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01.01.02 - 2008 , () - , .
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. . . .
27 2008 . 16 . 40 . .501.001.85 . . . : 119991, -1, , , . . . , - , 16-24.
- ( , 14 ).
23 2008 .
.501.001.85 - , . . .
:
n-y(n) + qi(x) y(i) + p(x) |y|k-1y = 0, (1) i=y(n)(x) = p x, y(x), y (x),..., y(n-1)(x) |y(x)|k-1y(x), (2) y(n) = p0|y(x)|k-1y(x), (3) y(n) + p(x) |y|k-1y = 0, (4) d d d rn(x)... r1(x) r0 (x) y... + |y|k = 0, (5) dx dx dx d d d rn(x)... r1(x) r0 (x) y... - |y|k = 0 (6) dx dx dx :
d d d rn(x)... r1(x) r0 (x) y... |y|k, (7) dx dx dx n-y(n) + aj(x) y(j) p |y|k, (8) j=n-y(n) + aj(x) y(j) -p |y|k, (9) j=n-y(n) + aj(x) y(j) -p |y|k, (10) j=n-y(n) + aj(x) y(j) p |y|k. (11) j= . (1) (6) y + x |y|k-1y = 0, (12) . 1 XX () , , , , .2 (12) 1 d d 2 + ||k-1 = 0, (13) 2 d d , , (())k , .
, .
(12) k . 3, . 4 . 5. (4) n = 2.
(4) n > 2 (2) , , , . . . . 6, . . B. C. 7, . . 8, . . 9, . . R. Emden. Gaskugeln. Leipzig, 1907.
.., .., ... . , , 1981.
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10, . . 11,12, . . 13 . , 1990 , . . . . 14. n = 2. , . . , (2) , n = 2 , . ( 16.4): , n > 2 .
(4) n = 2 p(x) < 0 . . . . 15.
. . 16, .
. , . . 17 . . . . . 18 (4), , , , Kozlov V. A. On Kneser solutions of higher order nonlinear ordinary dierential equations. Ark. Mat., 1999, v. 37, 2, p. 305а322.
. . . , . , 2001, . 65, 2, . 81а126.
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. . . . . 1969, . 5, 12, c. 2267а2268.
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. ., . . y = p(x)yk. . , , 1980, c. 134а141.
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.
, , . . , . . 19 , , y(n) q0|y|k, k > 1, q0 = const.
. 20 y(n) 1 2 m q1(t)|y|k +q2(t)|y|k + +qm(t)|y|k. . . 21 (4) .
. (1) qj(x) = 0, j = 0,..., n - 1. n = F. Atkinson22 .
(F. Atkinson). f(x) x 0 . k , 1. y + f(x)y2k-1 = , xf(x) dx = .
, , . ., . . . . . . , 2001, . 234, 383 c.
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, 2002, . 38, 3, c. 362а368.
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Atkinson F. V. On second order nonlinear sillations. Pacif. J. Math., 1955, v. 5, 1, p. 643а647.
. . 23,24,25, . . 26,27, D. L. Lovelady28,29, . . . . 30, .
y + p(x)f(y) = 0 y + g(x, y) = 0, , F. Atkinson, S. A. Belohorec31, . . 32, J. W. Masci and J. S. W. Wong33,34,35.
3- 4- . . . . 36, T. Kusano M. Naito37, . . . . , 1980, . 16, 3, c. 470а482 4, c. 635а644.
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. -, 1982, . 16, c. 3а72.
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. . y(n) - p(x)y = 0. , 1961, . 10, c. 419а436.
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Lovelady D. L. An asymptotic analysis of an odd order linear dierential equation.
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Belohorec S. A criterion for oscillation and nonoscillation. Acta F. R. N. Univ.
Comen. Math., 1969, v. 20, p. 75-79.
. . u + a(t)|u|n sgn u = 0. Cas. pest. mat., 1962, v. 87, 4, p. 492-495.
Masci J. W., Wong J. S. W. Oscillation of solutions to second-order nonlinear dierential equations. Pacif. J. Math., 1968, v. 24, 1, p. 111-117.
Wong J. S. W. A note on second order nonlinear oscillation. SIAM Review, 1968, v. 10, p. 88-91.
