Memoirs on Differential Equations and Mathematical Physics Volume 28, 2003, 89–107

Volume 28, 2003, 89–107

J. Mirzov

NONOSCILLATORY SOLUTIONS OF SOME DIFFERENTIAL SYSTEMS

ditions for non-existence of global solutions of constant sign in subcritical and critical cases. In the supercritical case we indicate one sufficient condi- tion for non-existence of global solutions of constant sign which is, possibly, also exact.

2000 Mathematics Subject Classification. 34C10.

Key words and phrases: System of differential inequalities, global solution of constant sign, non-existence.

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Introduction

The problem on asymptotic behavior of solutions of second order nonlin- ear nonautonomous ordinary differential equations attracted attention of a great number of mathematicians at the beginning of the twentieth century in connection with astrophysical investigations of R. Emden in which there appeared the equation of the type

u⁰⁰±t^σuⁿ= 0.

The detailed qualitative investigation of this equation, called subsequently the Emden-Fowler equation, for different values of parametersσandnwas carried out by R. Fowler.

The interest in the study of asymptotics of solutions of nonlinear second order equations has considerably increased after the appearance of the well- known R. Bellman’s monograph [5] in which the author stated all basic results concerning the Emden-Fowler equation.

The qualitative investigation of the Emden-Fowler type equation

u⁰⁰+a(t)|u|ⁿsignu= 0, (0.1)

wheren∈(0,+∞) and the functiona: [0,+∞)→Ris summable on each finite segment, was started by F.V. Atkinson [2]. He proved that ifa(t)≥0 andn >1, then the condition

+∞

Z

0

ta(t)dt= +∞

is necessary and sufficient for all proper solutions of the equation (0.1) to be oscillatory.

ˇS. Belohorec [6] proved that ifa(t)≥0, 0< n <1, then for all proper solutions of the equation (0.1) to be oscillatory, it is necessary and sufficient that

+∞

Z

0

tⁿa(t)dt= +∞.

The oscillation problem of solutions of the equation (0.1) in casea(t) is a function with alternating signs, has been studied by I.T. Kiguradze [25].

The efficient methods for investigating the asymptotic behavior of proper and singular solutions of the equation (0.1) have been proposed by M.M.

Aripov [1], L.A. Beklemisheva [4], ˇS. Belohorec [6,7], T.A. Chanturia [27,30], V.M. Evtukhov [20], A.G. Katranov [24], I.T. Kiguradze [25–28], L.B. Kle- banov [29], A.V. Kostin [32] and others.

The equations of the type

u⁰⁰+f(t, u) = 0.

are also studied in detail. In particular, there were obtained: sufficient conditions for the existence of proper solutions; sufficient conditions for the

boundedness and stability in one or another sense; necessary and sufficient conditions for all proper solutions to be oscillatory; sufficient conditions for the existence of at least one oscillatory solution; sufficient conditions for all proper solutions to be nonoscillatory; conditions for the solvability of various boundary value problems.

Similar problems for higher order nonlinear differential equations and systems of nonlinear nonautonomous differential equations have been stud- ied by M. Bartuˇshek [3], T.A. Chanturia [27,30], Z. Doˇsla [8,9], O. Doˇsly [10–14], A. Elbert [13, 17–19], J. Jaroˇs [22], I.T. Kiguradze [26–28], T. Ku- sano [19, 22, 33], A.G. Lomtatidze [14, 23, 34, 35], J.V. Manojloviˇc [36], J. D. Mirzov [37], I. Neˇcas [39], B. P˚uˇza [41–43], V.A. Rabtsevich [44], B. L. Shekhter [28], Ch.A. Skhalyakho [45, 46] etc.

In the last years a considerable progress has been made by many math- ematicians in investigation of problems connected with the existence (non- existence) of solutions of constant signs of nonlinear differential equations and systems of differential equations. Among them we can mention the works of M. Cecchi [8, 9], P. Drabek [15, 16], Yu.V. Egorov [21], V.A.

Galaktionov [21], V.A. Kondrat’ev [21], R.G. Koplatadze [30, 31], R. Man- asevich [16], M. Marini [8, 9], E. Mitidieri [38] and S.I. Pokhozhaev [38, 40].

Our work is devoted to systems of the type

u⁰₂signu1≤ −a2(t)|u1|^λ2 ≤0≤a1(t)|u2|^λ1 ≤u⁰₁signu2, (0.2)

−a2(t)|u1|^λ2≤u⁰₂signu1≤0≤u⁰₁signu2≤a1(t)|u2|^λ1, (0.3) where ai : (0,+∞)→[0,+∞) (i= 1,2) are the functions which are sum- mable on every finite segment from (0,+∞),λi>0 (i= 1,2).

