(τ)}{ξ(τ)} + β_{0} = ν exp(−H(τ))|c^{σ}(1 − λ) ×

× ∫_{γ}^{τ} θ(t)dt|^{\frac{1}{1−λ}} [1 + o(1)], τ → +∞,  (3.22)

where ν =−α_{0} sign(1 − λ), γ =+∞ if ∫_{b}^{+∞}θ(t)dt<∞ and ν =−α_{0} sign(1−λ),

γ = b otherwise, c_{0} ≠ 0.

Equation (3.1) has a proper solution ξ with property (3.4) which satisfies both asymptotic equalities (3.20), (3.21) if and only if

β_{0}S_{0} > 0, ∫_{b}^{+∞}θ(τ)dτ < +∞,  (3.23)

and equalities (3.20), (3.22) if and only if

σβ_{0}(1 − λ) > 0.  (3.24)

Proof. Let ξ be a proper solution of (3.1) with property (3.4). Set

u(τ) = \frac{ξ′(τ)}{ξ(τ)} + β_{0}, φ(τ) = ∫_{τ_{0}}^{τ} u(t)dt, Φ(τ) = ∫_{r}^{τ} θ(t) exp φ(t)dt,

where r = +∞ if ∫_{τ_{0}}^{+∞} θ(t) exp φ(t)dt converges, and r = τ_{0} otherwise. Then

lim_{τ→+∞} u(τ) = 0  (3.25)
ξ(τ) = ξ_{0} exp(−β_{0}τ + φ(τ)),  (3.26)

where ξ_{0} is a certain constant. Substituting (3.26) into the right-hand side of (3.1), we find

|u(τ)|^{1−λ} = exp(−(1 − λ)H(τ))[c_{1} + α_{0}ξ_{0}^{σ}(1 − λ)ν_{0}Φ(τ)], (3.27)

where ν_{0} = sign u(τ), c_{1} ∈ ℝ. It is clear from (3.27) that either

u(τ) = c_{0} exp(−H(τ))[1 + o(1)], τ → +∞,  (3.28)

where c_{0} ≠ 0 or

u(τ) ∼ ν exp(−H(τ))|ξ_{0}^{σ}(1 − λ)Φ(τ)|^{\frac{1}{1−λ}} , τ → +∞.  (3.29)

Moreover, (3.28) happens to be the case only if r = τ_{0}.

Assume that the solution of (3.1) in question satisfies (3.28). This does not contradict (3.25) only if β_{0}S_{0} > 0. It is easy to see that if this inequality holds, then ∫_{τ_{0}}^{+∞} θ(τ)dτ < ∞, and the solution ξ satisfies asymptotic equalities (3.20), (3.21) by (3.26), (3.28).

Suppose now that the solution in question satisfies (3.29). According to (3.25), since for any ρ ∈ [0, ρ^{∗}[, where ρ^{∗} = min \{|\frac{σ}{1−λ} |, |S_{0}|\}, φ(τ) = o(τ), τ → +∞, we have

lim_{τ→+∞} \frac{Φ(τ)}{exp((1 − λ)[H(τ) − ρτ])} = \begin{cases}0 if σβ_{0} > 0 \\ ±∞ if σβ_{0} < 0 \end{cases} , (3.30)

which (for ρ = 0) implies that (3.29) does not contradict (3.25) only if σβ_{0}(1 − λ) > 0. Moreover, if this inequality holds, then

[Φ(τ)]^{\frac{1}{1−λ}} = o(exp(H(τ) − ρτ)), τ → +∞,

and therefore

∫_{τ_{0}}^{+∞}exp(−H(τ))|Φ(τ)|^{\frac{1}{1−λ}} dτ < ∞.  (3.31)

Next, (3.26), (3.29), and (3.31) imply that the solution ξ admits representation (3.20), where c > 0 is a certain constant. Substituting (3.20) into the right-hand side of (3.1) and integrating the obtained equation, it is not difficult to make sure that ξ satisfies (3.22).