Wong J. S. W. On second-order nonlinear oscillation. Funkcialaj Ekvacioj, 1968, v. 11, p. 207-234.
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Kusano T., Naito M. Nonlinear oscillation of fourth-order dierential equations. Canad. J. Math., 1976, v. 28, 4, . 840-852.
D. L. Lovelady38, V. R. Taylor, Jr.39, P. Waltman40.
F. Atkinson y(n) + p(x)|y(x)|k sgn y = . . 41 . . 42.
(1) qj(x) = 0 43,44,45,46,47,48,49, .
. (1) (11), , , ; ; ; Lovelady D. L. An oscillation criterion for a fourth-order integrally superlinear dierential equation. Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Natur. 1975, (8) 58, 4, p. 531-536.
Taylor W. E., Jr. Oscillation criteria for certain nonlinear fourth order equations. Internat. J. Math., 1983, v. 6, 3, p. 551-557.
Waltman P.Oscillation criteria for third order nonlinear dierential equations. Pacif. J. Math, 1966, v. 18, p. 385-389.
. . dmu/dtm+a(t)|u|nsgnu = 0.
. ., 1964, . 65, 2, . 172-187.
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Kartsatos A. G. N th order oscillations with middle terms of order N - 2. Paciжc J. Math., 1976, v. 67, 2, p. 477-488.
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Kusano T., Naito M. Nonlinear oscillation of fourth-order dierential equations. Canad. J. Math., 1976, v. 28, 4, p.840-852.
Lovelady D. L. On the oscillatory behavior of bounded solutions of higher order differential equations. J. Di. Equations, 1975, v. 19, 1, p. 167-175.
Lovelady D. L. An oscillation criterion for a fourth-order integrally superlinear dierential equation. Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Natur, 1975, (8) 58, 4, p. 531-536.
Taylor W. E., Jr. Oscillation criteria for certain nonlinear fourth order equations.
Internat. J. Math., 1983, v. 6, 3, p. 551-557.
Waltman P.Oscillation criteria for third order nonlinear dierential equations. Pacif. J. Math, 1966, v. 18, p.385-389.
; ; .
. , .
(1) 1, (8) 2 (1) 3 dn n-1 dj L = + qj(x) dxn dxj j= d d d y[n](x) = rn(x)... r1(x) r0(x) y..., dx dx dx rj(x) .
G. Polya50, Ch. I. de la VallВ ee-Poussin51, A. 52 , , , . 2 , .
3 , x + 1, 3.
4а7 , n- G. PВ On the mean-value theorem corresponding to a given linear homogeneous olya dierential equation. Trans. Amer. Math. Soc., 1924, v. 24, p. 312а324.
Ch.I. de la VallВ equation diВ eaire eee-Poussin Sur lВ erentielle linВ du second ordre. DВ termination dune intВ par deux valeurs assignВ Extension aux В egrale ees. equations dordre n. Journ. Math. Pur. et Appl., 1929, v. 9, 8, p. 125а144.
.. x(n) +p1(t)x(n-1) + +pn(t)x = 0.
, 1969, . 24, . 2 (146), . 43а96.
(n - 1)- .
.
. .
:
Х (1), (5) (6) , ;
Х (1) (5) ( );
Х (8) (11) , ;
Х (2) , , , ; , ( . . ), ;
Х (4) (3) ;
Х .
. . , . .
. :
Х . . . . 1985, 1988, 1990.
Х . , 1993, 1994, 1995, 2000, 2002, 2004, 2006, 2007.
Х . , 2001, 2003.
Х . , 1995.
Х . , 1995, 1996, 2005, 2007.
Х The First International Scientiжc and Practical Conference дDierential Equations and Applications. Saint-Petersburg, 1996.
Х International Colloquium on Dierential Equations. Plovdiv, Bulgaria, 1996, 1997.
Х International Symposium дComplex Analysis and Related Topics.
Cuernavaca, Mexico, 1996.
Х International Symposium Dedicated to the 90th Birthday Anniversary of Academician I.Vekua. Tbilisi, 1997.
Х 4th Symposium on Mathematical Analysis and Its Applications. Arangelovac, Yugoslavia, 1997.
Х . , 1997.
Х Mark Krein International Conference дOperator Theory And Applications. Odessa, Ukraine, 1997.
Х Conference on Dierential Equations and Their Applications.