Below we will indicate exact conditions guaranteeing the non-existence of global solutions of constant signs of systems of the type (0.2), i.e. the solutions such thatu1(t)·u2(t)6= 0 for 0< t <+∞. The exactness of the conditions is understood in a sense that their violation leads to the existence of global solutions of constant sign of the system (0.3).

For the nonlinear system of the type

u⁰_i= (−1)ⁱ⁻¹a_i(t)|u3−i|^λisignu3−i,

whereai(t)≥0 (i= 1,2),λ1·λ2= 1, we present an original characteristic of the principal solution and new criteria for the non-existence of conjugate points.

The basic method of our investigation is the method of a priori estimates which is widely used by I.T. Kiguradze and his numerous followers.

The results obtained in the paper can be applied to the qualitative theory of ordinary differential equations, to the theory of boundary value problems, in investigating the behavior of solutions of partial differential equations withp-Laplacian, and so on.

It should be noted that the statements given in the present work can, with natural changes, be paraphrased for any interval of the type (a, b), where−∞ ≤a < b≤+∞.

1. Systems of Inequalities in the Subcritical Case

In this section, using the results of [37], we find exact conditions ensuring the non-existence of global solutions of constant signs of systems of the type (0.2) withλ1·λ2<1, and then we apply the obtained conditions to some partial differential equations.

Theorem 1.1. Letλ1·λ2<1and for somet0∈[0,+∞)

t0

Z

0

a1(t)dt <+∞,

t0

Z

0

a2(t)dt= +∞,

+∞

Z

t0

a1(t)dt= +∞,

+∞

Z

t0

a2(t)dt <+∞.

(1.1)

Then for the system (0.2) to have no global solution with the property u1(t)·u2(t)>0for 0< t <+∞, it is sufficient that the equality

t0

Z

0

a1(t) Z^t0

t

a2(τ)dτ λ1

dt+

+∞

Z

t0

a2(t) Z^t

t0

a1(τ)dτ λ2

dt= +∞ (1.2)

be fulfilled.

Proof. Assume the contrary, i.e., suppose that the first summand in (1.2) is equal to +∞and, nevertheless,u1(t)·u2(t)>0 fort∈(0,+∞) for some solutionu1(t),u2(t) of the system (0.2). Then for 0< t <+∞,

|u1|⁰≥a1(t)|u2|^λ1, |u2|⁰≤ −a2(t)|u1|^λ2. (1.3) Hence|u2(t)| ≥ |u1(t)|^λ2Rt0

t a2(τ)dτ for 0< t≤t0. Therefore

|u1(t)|^−λ1λ2|u1(t)|⁰ ≥a1(t) Z^t0

t

a2(τ)dτ λ1

for 0< t≤t0. Integrating the last inequality from t to t0 and passing to the limit as t→0+, we come to the contradiction.

Suppose now that the second summand in (1.2) is equal to +∞and, nev- ertheless,u1(t)·u2(t)>0 fort∈(0,+∞) for some solutionu1(t),u2(t) of the system (0.2). Then (1.3) holds. Consequently,|u1(t)| ≥ |u2(t)|^λ1Rt

t0a1(τ)dτ for t ≥ t0. Therefore |u2(t)|^−λ1·λ2|u2(t)|⁰ ≤ −a2(t) Rt

t0a1(τ)dτλ2

for t≥t0. Integrating the last inequality fromt0totand passing to the limit as

t→+∞, we arrive at the contradiction.

Theorem 1.2. Let λ1 ·λ2 < 1 and (1.1) be satisfied. Then for the system(0.3)to have no global solution with the propertyu1(t)·u2(t)>0for 0< t <+∞, it is necessary that the equality (1.2) be fulfilled.

Proof. Suppose that (1.2) is not fulfilled. Let us show that the system (0.3) has a solution u1(t), u2(t) defined on (0,+∞) with the property u1(t)·u2(t)>0 for 0< t <+∞.

Consider the solutionu1(t),u2(t) of the system (0.3) defined by the initial conditionsu1(t0) =u10,u2(t0) =u20,u10·u20>0, where the numbersu10

andu20will be chosen later on. In some neighbourhood oft0the inequalities u1(t)·u2(t)>0,

−a2(t)|u1(t)|^λ2≤ |u2(t)|⁰≤0≤ |u1(t)|⁰≤a1(t)|u2(t)|^λ1 (1.4) hold. Consequently, fort≥t0 in the neighbourhood oft0we have

|u20| −

t

Z

t0

a2(τ) |u10|+

τ

Z

t0

a1(s)|u2(s)|^λ1ds

λ2

dτ ≤ |u2(t)|.