Let conditions (3.23) be fulfilled, and let c > 0, c_{0} ≠ 0 be arbitrary fixed numbers. We shall prove that there exists at least one solution ξ of equation (3.1) satisfying representations (3.20), (3.21).

Using

\begin{matrix}ξ(τ) = c exp(−β_{0}τ)[1 + u_{1}(τ)], \\ \frac{ξ′(τ)}{ξ(τ)} + β_{0} = c_{0} exp(−H(τ))[1 + u_{2}(τ)], \end{matrix}  (3.32)

equation (3.1) is transformed into the differential system

\begin{cases}u′_{1} = c exp(−H(τ))[1 + u_{1} + u_{2} + X_{1}(u_{1}, u_{2})] \\ u′_{2} = mθ(τ)[1 + σu_{1} + λu_{2} + X_{2}(u_{1}, u_{2})] \end{cases}  (3.33)

where m = α_{0}c_{0}|c_{0}|^{λ−2}c^{σ}, X_{1}(u_{1}, u_{2}) = u_{1}u_{2}, X_{2}(u_{1}, u_{2}) = (1 + u_{1})^{σ} × |1 + u_{2}|^{λ} − 1 − σu_{1} − λu_{2}. Consider system (3.33) in the domain Ω (see (2.2), where T = b, 0 < δ < 1). The functions A_{i}, F_{i}, G_{i} (i = 1, 2) defined in §2 for system (3.33) are of the form

A_{2}(τ) = |mσ ∫_{α_{2}}^{τ} θ(t) exp(λm ∫_{t}^{τ} θ(s)ds dt|;

A_{1}(τ) = |c_{0} ∫_{α_{1}}^{τ} A_{2}(t) exp ( − H(t) + c_{0} ∫_{t}^{τ} exp(−H(s))ds)dt|;

F_{1}(τ) = |c_{0} ∫_{β_{1}}^{τ} exp ( − H(t) + c_{0} ∫_{t}^{τ} exp(−H(s))ds)dt| + \frac{1}{|σ|}A_{1}(τ);

F_{2}(τ) = G_{2}(τ) = \frac{1}{|σ|}A_{2}(τ); G_{1}(τ) = F_{1}(τ).

It follows from Lemma 2.1 and (3.23) that lim_{τ→+∞} A_{i}(τ) = lim_{τ→+∞} F_{i}(τ) = lim_{τ→+∞} G_{i}(τ) = 0 (i = 1, 2). Furthermore, partial derivatives of X_{i} (i = 1, 2) with respect to u_{1}, u_{2} are bounded in the domain Ω, and X_{i}(0, 0) = 0 (i = 1, 2). Thus the system (3.33) satisfies the conditions of Theorem 2.1 and has at least one real solution (u_{1}(τ), u_{2}(τ)) tending to zero as τ → +∞ to which, due to the transformation (3.32), there corresponds a proper solution ξ of (3.1) satisfying asymptotic equalities (3.20), (3.21).

Let now inequality (3.24) hold, and let c > 0 be an arbitrary fixed number. We shall prove that equation (3.1) has at least one solution ξ satisfying representations (3.20), (3.22).

Applying the transformation

\begin{matrix}ξ(τ) = c exp(−β_{0}τ)[1 + u_{1}(τ)], \\ \frac{ξ′(τ)}{ξ(τ)} + β_{0} = N(τ)[1 + hu_{1}(τ) + u_{2}(τ)], \end{matrix}  (3.34)

where N(τ) = ν exp(−H(τ))|c^{σ}(1 − λ) ∫_{γ}^{τ} θ(t)dt|^{\frac{1}{1−λ}}, h = σ/(1 − λ), we get the system

\begin{cases}u′_{1} = N(τ)[1 + (h + 1)u_{1} + u_{2} + X_{1}(u_{1}, u_{2}) \\ u′_{2} = −hN(τ) − h(h + 1)N(τ)u_{1} − [hN(τ)+ \\ +(1 − λ)Q(τ)]u_{2} + Q(τ)X_{2}(τ, u_{1}, u_{2}), \end{cases}  (3.35)

where Q(τ) = θ(τ)[(1 − λ) ∫_{γ}^{τ} θ(t)dt]^{−1} (ν, γ are the same as in (3.22))

