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ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ, ДИНАМИЧЕСКИЕ СИСТЕМЫ И ОПТИМАЛЬНОЕ УПРАВЛЕНИЕ
УДК 517.91
ОБ АСИМПТОТИЧЕСКОЙ КЛАССИФИКАЦИИ РЕШЕНИЙ НЕЛИНЕЙНЫХ УРАВНЕНИЙ ТРЕТЬЕГО И ЧЕТВЕРТОГО ПОРЯДКОВ СО СТЕПЕННОЙ НЕЛИНЕЙНОСТЬЮ
И.В. Асташова
Московский государственный университет им. М.В. Ломоносова, Москва, Российская Федерация
Московский государственный университет экономики, статистики и информатики (МЭСИ), Москва, Российская Федерация e-mail: ast@diffiety.ac.ru
Исследовано асимптотическое поведение всех решений нелинейных дифференциальных уравнений типа Эмдена - Фаулера третьего и четвертого порядков. Приведены ранее полученные автором настоящей статьи результаты. Уравнение n-го порядка сведено к системе на (n — 1)-мерной сфере. С помощью исследования асимптотического поведения всех возможных траекторий системы получена асимптотическая классификация решений исходного уравнения.
Ключевые слова: нелинейное дифференциальное уравнение высокого порядка, асимптотическое поведение решений, качественные свойства, асимптотическая классификация решений.
ON ASYMPTOTIC CLASSIFICATION OF SOLUTIONS TO NONLINEAR THIRD- AND FOURTH-ORDER DIFFERENTIAL EQUATIONS WITH POWER NONLINEARITY
I.V. Astashova
Lomonosov Moscow State University, Moscow, Russian Federation Moscow State University of Economics, Statistics and Informatics (MESI), Moscow, Russian Federation e-mail: ast@diffiety.ac.ru
The asymptotic behavior of all solutions to the fourth and the third order Emden -Fowler type differential equation is investigated. The author's previously obtained results are supplemented. The equation of the n-th order is transformed into a system on the (n — 1)-dimensional sphere. By the investigation of asymptotic behavior to all possible trajectories of this system the asymptotic classification of all solutions to the equation is obtained.
Keywords: nonlinear higher-order ordinary differential equation, asymptotic behavior, qualitative properties, asymptotic classification of solutions.
Introduction. The investigation of asymptotic behavior of solutions to nonlinear differential equations near the boundaries of their domain and the classification of all possible solutions to this equations is one of the major problems in qualitative theory of differential equations. This problem is one of the most important because there are no general
methods for investigation of qualitative properties of solutions to nonlinear differential equations. Note that Emden-Fowler equation appears for the first time in [1]. Its physical origin is also described in [2]. This equation was investigated in detail in the books [3, 4], and later in [5]. See also [6, 7] and references. Asymptotic properties of solutions to different generalizations of this equation were investigated in [8-35]. The results concerning asymptotic behavior of solutions to nonlinear ordinary differential equations is used to describe the properties of solutions to nonlinear partial differential equations. See, for example, [36-40].
In this article the asymptotic classification of all possible solutions to the fourth order Emden - Fowler type differential equations
yIV(x) + Po |y|k-1 y(x) = 0, k > 1, po > 0 (1)
and
yIV(x) - po |y|k-1 y(x) = 0, k> 1, po > 0 (2)
is given.
The asymptotic classification of all possible solutions to the third order Emden - Fowler type differential equations
yIII(x)+ p(x) |y|k-1 y(x) = 0, k > 1, p(x) > 0 (3)
is described.
For fourth-order nonlinear equations, the oscillatory problem was investigated in [10, 13, 14, 17, 21, 28, 29, 31, 33, 35], in linear case — in [41].
Phase Sphere. Note that if a function y(x) is a solution to equation (1), the same is true for the function
z(x) = Ay(Bx + C), (4)
where A = 0, B > 0, and C are any constants satisfying
|A|k-1 = B4. (5)
Indeed, we have
zIV(x) + po |z|k-1 z(x)= AB4yIV(Bx + C) +
+Po |Ay(Bx + C)|k-1 Ay(Bx + C) =
= AyIV(Bx + C) [b4 - |A|k-^ = 0.
Any non-trivial solution y(x) to equation (1) generates a curve (y(x),y'(x),y"(x),y"'(x)) in R4\{0}. Let us introduce in R4\{0} an equivalence relation such that two solutions connected by (4), (5) generate equivalent curves, i.e. the curves passing through equivalent points (may be for different x).
We assume that points (y0, yi, y2, y3) and (z0, z3, z2, z3) in R4\{0} are equivalent if there exists a positive constant A such that Zj = A4+j(k_1)yj, j = 0,1, 2,3.