Since (x+y)^α≤2^α(x^α+y^α) for anyx >0,y >0,α≥0, the last inequality yields

|u20| −2^λ2|u10|^λ2

t

Z

t0

a2(τ)dτ−

−2^λ2|u20|^λ1·λ2

t

Z

t0

a2(τ) Z^τ

t0

a1(s)ds λ2

dτ ≤ |u2(t)|.

Thus we see that|u2(t)|>0 for allt≥t0if

|u20| −2^λ2|u10|^λ2

+∞

Z

t0

a2(t)dt−

−2^λ2|u20|^λ1·λ2

+∞

Z

t0

a2(t) Z^t

t0

a1(τ)dτ λ2

dt >0. (1.5)

Fort≤t0, in the neighbourhood of t0 according to (1.4) we have

|u10| ≤ |u1(t)|+

t0

Z

t

a1(τ)

|u20|+

t0

Z

τ

a2(s)|u1(s)|^λ2ds

λ1

dτ ≤ |u1(t)|+

+2^λ1|u20|^λ1

t0

Z

t

a1(τ)dτ + 2^λ1|u10|^λ1·λ2

t0

Z

t

a1(τ) Z^t0

τ

a2(s)ds λ1

dτ.

Hence, if

|u10| −2^λ1|u20|^λ1

t0

Z

0

a1(t)dt−

−2^λ1|u10|^λ1·λ2

t0

Z

0

a1(t) Z^t0

t

a2(s)ds λ1

dt >0, (1.6)

then|u1(t)|>0 for allt∈(0, t0].

To complete the proof of the theorem we have to show that the system of inequalities (1.5), (1.6) has at least one solution |u10|, |u20|. Suppose

|u20| =|u10|^γ, where λ2 < γ < _λ¹

1. It is obvious that for sufficiently large

|u10| the inequalities (1.5), (1.6) are fulfilled. Therefore for any solution of the system (0.3) defined by the above-mentioned initial values we have

u1(t)·u2(t)>0 for 0< t <+∞.

From Theorems 1.1 and 1.2 we immediately arrive at

Theorem 1.3. Letλ1·λ2<1and(1.1)be satisfied. Then for the system a1(t)|u2|^λ1 ≤u⁰₁signu2≤M a1(t)|u2|^λ1,

−M a2(t)|u1|^λ2 ≤u⁰₂signu1≤ −a2(t)|u1|^λ2, (1.7) whereM ≥1, to have no global solution with the propertyu1(t)·u2(t)>0 for 0< t < +∞, it is necessary and sufficient that the condition (1.2) be fulfilled.

Theorem 1.4. Letλ1·λ2<1and for somet0∈(0,+∞),

t0

Z

0

a1(t)dt= +∞,

t0

Z

0

a2(t)dt <+∞,

+∞

Z

t0

a1(t)dt <+∞,

+∞

Z

t0

a2(t)dt= +∞.

(1.8)

Then for the system (0.2) to have no global solution with the property u1(t)·u2(t)<0for 0< t <+∞, it is sufficient that the equality

t0

Z

0

a2(t) Z^t0

t

a1(τ)dτ λ2

dt+

+∞

Z

t0

a1(t) Z^t

t0

a2(τ)dτ λ1

dt= +∞ (1.9)

be fulfilled.

Proof. Assumeai =b3−i, λi =µ3−i,ui= (−1)ⁱ⁻¹v3−i (i= 1,2) and make

use of Theorem 1.1.

Theorem 1.5. Let λ1 ·λ2 < 1 and (1.8) be satisfied. Then for the system(0.3)to have no global solution with the propertyu1(t)·u2(t)<0for 0< t <+∞, it is necessary that the condition (1.9) hold.

The proof is similar to that of the previous theorem and follows directly from Theorem 1.2.

From Theorems 1.4 and 1.5 follows

Theorem 1.6. Letλ1·λ2<1and(1.8)be satisfied. Then for the system (1.7)to have no solution with the property u1(t)·u2(t)<0for0< t <+∞, it is necessary and sufficient that the condition (1.9) hold.

At the end of this section we give some applications of the obtained results to the partial differential equations of the type

div(| 5u|^m−25u) +f(|x|)|u|ⁿ⁻²u= 0, (1.10) where the function f : (0,+∞) → [0,+∞) is summable on every finite segment of the interval (0,+∞), m >1,n >1,x= (x1, x2, . . . , xN)∈R^N, N ≥2,5u=

∂u

∂x1,_∂x^∂u₂, . . . ,_∂x^∂u_N .

