Ukrainian Mathematical Journal, Vol. 61, No. 12, 2009
ASYMPTOTIC REPRESENTATIONS OF SOLUTIONS OF ESSENTIALLY NONLINEAR TWO-DIMENSIONAL SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS V. M. Evtukhov 1 and E. S. Vladova 1
UDC 517.925.44
We establish asymptotic representations for one class of solutions of two-dimensional systems of ordinary differential equations that are more general than systems of the Emden–Fowler type.
1. Statement of the Problem and Main Result Consider the system of differential equations yi′ = α i pi (t ) ϕ 3 − i ( y3 − i ) ,
i = 1, 2,
(1.1)
where α i ∈ {−1, 1} , pi (t ): [ a, ω [ → ] 0, +∞ [ are continuous functions, – ∞ < a < ω ≤ + ∞, 2 ϕ i : Δ(Yi0 ) →
] 0, +∞ [
are continuously differentiable functions of orders σ i , σ1σ 2 ≠ 1 , that vary regularly as y → Yi0 ,
Δ (Yi0 ) is a certain one-sided neighborhood of the point Yi0 , and Yi0 is equal to either 0 or ±∞ . σ
In the case where ϕ i ( y ) = y i , this system of equations is called a system of the Emden – Fowler type. The asymptotic properties of its solutions were studied in detail in [1 – 4]. In the present paper, we omit the assumption that ϕ i ( y) , i = 1, 2, are power functions and assume that these functions are close to power ones in the neighborhoods of the points Yi0 in the sense of the definition of regularly varying functions [4]. 2 of system (1.1) defined on an interval [ t 0 , ω [ ⊂ [ a, ω [ is called a Pω (Y10 , Y20 , λ1, λ 2 ) A solution ( yi )i=1 solution if it satisfies the following conditions:
yi (t ) ∈ Δ (Yi0 )
for t ∈ [ t 0 , ω [ ,
lim yi (t ) = Yi0 ,
t ↑ω
(1.2) π (t ) yi′ (t ) = λi , lim ω yi (t ) t ↑ω
i = 1, 2,
where 1 Odessa National University, Odessa, Ukraine. 2 For ω = +∞ , we assume that a > 0.
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 12, pp. 1597 – 1611, December, 2009. Original article submitted April 27, 2009. 0041–5995/09/6112–1877
© 2009
Springer Science+Business Media, Inc.
1877
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V. M. EVTUKHOV
⎧⎪ t
if ω = +∞,
⎩⎪ t − ω
if ω < +∞.
π ω (t ) = ⎨
AND
E. S. VLADOVA
The aim of the present paper is to establish, in the case of nonzero real constants λ1 and λ 2 , necessary and sufficient conditions for the existence of Pω (Y10 , Y20 , λ1, λ 2 ) -solutions of the system of differential equations (1.1) and to find asymptotic relations for these solutions as t ↑ ω . We set ⎧1 ⎪ μi = ⎨ ⎪⎩ −1
if Yi0 = +∞ or Yi0 = 0 and Δ(Yi0 ) is the right neiighborhood of 0, if Yi0 = −∞ or Yi0 = 0 and Δ(Yi0 ) is the left neighborhood of 0
and note that the numbers μ i , i = 1, 2, determine the signs of the components of the Pω (Y10 , Y20 , λ1 , λ 2 ) -solution in a certain left neighborhood of ω. We also set t
I i1 (t ) =
∫
t
pi (τ) d τ ,
I i 2 (t ) =
Ai1
∫
π ω (τ ) pi (τ ) d τ ,
i = 1, 2,
Ai 2
where
Ai1
Ai 2
⎧ω ⎪ = ⎨ ⎪a ⎩
⎧ω ⎪ = ⎨ ⎪a ⎩
ω
if
∫a
if
∫a
ω
if
∫a
if
∫a
ω
ω
pi (τ )d τ < +∞, pi (τ )d τ = +∞,
π ω (τ ) pi (τ )d τ < +∞, π ω (τ ) pi (τ )d τ = +∞.