X_{1}(u_{1}, u_{2}) = hu_{1}^{2} + u_{1}u_{2}, X_{2}(τ, u_{1}, u_{2}) = |1 + hu_{1} + u_{2}|^{λ}(1 + u_{1})^{σ} − 1 − (hλ + σ)u_{1} − λu_{2} − hN(τ)Q^{−1}(τ)X_{1}(u_{1}, u_{2}).

Consider system (3.35) in the domain Ω (see (2.2), where T = b 0 < δ < 1/(|h| + 1)). The functions a_{i}, A_{i}, F_{i}, G_{i} (i = 1, 2) defined in §2 for system (3.35) are of the form

a_{2}(τ, t) = exp ∫_{t}^{τ} [−hN(s) − (1 − λ)Q(s)]ds;

a_{1}(τ, t) = exp ((h + 1) ∫_{t}^{τ} N(s)ds);

A_{2}(τ) = |h(h + 1) ∫_{α_{2}}^{τ} N(t)a_{2}(τ, t)dt|; F_{2}(τ) = \frac{1}{|h + 1|}A_{2}(τ);

G_{2}(τ) = | ∫_{γ_{2}}^{τ} Q(t)a_{2}(τ, t)dt|; A_{1}(τ) = | ∫_{α_{1}}^{τ} N(t)A_{2}(t)a_{1}(τ, t)dt|;

F_{1}(τ) = | ∫_{β_{1}}^{τ} N(t)[1 + F_{2}(t)]a_{1}(τ, t)dt|;

G_{1}(τ) = | ∫_{γ_{1}}^{τ} N(t)[1 + G_{2}(t)]a_{1}(τ, t)dt|.

Since (3.30) is fulfilled for any function φ(τ) = o(τ), τ → +∞, we have

N(τ) = o(exp(ρ_{0}τ)), τ → +∞  (3.36)

for arbitrary ρ_{0} ∈]0, ρ^{∗}[. Using L’Hospital’s rule it is easy to make sure that

lim_{τ→+∞} ^{\frac{∫_{b}^{τ} θ(t)dt}{θ(τ) exp(ρ_{0}τ)}} = 0. Therefore, taking into consideration (3.36), we have

lim_{τ→+∞} N(τ)Q^{−1}(τ) = 0.  (3.37)

It follows from Lemmas 2.1, 2.2 and (3.36), (3.37) that lim_{τ→+∞} A_{i}(τ) = lim_{τ→+∞} F_{i}(τ) = lim_{τ→+∞} G_{1}(τ) = 0 (i = 1, 2),

lim_{τ→+∞} G_{2}(τ) = \{0_{\frac{1}{|1−λ|}} if ∫_{b}^{+∞}θ(τ)dτ \begin{matrix}< +∞ \\ = +∞ \end{matrix} .

Partial derivatives \frac{∂X_{i}}{∂u_{k}} (i, k = 1, 2) tend to zero as |u_{1}|+|u_{2}| → 0 uniformly with respect to τ ∈ [T, +∞[. Furthermore, X_{2}(τ, 0, 0) ≡ 0 for τ ≥ T, X_{1}(0, 0) = 0.

Thus by Theorem 2.2 system (3.35) has at least one real solution (u_{1}(τ), u_{2}(τ)) tending to zero as τ → +∞ to which, due to transformation (3.34), there corresponds a proper solution ξ of (3.1) satisfying (3.20), (3.22).

3.3. On the Proper Solutions of Equation (3.1) with the Property (3.5). Below we shall use the following simple statement whose validity can be easily verified.