It is known that the function u(x) =y(|x|) =y(t) is a solution of (1.10) if and only ify(t) satisfies the ordinary differential equation [33]

(t^N−1|y⁰|^m−2y⁰)⁰+t^N−1f(t)|y|ⁿ⁻²y= 0. (1.11) Theorem 1.7. Letn < m, N < mand for somet0∈(0,+∞),

t0

Z

0

t^N−1f(t)dt= +∞,

+∞

Z

t0

t^N−1f(t)dt <+∞.

Then for the equation (1.10) to have no global radial positive (negative), increasing(decreasing)with respect to the radial variable solution, it is nec- essary and sufficient that

t0

Z

0

t⁻

N−1 m−1

Z^t0

t

τ^N−1f(τ)dτ m−¹1

dt+

+∞

Z

t0

t^{N−1+(m−N)(n−1)m−1} f(t)dt= +∞.

Proof. We rewrite the equation (1.11) in the form of the system u⁰₁=t⁻

N−1

m−1|u2|^m−11signu2, u⁰₂=−t^N−1f(t)|u1|ⁿ⁻¹signu1

and make use of Theorem 1.3.

Theorem 1.8. Letn < m, N > mand for somet0∈(0,+∞),

t0

Z

0

t^N−1f(t)dt <+∞,

+∞

Z

t0

t^N−1f(t)dt= +∞.

Then for the equation (1.10) to have no global radial positive (negative), decreasing(increasing)with respect to the radial variable solution, it is nec- essary and sufficient that

t0

Z

0

t^{N−1+(m−N)(n−m−1 1)}f(t)dt+

+∞

Z

t0

t⁻

N−1 m−1

Z^t

t0

τ^N−1f(τ)dτ m−1¹

dt= +∞.

The proof follows directly from Theorem 1.6.

2. Systems of Inequalities in the Critical Case

In this section we obtain exact conditions for the non-existence of global solution of constant sign of the system (0.2) forλ1·λ2= 1.

First we give some auxiliary statements.

Lemma 2.1. Letλ1·λ2= 1, σ∈ {−1,1}the system(0.2)have a solution defined on [t0,+∞) ((0, t0]) and possessing the property σu1(t)·u2(t) >0 for t≥t0 (t≤t0). Then the system

v₁⁰ =a1(t)|v2|^λ1signv2, v⁰₂=−a2(t)|v1|^λ2signv1 (2.1) has also a solution defined on [t0,+∞] ((0, t0])and possessing the property σv1(t)·v2(t)>0fort≥t0 (t≤t0).

Proof. For the sake of definiteness, we assume thatσ= 1. Suppose that the system (0.2) has the solutionu1(t),u2(t), defined on [t0,+∞) and possessing the propertyu1(t)·u2(t)>0 fort≥t0. Consider a solutionv1(t),v2(t) of the system (2.1) whose initial values satisfy the inequality

v2(t0) signv1(t0)

|v1(t0)|^λ1 ≥ u2(t0) signu1(t0)

|u1(t0)|^λ2 . From (0.2) it follows that

u2signu1

|u1|^λ2 0

≤ −λ2a1(t) |u2|

|u1|^λ2 1+λ1

−a2(t) for t≥t0.

Therefore, by virtue of the lemma on differential inequalities (see, e.g., [26], p. 42), we obtain

v2(t) signv1(t)

|v1(t)|^λ2 ≥ u2(t) signu1(t)

|u1(t)|^λ2 for t≥t0,

where it is clear that the function appearing in the left-hand side of the inequality is non-increasing on [t0,+∞).

Let us prove the second part of the statement of the lemma. Suppose that the system (0.2) has a solutionu1(t),u2(t) defined on (0, t0] and possessing the propertyu1(t)·u2(t)>0 fort≤t0. Consider the solutionv1(t),v2(t) of the system (2.1) whose initial values satisfy the inequality

v1(t0) signv2(t0)

|v2(t0)|^λ1 ≥ u1(t0) signu2(t0)

|u2(t0)|^λ1 .

From (0.2) it follows that u1signu2

|u2|^λ1 0

≥λ1a2(t) |u1|

|u2|^λ1 1+λ2

+a1(t) for 0< t≤t0. Therefore by virtue of the above-mentioned lemma on differential inequali- ties, we get

v1(t) signv2(t)

|v2(t)|^λ1 ≥u1(t) signu2(t)

|u2(t)|^λ1 for 0< t≤t0,

where the function appearing in the left-hand side of the inequality does not decrease on (0, t0]. The case where σ = −1 is reduced by means of the substitutionu_i = (−1)ⁱ⁻¹w3−i,v_i = (−1)ⁱ⁻¹z3−i, a_i=b3−i,λ_i =µ3−i

(i= 1,2) (since the systems (0.2), (0.3) and (2.1) are invariant with respect to the above-mentioned substitution) to the previous one.