The factor space obtained is homeomorphic to the three-dimensional sphere S3 = {y e R4 : y0 + y2 + y2 + y2 = 1} . On this sphere there is exactly one representative of each equivalence class because for any point (y0,y1,y2,y3) e R4\{0} the equation A8y0 + A2k+6y2 + A4k+4y2 + + A6k+2yf = 1 has exactly one positive root A.
It is possible to construct another hyper-surface in R4 with a single representative of each equivalence class, namely,
E = jy e R4 : ^ |yj|^^ = 1 j . (6)
We define $s : R4\{0} ^ S3 and $E : R4\{0} ^ E as mappings taking each point in R4\{0} to the equivalent point in S3 or E. Note that the restrictions \E and \S3 are inverse homeomorphisms.
Lemma 1. There is a dynamical system on the sphere S3 such that all its trajectories can be obtained by the mapping from the curves generated in R4\{0} by nontrivial solutions to equation (1). Conversely, any nontrivial solution to equation (1) generates in R4\{0} a curve whose image under is a trajectory of the above dynamical system.
< First we define on the sphere S3 a smooth structure using an atlas consisting of eight charts.
The two semi-spheres defined by the inequalities y0 > 0 and y0 < 0 are covered by the charts with the coordinate functions (respectively u+, u+,
4+j(k-1)
u+ and u-, u-, u-) defined by the formulae u± = yj \y0\ 4 sgny0, j = 1, 2,33.
The semi-spheres defined by the inequalities y3 > 0 and y3 < 0 are covered by the charts with the coordinate functions (respectively v+, v+, v+ and v-, v-, v-) defined as
4+j(fe-1)
vj = yj \yi\ k+3 sgnУ1, j = 0, 2 3.
The semi-spheres defined by the inequalities y2 > 0 and y2 < 0 are covered by the charts with the coordinate functions (respectively w+, w+,
4+j(fe-1)
w+ and w-, w-, w-) defined as w± = yj \y2\ 2k+2 sgny2, j = 0,1,3.
Finally, the semi-spheres defined by the inequalities y3 > 0 and y3 < 0 are covered by the charts with the coordinate functions (respectively ,
11 _i_ 4+j(k-1)
g+,g+ and g- g- g2_) defined as gj = yj \y3\ 3k+1 sgny3, j = 0, 1 2.
Note that each of these coordinate functions can be defined by its own formula on the whole corresponding semi-space (yj^0) and it takes
equivalent points to the same value. This fact facilitates description of the trajectories generated on S3 by solutions to equation (1). To be more precise, by their restrictions on the intervals where some derivative has constant sign.
E.g., when a solution is positive, the trajectory generated can be described by the following differential equations:
+
du+ dx
_k+3 k + 3 ,2, ,_k±I
= y"\y\ 4 sgny -
-y
4 =
( + k + 3
du+ dx
= y"
du+ dx
u+ —
2k+2
4
-u
+2
2k+6 " 4 =
k_1 IT
= -P0
2k + 2
4 sgn y--4— y y
+ 2k + 2 + +
u3 4 ul u2 I ;
,k_3k+1 3k + 1 . .... ,_3k+5 \k 4--y' y'" \y\ 4 =
4
k_1 ( 3k + 1
4
-Po
4
u+u+
Parameterizing it by tu =
kl
4 dx, we obtain its internal description
in terms of u
+
xo
du+ dtu du2+ dtu du3+ dt„,
u+
k+3
u
u+
-Po
4
2k + 2
+2 1 ;
++
4
3k + 1 1
uu
l2
u+u+.
The same equations appear for (u-, u2, u3). Similar calculations yield equations for other charts:
dv±
dtv
dv±
dtv
dv±
dtv
=1
v3± -
4
k+3 2k + 2 2 ТГ3 ;
I ±|k ±
-Po Iv0 I sgn vo 3k + 1 k + 3 2 3 '
dw± dt
dw± dt
dw±
dt
Uybw
= —
4
2k + 2
=1
k+3
тл--Wi ;
2k + 2 1 3
II II k = —Po I Wo I sgnw3
3k + 1 2k + 2
w±2,
1
x
dg,
dtq dg±
= g± +
4
3k + 1
I ±|k+l
Po |g0 I
dtq dg±
dt,
q
± k + 3 ± = g0 + 3k+îPog 0
1 2k + 2 ± 1 = 1 + 3k+i Pogo
± |k ± g± | sgn g± ;
±ik ± g± | sgn go .
Using a partition of unity one can obtain a dynamical system on the whole sphere S3 to describe all trajectories generated by nontrivial solutions to equation (1). ►
Typical and Non-Typical Solutions. Now we consider the space R4 as the union of its 16 = 24 closed subsets defined according to different
combinations of signs of the four coordinates. Denote these sets by
±
± ± ±
С R4, where each sign ± can be substituted by +, or —, or 0 (for
boundary points). For example,
+ +
0
= {y G R4 : yo > 0, y I > 0, y 2 = 0,У3 < 0 J .