Also note that, by virtue of properties of regularly varying functions [4], we have lim
z → Yi z ∈ Δ (Yi0 )
zϕ′i ( z ) = σi , ϕi (z)
ϕ i (z ) =
z
σi
θi ( z ) ,
i = 1, 2,
(1.3)
where θi ( z ) is a function slowly varying as z → Yi0 . Theorem 1. Let λ i ∈ R\ {0} , i = 1, 2. Then, for the existence of Pω (Y10 , Y20 , λ1, λ 2 ) -solutions of the system of differential equations (1.1), it is necessary, and sufficient if either λ1 + λ 2 ≠ 0 or λ1 + λ 2 = 0
and σ1σ 2 < 1 ,
(1.4)
ASYMPTOTIC REPRESENTATIONS
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that, for each i ∈ {1, 2} , lim
t ↑ω
π ω (t ) pi (t ) = λi − σ 3 − i λ 3 − i I i1 (t )
lim
t ↑ω
π ω2 (t ) pi (t ) = 1 I i 2 (t )
if λ i − σ 3 − i λ 3 − i = 0 ,
(1.5)
if λ i − σ 3 − i λ 3 − i = 0 ,
(1.6)
and the sign conditions λ i π ω (t ) > 0
for Yi0 = ±∞ ,
λ i π ω (t ) < 0
for Yi0 = 0 ,
(1.7)
α i sign [ λ i π ω (t ) ] = μ i
(1.8)
be satisfied. Each of these solutions admits the following asymptotic representation as t ↑ ω : yi (t ) ϕ 3 − i y3 − i (t )
(
)
=
α i π ω (t ) pi (t ) [ 1 + o(1) ] , λi
i = 1, 2;
(1.9)
moreover, there exists a one-parameter family of these solutions if λ1λ 2 (1 − σ1σ 2 ) < 0, and there exists a two-parameter family of these solutions if (λ1 + λ 2 ) π ω (t ) > 0 for t ∈ [ a, ω [ and λ1λ 2 (1 − σ1σ 2 ) > 0. Remark 1. According to the conditions λ i ∈R\{0} , i = 1, 2, and σ1σ 2 ≠ 1 , the difference λ i – λ 3− i σ 3− i can be equal to zero only for one of the values i ∈ {1, 2} . Proof of Theorem 1. Necessity. Let yi : [ t 0 , ω [ → Δ(Yi0 ) , i = 1, 2, be an arbitrary Pω (Y10 , Y20 , λ1, λ 2 ) solution of the system of differential equations (1.1). Then, by the definition of Pω -solution [namely, by virtue of (1.2)], taking into account the introduced numbers μ i , i = 1, 2, we get yi (t ) = μ i π ω (t )
λ i + o(1)
,
i = 1, 2,
for
t ↑ ω.
λ
Since λ i ≠ 0 , we conclude that π ω (t ) i → Yi , i = 1, 2, as t ↑ ω , and, hence, conditions (1.7) are satisfied. By virtue of the third condition in (1.2), it follows from (1.1) that
(
)
λ i yi (t ) = α i π ω (t ) pi (t ) ϕ 3 − i y3 − i (t ) [1 + o(1)] ,
i = 1, 2,
as t ↑ ω .
(1.10)
This yields the asymptotic representations (1.9); furthermore, with regard for the positivity of the functions pi , i = 1, 2, on the intervals of their definition, we obtain relations (1.8).
ϕi ,
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AND
E. S. VLADOVA
Taking conditions (1.2) and (1.3) into account, we get ⎛ ⎞′ yi (t ) π ω (t ) ⎜ ⎟ ⎝ π ω (t ) ϕ 3 − i y3 − i (t ) ⎠ lim yi (t ) t ↑ω π ω (t ) ϕ 3 − i y3 − i (t )
(
)
(
)
(
)
⎛ π (t ) yi′ (t ) π ω (t ) y′3 − i (t ) y3 − i (t ) ϕ′3 − i y3 − i (t ) ⎞ = lim ⎜ ω −1– ⎟ yi (t ) y3 − i (t ) t ↑ω ⎝ ϕ 3 − i y3 − i (t ) ⎠
(
= λ i − 1 − λ 3 − i σ 3 − i ≠ −1
)
if λ i − λ 3 − i σ 3 − i ≠ 0
and ⎛ ⎞′ yi (t ) π ω (t ) ⎜ ⎟ ⎛ π (t ) yi′ (t ) π ω (t ) y′3 − i (t ) y3 − i (t ) ϕ′3 − i y3 − i (t ) ⎞ ⎝ ϕ 3 − i y3 − i (t ) ⎠ – = lim ⎜ ω lim ⎟ = 0 yi (t ) yi (t ) y3 − i (t ) t ↑ω t ↑ω ⎝ ϕ 3 − i y3 − i (t ) ⎠ ϕ 3 − i y3 − i (t )
(
(
)
(
)
(
)
)
if λ i − λ 3 − i σ 3 − i = 0 . By virtue of these limit relations, we obtain t
yi (τ ) d τ
∫ π ω (τ) ϕ 3 − i (y3 − i (τ))
=
Ai
t
yi (τ ) d τ
∫ ϕ 3 − i (y3 − i (τ))
=
Ai
yi (t ) [1 + o(1)]
(λ i − λ 3 − i σ 3 − i ) ϕ 3 − i (y3 − i (t )) π ω (t ) yi (t ) [ 1 + o(1) ] ϕ 3 − i y3 − i (t )
(
)
as t ↑ ω
as t ↑ ω
if λ i − λ 3 − i σ 3 − i ≠ 0 ,
if λ i − λ 3 − i σ 3 − i = 0 ,
where the limit of integration A i is equal to either ω or t 0 and is chosen so that the corresponding integral tends either to zero or to ± ∞ as t ↑ ω . Rewriting (1.10) in the form yi (t ) π ω (t ) ϕ 3 − i y3 − i (t )
= α i λ i−1 pi (t ) [ 1 + o(1) ]
as t ↑ ω
if λ i − λ 3 − i σ 3 − i ≠ 0 ,
yi (t ) −1 = α i λ i π ω (t ) pi (t ) [ 1 + o(1) ] ϕ 3 − i y3 − i (t )
as t ↑ ω
if λ i − λ 3 − i σ 3 − i = 0
(
(
)
)
ASYMPTOTIC REPRESENTATIONS
OF
SOLUTIONS OF ESSENTIALLY NONLINEAR TWO-DIMENSIONAL SYSTEMS
1881
and integrating from A i to t, we obtain the following asymptotic relations for each i ∈ {1, 2} as t ↑ ω : α i (λ i − σ 3 − i λ 3 − i ) yi (t ) = I i1 (t ) [ 1 + o(1) ] λi ϕ 3 − i y3 − i (t )
(
if λ i − σ 3 − i λ 3 − i ≠ 0 ,
)
yi (t ) α i I i 2 (t ) = [ 1 + o(1) ] λ i π ω (t ) ϕ 3 − i y3 − i (t )
(
)
if λ i − σ 3 − i λ 3 − i = 0 .