Lemma 3.1. Let f : [T, +∞[→ ℝ be a continuously differentiable function such that lim_{t→+∞} |f(t)| = +∞. If for some ε > 0 there exists lim_{t→+∞} f′(t)/|f(t)|^{1+ε}, then this limit equals zero.

Consider first the solutions of (3.1) for which

lim_{τ→+∞} ξ′(τ)/ξ(τ) = +∞.  (3.38)

Lemma 3.2. Let ξ be a proper solution of (3.1) with the property (3.33). Then

lim_{τ→+∞} ξ^{σ}(τ)u^{λ−2}(τ) = +∞ when σ > 0.  (3.39)

and

lim_{τ→+∞} ξ^{σ}(τ)u^{λ−1}(τ) = 0 when σ < 0.  (3.40)

Proof. Let ξ be a proper solution of (3.1) with property (3.38). Obviously,

lim_{τ→+∞} ξ(τ) = +∞.  (3.41)

First we shall show that for any ε > 0 the function z(τ) = u(τ)ξ^{−ε}(τ) has the limit as τ → +∞, and this limit equals zero.

Assume on the contrary that lim_{τ→+∞} z(τ) does not exist. Then there exists a constant \overline{c} different from zero and ε^{\frac{1}{λ−2}} , such that the graph of the function z = z(τ) intersects the straight line z = \overline{c} at τ = t_{k}, k = 1, 2, …, and the sequence {t_{k}}_{k=1}^{∞} tends to infinity. Since by (3.1), z′(τ) ≡ z(τ)β_{0}[ε− S(τ)] for τ ≥ t_{0}, this implies that due to (3.2) and (3.41) the values z′(t_{k}), k = N, N+1, … for some N are of the same sign, which is impossible. Hence lim_{τ→+∞} z(τ) exists, and because of the fact that z(τ) ∼ ξ′(τ)/ξ^{1+ε}(τ) as τ → +∞, (3.41), and Lemma 3.1, we have

lim_{τ→+∞} z(τ) = 0.  (3.42)

By virtue of (3.38) and (3.41) the validity of (3.39) and (3.40) is obvious if λ > 2 and λ < 1, respectively.

Let σ > 0 and λ < 2. Choosing ε such that σ + (λ − 2)ε > 0 and taking into account (3.38), (4.42), we obtain

lim_{τ→+∞} ξ^{σ}(τ)u^{λ−2}(τ) = lim_{τ→+∞} ξ^{σ+(λ−r)ε}(τ)z^{λ−2}(τ) = +∞,

i.e., (3.39) holds when λ < 2.

If σ < 0 and λ > 1 we choose ε so that σ + (λ − 1)ε < 0. Then because of (3.38), (3.42) we have

lim_{τ→+∞} ξ^{σ}(τ)u^{λ−1}(τ) = lim_{τ→+∞} ξ^{σ+(λ−1)ε}(τ)z^{λ−1}(τ) = 0,

i.e., (3.40) holds when λ > 1. Thus Lemma 3.2 is proved.

Theorem 3.4. Equation (3.1) has solutions with the property (3.38) if and only if

σ < 0, β_{0}S_{0} < 0.  (3.43)

Furthermore, each of such solutions admits asymptotic representations

ξ(τ) = c exp ( − β_{0}τ + c_{0} ∫_{b}^{τ} exp(−H(t))dt)[1 + o(1)], τ → +∞, (3.44)

\frac{ξ′(τ)}{ξ(τ)} + β_{0} = c_{0} exp(−H(τ))[1 + o(exp(−kτ))], τ → +∞, (3.45)

where c > 0, c_{0} > 0, k > 0.