Lemma 2.2. Letλ1·λ2= 1,σ ∈ {−1,1}, the system(2.1)have a solution defined on [t0,+∞) ((0, t0]) and possessing the property σv1(t)·v2(t)> 0 for t ≥t0 (t ≤t0). Then the system (0.3) has also a solution defined on [t0,+∞) ((0, t0]) and possessing the property σu1(t)·u2(t)>0 for t ≥t0

(t≤t0).

Proof. For the sake of definiteness, we assume σ = 1. Suppose that the system (2.1) has a solution v1(t),v2(t) defined on [t0,+∞) and possessing the propertyv1(t)·v2(t)>0 fort≥t0. Consider the solutionu1(t), u2(t) of the system (0.3) whose initial values satisfy the inequality

u2(t0) signu1(t0)

|u1(t0)|^λ2 ≥ v2(t0) signv1(t0)

|v1(t0)|^λ2 . It follows from (0.3) that

u2signu1

|u1|^λ2 0

≥ −λ2a1(t) |u2|

|u1|^λ2 1+λ1

−a2(t)

in some right half-neighbourhood oft0, and the function appearing under the sign of the derivative does not increase in the above-mentioned half- neighbourhood. According to the lemma on differential inequalities,

u2(t) signu1(t)

|u1(t)|^λ2 ≥v2(t) signv1(t)

|v1(t)|^λ2 for allt≥t0.

Let us prove the second part of the lemma. Let the system (2.1) have a so- lution v1(t), v2(t) defined on (0, t0] and possessing the property v1(t)·v2(t)>0 fort≤t0. Consider the solutionu1(t),u2(t) of the system (0.3) whose initial values satisfy the inequality

u1(t0) signu2(t0)

|u2(t0)|^λ1 ≥ v1(t0) signv2(t0)

|v2(t0)|^λ1 .

It follows from (0.3) that u1signu2

|u2|^λ1 0

≤λ1a2(t) |u1|

|u2|^λ1 1+λ2

+a1(t)

in some left half-neighbourhood of t0, and the function appearing under the sign of the derivative does not decrease in the above-mentioned half- neighbourhood. According to the lemma on differential inequalities,

u1(t) signu2(t)

|u2(t)|^λ1 ≥v1(t) signv2(t)

|v2(t)|^λ1

for allt∈(0, t0]. Forσ=−1, the statement of the lemma remains valid.

Theorem 2.1. Suppose that λ1·λ2= 1, (1.1) holds and the condition

n→+∞lim Z^t0

0

a2(t)ρ^1+λ_n ²(t)dt+

+∞

Z

t0

a1(t)r_n^1+λ1(t)dt

<+∞ (2.2) is satisfied, where

r0(t) =

+∞

Z

t

a2(τ)dτ >0, rn(t) =

=λ2 +∞

Z

t

a1(τ)r^1+λ_n−1¹(τ)dτ+

+∞

Z

t

a2(τ)dτ for t0≤t <+∞, (2.3)

ρ0(t) = Zt

0

a1(τ)dτ >0, ρn(t) =

=λ1 t

Z

0

a2(τ)ρ^1+λ_n−1²(τ)dτ +

t

Z

0

a1(τ)dτ for 0< t≤t0. (2.4) Then for the system(2.1)to have no global solution with the propertyv1(t)· v2(t)>0for0< t <+∞, it is necessary and sufficient that

r(t0)·ρ^λ2(t0)>1, (2.5) wherer(t) = limn→+∞rn(t)andρ(t) = limn→+∞ρn(t).

Proof. The necessity. Let the system (2.1) have no solution with the prop- ertyv1(t)·v2(t)>0 for 0< t <+∞. It follows from (2.2) that there exist the limits

r(t) = lim

n→+∞rn(t) for t0≤t <+∞, ρ(t) = lim

n→+∞ρn(t) for 0< t≤t0. Clearly,r(t) is a minimal solution of the equation

r⁰=−λ2a1(t)|r|^1+λ1−a2(t) (2.6)

defined on [t0,+∞) (see [37], p. 65), andρ(t) is a minimal solution of the equation

ρ⁰ =λ1a2(t)|ρ|^1+λ2+a1(t) (2.7) defined on (0, t0]; note that r(t) > 0 for t0 ≤ t <+∞ and ρ(t) > 0 for 0 < t ≤ t0. Consequently, ρ^−λ2(t) is a maximal solution of the equation (2.6) defined on (0, t0]. As far as we have assumed that the system (2.1) has no solution defined on (0,+∞) and possessing the propertyv1(t)·v2(t)>0 for 0 < t < +∞, the equation (2.6) has no solution defined on (0,+∞).