By virtue of the definition of Pω (Y10 , Y20 , λ1, λ 2 ) -solution, these asymptotic relations and equality (1.10) yield conditions (1.5) and (1.6). Sufficiency. Assume that λ i ∈ R\ {0} , i = 1, 2 , and, in addition to conditions (1.5) – (1.8), one of conditions (1.4) is satisfied. We show that, in this case, the system of differential equations (1.1) has at least one Pω (Y10 , Y20 , λ1, λ 2 ) -solution that admits the asymptotic relations (1.9) as t ↑ ω . First, we consider the system of relations yi = Qi (t ) [1 + vi ] , ϕ 3 − i ( y3 − i )
i = 1, 2 ,
(1.11)
where ⎧ α i ( λ i − λ 3− i σ 3− i ) I i1 (t ) ⎪ λi ⎪ Qi (t ) = ⎨ ⎪ α i I i 2 (t ) ⎪ λ π (t ) ⎩ i ω
if λ i − σ 3− i λ 3− i ≠ 0, if λ i − σ 3− i λ 3− i = 0,
and establish that it uniquely determines continuously differentiable implicit functions yi = Yi (t , v1, v2 ) defined on the set D0 = [ t 0 , ω [ × V0 , where t 0 ∈ [ a, ω [ and V0 = {( v1, v2 ) : vi ≤ 1 / 2, i = 1, 2} , and having the form Yi (t , v1, v2 ) = μ i π ω (t )
λ i + zi (t , v1 , v2 )
,
i = 1, 2 ,
(1.12)
where the functions zi are such that zi (t , v1, v2 )
≤
λi 2
for (t , v1, v2 ) ∈ D0
(1.13)
uniformly in ( v1, v2 ) ∈ V0 .
(1.14)
and lim zi (t , v1, v2 ) = 0
t ↑ω
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V. M. EVTUKHOV
E. S. VLADOVA
AND
For this purpose, setting yi = μ i π ω (t )
λ i + zi
i = 1, 2 ,
,
(1.15)
in (1.11) and taking (1.3) and (1.5) – (1.8) into account, we obtain the system of relations π ω (t )
λ i + zi − σ 3 − i λ 3 − i − σ 3 − i z
(
= Qi (t ) θ 3 − i μ 3 − i π ω (t )
λ 3 − i + z3 − i
) (1 + v ) ,
i = 1, 2 ,
i
on the set Ω 0 = [ t1, ω [ × Z 0 × V0 , where t1 is a certain number from the interval
{(z1, z2 ) :
[ a, ω [
zi ≤ λ i /2} . Hence,
zi − σ 3 − i z 3 − i = σ 3 − i λ 3 − i
(
ln ⎡ Qi (t ) θ 3 − i μ 3 − i π ω (t ) ⎣ − λi + ln π ω (t )
λ 3 − i + z3 − i
) (1 + v ) ⎤⎦ , i
and Z 0 =
i = 1, 2 .
Partially solving this system with respect to z1 and z2 , we obtain zi = − λ i + ai (t ) + bi (t , v1, v2 ) + Z i (t , z1, z2 ) ,
i = 1, 2 ,
(1.16)
where
ai (t ) =
(
ln Qi (t ) Q3 − i (t )
σ3 − i
(1 − σ1σ 2 ) ln π ω (t )
), σ
ln ⎡(1 + vi ) (1 + v 3 − i ) 3 − i ⎤ ⎣ ⎦, bi (t , v1, v2 ) = (1 − σ1σ 2 ) ln π ω (t )
(
λ
+z
)
σ
(
ln ⎡ θ 3 − i μ 3 − i π ω (t ) 3 − i 3 − i θi 3 − i μ i π ω (t ) ⎣ Z i (t , z1, z2 ) = (1 − σ1σ 2 ) ln π ω (t )
λ i + zi
) ⎤⎦ ,
i = 1, 2 .
Here, lim bi (t , v1, v2 ) = 0 ,
t ↑ω
i = 1, 2 ,
uniformly in ( v1, v2 ) ∈ V0
(1.17)
and, by virtue of properties of slowly varying functions (see [4]), lim Z i (t , z1, z2 ) = 0 ,
t ↑ω
i = 1, 2 ,
uniformly in ( z1, z2 ) ∈ Z 0 .
Since conditions (1.5) and (1.6) are satisfied, the following asymptotic representations are true:
(1.18)
ASYMPTOTIC REPRESENTATIONS
OF
SOLUTIONS OF ESSENTIALLY NONLINEAR TWO-DIMENSIONAL SYSTEMS
Qi (t ) Q3 − i (t )
σ3 − i
= Ci π ω (t )
λ i (1 − σ1σ 2 ) + o(1)
i = 1, 2 ,
,
1883
as t ↑ ω ,
where C1 and C 2 are certain positive constants. Therefore, lim ai (t ) = λ i ,
t ↑ω
i = 1, 2 .