Besides, let П_ and denote respectively
+ + + + + + 1 1 1 —
+ и + + и + и и + + и + и + + и +
and
+ + + + — — — —
+ + + — — — — +
+ и + и — и — и — и — и + и +
+ — — — — + + +
Note, that the sets Q- and cover the whole space R4, intersect only along their common boundary, and can be obtained from each other using the mapping (y0,yi,y2,ys) G R4 ^ (yo, -yi,y2, -y3) G R4, which corresponds to changing the sign of the independent variable (x ^ —x).
Lemma 2. The sets Q- n S3, n S3, Q- n E, and n E are homeomorphic to the solid torus.
< It is sufficient to consider n S3. The set is the union of its two homeomorphic subsets
Q++ =
and
=
+ + + +
+ + + -
+ и + и - и —
+
- и - и + и + +
+ + +
In order to describe the set П$3, we use the stereographic projection S3\{(-1,0,0,0)} ^ R3 (Fig. 1).
The image of Q++ П S3 under this projection is contained in the ball of radius 2 and is equal to the union of its two quarters, which is homeomorphic to the 3-dimensional ball. The same is true for П S3.
The intersection (П++ П S3)n(fi+_ П S3) =
V
0 0
+ —
+ и —
+ —
\
/
S3
maps to the disjoint union of two spherical triangles (2-dimensional figures,
Fig. 1. Stereographic projection and its image of П++ П S3
not their boundaries). Thus, the set n S3 is homeomorphic to the pair of two balls glued along two disjoint triangles, which is equivalent to the solid torus. ►
Lemma 3. Any trajectory in R4 generated by a non-trivial solution to (1) either completely lies inside one of the sets Q_ and (i.e., in their interior), or consists of two parts, first inside Q_ and another inside with a single point in their common boundary.
< For the trajectories generated by solutions to equation (1), consider
±
all possible passages between the sets
± ± ±
Inside the only possible passages are
+ + + +
+ + + —
+ + — —
+ — — —
t г 1 г 1 1
+ + + _ _
+ + + —
ey are
" + " " + " " + " " + "
— — — +
+ + — —
— + + +
1 г 1 г 1 t
+ + + + +
+
(7)
(8)
and the only possible passages between and are
" + " " + " " + " — " + " " + "
+ — — — — +
+ + — , + + +
— — — + + +
— " + " " + " " + " " + " " + "
— — — + + +
+ + — + + + —
— + + " + " + + —
+ + + + — — —
" + " + + + — — —
+ + + , — + +
— — + — — +
always from to Q+.
So, any trajectory generated by a non-trivial solution can perform only one passage between and Q+, which can be only from Q- to Q+. ►
Lemma 4. There exist trajectories of all three types mentioned in Lemma 3, namely
• trajectories lying completely in
• trajectories lying completely in
• trajectories with a single passage ^ Q+.
M Any solution to (1) with initial data corresponding to a point from
H generates a trajectory of the 3rd type. E.g., the solution with initial data y'(0) = 0, y(0) = y"(0) = y'"(0) = 1 generates a trajectory with the passage
" + " " + "
— +
+ С П- ^ +
+ +
С П
If there exists a solution y(x) to (1) generating a trajectory lying completely in Q_, then the function z(x) = y(-x) is also a solution
to (1) and generates a trajectory completely lying in Q+. So, we have to prove existence of a trajectory of the first type.
Assume the converse. Then any trajectory passing through a point s e n S3 must reach the boundary n S3. Thus we obtain the mapping tt_ n S3 — dQ_ n S3.
To prove its continuity we represent it as se^_nS3 — Trajp0 (s, £(s)) e e n S3.
Here Trajp0 (s, t) is the point in S3 reached at the time t by the trajectory of the dynamical system on the sphere that passed s at the time 0. The mapping Trajpo : S3 x R — S3 is continuous according to the general properties of differential equations.
The function £ : n S3 — R gives the time t at which the trajectory passing through the given point of at t0 = 0 reaches dQ_. Now we prove continuity of £.
Suppose £(si) = ti and e > 0. Then, since Trajpo(si,ti + e) is inside Q+, there exists a neighborhood U+ of s1 such that for any s e U+ the point Trajp0 (s, t1 + e) is also inside Q+. So, we have £(s) < t1 + e for all s e U+.
Similarly, since Trajp0 (s1 ,t1 — e) is inside Q_, there exists a neighborhood U_ of s1 such that for any s e U_ the point Trajp0 (s,t1 — e) is also inside Q_, whence £(s) > t1 — e.
So, for all s e U_ nU+ we have |£(s) — t11 < e. Thus £(s) is continuous on Q_ n S3 and we have the continuous mapping Q_ n S3 — dn S3 whose restriction to dn S3 is the identity map. In other words, we have the composition dQ_ n S3 — Q_ n S3 — dQ_ n S3, which is the identity map, inducing the identity map on the homology groups:
H2(dn S3) — H2(n_ n S3) — H2(dn S3).
