CC BY
13
ISSN 1998-4812
BecTHHK EamKHpcKoro yHHBepcHTeTa. 2016. T. 21. №3
549
UDC 517.4
Communication
ON ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF THE FIRST-ORDER SYSTEM OF DIFFERENTIAL EQUATIONS WITH A STRONGLY OSCILLATING COEFFICIENT
© O. V. Myakinova*, E. A. Nazirova, L. R. Valeyeva
Bashkir State University 32 Zaki Validi St., 450076 Ufa, Republic of Bashkortostan, Russia.
*Email: myakinovaov@gmail. com
It is well known that the asymptotic of fundamental system of solutions as x ^ œ of L - diagonal systems u/ = WU + CU, where W is diagonal matrix and C is the matrix with summable coefficients, mainly depends on the elements of matrix W. This result was obtained by Levinson, and it lies at the base ofmany investigations of the asymptotic behavior ofsolutions ofa singular high-order differential equations and systems of any order. In our communication, we consider the system in case when the coefficients of matrix C are not summable and represent the oscillating functions. For this case, the analogue of the Levinson's theorem is proved. The asymptotic formulas for the solutions of the system as x ^ œ with the weaker conditions on the coefficients of matrix C are obtained. The received formulas can be used for the investigation of the asymptotic behavior of the solutions of the singular differential equations.
Keywords: asymptotic behavior of solutions of the system, oscillating functions, integrable func-
tions.
It is well known [1, 2] that the most effective way to study the asymototic behavior of solution of the system
dy
dx
= A(x)y, x e [0, +œ)
(1)
where y(x) = colon(y1(x),y2(x), ...yn(x)) is the unknown vector-function, A(x) is square matrix of nth order, under various assumptions on its elements a¿j(x) or its eigenvalues, is to reduce system (1) to the so-called L - diagonal form [1].
We formulate the most general theorem in case n = 2.
Theorem 1. Suppose that
function m(x) is integrable on every finite interval
[0,6], b > 0,
iuncti°ns Cii(X),Ci2 (x) ,C21(x),C22(x) are integrable on the interval [0, +»),
for sufficiently large x, the function Re w(x) does not change sign.
Then the system
dyi
dx
= c11(x)y1 + C12(x)y2
= M(X) + c21(x)y1 + c22(x)y2
(2)
has the solution that satisfies the following condition lim y:L(x) = 1, lim y2(x) = 0. (3)
Note that the conditions on the coefficients Cij (x)are rather restrictive.
We will show that the condition of summabil-ity on Cij (x) can be replaced by weaker conditions. For simplicity we consider a system of equations in which one of the coefficients is the oscillating function sin(ex). We obtained the following theorem.
Theorem 2. Suppose that
function œ(x) is integrable on every finite interval [0, b], b>0,
functions Cii(X),Ci2
(X) ,C21(x),C22(x) are integrable on the interval [0, +œ),
for sufficiently large x, the function Re m(x) does not change sign.
Then the system
'^t = sin(eX)yi(X) + Ci2(x)y2(x).
^ = M(X)y2(X) + C2i(X)yi(X) +
(4)
+c22(x)y2(x) has the solution that satisfies the following condition (3).
Proof. Let x > 0 be such that the function Re w(x) does not change sign on (x0, +m). As in [1], we write out the system of integral equations equivalent to system (4)
Vi(x) = 1 - .jTM^) ydO + 0^(072(0] d( V2(x) = - eJ-^l)dHc2l (f) yi(0 + C22(OV2(OW
if > —m, (the first case)
V2(x) = + ejX"l((l)d(l[C2i (O yi(0 + C22(Oy2(.OW.
if = —m, (the second case),
(5)*°
where the numbers T and T1 are constants (T > 0, T1 > 0).