(EQUADIFF -9) Brno, Czech Republic, 1997.
Х . --, 1999.
Х Diйety School. School in Geometry of Partial Dierential Equations, S. Stefano Del Sole, Avellino, Italy, 2002.
Х International Petrovskii Conference дDierential Equations and Related Topics. Moscow, 1996, 2001, 2004, 2007.
Х 3rd ISAAC Congress. Berlin, Germany, 2001.
Х . , 2002, 2004, 2006.
Х International Conference дFunction Spaces, Approximation Theory, Nonlinear Analysis dedicated to the centennial of S. M. Nikolskii.
Moscow, 2005.
Х .. . , 2006.
Х , , . 2006.
Х , , 100- . . . . 2007.
Х Conference on Dierential Equations and their applications (EQUADIFF2007). Vienna, Austria, 2007.
Х 14- . . . . . . 2008.
Х .
. . . . . 2008.
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Х - . . . , . . . , . . . 1986, 1996, 1998, 2004, 2005, 2006, 2007, 2008.
Х - / . . . , . . . 1996, 2001, 2005.
Х - / . . . , . . . , . . . 2005.
Х . . . , . . . 2005.
Х . / . . . 2005, 2007.
Х . . . . . . 2004, 2006.
Х - . . . . . . 2007а2008.
Х , () 2002а2008.
. 32 (14 , ), 2 . .
. , , , . 240 , 136 . 12 . , , : , : , . , .
.
[j] j- :
d d d y[j](x) = rj(x)... r1(x) r0(x) y..., dx dx dx rj(x) .
, y[0](x) = r0(x) y(x), j > 0 y[j](x) = rj(x) y[j-1] (x).
, rj(x), j mj = inf rl(x) : x [a, b], i l=i j Mij = sup rl(x) : x [a, b], l=i Mij j =, i mj i , 0 < mj Mij, j 1.
i i [a, b] dn n-1 dj L = + qj(x) (14) dxn dxj j= deg L-j b - a QL = sup qj(x) : x [a, b], 0 j < deg L.
n = (15) k - 2n n+1+ k-Ynk = 2. (16) 1 (1):
n-y(n) + qi(x) y(i) + p(x) |y|k-1y = 0, i= n 1, k > 1, p(x) qi(x) , |p(x)| p > 0, (, (5) (6)) y[n] + |y|k-1y = 0, y[n] - |y|k-1y = 0, [j] j- :
d d d y[j](x) = rj(x)... r1(x) r0(x) y...
dx dx dx rj(x).
, . .
(1.1). y(x) [a, b] (5) (6). x (a, b) n k-y(x) C1 1, n-2n i i+1+ 1 ( ) k-n n k-C1 = (Ynk M0 ) i 2, i=b - a 1 = min x - a, b - x,.
(1.1.1). rj(x), j = 0,..., n, 0 < m < rj(x), j = 0,..., n - 1, rj(x) < M < +, j = 0,..., n.
(5) (6).
(1.2). [a, b] y(x) (5) n k-y(x) C2 (x - a), x (a, b], n-1 n-2n i i+1+ 1 ( ) 2(n+1)i i k-n n k-C2 = (3n Ynk M0 ) i 2 .
i! i=0 i= (1.2.1). [a, b] y(x) (6) n n k-y(x) C2 (b - x), x [a, b), C2 , 1.2.
(1.2.2). [a, b] y(x) (5) n n n k-1 k-y(x) 2 C2 (b - a) x [a, b], C2 , 1.2.
. , n (5), , . > 0. [0, 1] n k-y(x) = (x + ) n n--n j + y(n) + |y|k-1y = 0.
k - j= y(0) + 0.
(1.2.3). rj(x), j = 0,..., n, 1.1.1. (5).
. , rj(x) < M < + . |x|n+1-k |x + 1|k y(n) + |y|k-1y = 0, n! (5), , (-, -1) y(x) = 1 + 1/x.
(1.2.4). rj(x), j = 0,..., n, 1.1.1 . (5) n (6) n.
. , (5) (6) n. , n--n j + y(n) + (-1)n+1 |y|k-1y = 0, k - j=n k- y(x) = x-, (0, ).
(1).