(1.19)
Furthermore,
(
λ +z
i i θ′i μ i π ω (t ) σ 3 − i μ i π ω (t ) ∂Z i (t , z1, z2 ) = λ +z ∂zi 1 − σ1σ 2 θi μ i π ω (t ) i i
(
λ
)
(
+z
μ 3 − i π ω (t ) 3 − i 3 − i θ′3 − i μ 3 − i π ω (t ) ∂Z i (t , z1, z2 ) 1 = +z λ ∂z 3 − i 1 − σ1σ 2 θ 3 − i μ 3 − i π ω (t ) 3 − i 3 − i
(
λ i + zi
),
λ 3 − i + z3 − i
)
),
i = 1, 2 .
Taking into account that the slowly varying functions θi , i = 1, 2 , satisfy the conditions lim
z → Yi
0
zθ′i ( z ) = 0, θi ( z )
i = 1, 2 ,
we conclude that
lim
t ↑ω
∂Z i (t , z1, z2 ) = 0, ∂z k
i, k = 1, 2 ,
uniformly in ( z1, z2 ) ∈ Z 0 .
By virtue of the limit relations presented above, there exists a number t 0 ∈ [ t1, ω [ such that the inequalities i = 1, 2 ,
(1.20)
z k − zk ,
i = 1, 2 ,
(1.21)
≤
1 min { λ1 , λ 2 2
},
λ0 = hold on the set
λ0 , 2
ai (t ) + bi (t , v1, v2 ) + Z i (t , z1, z2 ) − λ i
[ t0 , ω [ ×
Z 0 × V0 , and the Lipschitz conditions
Z i (t , z1, z2 ) − Z i (t , z1, z2 ) ≤
1 3
2
∑
k =1
are satisfied for t ∈ [ t 0 , ω [ and any ( z1, z2 ) , ( z1, z2 ) ∈ Z 0 .
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V. M. EVTUKHOV
For the number t 0 thus chosen, we denote by B the Banach space of vector functions R 2 continuous and bounded on the set Ω = [ t 0 , ω [ × V0 with the norm
AND
E. S. VLADOVA
z = ( zi )i2=1: Ω →
= sup { z1 (t , v1, v2 ) + z2 (t , v1, v2 ) :(t , v1, v2 ) ∈ Ω} .
z
In the space B, we select the subspace B 0 of functions for which
z ≤ λ 0 . On B 0 , taking an arbitrary
number ν ∈ (0, 1) , we consider the operator Φ = (Φ i )i2= 1 defined by the relations Φ i ( z ) (t , v1, v2 ) = zi (t , v1, v2 ) – ν[ zi (t , v1, v2 ) + λ i − ai (t ) – bi (t , v1, v2 ) – Z i (t , z1 (t , v1, v2 ), z2 (t , v1, v2 )) ] ,
i = 1, 2 .
For any z ∈ B 0 , by virtue of conditions (1.20), we get Φ i ( z ) (t , v1, v2 ) ≤ (1 − ν) zi (t , v1, v2 ) +
νλ 0 , 2
i = 1, 2 ,
for (t , v1, v2 ) ∈ Ω .
Therefore, on the set Ω, we have 2
∑
i =1
2
Φ i ( z )(t , v1, v2 ) ≤ (1 − ν)∑ zi (t , v1, v2 ) + ν λ 0 i =1
≤ (1 − ν) z + νλ 0 ≤ (1 − ν) λ 0 + ν λ 0 = λ 0 . This yields Φ ( z ) ≤ λ 0 , i.e., Φ(B 0 ) ⊂ B 0 . Now let z, z ∈ B 0 . Then, by virtue of (1.21), the following relation holds for (t , v1, v2 ) ∈ Ω : Φ i ( z ) (t , v1, v2 ) − Φ i ( z ) (t , v1, v2 ) ≤ (1 − ν) zi (t , v1, v2 ) − zi (t , v1, v2 ) + ν Z i (t , z1 (t , v1, v2 ), z2 (t , v1, v2 )) − Z i (t , z1 (t , v1, v2 ), z2 (t , v1, v2 )) ≤ (1 − ν) zi (t , v1, v2 ) − zi (t , v1, v2 )
+
ν 3
2
∑
k =1
Therefore, on the set Ω, we have
z k (t , v1, v2 ) − zk (t , v1, v2 ) ,
i = 1, 2 .