The second case is easier to study. It is easy to notice that in this case, T and T1 sufficiently large y2(x)=o(1), xe ^ +m if y1(x)=1+o(1), xe ^ +m. Examine the first case.
y^x) = 1 — I sin(el) y1(t)dt— I c12(t)y2(t) dt,
Jx Jx
Jf+" t
e-ixl0(tl)atl[C2i (t) yi(t) + C22(t)y2(t)]dt
x
We apply the method of successive approximations. Denote
and put
^(x, t) = e
-SxM(ti)dti
y(0) = 1 yf = 0
550
МАТЕМАТИКА и МЕХАНИКА
Л
(П+1)
Jr + W z^ + W
sm(e4) y,(n) (t)dt - I c12 (t)y2(n) (t) dt,
y>
(n+1)
Jr+W
^(x,t)[C2i (t)yi(n)(t) + C22(t)y2n)(t)]dt
r
(1)
Then integrating by parts, we obtain for yx (x):
yi(1)(x)
sin(ef) dt
= 1 -f
Jx
Jr + ra
e-tcos(ef) dt
X
Which implies that
|y1(1) - yx(0)| < 2e-x, for x > 0
Note that for t > x > x0
|^(x,t)| = < e«,
where a = 0 if >0 on (x0, +ro) and a =
— Jx+ra^(w(x))dx < 0 if ^O(x)) < 0
Hence
Ь:
(1)
:
y2(0)| <ec
Jr + W
|c:i (t)|dt.
r
Next
yi(2)-yi(1) =
Jr+W
e-t cos(ef) sin(et)dt
X
Jr + W z'+W
sin^) I e-s cos(es)dsdt + X Л
J, + W /"+W
ci2(0 I ^(t, 5)c2i (s)dsdt, X Л
yf-y2(i) =
+W
e tcos(et) dt
Jr + ra
^(t,s)C2i(t)
X
r + ra
— I e-scos(es)ds ■'t
/* + ra r+ra
+ | ^(t,s)C22(t) | ^(t,s)C2i (t)dS
If we denote
+ra
max ea I |c,;(t)| dt = c,
1<U<2 J 1 ^ 1
T
where T is chosen such that c < 1, we obtain
|yx(1)— yx(0)| <2e-*,
dt.
1У!3)-УГ1 =
bi -У1Ч < 4e-2x + c2,
|y2(i)-y20)|<C,
|у2(2) -y2(i)| < c2 +2ce-x,
Jr+W r + W
stn(ef) I cos(es)stn(es)ds dt
X Л
Jr+W z"+W r+W
stn(ef) I I e-u cos(eu)duds dt
X "'t •'s
Jr+W Z" + W Z" + W
Ci:(t) I ci2(s) I c2i (w)^(u,s)duds dt
x Л J.4
which implies that
,(3)
У!
|yi
Similarly
|y2(3)-y2(2)| <c3 + 2c2e-*,
,(2)| < 8e-3x + c3.
and so on. Thus, the series
(0) + (yi(i)-yi0)} + -+(yi(n+i)
Ух
(6)
(0)
УГ) +
yr + (y2(1)— y2(0)) + -+(y2(n+1)— yn + - (7)
converge uniformly on the interval x £ (0, +ro). It is easy to deduce that the sums of series (6), (7) are solutions to system (5).
The case when the integral
Jr+W
^(w(x))dx = -те
rn
■'Xq
is more lightweight and investigated similarly. In this case, T and T1 are selected from the conditions of the convergence of majorizing series.
Given the conditions 1) - 3) of Theorem 2, we conclude that considering system has the solution that satisfies the condition (3).
Remark 1. An analogue of Theorem 2 holds for a system of any order with oscillating coefficients.
Remark 2. Since ordinary differential equations can be reduced to systems of type (1), the Theorem 2 allows us to investigate the asymptotic behavior of the solutions of such equations, which will be discussed in a separate article.
The work has been financially supported by Russian Foundation for Basic Research (project 15-01-01095-A "The direct and inverse spectral problems of the theory of differential operators").
REFERENCES
1. M. V. Fedoryuk, Asymptotic methods for linear ordinary differential equations, Xan lea. Moscow, 1983.
2. M. A. Naimark, Linear differential operators, Moscow, 1969.
Received 20.08.2016