Proof. Let ξ be a proper solution of (3.1) with property (3.38) and u(τ) = β_{0}+ξ′(τ)/ξ(τ). When σ > 0, it follows from (3.1), (3.2), (3.38), and Lemma 3.2 that

lim_{τ→+∞} \frac{α_{0}u′(τ)}{u^{2}(τ)} = lim_{τ→+∞} [ − α_{0}β_{0}S(τ)u^{−1}(τ) + u^{λ−2}(τ)ξ^{σ}(τ)] = +∞,

which contradicts Lemma 3.1.

When σ < 0 and β_{0}S_{0} > 0, it follows from (3.1), (3.2) and Lemma 3.2 that lim_{τ→+∞} \frac{u′(τ)}{u(τ)} = −β_{0}S_{0} < 0, which contradicts (3.38).

Thus equation (3.1) can have a proper solution with property (3.38) provided only that (3.43) holds. Let inequalities (3.43) be fulfilled, and let ξ be such a solution. Put ε(τ) = β_{0}S(τ) + u′(τ)/u(τ), ψ(τ) = ∫_{τ_{0}}^{τ} ε(t)dt. Then u(τ) = c_{1} exp(−H(τ) + ψ(τ)),

ξ(τ) = ξ_{0} exp ( − β_{0}τ + c_{1} ∫_{τ_{0}}^{τ} exp(−H(t) + ψ(t))dt),  (3.46)

where ε_{0} > 0, c_{1} > 0. It follows from (3.1), (3.2), (3.43), and Lemma 3.2 that

lim_{τ→+∞} ε(τ) = 0.  (3.47)

Substituting (3.46) into the right-hand side of (3.1) and taking into account (3.38), (3.43), and (3.47), we find that

u(τ) = exp(−H(τ))[\overline{c}_{1} + (1 − λ)α_{0}ξ_{0}^{σ} ×

× ∫_{∞}^{τ} θ(t) exp (σc_{1} ∫_{τ_{0}}^{t} exp(−H(s) + ψ(s))ds)dt]^{\frac{1}{1−λ}} ,  (3.48)

where \overline{c}_{1} ≥ 0. Because of (3.42), (3.46)–(3.48) by using L’Hospital’s rule, it is not difficult to verify that if \overline{c}_{1} = 0, then lim_{τ→+∞} u(τ) = 0 when λ < 1, and

lim_{τ→+∞} ξ^{σ}(τ)u^{λ−1}(τ) = +∞ when λ > 1,

which contradicts (3.38) and (3.39), respectively. Consequently, \overline{c}_{1} > 0. Note that owing to (3.43) and (3.47),

∫_{τ}^{+∞}θ(t)exp(σc_{1}∫_{τ_{0}}^{t}exp(−H(s)+ψ(s))ds)dt=o(exp(−kτ)), τ →+∞, (3.49)

for any k > 0. Therefore, representation (3.48) can be expressed in the form (3.45) which implies that ξ satisfies (3.44) with certain constants c > 0, c_{0} > 0, k > 0.

Next we shall prove that conditions (3.43) are sufficient for (3.1) to have a proper solution ξ satisfying (3.44), (3.45).

Let c, c_{0}, k be arbitrary fixed numbers satisfying inequalities c>0, c_{0} >0,

k > −β_{0}S_{0}.  (3.50)

Applying to (3.1) the transformation

ξ(τ) = c exp ( − β_{0}τ + c_{0} ∫_{b}^{τ} exp(−H(t))dt)[1 + u_{1}(τ)],

\frac{ξ′(τ)}{ξ(τ)} + β_{0} = c_{0} exp(−H(τ))[1 + exp(−kτ)u_{2}(τ)],

(3.51)

we obtain the system

\begin{cases}u′_{1} = c_{0} exp(−H(τ) − kτ)[u_{2} + X_{1}(u_{1}, u_{2})] \\ u′_{2} = ku_{2} + α_{0}c_{0}^{λ−1}c^{σ}θ(τ)× \\ × exp (σc_{1} ∫_{b}^{τ} exp(−H(t))dt + kτ)[1 + X_{2}(τ, u_{1}, u_{2})], \end{cases} (3.52)

where X_{1}(u_{1}, u_{2}) = u_{1}u_{2}, X_{2}(τ, u_{1}, u_{2}) = (1 + u_{1})^{σ}|1 + exp(−kτ)u_{2}|^{λ} − 1.