Since n S3 and dn S3 are homeomorphic to the solid torus and the torus surface respectively, the above composition can be written as Z — 0 — Z, which cannot be the identity mapping. This contradiction proves the lemma. ►
Lemma 5. Suppose y(x) is a non-trivial solution to equation (1) maximally extended to the right. Then neither y (x) nor any of its derivatives y'(x), y"(x), y'"(x) can have constant sign near the right boundary of their domain.
< We prove it for y(x). For the derivatives the proof is just similar.
Suppose y(x) is defined on an interval (x_, x+), bounded or not, and is positive in a neighborhood of x+. Then y'"(x), due to (1), is monotonically decreasing to a finite or infinite limit as x —y x+. Then y'"(x) ultimately has a constant sign. In the same way, y"(x), y'(x), and y(x) itself are all ultimately monotone and have finite or infinite limits as x — x + .
Suppose < If either of the limits mentioned is finite, then all other limits are finite, too, which is impossible for a maximally extended solution. If all limits are infinite, they must have the same sign, which contradicts to equation (1).
Now suppose = If either of the limits mentioned is nonzero, then all limits must be infinite and have the same sign, which contradicts to equation (1). If all these limits are zero, then y(x), which is ultimately positive, is decreasing to 0. Hence, y'(x) is ultimately negative and increasing to 0. Similarly, y"(x) is ultimately positive and decreasing to 0, y'"(x) is ultimately negative and increasing to 0, which contradicts to equation (1), since y(x) is ultimately positive. These contradictions prove the lemma. ►
Thus, no trajectory generated in R4 by a non-trivial solution to (1) can
±
± ± ±
ultimately rest in one of the sets
Corollary 1. All maximally extended solutions to equation (1), as well as their derivatives, are oscillatory near both boundaries of their domains.
Note that according to Lemma 3 we can distinguish two types of asymptotic behavior of oscillatory solutions to equation (1), near the right boundaries of their domains.
Definition 1. An oscillatory solution to equation (1) is called typical (to the right) if ultimately this solution and its derivatives change their signs according to scheme (7), and non-typical if according to (8).
Asymptotic Behavior of Typical Solutions. This section is devoted to the asymptotic behavior of typical (to the right) solutions to equation (1), i.e. those generating trajectories ultimately lying inside Q+.
Since such a trajectory ultimately admits only the passages shown in (1), there exists an increasing sequence of the points x0'' < x0' < x0 < x0 < < x"' < x'3 < xi < x3 < ... such that y(xj) = y'(xj) = y''(xj') = = y'''(xj'') = 0 (j = 1, 2,...), and each point is a zero only for one of the functions y(x), y'(x), y''(x), y'''(x) (Fig. 2). The points xj, xj, xj', xj'' will be called the nodes of the solution y(x).
For solutions generating trajectories completely lying inside Q+, the sequences of their nodes can be indexed by all integers (negative ones, too).
Lemma 6. Any typical solution y(x) to equation (1) satisfies at its nodes the following inequalities:
|y(xj)| < |y(xj'+1)| < |y(xj+ i)| < |y(xj+i)| ; (9)
Fig. 2. Zeroes of the derivatives of a typical solution
| y' (xj )|< |y' (Xj )| <| y' (xj+! )|<|y' j) |; \y'' W )| < |y'' (xj )| <|y" (xj )| <|y'' (xj+ 1 )|;
|y(sj)l < |y'''j)| < |y'"(xj+i)| < |yW(xj+i)|.
< Indeed,
(10) (ii) (12)
"з+i
Po k + 1
xj )lfc+1 - |y (xj'+1 )|к+1) = - po / y'(x) |y (x)|k 1У(x) dx =
"з+i
"з+i
= I y'(x)yIV(x)dx = y'(x)V''(x)
3+1
y''(x)y'''(x) dx < 0,
since y"(x)y'''(x) > 0 for all x G [xj , xj'+ ^ and y'(xj) = y'"(xj+1) = 0. This gives the first of inequalities (9)), whereas the rest inequalities follow from y(x)y'(x) > 0 on the interval [xj'+1, xj+1) .