(1.4). y(x) [ a, b ] (1), |p(x)| p | qj(x)| Qn-j, j = 0,..., n - 1, p > 0 Q > 0.
x ( a, b ) n k-y(x) C3 3, n-2(ni+1) n k-i2+i+2+ Ynk k-C3 = 2, p i=b - a 2-n -n+3 = min x - a, b - x,,.
3 3Q (1.5). y(x) [ a, b ] (1), p(x) p | qj(x)| Qn-j, j = 0,..., n - 1, p > 0 Q > 0.
x ( a, b ] n k-y(x) C4 4, n-1 n-2n k-i i+1+ ( ) 4 (3 Ynk)n 2(n+1)i k-C4 = 16 2 , p i! i=0 i=2-n -n+4 = min x - a,.
Q (1.5.1). [a, b] y(x) (1) n p(x) -p < | qj(x)| Qn-j, j = 0,..., n - 1, n k-y(x) C4 5, x [ a, b ), 2-n -n+5 = min b - x,, Q C4 , 1.5.
(1.5.2). [a, b] y(x) (1) n p(x) p > 0 | qj(x)| Qn-j, j = 0,..., n - 1, y(x) C5, x [ a, b ], n k-2-n -n+C5 = C4 min b - a,, Q C4 , 1.5.
. 1.4 1.5 3 4, (1) , 1.1.1, 1.2.3 1.2.4 (5) (6), . , (1) qj(x), . , y(n) - 2y + y3 = 0 y(n) + 2y - y3 = 0 y(x) = .
2 (8):
n-y(n) + aj(x) y(j) p |y|k, j= aj(x) , p > 0, n 1, k > 1, (7):
d d d rn(x)... r1(x) r0 (x) y... |y|k, dx dx dx rj(x) . , .
(2.1). [ a, b ] y(x) (7) |y(x)| C1 min{x - a, b - x}-n/(k-1), x (a, b), C1 = C1 ( n, k, inf rj(x), sup rj(x) ), inf rj(x) x [ a, b ] j = 0,..., n-1, sup rj(x) x [ a, b ] j = 0,..., n.
(2.1.1). rj(x), j = 0,..., n, 0 < m < rj(x) < M < +. (7).
(2.2). k > 1, p > 0, Q > 0, n 1 > 0 C2 > 0, a0(x),..., an-1(x), [ a, b ] sup | aj(x) |1/(n-j) : x [ a, b ], j = 0,..., n - 1 Q, [ a, b ] (8) |y(x)| C2 min { , x - a, b - x }-n/(k-1), x (a, b).
(9) (8) , (9) ( 2.2.1).
1. , 2.2 , 2.1.1. y(n) + y |y|k, y(x) 1/(k-1).
2. (10) (11):
n-y(n) + ai(x) y(i) p |y|k, i=n-y(n) + ai(x) y(i) -p |y|k i= ai(x), p, n k , , (8) (9).
3 (1), qj(x) , xn-j-1 |qj(x)| dx, j = 0,..., n - 1.
x p(x) , (1) x +. p(x) > 0 , . n , F. Atkinson (1).
(3.1). (1) p(x) qj(x), j = 0, 1,..., n - 1, xn-1 |p(x)| dx < , (17) x xn-j-1 |qj(x)| dx < . (18) x h = 0 (1) + y(x), x h, xj-1 y(j)(x) dx < , j = 1,..., n. (19) x (3.3). (1) p(x) , qj(x), j = 0,..., n - 1, (18).
:
(i) p(x) (17), (ii) (1) + y(x), x .
( ). (1) n p(x) , qj(x), j = 0,..., n - 1, (18).
:
(i) xn-1 |p(x)| dx = , x (ii) (1), +, .
4 (2). n 2 k > 1 , . 2 n 13 (n - 1)- . n , , . n = 3, 4 k > 1 , , , , n = 4 .
(2), k > 0, p(x, y0, y1,..., yn-1) y0, y1,..., yn-1.
, (2) p(x, y0,..., yn-1) p0 > 0 x x - 0, y0 ,..., yn-1 , > 0 n-p(x, y0,..., yn-1) - p0 = O |x - x| + |yj|-. (20) j= , x y0,..., yn-1, z0,..., zn-1 p(x, y0,..., yn-1) - p(x, z0,..., zn-1) (21) K1 max |yj|- - |zj|- j K1 > 0 > 0.