Thereforeρ^−λ2(t0)< r(t0), i.e. (2.5) holds.

The sufficiency. Let (2.5) hold. Let us show that the system (2.1) has no global solution possessing the propertyv1(t)·v2(t)>0 for 0< t <+∞.

Suppose that this is not the case, i.e., we suppose that the system (2.1) has a solution with the above-mentioned property. Then the function ^{v2(t) sign}_|v ^v1(t)

1(t)|^λ2

is a solution of the equation (2.6), defined on (0,+∞). As far as r(t) is a minimal solution of the equation (2.6) on [t0,+∞), we get

v2(t0) signv1(t0)

|v1(t0)|^λ2 ≥r(t0),

and also, taking into account the fact thatρ^−λ2(t) is a maximal solution of the equation (2.6), defined on (0, t0], we have

ρ^−λ2(t0)≥ v2(t0) signv1(t0)

|v1(t0)|^λ2 .

Consequently,r(t0)·ρ^λ2(t0)≤1, which contradicts (2.5).

Theorem 2.2. Suppose that λ1·λ2= 1, (1.8) holds and the condition

n→+∞lim Z^t0

0

a1(t)ρ^1+λ_n ¹(t)dt+

+∞

Z

t0

a2(t)r^1+λ_n ²(t)dt

<+∞, (2.8) is satisfied, where

r0(t) =

+∞

Z

t

a1(τ)dτ >0, rn(t) =λ1 +∞

Z

t

a2(τ)r_n−1^1+λ2(τ)dτ+

+

+∞

Z

t

a1(τ)dτ for t0≤t <+∞ (2.9)

and

ρ0(t) =

t

Z

0

a2(τ)dτ >0, ρn(t) =λ2 t

Z

0

a1(τ)ρ^1+λ_n−1¹(τ)dτ+

+

t

Z

0

a2(τ)dτ for 0< t≤t0. (2.10)

Then for the system (2.1) to have no global solution with the property v1(t)·v2(t)<0for0< t <+∞, it is necessary and sufficient that

r(t0)·ρ^λ1(t0)>1, (2.11) wherer(t) = limn→+∞rn(t)andρ(t) = limn→+∞ρn(t).

Proof. We putvi = (−1)ⁱ⁻¹w3−i,ai=b3−i,λi=µ3−i (i= 1,2) and make

use of Theorem 2.1.

Theorem 2.3. Letλ1·λ2 = 1 and the conditions (1.1), (2.2)–(2.4) be satisfied. If the inequality (2.5) is fulfilled, then the system (0.2) has no global solution with the property u1(t)·u2(t) >0 for 0< t < +∞; if the system (0.3) has no global solution with the property u1(t)·u2(t) > 0for 0< t <+∞, then the inequality (2.5) holds.

Proof. Let the inequality (2.5) hold. Make sure that the system (0.2) has no solution with the propertyu1(t)·u2(t)>0 for 0< t <+∞. Indeed, if we assume the contrary that the system (0.2) has a solution with the property u1(t)·u2(t) > 0 for 0 < t < +∞, then this by Lemma 2.1 will mean that the system (2.1) has a solution with the property v1(t)·v2(t)>0 for 0< t <+∞. But this is impossible because by Theorem 2.1 the condition (2.5) is sufficient for the non-existence of solutions with the above-mentioned property.

Let the system (0.3) have no solution with the propertyu1(t)·u2(t)>0 for 0 < t < +∞. Let us show that in this case the inequality (2.5) is satisfied. Indeed, if this is not the case, then by Theorem 2.1 system (2.1) has a solution with the property v1(t)·v2(t) > 0 for 0 < t < +∞. But then by Lemma 2.1, the system (0.3) has also a solution with the property u1(t)·u2(t)>0 for 0< t <+∞, which contradicts our supposition.

Theorem 2.4. Letλ1·λ2= 1, and let the conditions(1.8),(2.8)–(2.10) be satisfied. If the inequality(2.11)is fulfilled, then the system(0.2)has no global solution with the property u1(t)·u2(t) <0 for 0< t < +∞; if the system (0.3) has no global solution with the property u1(t)·u2(t) < 0for 0< t <+∞, then the inequality (2.11)holds.

Proof. Putui= (−1)ⁱ⁻¹v3−i,ai=b3−i,λi=µ3−i (i= 1,2) and make use

of Theorem 2.3.

In conclusion, we present new criteria for the non-existence of conju- gate points and an original characteristic of the principal solution (for the definition see [37], p. 95) of the system (2.1).