Similarly, for the first of inequalities (10) we have y' (xj')2 — y' (xj )2 =
= —2 / y'(x)y''(x) dx = —2y(x)y''(x)
+ 2
"з
J y (x)y''' (x) dx
< 0, since
y(xj) = y''(xj') = 0 and y(x)y///(x) < 0 on [xj',xj) . The rest ones follow from the inequality y'(x)y''(x) > 0 on [xj ,xj'+1) . In the same way, for the first of (11) we have
y'' (xj'' )2 — y'' (xj )2 = — 2 y'' (x)y''' (x) dx =
= —2y'(x)y'''(x) 3 +2 I y'(x)yIV(x)dx< 0,
X
з
X
X
since y'(x)yIV(x) = —p0 |y|k-1 y(x)y'(x) < 0 on [x"",x") and y'(x") = = y'"(x"-') = 0. The rest ones follow from y"(x)y///(x) > 0 on |x", x"+1 Finally, for the first of (12) we have
■"j+1
V"\xj)2 - y"/(xj'+1)2 = —2 / y,"(x)yIV(x) dx =
r"
= 2pW y"/(x)y(x) |y(x)|k 1 dx =2Poy"(x)y(x) |y(x)|k 1
rj+i Xj
Г j+i k~1
—2kp0 / y"(x)y'(x) |y(x)| dx < 0,
since y'(x)y"(x) > 0 on [x", xj'+J and y(x") = y"(x"'+1) = 0, whereas the rest inequalities follow from y'"(x)yIV(x) > 0 on [x"'+1, x^) . ►
So, the absolute values of the local extrema of any typical solution to equation (1) form a strictly increasing sequence. The same holds for its first, second, and third derivatives.
Hereafter we need some extra notations. Put Q+=Traj1(n+nS3,1)cS3. This is a compact subset of the interior of Q+ containing ultimate parts of all trajectories generated by maximally extended typical solutions to equation (1) with p0 = 1. As for solutions generating the curves in R4 completely lying in Q+, the trajectories related completely lie in Q+.
Besides, we define the compact sets K = {a e Q+ : a = 0} and the functions " : R4\{0} ^ R, j = 0,1, 2, 3, taking each a e R4\{0} to the minimal positive zero of the derivative y(j)(x) of the solution to the initial data problem
yIV(x) + y(x) |y(x)|k-1 = 0; (13)
y(")(0) = a", j =0,1,2,3. V ;
Further, to each solution y(x) to equation (1) we associate the function
3 1 1
Fy(x) = |py°°(x)| "(k-1)+4 with p = pk-1. The notation Fy does not
"=0
use p0, since non-trivial Unctions cannot be solutions to equation (1) with different p0.
Lemma 7. The restrictions , i, j = 0,1, 2, 3, are continuous.
< First we prove continuity of at a e Q+ with a,t > 0. Suppose £i(a) = xi and e > 0.
We can assume that e is sufficiently small to be less than xi and to provide, for the solution y(x) to (13), the inequalities y(i)(x — e) > 0 on [0, xi — e] and y(i)(xi + e) < 0. In this case the point a has a neighborhood
j
U C such that the above inequalities are satisfied for all solutions to (13) with initial data a' G U. Hence, |&(a') — xi| < e. Continuity of ^ at a G with ai > 0 is proved.
In the same way it is proved at a G with ai < 0. Since ai = 0 if a G Kj, i = j, we have proved continuity of the restriction in the case i = j.
As for , note that between two zeros of y(i)(x) there exists a zero Xj of another derivative y(j)(x). The values y(m)(xj), m = 0,1,2,3, due to continuity of |Ki, depend continuously on a G Ki, whereas the restriction depends continuously on these values. This proves continuity of the restriction . ►
Lemma 8. For any k > 1 there exist Q > q > 1 such that for any typical solution y(x) to equation (1) the values of all expressions
1 )
y(xj) y/(xj)
y'(xj')
i
fe+3
y(xj')
y'j1)
y(xj)
1
fe+3
y(xj) y'(xj')
y'(xj")
1
fe+3
y"(xj) 2k+2 y"(xj) 2fc+2 y"(xj'+1) 2fc+2
y"(xj") y"(xj) y"(xj)
У'К-н)
1
3fe+1
/'(xj )
1
3fe+1
i
3k + 1
with sufficiently large j are contained in the segment [q, Q].
< Let us define the continuous functions : K ^ R (all indices i,j,1 are from 0 to 3 and pairwise different) taking each point a G K to the ratio of the absolute values of the j-th derivative of the solution y(x) to (13) at 0 and at the next point where the l-th derivative vanishes,
(both the numerator and the denominator are
Le- **(a) = (a))
non-zero if a G Ki).
Due to Lemma 6, each function ^7 at all points of the compact set Ki is positive and less than 1. Hence 0 < inf ^7 (a) < sup ^7 (a) < 1.