, p p0 = const > 0, (2) (3), , y(x) = C(x - x)-, x < x, k-n ( + 1)... ( + n - 1) =, C = (22) k - 1 p, (2) y(x) = C(x - x)- (1 + o(1)), x x - 0, (23) C (22).
, 3 n 13 (n - 1)- (2) .
(2) n.
, p(x, y0,..., yn-1) p0 = const > 0 x , y0 0,..., yn-1 0, > 0 n-p(x, y0,..., yn-1) - p0 = O |x|- + |yj|. (24) j= , x , y0 0,..., yn-1 0, z0 0,..., zn-1 p(x, y0,..., yn-1) - p(x, z0,..., zn-1) (25) K2 max |yj| - |zj| j K2 > 0 > 0.
(3) n y(x) = C(x - x)-, x > x, (26) C (22). (x, ) x .
, (2) y(x) = Cx- (1 + o(1)), x , (27) C (22).
(4.1). (2) p(x, y0,..., yn-1) x x - 0, y0 ,..., yn-1 p0 = const > 0, (20), (21). x (2) (23)а(22).
(4.2). 3 n 13, p(x, y0,..., yn-1) x x - 0, y0 ,..., yn-1 p0 > 0, (20), (21). (n - 1) (2), (23)а(22).
y(x) (2), [x0, ) , (-1)iy(i)(x) > 0, x x0, i = 0,..., n - 1.
(4.3). x , y0 0,..., yn-1 0 p(x, y0,..., yn-1) p0 > 0, (24) (25), (2) n (23), C (22).
n = 3 n = 4 p(x, y0,..., yn-1) , .
(4.5). (2) n = 3 n = 4, p(x, y0,..., yn-1) y0,..., yn-1 p0 > 0 x x - 0, y0 ,..., yn-1 . (2) x = x (23) C, (22).
(2) 0 < pmin p(x, y0,..., yn-1) pmax < +. (28) (4.6). (2), , , , n.
n, y(j)(x), j = 1,..., n - 1 , y(x), j , , j .
(4.7). (3) n = 4. (3) y(x) = C(x - x)-, x > x, C (22), x ( ).
(4.8). (2) n = 4, p(x, y0, y1, y2, y3) y0, y1, y2, y3 . (2).
(4.9). n = 4, p(x, y0, y1, y2, y3) 4.8 (28). , x +, y0 0,..., y3 0 p(x, y0, y1, y2, y3), p0 > 0. (2) y(x) = Cx- (1 + o(1)), x +, C (22).
(2) x.
n x = -x (2) , , x.
(4.10). n = 4 , p(x, y0, y1,..., yn-1) p0 > 0 x x +0, (-1)iyi +, i = 0, 1,..., n-1, y0, y1, y2, y3, (2), (x, x1) x = x, y(x) = C(x - x)-(1 + o(1)), x x + 0, C (22).
, n (2) p(x, y0, y1,..., yn-1) , .
(2). , (-, x0], , n .
(4.11). n = 3 n = 4 ( ) (3) y(x) = C(x - x)-, x < x, C (22).
(4.12). n = 3 n = 4, p(x, y0,..., yn-1) , y0,..., yn-1. ( ) (2).
(4.13). n = 3 n = 4. , p(x, y0,..., yn-1) 4.12, (28) p(x, y0,..., yn-1) x -, y0 0,..., yn-1 0, p0 > 0. ( ) (2) y(x) = C |x|-, x -, C (22).
n > 2 (2) n = 3, 4. ( ).
(5.1). n > 2 (2), p(x, y0,..., yn-1) (28) y0,..., yn-1, .
n = 3, (3) .
x1 < x2 < < xi <... , y(xi) = 0, i = 1, 2,..., y(x) = 0 x (xi, xi+1), x 1 < x 2 < < x i <... , y (x i) = 0, (x i, x i+1), i = 1, 2,..., y(x) .
(5.2). n = 3 B (0, 1), p0 k, y(x) (3) :
xi+1 - xi 1) = B-1, i = 2, 3,..., (29) xi - xi-y(x i+1) 2) = -B, i = 1, 2, 3,..., (30) y(x i) y (xi+1) 3) = -B+1, i = 1, 2, 3,..., (31) y (xi) 4) |y(x i)| = M(x i - x)-, i = 1, 2, 3,... (32) M > 0 x, M p0 m0.