(1.22)
ASYMPTOTIC REPRESENTATIONS 2
∑
k =1
OF
SOLUTIONS OF ESSENTIALLY NONLINEAR TWO-DIMENSIONAL SYSTEMS
1885
Φ k ( z ) (t , v1, v2 ) − Φ k ( z ) (t , v1, v2 )
ν⎞ ⎛ ≤ ⎜1 − ⎟ ⎝ 3⎠
2
∑
k =1
z k (t , v1, v2 ) − zk (t , v1, v2 )
ν⎞ ⎛ ≤ ⎜1 − ⎟ z − z , ⎝ 3⎠
whence Φ( z ) − Φ( z )
ν⎞ ⎛ ≤ ⎜1 − ⎟ z − z . ⎝ 3⎠
Thus, the operator Φ maps the space B 0 into itself and is a contracting operator in this space. According to the contracting-mapping principle, there exists a unique vector function z ∈ B 0 such that z = Φ( z ) . By virtue of (1.22), this vector function, continuous on the set Ω, is the unique solution of system (1.16) that satisfies the condition z ≤ λ 0 . By virtue of this condition and relations (1.17) – (1.19), it follows from (1.16) that the components of this solution tend to zero as t ↑ ω uniformly in ( v1, v2 ) ∈ V0 . The continuous differentiability of this solution on the set Ω follows directly from the known local theorem on the existence of implicit functions defined by a system of relations. By virtue of (1.15), the obtained vector function z = ( zi )i2= 1 is associated with a vector function (Yi )i2= 1 with components of the form (1.12), which is a solution of system (1.11). Applying the transformation yi (t ) = Yi (t , v1 ( x ), v2 ( x )) ,
i = 1, 2 ,
x = β ln π ω (t ) ,
(1.23)
where ⎧⎪ 1 β = ⎨ ⎪⎩ −1
if ω = +∞, if ω < +∞,
to the system of differential equations (1.1) and taking into account that, for t ∈ [ t 0 , ω [ and ( v1 ( x ), v2 ( x )) ∈ V0 , the vector function (Yi (t , v1 ( x ), v2 ( x )))i = 1 is a solution of the system of equations 2
yi (t ) ϕ 3 − i y3 − i (t )
(
)
= Qi (t ) [ 1 + vi ( x ) ] ,
i = 1, 2 ,
(1.24)
we obtain the system of differential equations v′i = β hi ( x ) − β h3 − i ( x ) ξ 3 − i ( x, v1, v2 )
1 + vi – βgi ( x ) [ 1 + vi ] , 1 + v3 − i
i = 1, 2 ,
(1.25)
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where λi π ω (t ) pi (t ) ⎧ ⎪λ −σ λ I i1 (t ) 3− i 3− i ⎪ i hi ( x ) = hi ( x (t )) = ⎨ ⎪ λ i π ω2 (t ) pi (t ) ⎪ I i 2 (t ) ⎩ ⎧π ω (t ) pi (t ) ⎪ I (t ) i1 ⎪ gi ( x(t )) = ⎨ 2 ⎪π ω (t ) pi (t ) ⎪ I (t ) − 1 ⎩ i2 ξ i (x, v1, v2 ) = ξ(x(t ), v1, v2 ) =
if λ i − σ 3− i λ 3− i ≠ 0, if λ i − σ 3− i λ 3− i = 0,
if λ i − σ 3− i λ 3− i ≠ 0, if λ i − σ 3− i λ 3− i = 0, Yi (t , v1, v2 ) ϕ′i (Yi (t , v1, v2 )) . Yi (t , v1, v2 )
Here, by virtue of conditions (1.5) and (1.6), we have lim hi ( x ) = lim hi (x(t )) = λ i ,
x → +∞
t ↑ω
lim gi ( x ) = lim gi (x (t )) = λ i − σ 3 − i λ 3 − i ,
x → +∞
i = 1, 2 .
t ↑ω
(1.26)
Since lim Yi (t , v1, v2 ) = Yi0 ,
t ↑ω
i = 1, 2 ,
uniformly in ( v1, v2 ) ∈ V0
and the first condition in (1.3) is satisfied, the following representation is true: ξ i (x, v1, v2 ) = σ i + Ri1 ( x, v1, v2 ) ,
i = 1, 2 ,
where Ri1 ( x, v1, v2 ) → 0 as x → +∞
uniformly in ( v1, v2 ) ∈ V0 .
(1.27)
Taking into account these representations and the representations 1 + v3 – i
= 1 + v 3 – i − vi + Ri 2 ( v1, v2 ) ,
1 + vi
i = 1, 2 ,
where the functions Ri2 are such that lim
v1 + v2
∂Ri 2 ( v1, v2 ) = 0, →0 ∂v k
i, k = 1, 2,
(1.28)
ASYMPTOTIC REPRESENTATIONS
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SOLUTIONS OF ESSENTIALLY NONLINEAR TWO-DIMENSIONAL SYSTEMS
1887
we rewrite the system of differential equations (1.25) in the form v′i = fi ( x ) + pi1 ( x ) v1 + pi 2 ( x ) v2 + Vi1 ( x, v1, v2 ) + Vi 2 ( x, v1, v2 ) ,
i = 1, 2,
(1.29)
where fi ( x ) = β ⎡⎣ hi ( x ) − σ 3 − i h3 − i ( x ) − gi ( x ) ⎤⎦ , pii ( x ) = −β ⎡⎣ σ 3 − i h3 − i ( x ) + gi ( x ) ⎤⎦ ,
pi 3 − i ( x ) = β h3 − i ( x ) ,
Vi k ( x, v1, v2 ) = −β h3 − i ( x ) R3 − i k ( x, v1, v2 ) ,
i, k = 1, 2.