Theorem 2.5. Letλ1·λ2= 1, and let(1.1), (2.2)–(2.4)hold. Then for every solutionv1(t),v2(t)of the system(2.1)the componentsv_i(t) (i= 1,2) have on (0,+∞)not more than one zero if and only if

r(t0)·ρ^λ2(t0)≤1. (2.12)

Proof. The necessity. Let for every solution v1(t), v2(t) of the system (2.1) the componentsvi(t) (i = 1,2) have on (0,+∞) not more than one zero. Let us show that (2.12) holds. Indeed, if we assume that this is not the case, then (2.5) is fulfilled. Let us take α ∈ (ρ^−λ2(t0), r(t0)) and consider the solutionx(t) of the equation (2.6) which is defined by the initial conditionx(t0) =α. Sincer(t) is a minimal solution of the equation (2.6) defined on [t0,+∞), andρ^−λ2(t) is a maximal solution of the equation (2.6) defined on (0, t0], there exist numbers t1 ∈(0, t0) and t2 ∈ (t0,+∞) such that limt→t1+x(t) = +∞and limt→t2−x(t) =−∞. The latter means that for the solution v1(t), v2(t) of the system (2.1) which corresponds to the solution x(t) of the equation (2.6), the component v1(t) has two zeros t1

andt2, which contradicts our assumption.

Analogously, let us takeβ ∈(r^−λ1(t0), ρ(t0)) and consider the solution y(t) of the equation (2.7) defined by the initial conditiony(t0) =β. Since r^−λ1(t) is a maximal solution of the equation (2.7) defined on [t0,+∞), andρ(t) is a minimal solution of the same equation defined on (0, t0], there exist numberst1∈(0, t0) andt2∈(t0,+∞) such that limt→t1+y(t) =−∞

and limt→t2−y(t) = +∞. But this implies that for the solutionv1(t),v2(t) of the system (2.1) which corresponds to the solutiony(t) of the equation (2.7), the componentv2(t) has two zerost1 and t2, which contradicts our assumption.

The sufficiency. Let (2.12) hold. Then the solution x(t) of the equation (2.6) defined by the initial value x(t0) ∈ [r(t0), ρ^−λ1(t0)] will be given on (0,+∞). If we assume that for some solution v1(t), v2(t) of the system (2.1) the component v1(t) has two different zeros t1 < t2 on the interval (0,+∞), then this will imply that the function ^{v2(t) sign}_|v1(t)|^λv₂^1(t) is a solution of the equation (2.6) and

t→tlim1+

v2(t) signv1(t)

|v1(t)|^λ2 = +∞, lim

t→t2−

v2(t) signv1(t)

|v1(t)|^λ2 =−∞.

But then there exists a point τ ∈(t1, t2) such that x(τ) =v2(τ) signv1(τ)

|v1(τ)|^λ2 .

This contradicts the uniqueness of a solution of the Cauchy problem for the equation (2.6).

Analogously, if we assume that for some solution of the system (2.1) the component v2(t) has two different zeros t1 < t2 on the interval (0,+∞), then the function ^{v1(t) sign}_|v ^v2(t)

2(t)|^λ1 is a solution of the equation (2.7) and

t→tlim1+

v1(t) signv2(t)

|v2(t)|^λ1 =−∞, lim

t→t2−

v1(t) signv2(t)

|v2(t)|^λ1 = +∞.

Therefore for the solution y(t) of the equation (2.7) defined by the initial value y(t0) ∈ [ρ(t0), r^−λ2(t0)] and given on (0,+∞) there exists a point

τ ∈(t1, t2) such that

y(τ) = v1(τ) signv2(τ)

|v2(τ)|^λ1 .

As before, we have a contradiction since the solution of the Cauchy problem for the equation (2.7) is unique.

Remark. Note that (2.12) is a necessary and sufficient condition for any component of each solution of the system (2.1) to have not more than one zero on (0,+∞).

Theorem 2.6. Let λ1·λ2 = 1and (1.8), (2.8)–(2.10) hold. Then for every solutionv1(t),v2(t)of the system(2.1)the componentsvi(t) (i= 1,2) have on (0,+∞)not more than one zero if and only if

r(t0)·ρ^λ1(t0)≤1.

Proof follows from Theorem 2.5 if we make the substitutionvi= (−1)ⁱ⁻¹× w3−i,ai=b3−i,λi =µ3−i (i= 1,2).