(5.3). p(x, y0, y1, y2) > 0 , y0, y1, y2 y0, y1, y2 p0 > 0 x . y(x) (2), x1 < x2 <... x 1 < x 2 <... . B (0, 1) , 5.2. i :
xi+1 - xi y(x i+1) 1) B, 2) -B, xi+2 - xi+1 y(x i) y (xi+1) 3) -B+1, 4) |y(x i)| = (x i)-+o(1).
y (xi) (2) , (x, x0), - x < x0 , (x, x1), x < x1 < x0.
(5.4). p(x, y0, y1, y2) y0, y1, y2. , p(x, y0, y1, y2) p0 > 0 x x + 0 y0, y1, y2.
n = 3 B (0, 1), (2), (x, x0), - x < x0 , xi+1 - xi+1) B, i , xi - xi+y (xi) 2) -B+1, i , y (xi+1) y(x i) 3) -B, i , y(x i+1) 4) y(x i) = |x - x i|-+o(1), i , x1 > x2 >... > xi >... x 1 > x 2 >... > x i >... , y(xi) = 0, y(x) = 0 xj+1 < x < xi, y (x i) = 0 y (x) = 0 x j+1 < x < x i.
(5.5). n = 4. y(x) (3) , , min max y(x) , , , min.
max (5.6). h > 0 (3) n = 4, h.
, h > 0 OX.
(5.7). y(x) , (3) n = 4. x1 < x2 <... < xi <... y(x), y(xi) = 0, i = 1, 2,... y(x) = 0 x (xi, xi+1), i = 1, 2,..., x 1 < x 2 <... < x i <... y(x), y (x i) = 0 y(x) x (xi, xi+1), i = 1, 2,....
, (xi+1 - xi), |y(x i)|, |y (xi)|, |y (x i)| |y (xi)|, y (xi) y (x i) .
6 (3) (4) n = 3, 4. (k > 1), (0 < k < 1).
(6.1). k > 1, p(x) , p p x - x +. y + p(x) |y|k-1y = .
0. y(x) 0.
1а2. (b, +) ( ) ( ):
k-y(x) = C3k(p(b)) (x - b)- (1 + o(1)), x b + 0, k-y(x) = C3k(p) x- (1 + o(1)), x +, k-3(k + 2)(2k + 1) C3k(p) =.
p (k - 1)3. (-, b) , . . lim y(j)(x) = 0, lim y(j)(x) = , j = 0, 1, 2, x- xb k-|y(x )| = |x |- +o(1), x -, k-|y(x )| = |b - x |- +o(1), x b + 0.
4а5. (b, b ) , . ( ):
k-y(x) = C3k(p(b )) (x - b )- (1 + o(1)), x b + 0, lim y(j)(x) = , j = 0, 1, 2, xb k-|y(x )| = |b - x |- +o(1), x b - 0.
(6.2). k > 1 p0 > 0 yIV(x) + p0 |y|k-1y = .
0. y(x) 0.
1. (-, b) . . lim y(j)(x) = 0, lim y(j)(x) = , j = 0, 1, 2, 3, x- xb 4 k-1 k-C1 |x - b|- |y(x)| C2 |x - b|- (33) k p0 C1 C2.
2. (b, +) . . lim y(j)(x) = , lim y(j)(x) = 0, j = 0, 1, 2, 3, xb x+ (33) k p0 C1 C2.
3. , (b, b ). lim y(j)(x) = lim y(j)(x) = , j = 0, 1, 2, 3, xb xb , - (33) b = b b = b k p0 C1 C2.
(6.3). k > 1 p0 > 0 yIV(x) - p0 |y|k-1y = .
0. y(x) 0.
1а2. (b, +) ( ) ( ):
k-y(x) = C4k(p(b)) (x - b)- (1 + o(1)), x b + 0, k-y(x) = C4k(p) x- (1 + o(1)), x +, k-4(k + 3)(2k + 2)(3k + 1) C4k(p) =.
p (k - 1)3а4. (-, b) ( ) ( ):
k-y(x) = C4k(p) |x|- (1 + o(1)), x -, k-y(x) = C4k(p(b)) (b - x)- (1 + o(1)), x b - 0.