Consider system (3.52) in the domain Ω (see (2.2), where T = b, 0 < δ < min{1, exp(kT)}). Partial derivatives of X_{i} (i = 1, 2) with respect to u_{i}, u_{2} are bounded in the domain Ω, and X_{1}(0, 0) = 0, X_{2}(τ, 0, 0) ≡ 0 for τ ≥ T. The functions A_{i}, F_{i}, G_{i} (i = 1, 2) defined in §2 for system (3.52) are of the form

A_{2}(τ) ≡ A_{1}(τ) ≡ 0; F_{2}(τ) = c_{0}^{λ−1}c^{σ} exp(kτ)| ∫_{β_{2}}^{τ} θ(τ) ×

× exp (σc_{0} ∫_{b}^{t} exp(−H(s))ds)dt|;

F_{1}(τ) = c_{0}| ∫_{β_{12}}^{τ} exp(−H(t) − kt)F_{2}(t)dt|; G_{2}(τ) ≡ F_{2}(τ);

G_{1}(τ) = c_{0}| ∫_{γ_{1}}^{τ} exp(−H(t) − kt)dt| + F_{1}(τ).

It is easily seen that asymptotic equality (3.49) under the conditions (3.43) remains true if we set φ(τ) ≡ 0. It follows that lim_{τ→+∞} F_{2}(τ) = lim_{τ→+∞} G_{2}(τ) = 0. This implies lim_{τ→+∞} F_{1}(τ) = lim_{τ→+∞} G_{1}(τ) = 0 due to (3.51). Thus, by Theorem 2.1 system (3.52) has at least one real solution (u_{1}(τ), u_{2}(τ)) tending to zero as τ → +∞. Taking into account transformation (3.51), we complete the proof of the theorem.

Consider now the solutions of (3.1) satisfying

lim_{τ→+∞} ξ′(τ)/ξ(τ) = −∞.  (3.53)

We make the substitution 1/ξ(τ) = μ(τ) to obtain the equation

(\frac{μ′(τ)}{μ(τ)} − β_{0})′ + β_{0}S(τ)(\frac{μ′(τ)}{μ(τ)} − β_{0}) = −α_{0}μ^{−σ}(τ)|\frac{μ′(τ)}{μ(τ)} − β_{0}|^{λ}. (3.54)

Clearly, a proper solution ξ of (3.1) with property (3.53) corresponds to the solution μ of (3.54) with the property lim_{τ→+∞} μ′(τ)/μ(τ) = +∞, and vice versa. Since equations (3.1) and (3.54) are of the same form, using the above arguments it is not difficult to see that the following statement is true.

Theorem 3.5. Equation (3.10) has solutions with the property (3.53) if and only if σ > 0, β_{0}S_{0} < 0. Furthermore, each of such solutions ξ admits asymptotic representations

ξ(τ) = c exp ( − β_{0}τ − c_{0} ∫_{b}^{τ} exp(−H(t))dt)[1 + o(1)], τ → +∞,

\frac{ξ′(τ)}{ξ(τ)} + β_{0} = −c_{0} exp(−H(τ))[1 + (exp(−kτ))], τ → +∞,

where c > 0, c_{0} > 0, k > 0.