K Ki
Now consider an arbitrary typical solution y(x) to (1) and two its nodes, say xj and xj+1, with sufficiently large numbers such that the related points in S3 belong to Q+. In this case we can choose constants A = 0 and B > 0 such that the function z(x) = Ay(Bx + xj) is a solution to (13) with a G Ki. Indeed, this is equivalent to existence of A = 0 and B > 0 such
4
1
1
1
that
|A|k-1 = B4po; £ (ABmy<™)(x;. ))2 = 1,
m=0,2,3
which follows from existence of a root A to the equation
3 3^+1
(y(xj))2 A2 + (y"(xj))2p-1 |A|k+1 + (y'"(xj))2p-2 |A|-+- = 1. y(xj'+1) 1
The value
y(X )
is equal to this for z(x) at Ç3(a) and 0, where
1
ao = |A|, a1 = 0, a2 = |A| B2, «3 = |A| B3, i.e. equal to ^>103 (a) 4. Put _ 1 _ 1
q = ^sup^103(a)^ , Q = ^inf -^103(a)^ and obtain the statement of
the lemma for the first ratio. The same procedure can be used for others. Then we just choose the minimum of 12 values of q and the maximum of 12 values of Q. ►
Lemma 9. The domain of any typical (to the right) solution y(x) to equation (1) is right-bounded. If x* is its right boundary, then
lim |y(n)(x)| = n = 0,1,2,3. (14)
x—>-x*
M It follows from Lemma 8 that the absolute values of the neighboring local extrema of any typical solution for sufficiently large number, say for j > J, satisfy the inequality |y(xj+1)| > q12 |y(xj )| with some q > 1, whence
|y(xj)|> q12(j-J) |y(xJ)| . (15)
In particular, this yields (14) for n = 0. Other n are treated similarly. It is proved in [7] that there exists a constant C > 0 depending only on k and p0 such that all positive solutions to equation (1) defined on
4
a segment [a, 6] satisfy the inequality |y(x)| < C |b — a| k. The same
holds for negative ones. Hence the local extrema satisfy the estimate
_ 4
|y(xj)| < C (xj — Xj-1) k-1 , which yields, together with (15), the
inequality (xj — x^) < Q-3(k-1)(j-J)
с
V(x'j)
It follows from Q > 1 that Q-3(k-1) < 1, ^ (xj — xj-1) < to, and
j=J
the domain is right-bounded. ►
Lemma 10. For any k > 1 there exists positive constants m < M such that for any typical solution y(x) to equation (1) the distance between
its neighboring points of local extremum, xj and xj+, ultimately satisfies the estimates
m < (xj+1 - xj)Fy(xj)k-1 < M. (16)
< Put E+ = $E (Q+) . It is a compact subset of the set E defined by (6) and lying inside Q+. Put
m = inf |^1(a) : a £ E+, a1 = 0} > 0;
M = sup |^1(a) : a £ E+,a1 =0} < to.
Let y(x) be a typical solution to equation (1), xj and xj+1 be neighboring points of its local extremum. We can choose positive constants A and B such that the function z(x) = Ay(Bx+xj) is a solution to equation (1) with p0 = 1 and its data at zero correspond to some point in E, i.e. Fz(0) = 1. It is sufficient for this to find a positive solution to the system
Ak-1 = B4po;
3 1
Y^ |ABmy(m)(xj)| m(k-1)+4 = 1,
m=0
namely
A =
B =
-4
y(m)(xj) m(k-1)+4
4 Po4
1
-(k-1)
РУ
(m) I
(xj)l m(k-1)+4
= Fy (Xj )-(k-1).
Moreover, for local extrema with sufficiently large numbers, the point defined in R4 by the data of the function z(x) at zero belongs to E+. Hence the first positive point L of local extremum of z(x) belongs to [m, M], whence the difference xj+1 — xj is equal to LB and satisfies (16). ►
Lemma 11. For any k > 1 and p0 > 0 there exists a constant 9 > 0 such that local extrema of any typical solution y(x) to equation (1), ultimately satisfy the inequality |y(xj)| > 9Fy(xj)4.
< Let y(x) be a typical solution to equation (1) and xj be its local extremum point with sufficiently large number. Put 9 = inf{|a0| : a £ E+,a1 = 0} > 0 and choose a constant A > 0 such that the data at zero for the solution z(x) = A4y(Ak-1x + xj) correspond to some
point in E+. Then Fz(0) = 1 and |z(0)| > 9. Since z(0) = A4y(xj) and Fz(0) = AFy(xj), the lemma is proved. ►
Remark 1. For typical solutions to (1) with their corresponding curves lying completely in Q+, the statements of Lemmas 8, 10, and 11 hold in the whole domain, not only ultimately.
1
Theorem 1. For any real k > 1 and > 0 there exist positive constants Ci and C2 such that local extrema of any typical maximally extended to the right solution y(x) to equation (1) in some neighborhood of
_ 4
the right bound x* of its domain satisfy the inequalities Ci (x* — xj) k-i <
4
< |y(xj)| < C2(x* — xj) k-i .