In this system, by virtue of conditions (1.26) – (1.28), we have lim fi ( x ) = 0 ,
x → +∞
P =
⎛ p11 ( x ) lim ⎜ x → +∞ ⎝ p ( x ) 21
i = 1, 2,
p12 ( x )⎞ ⎛ −β λ1 ⎟ = ⎜ ⎝ β σ 1 λ1 p22 ( x )⎠
β σ 2 λ 2⎞ ⎟, −β λ 2 ⎠
lim Vi1 ( x, v1, v2 ) = 0 , i = 1, 2, uniformly in ( v1, v2 ) ∈ V0 ,
x → +∞
lim
v1 + v2 → 0
Vi 2 ( x, v1, v2 ) = 0 , i = 1, 2, uniformly in t ∈ [ t 0 , ω [ .
The characteristic equation det [ P − νE2 ] = 0, where E2 is the second-order identity matrix, for the limit coefficient matrix of the linear part of the system has the form ν 2 + β(λ1 + λ 2 ) + λ1λ 2 (1 − σ1σ 2 ) = 0 .
(1.30)
By virtue of the conditions λ i ∈ R\ {0} , i = 1, 2, and σ1σ 2 ≠ 1 , and the fact that one of conditions (1.4) is satisfied, this equation does not have roots with zero real part. Consequently, all conditions of Lemma 1 in [5] are satisfied for the system of differential equations (1.29). According to this lemma, the system of differential equations (1.29) has at least one solution { vi }i2=1 : [ x1, +∞ [ → R 2 ( x1 ≥ x0 = β ln π ω (t 0 )
)
that tends to zero as x → +∞ . Moreover, there exists a one-parameter family of these solutions if there exists only one root of Eq. (1.30) with negative real part, i.e., λ1λ 2 (1 − σ1σ 2 ) < 0, and there exists a two-parameter family of these solutions if all roots of this equation have negative real parts, i.e., one has β(λ1 + λ 2 ) < 0 and λ1λ 2 (1 − σ1σ 2 ) > 0. By virtue of transformation (1.23) and the system of relations (1.24) for the functions Yi (t , v1 ( x (t )) , v2 ( x(t ))) , i = 1, 2 , these solutions of system (1.29) are associated with solutions ( y1, y2 ) of the system of differential equations (1.1) that admit the asymptotic representations
1888
V. M. EVTUKHOV
yi (t ) ϕ 3 − i y3 − i (t )
(
)
= Qi (t ) [1 + o(1)] ,
i = 1, 2 ,
AND
E. S. VLADOVA
as t ↑ ω .
By virtue of (1.5) and (1.6), these asymptotic representations can be rewritten in the form (1.9). It remains to verify that each aforementioned solution of the system of differential equations (1.1) is a Pω (Y10 , Y20 , λ1, λ 2 ) -solution. Since these solutions are associated with solutions (v1 ( x ), v2 ( x )) of system (1.29) that tend to zero as x → +∞ , by virtue of the properties of the functions Yi (t , v1, v2 ) , i = 1, 2 , established above the first two conditions in (1.2) are satisfied. Moreover, taking (1.24) and (1.26) into account, we obtain the following relations for these solutions of system (1.1):
lim
t ↑ω
(
)
α i π ω (t ) pi (t ) ϕ 3 − i y3 − i (t ) π ω (t ) yi′ (t ) α π (t ) pi (t ) = lim = lim hi (x (t )) = λ i . = lim i ω yi (t ) Oi (t ) t ↑ω t ↑ω t ↑ω yi (t )
Thus, the third condition in the definition of Pω (Y10 , Y20 , λ1 , λ 2 ) -solution [the third condition in (1.2)] is also satisfied. The theorem is proved. 2. Examples First, as an example that illustrates the result obtained, we consider the system of differential equations yi′ = α i pi (t ) y3 − i
σ3 − i
ln y3 − i
γ3−i
sign y3 − i ,
i = 1, 2 ,
(2.1)
where α i , σ i , and pi are the same as in system (1.1) and γ i ∈R . Let λ i ∈ R\ {0} , i = 1, 2, and let Yi0 be equal to either zero or ± ∞ for each value of i ∈ {1, 2} . Since each component of any Pω (Y10 , Y20 , λ1 , λ 2 ) -solution ( y1, y2 ) of the system of equations (2.1) is alternating in a certain left neighborhood of ω, in this neighborhood we have sign yi (t ) = μ i , i = 1, 2, μ i = 1 if Yi0 = + ∞, and μ i = – 1 if Yi0 = – ∞. If Yi0 = 0, then μ i can be equal to + 1 or – 1. In this case, ϕ i ( yi ) = yi
σi
ln yi
γi
, i ∈ {1, 2} . This is a regularly varying function of order σ i for
both yi → 0 and yi → ± ∞. By virtue of Theorem 1, for the existence of Pω (Y10 , Y20 , λ1, λ 2 ) -solutions of the system of differential equations (2.1) it is necessary, and sufficient if one of conditions (1.4) is satisfied, that, for each i ∈ {1, 2} , the limit relations (1.5) and (1.6) be true, the sign conditions (1.7) be satisfied, and α i sign [ λ i π ω (t ) ] = μ i ; furthermore, every solution ( y1, y2 ) of this sort admits the asymptotic representation yi (t ) y3 − i (t )
σ3 − i
ln y3 − i (t )
γ3−i
=
α i μ 3 − i π ω (t ) pi (t ) λi
[1 + o(1)] ,
i = 1, 2 ,
as t ↑ ω .