Theorem 2.7. Letλ1·λ2= 1 and the conditions

+∞

Z

t0

a1(t)dt= +∞, 0≤

+∞

Z

t

a2(τ)dτ <+∞,

r0(t) =

+∞

Z

t

a2(τ)dτ, rn(t) =λ2 +∞

Z

t

a1(τ)r_n−1^1+λ1(τ)dτ+

+∞

Z

t

a2(τ)dτ,

n→+∞lim rn(t) =r(t) for t≥t0

be satisfied(note that in this theorema2(t)may be a function with alternat- ing signs on[t0,+∞)). Then the solutionv1(t), v2(t)of the system(2.1)is principal if and only if the equality

v2(t) signv1(t)

|v1(t)|^λ2 =r(t) for t≥t0 (2.13) is fulfilled or, what comes to the same thing, that

v1(t) =v1(t0) exp

t

Z

t0

a1(τ)r^λ1(τ)dτ for t≥t0. (2.14)

Proof. The functionr(t) is a minimal solution of the equation (2.6) given on [t0,+∞), and hence the bounding solution of that equation [37]. Conse- quently, the solutionv1(t),v2(t) of the system (2.1) is principal if and only if (2.13) holds. By virtue of (2.1), the equality (2.14) follows from (2.13).

3. Systems of Inequalities in the Supercritical Case In this section we present sufficient conditions for the non-existence of global solutions of constant sign of the system (0.2) in the caseλ1·λ2>1.

Theorem 3.1. Let λ1 ·λ2 > 1 and (1.1) be satisfied. Then for the system(0.2) to have no global solution with the propertyu1(t)u2(t)>0for 0< t <+∞, it is sufficient that the equality

t0

Z

0

a2(t) Z^t

0

a1(τ)dτ λ2

dt+

+

+∞

Z

t0

a1(t) ^+∞Z

t

a2(τ)dτ λ1

dt= +∞ (3.1)

be fulfilled.

Proof. Assume the contrary, i.e., suppose that the first summand in (3.1) is equal to +∞and, nevertheless,u1(t)·u2(t)>0 for 0< t <+∞for some solutionu1(t),u2(t) of the system (0.2). Then (1.3) holds. Therefore

|u2(t)|^−λ1·λ2|u2(t)|⁰ ≤ −a2(t) Z^t

0

a1(τ)dτ λ2

for 0< t≤t0. Integrating the last inequality from t to t0 and passing to the limit as t→0+, we come to the contradiction.

Assume now that the second summand in (3.1) is equal to +∞ and, nevertheless, u1(t)·u2(t) > 0 for 0 < t < +∞ for some solution of the system (0.2). Then from (1.3) it follows that

|u1(t)|^−λ1·λ2|u1(t)|⁰≥a1(t) ^+∞Z

t

a2(τ)dτ λ1

for t≥t0.

Integrating the last inequality fromt0totand passing to limit ast→+∞,

we arrive at the contradiction.

Theorem 3.2. Let λ1 ·λ2 > 1 and (1.8) be satisfied. Then for the system(0.2) to have no global solutions with the property u1(t)·u2(t)<0 for 0< t <+∞, it is sufficient that

t0

Z

0

a1(t) Z^t

0

a2(τ)dτ λ1

dt+

+∞

Z

t0

a2(t) Z^+∞

t

a1(τ)dτ λ2

dt= +∞. (3.2)

Proof. Putui= (−1)ⁱ⁻¹v3−i,ai=b3−i,λi=µ3−i (i= 1,2) and make use

of Theorem 3.1.

We do not know whether the conditions (3.1) or (3.2) are necessary for the system (0.3) to have no global solution of constant sign on (0,+∞).

Note that Theorems 1.2 and 1.5 hold true forλ1·λ2>1 as well. To see that this is so, we have to repeat the proofs of these theorems by putting

1

λ1 < γ < λ2and choosing|u10| sufficiently small.

It should also be noted that the conditions (1.2) or (1.9) in the case λ1·λ2>1 are not sufficient for the system (0.2) to have no global solutions of constant sign on (0,+∞).

As an example, consider the system

u⁰₁=t⁻²u2, u⁰₂=−3t²|u1|⁵signu1

for which the conditions (1.8) and (1.9) are fulfilled. Nevertheless, this system has a global solution of constant sign on (0,+∞):

u1(t) = (1 +t²)⁻¹², u2(t) =−t³(1 +t²)⁻³². Acknowledgement

The author should like to express his sincere gratitude to Z. Doˇsla, O.

Doˇsly, A. Lomtatidze, B. P˚uˇza and other participants of the Seminar on the Theory of Differential Equations of Masaryk University in Brno for valuable discussions on obtained results.

The present paper has been written under the financial support of NATO Science Fellowships Programme.

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(Received 15.08.2002) Author’s address:

Faculty of Mathematics Adyghe State University

208 Universitetskaya St., Maykop 385000 Adyghe Republic, Russia