M Let xJ and xJ+i be two neighboring points of local extremum of a solution y(x) such that the statements of Lemmas 8, 10, and 11 hold. According to these Lemmas, for all j > J we have
xj+i — xj < MFy{xj)-(k-i) < MFy(xJ)-(k-i)q-3(k-i)(j-J),
i-i- i- * / ^ (, /w MFy (xJ )-(k-i) J which implies x* — xJ = ^ (xj+i — x^ < ^ -3(k-i) and
j=J
4
■ , /41/ * Fy (xJ )4 / MFy (xJ )-(k-1)\ k-i
|y(xJ)| (x* - xJ)k-1 < ( Jk-1) )
Mp 4
4
Д \ k- 1
1 - q-3(k-1)
On the other hand, xj+1—xj > mFy (xJ)-(k-1)Q-3(k-1)(j-J), which implies
4
4 /mF (r'T)-(k-1)\ k-1
(x'7)| (x* - x'7) k^ ^ V ' mFy (xJ) \
J)l (x* — xJ)^ > 9Fy(xJ)4 ("1F— J-k--)1 )
1 — Q-3(k-1)
m \ k—1
= 0' m x
1 — Q-3(k-i)
Asymptotic Classification of the Solutions to the Fourth-Order Equation (1). In this part we consider the asymptotic behavior of nontrivial solutions to equation (1) in the cases not previously considered. Then asymptotic classification of all maximally extended solutions to equation (1) will be given.
First for solutions to equation (1) generating in R4 curves lying entirely in Q+, we describe their asymptotic behavior near the left boundary of the domain.
Lemma 12. Suppose y (x) is a maximally extended to the left nontrivial solution to equation (1) with derivatives changing their signs according to scheme (7). Then the domain of y(x) is unbounded to the left, the functions y(x),y'(x),y"(x),y'"(x) tend to zero as x ^ —to, and the distance between its neighboring zeros tends monotonically to to as x ^ —to.
Using the substitution x ^ —x we can describe the asymptotic behavior of non-typical solutions near the right boundaries of their domains. Combining these results we obtain the following theorem.
Theorem 2. Suppose k > 1 and p0 > 0. Then all maximally extended solutions to equation (1) are divided into the following four types according to their asymptotic behavior (Fig. 3).
0. The trivial solution y (x) = 0.
1. Oscillatory solutions defined on (—ro, b). The distance between their neighboring zeros infinitely increases near the left boundary of the domain and tends to zero near the right one. The solutions and their derivatives satisfy the relations lim y)(x) = 0, lim |y)(x) I = ro for j = 0,1, 2, 3.
x^—tt x^b
At the points of local extremum the following estimates hold:
C1 |x - b| k—1 < |y(x)| < C2 |x - b| k-1
(17)
with the positive constants C1 and C2 depending only on k and p0.
2. Oscillatory solutions defined on (b, +ro). The distance between their neighboring zeros tends to zero near the left boundary of the domain and infinitely increases near the right one. The solutions and their derivatives satisfy the relations lim y)(x) = 0, lim |y) (x)1 = ro for j = 0,1, 2, 3.
x^+tt x^b
At the points of local extremum estimates (17) hold with the positive constants Ci and C2 depending only on k and p0.
3. Oscillatory solutions, defined on bounded intervals (b7, b"). All their derivatives y), with j = 0,1, 2, 3,4 satisfy lim |y) (x)1 = lim |y) (x)1 =
x^b' 1 1 x^b" 1 1
= ro. At the points of local extremum sufficiently close to any boundary of the domain, estimates (17) hold respectively with b = bf or b = b" and the positive constants C1 and C2 depending only on k and p0.
Fig. 3. Solutions to equation (1)
Fig. 4. Solution to equation (2)
Asymptotic Classification of the Solutions to the Fourth-Order Equation (2). In this section previously obtained results on the asymptotic behavior of solutions to equation (2) are formulated [7, 28].
Theorem 3. Suppose k > 1 and p0 > 0. Then all maximally extended solutions to equation (2) are divided into the following fourteen types according to their asymptotic behavior (Fig. 4).
0. The trivial solution y(x) = 0.
1-2. Defined on (b, +rc>) Kneser (up to the sign) solutions (see definition in [5]) with the power asymptotic behavior near the boundaries of the domain (with the relative signs ±):
_ 4
y(x)~± C4k(x - b) k-1, x ^ b + 0;
y(x)~± C4kx k-1,
x ^
where C4k =
'4(fc + 3)(2k + 2)(3k + 1)4 k-1
po (k -1)4 J '
3-4. Defined on semi-axes (-rc>,b) Kneser (up to the sign) solutions with the power asymptotic behavior near the boundaries of the domain (with the relative signs ±): 4
y(x)~± C4k |x| k-1
x —^ — oo;
4
y(x)~ ± C4k(b - x)-k-!, x ^ b - 0.
5. Defined on the whole axis periodic oscillatory solutions. All of them can be received from one, say z(x), by the relation y(x) = A4z(Ak-1x + + x0) with arbitrary A > 0 and x0. So, there exists such a solution with any maximum h > 0 and with any period T > 0, but not with any pair (h, T).