(2.2)
ASYMPTOTIC REPRESENTATIONS
OF
SOLUTIONS OF ESSENTIALLY NONLINEAR TWO-DIMENSIONAL SYSTEMS
1889
According to the third condition in (1.2), we have yi (t ) = μ i π ω (t )
λ i + o(1)
i = 1, 2 ,
,
as t ↑ ω .
Therefore, ln yi (t ) =
[ λi
+ o(1) ] ln yi (t ) ,
i = 1, 2 ,
as t ↑ ω ,
and the asymptotic representations (2.2) can be rewritten in the form
yi (t ) y3 − i (t )
σ3 − i
=
γ3−i
α i μ 3 − i π ω (t ) λ 3 − i ln π ω (t )
pi (t )
λi
[1 + o(1)] ,
i = 1, 2 , as t ↑ ω .
Using these relations, one can easily obtain the following asymptotic (as t ↑ ω ) formulas for the components of the Pω (Y10 , Y20 , λ1, λ 2 ) -solution: yi (t ) = ci π ω (t )
(1 + σ 3 − i )/(1 − σ1σ 2 )
ln π ω (t ) σ
× pi1/(1 − σ1σ 2 ) (t ) p3 −3 −i i
( γ 3 − i + σ 3 − i γ i ) /(1 − σ1σ 2 )
/(1 − σ1σ 2 )
(t ) [1 + o(1)] ,
i = 1, 2 ,
(2.3)
where ci = μ i λ i
( σ 3 − i γ i − 1)/(1 − σ1σ 2 )
λ3 − i
( γ 3 − i − σ 3 − i )/(1 − σ1σ 2 )
.
These asymptotic representations are new even for systems of the Emden – Fowler type, i.e., for γ 1 = γ 2 = 0 . Further, we consider the equation u ′′ = α 0 p(t ) ϕ1 (u ) ϕ 2 (u ′ ) , where α 0 ∈ {−1, 1} , p : [ a, ω [ →
] 0, +∞ [
(2.4)
is a continuous function, ϕ i : Δ (U i0 ) → ] 0, +∞ [ , i = 1, 2 ,
are continuously differentiable functions of orders σ i , σ 2 ≠ 1 , σ1 + σ 2 ≠ 1, that vary regularly as z → U i0 , Δ (U i0 ) is a certain one-sided neighborhood of the point U i0 , and U i0 is equal to either 0 or ± ∞. A solution u of Eq. (2.4) is called a Pω (λ 0 ) -solution (– ∞ ≤ λ 0 ≤ + ∞) if it is defined on a certain interval [ t 0 , ω [ ⊂ [ a, ω [ and satisfies the following conditions: u (i − 1) (t ) ∈ Δ (U i0 )
for t ∈ [ t 0 , ω [ ,
lim
t ↑ω
lim u (i − 1) (t ) = U i0 ,
t ↑ω
π ω (t ) u ′′(t ) = λ0 . u ′(t )
i = 1, 2 ,
1890
V. M. EVTUKHOV
AND
E. S. VLADOVA
It is easy to see that, for every Pω (λ 0 ) -solution of Eq. (2.4), one has lim
t ↑ω
π ω (t ) u ′(t ) = 1 + λ0 . u (t )
We introduce the function
z
⎧ 0 ⎪U2 ⎪ B = ⎨ ⎪ ⎪b ⎩
ds
∫ ϕ 2 (s) ,
ψ (z) =
B
U 20
ds converges, ϕ 2 (s)
U 20
ds diverges, ϕ 2 (s)
∫b
if
∫b
if
where b is an arbitrary number from the interval Δ (U 20 ) . Since ψ ′( z ) > 0 for z ∈ Δ (U 20 ) , we can conclude that ψ : Δ (U 20 ) → Δ (Y20 ) , where Δ (Y20 ) is a onesided neighborhood of Y20 , and Y20 is equal to either zero or ± ∞. Using properties of regularly varying functions and the l’Hospital rule, we get ⎛ z ⎞′ ⎜⎝ ϕ ( z )⎟⎠ z 2 = 1 − σ2 . lim = lim z → U 20 ψ ( z ) ϕ 2 ( z ) z → U 20 ψ ′( z )
(2.5)
Using the transformation u = y1 ,
ψ(u ′ ) = y2 ,
(2.6)
we reduce Eq. (2.4) to the system of equations y1′ = ψ −1 ( y2 ) , (2.7) y2′ = α 0 p(t ) ϕ1 ( y1 ) ; in this case, ψ −1 is a function of order
lim
y2 → Y20
(
)
y2 ψ −1 ( y2 ) ′ −1
ψ ( y2 )
=
1 that varies regularly as y2 → Y20 because 1 − σ2
lim
y2 → Y20
(
)
y2 ϕ 2 ψ −1 ( y2 ) −1
ψ ( y2 )
=
lim
z → U 20
1 ψ (z) ϕ 2 (z) = . 1 − σ2 z
It is also easy to see that a solution u of Eq. (2.4) is a Pω (λ 0 ) -solution if and only if the solution ( y1, y2 )
(
)
of system (2.7) associated with u by transformation (2.6) is a Pω U10 , Y20 , λ 0 + 1, (1 – σ 2 ) λ 0 -solution.