§ 4. Proofs of Theorems 1.1–1.5

Applying to (0.1) the transformation

y(t) = ln[V (t)ξ(τ)], τ = β_{0} ln V (t),  (4.1)

we get τ′(t) > 0 for t ∈ [a_{1}, ω[ (a_{1} is a certain number in the interval ]a, ω[), and lim_{t↑ω} τ(t) = +∞. The transformation (4.1) yields equation (3.1) in which S(τ) = S(τ(t)) = α_{0}Γ(t), b = β_{0} ln V (a). Moreover, proper solution ξ of (3.1) corresponds to each proper solution y of (0.1), and vice versa. Taking into account transformation (4.1) and the notation introduced in §§1 and 3, it is easy to see that

H(τ) = H(τ(t)) = ln |\frac{V′(t)}{V (t)}| + ln |\frac{V (a)}{V′(a)}|,

∫_{γ}^{τ(t)}θ(s)ds = |\frac{V′(a)}{V (a)}|^{1−λ} ∫_{ρ}^{t} p(s)ds,

where ρ = ω if ∫_{a}^{ω} p(t)dt < +∞, and ρ = a otherwise,

∫_{b}^{+∞}exp ( − β_{0} ∫_{b}^{τ} S(t)dt)dτ = |\frac{V′(a)}{V (a)}| ∫_{a}^{ω} dt.  (4.2)

Because of (1.1), the function S satisfies the condition (3.2), and S_{0} = α_{0}Γ_{0}. Therefore it follows from (4.2) that if either α_{0}β_{0}Γ_{0} > 0 or α_{0}β_{0}Γ_{0} < 0, then ω < +∞ or ω = +∞, respectively.

Taking into consideration the above arguments, it is easily seen that Theorems 3.1–3.6 result in Theorems 1.1–1.5.

Remark 1. The results dealing with the asymptotic behavior of proper solutions of (0.1) in the case λ = 1 may be found in [9].

In the case λ = 2, Theorems 1.1–1.5 in which

Γ_{0} = −α_{0}σ lim_{t↑ω} \frac{p(t)p′′(t)}{[p′(t)]^{2}}, V (t) = p^{− \frac{1}{σ}} (t)

are valid under an additional assumption that p is twice continuously differentiable function satisfying one of the conditions lim_{t↑ω} p(t) = 0 or lim_{t↑ω} p(t) = +∞.

Remark 2. Paper [10] contains results on the asymptotic properties of proper solutions of (0.1) in the case where Γ_{0} = ±∞.

References

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3. V. M. Evtukhov, On a second-order nonlinear differential equation. (Russian) Dokl. Akad. Nauk SSSR 233(1977), No. 4, 531–534.

4. V. M. Evtukhov, Asymptotic properties of solutions of a certain class of second-order differential equations. (Russian) Math. Nachr. 115(1984), 215–236.

5. V. M. Evtukhov, Asymptotic representations of solutions of a certain class of second-order nonlinear differential equations. (Russian) Bull. Acad. Sci. Georgian SSR 106(1982), No. 3, 473–476.

6. V. M. Evtukhov, Asymptotic behavior of solutions of a half-linear differential equation of the second-order. (Russian) Differentsial’nye Uravneniya 26(1990), No. 5, 776–787.

7. V. M. Evtukhov and N. G. Drik, Asymptotic representations of solutions of a nonlinear differential equation of the second-order. (Russian) Reports of Enlarged Sessions of Seminar of the I. N. Vekua Institute of Applied Mathematics 7(1992), No. 3, 39–42.

8. A. V. Kostin, On the existence of boundary particular solutions tending to zero as t → +∞ of systems of ordinary differential equations. (Russian) Differentsial’nye Uravneniya 1(1965), No. 5, 585–604.

9. V. M. Evtukhov and N. G. Drik, Asymptotic representations of solutions of a certain class of second-order nonlinear differential equations. (Russian) Bull. Acad. Sci. Georgian SSR 133(1989), No. 1, 29–32.

10. N. G. Drik, Asymptotic behavior of solutions of a second-order nonlinear differential equation in a special case. (Russian) Differentsial’nye Uravneniya 25(1989), No. 1, 1071–1072.

(Received 17.08.1994)

Authors’ address:

Faculty of Mechanics and Mathematics

I. Mechnikov Odessa State University

2, Petra Velikogo St., Odessa 270057

Ukraine