6-9. Defined on bounded intervals (6', 6'') solutions with the power asymptotic behavior near the boundaries of the domain (with the independent signs ±):
y(x)~± C4k(p(6'))(x - 6')-k-1, x ^ 6' + 0;
4
y(x)~ ± C4k(p(6'')) (6'' - x)-k-1, x ^ 6'' - 0.
10-11. Defined on semi-axes (—œ,6) solutions which are oscillatory as x ^ —œ and have the power asymptotic behavior near the right
_ 4
boundary of the domain: y(x)~ ± C4k (p(6))(6 — x) k-1, x ^ 6 — 0. For each solution a finite limit of the absolute values of its local extrema exists
as x ^ —œ.
12-13. Defined on semi-axes (6, +œ) solutions which are oscillatory as x ^ +œ and have the power asymptotic behavior near the left boundary of
_ 4
the domain: y(x)~ ± C4k (p(6))(x — 6) k-1, x ^ 6 + 0. For each solution a finite limit of the absolute values of its local extrema exists as x ^ +œ.
Asymptotic classification of the solutions to the third-order equation (3). In this section previously obtained results on the asymptotic behavior of solutions to equation (3) are formulated [7, 28].
Theorem 4. Suppose k > 1, and p(x) is a globally defined positive continuous function with positive limits p* and p* as x ^ ±œ. Then any nontrivial non-extensible solution to (3) is either (Fig. 5): 1-2) a Kneser solution on a semi-axis (6, +œ) satisfying
3
y(x) = ±C3fc(p(6)) (x — 6) (1 + o(1)) as x ^ 6 + 0,
_ 3
y(x) = ±C3k(p*) x k-1 (1 + o(1)) as x ^ +œ,
1
f 3(k + 2)(2k + 1)^ Where C3k(p)=^ p(k — 1)3 J ;
3) an oscillating, in both directions, solution on a semi-axis (—œ,6)
satisfying, at its local extremum points,
3
|y(x')| = |x'|-k-1+o(1) as x' ^ —œ,
3
|y(x')| = |6 — x'|-k-T+o(1) as x' ^ 6 + 0;
4-5) an oscillating near the right boundary and non-vanishing near the left one solution on a bounded interval (6', 6'') satisfying
y(x) = ±C3k(p(b))(x - b')-k-i (1 + o(1)) as x ^ b' + 0, and, at its local extremum points,
|y(x')| = |b'' - x'|-k-1+o(1)
as x' ^ b'' - 0.
Conclusion. Note that oscillatory solutions of equations (1) and (3) defined on (-œ; x*) or (x* ; +œ), are the solutions of the form
У (x) = |po | k-1 |x — x* | a h(log |x - x* |), a =
n
k- 1
(18)
with n = 4 and n = 3 respectively and an oscillatory periodic function
h : R ^ R.
Indeed more general result takes place. Thus, for the equation
+ Po |У| У(x) = 0, n> 2, k G R, k > 1, po = 0, (19)
the existence of oscillatory solutions of the type (18) is proved.
Theorem 5. For any integer n > 2 and real k > 1 there exists a non-constant oscillatory periodic function h(s) such that for any p0 > 0 and x* G R the function
y (x) = p0 k 1 (x*—x) a h(log(x*—x)), —œ < x < x*, a = is a solution to equation (19).
n
k1
, (20)
Fig. 5. Solution to equation (3)
1
Corollaries from this theorem for even and odd n are also proved for solutions defined near [42].
This work was supported by the RFBR Grant (no. 11-01-00989).
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The original manuscript was received by the editors in 23.06.2014
Асташова Ирина Викторовна — д-р физ.-мат. наук, профессор кафедры дифференциальных уравнений механико-математического факультета МГУ им. М.В. Ломоносова, профессор кафедры высшей математики Московского государственного университета экономики, статистики и информатики (МЭСИ). Автор 114 научных работ, в том числе трех монографий, в области качественной теории обыкновенных дифференциальных уравнений.
МГУ им. М.В. Ломоносова, Российская Федерация, 119991, Москва, Ленинские горы, д. 1.
МЭСИ, Российская Федерация, 119501, Москва, ул. Нежинская, д. 7.
Astashova I.V. — Dr. Sci. (Phys.-Math.), professor of Differential Equations department, Mechanical-Mathematical Faculty of the Lomonosov Moscow State University, professor of the Higher Mathematics department of the Moscow State University of Economics, Statistics and Informatics (MESI). Author of 114 publications, including 3 monographs, in the field of qualitative theory of ordinary differential equations. Lomonosov Moscow State University, Leninskie Gory 1, Moscow, 119991 Russian Federation.
Moscow State University of Economics, Statistics and Informatics (MESI), Nezhinskaya ul. 7, Moscow, 119501 Russian Federation.