ASYMPTOTIC REPRESENTATIONS
SOLUTIONS OF ESSENTIALLY NONLINEAR TWO-DIMENSIONAL SYSTEMS
OF
1891
Introducing numbers μ i0 for U i0 and Δ (U i0 ) , i = 1, 2 , by analogy with the definition of the numbers μ i for Yi0 and Δ (Yi0 ) , i = 1, 2 , and using the mapping ψ : Δ (U 20 ) → Δ (Y20 ) , we get
μ 02
⎧ (1 − σ 2 )μ 2 ⎪ = ⎨ ⎪⎩ (σ 2 − 1)μ 2
We can assume here that μ10 = μ1 .
if U 20 = ±∞, if U 20 = 0.
(
)
By virtue of Theorem 1, for the existence of Pω U10 , Y20 , λ 0 + 1, (1 – σ 2 ) λ 0 -solutions of system (2.4) in the case where λ 0 ∈ R\ {0, −1} and σ 2 ≠ 1 it is necessary, and sufficient if either
λ 0 (σ 2 − 2 ) ≠ 1
λ 0 (σ 2 − 2 ) = 1 and (σ 2 − 1) (σ1 + σ 2 − 1) > 0 ,
or
that lim
t ↑ω
π ω (t ) p(t ) t
∫A
1
p(τ ) d τ
lim
t ↑ω
= λ 0 (1 − σ 2 − σ1 ) − σ1
π ω2 (t ) p(t ) t
∫A
2
π ω (τ ) p(τ ) d τ
= 1
if
if
λ 0 (1 − σ 2 − σ1 ) ≠ σ1 ,
λ 0 (1 − σ 2 − σ1 ) = σ1 ,
and the sign conditions (1 + λ 0 ) π ω (t ) > 0 (1 + λ 0 ) π ω (t ) < 0 μ10 μ 02 (1 + λ 0 ) π ω (t ) > 0 ,
λ 0 π ω (t ) > 0 ,
λ 0 π ω (t ) < 0
and
for U10 = ±∞ , for U10 = 0 , and
α 0 μ 20 = −1
α 0 μ 02 = 1
for U 20 = ±∞ ,
for U 20 = 0
be satisfied. Furthermore, every solution of this sort admits the following asymptotic representation as t ↑ ω : y1 (t ) −1 ψ y2 (t )
(
)
=
π ω (t ) [1 + o(1)] , 1 + λ0
y2 (t ) α 0 π ω (t ) p(t ) = [1 + o(1)] . ϕ1(y1 (t )) (1 − σ 2 ) λ 0
1892
V. M. EVTUKHOV
AND
E. S. VLADOVA
Moreover, there exists a one-parameter family of these solutions if λ 0 (1 + λ 0 )(1 − σ1 − σ 2 ) < 0 , and there exists a two-parameter family of these solutions if λ 0 (1 + λ 0 )(1 − σ1 − σ 2 ) > 0 and [ 1 + (2 − σ 2 )λ 0 ] π ω (t ) > 0 . Using transformation (2.6) and taking (2.5) into account, we rewrite these asymptotic relations in the form π ω (t ) u ′(t ) = 1 + λ 0 + o(1) , u (t )
u ′(t ) α π (t ) p(t ) = 0 ω [1 + o(1)] . ϕ1(u (t )) ϕ 2 (u ′(t )) λ0
In contrast to [6] (Theorem 1.1), we establish here the existence and asymptotic behavior of Pω (λ 0 ) -solutions of Eq. (2.4) for λ 0 ∈ R\ {0, −1} without additional restrictions on the function ϕ1 . 3. Conclusions In the present paper, we have separated the class of Pω (Y10 , Y20 , λ1, λ 2 ) -solutions of the system of differential equations (1.1); for λ i ∈ R\ {0} , we have established necessary and sufficient conditions for the existence of these solutions and found asymptotic (as t ↑ ω ) relations for their components. We have also solved the problem of the number of these solutions. In the case of particular nonlinearities, the implicit asymptotic relations obtained here enable one to find (see the first example) explicit asymptotic representations for both components of Pω (Y10 , Y20 , λ1, λ 2 ) -solutions. By virtue of the arbitrariness of the choice of ω ≤ + ∞, the main result enables one to clarify the question of the existence of not only regular solutions but also various singular solutions of system (1.1). REFERENCES 1. D. D. Mirzov, “On asymptotic properties of solutions of one system of the Emden–Fowler type,” Differents. Uravn., 21, No. 9, 1498–1504 (1985). 2. D. D. Mirzov, “On some asymptotic properties of solutions of one system of the Emden–Fowler type,” Differents. Uravn., 23, No. 9, 1519–1532 (1987). 3. V. M. Evtukhov, “Asymptotic representations of regular solutions of one two-dimensional system of differential equations,” Dopov. Nats. Akad. Nauk Ukr., No. 4, 11–17 (2002). 4. E. Seneta, Regularly Varying Functions, Springer, Berlin (1976). 5. V. M. Evtukhov and V. M. Khar’kov, “Asymptotic representations of solutions of essentially nonlinear differential equations of the second order,” Differents. Uravn., 43, No. 10, 1311–1323 (2007). 6. V. M. Evtukhov and M. A. Belozerova, “Asymptotic representations of solutions of essentially nonlinear differential equations of the second order,” Ukr. Mat. Zh., 60, No. 3, 310–331 (2008).