5. . , , z(x), y(x) = 4z(k-1x + x0) > 0 x0. , h > 0 c T > 0, c (h, T ).
6а9. (b, b ) ( ):
k-y(x) = C4k(p(b )) (x - b )- (1 + o(1)), x b + 0, k-y(x) = C4k(p(b )) (b - x)- (1 + o(1)), x b - 0.
10а11. (-, b) , x - , :
k-y(x) = C4k(p(b)) (b - x)- (1 + o(1)), x b - 0.
x -.
12а13. (b, +) , x + , :
k-y(x) = C4k(p(b)) (x - b)- (1 + o(1)), x b + 0.
x +.
y + p(x, y, y, y ) |y|k-1y = 0, (34) k > 1, p : R R3 R , 0 < m p(x, y0, y1, y2) M < , (35) , .
, y(x) x = x, lim y(x) = +, lim y(x) = -.
xx xx (6.4). k > 1, p(x, y0, y1, y2) , (35) . y(x) (34) x = x. x = x .
(6.5). 6.4, (34). x < x , (x, x), x = x x = x.
(6.6). 6.4 x < x (34), (x, x).
(2) 0 < k < 1 . :
(6.7). p(x, y0,..., yn-1) x y0,..., yn-1. 0 0 x0, y0,..., yn-1, yi 0, .
(2) n = 3, 0 < k < 1.
(6.8). n = 3, 0 < k < 1, 6.7 p(x, y0, y1, y2) x + p > 0 y0, y1, y2. (2) + 0 x, 1-k y(x) = Cx (1 + o(1)), x +, 1-k p(1 - k) C =.
3(k + 2)(2k + 1) (6.9). n = 3, 0 < k < 1, 6.7 p(x, y0, y1, y2) x - p y0, y1, y2.. (2) - 0 x, .
, x1 > x2 >... - , y(xi) = 0, y(x) = 0 x (xi+1, xi), i = 1, 2,..., x 1 > x 2 >... - , y (x i) = 0, y (x) = 0 x (x i+1, x i), i = 1, 2,..., xi - xi+1 y(x i+1) 1-k B, -B, i , xi-1 - xi y(x i) B > 1, k p.
(6.10). 0 < k < 1 p(x) y(x) y = p(x) |y|k-1y a1 a2, y(ai) = y (ai) = y (ai) = 0, i = 1, 2, .
a1 0, x1 < x2 <... a1 - 0 , y(xi) = 0, y(x) = x (xi, xi+1), i = 1, 2,..., x 1 < x 2 <... a1 - 0 , y (x i) = 0, y (x) = 0 x (x i, x i+1), i = 1, 2,..., xi - xi-1 y(x i) 1-k B, -B (i ) xi+1 - xi y(x i+1) B > 1, k p(a1).
a2 0, 1-k y(x) = C(x - a2) (1 + o(1)), x a2 + 0, C o , 6.8, p = p(a2).
[a1, a2] (, ) .
7 y (x) = p(x)|y(x)|my(x), (36) m > 0, x R, p(x) .
.
p(x) p0 = const C\R Y (x), (0, +), |Y (x)| = C1x-2/m, arg Y (x) = C2 ln x 1 + 4/m m C1 = Q, Im p1 + 4/m C2 = -Q, Im p8(m + 2) -Re p0 + (Re p0)2 + (Im p0)(m + 4)Q =.
(7.1). m > 0 p(x) p0 = const C \ R. (36) :
1. , (-, x0) (x0, +), :
|y(x)| = | Y (|x - x0|) |, arg y(x) = arg Y (|x - x0|) + x0 0.
2. , (x1, x2), |y(x)| = |Y (|x - xk|)| (1 + o(1)), arg y(x) = arg Y (|x - xk|) (1 + o(1)) x xk, k = 1, 2.
(7.2). p(x) , m > 0 p(x0) = p0 C \ R. y(x) (36), (x1, x0) (x0, x2) - x1 < x0 < x2 +. |y(x)| = |Y (|x - x0|)| (1 + o(1)), arg y(x) = arg Y (|x - x0|) (1 + o(1)), x x0.
(7.3). p(x) , = 1, m > 0, p(x) p0 C\R x . y(x) (36), . |y(x)| = |Y (|x|)| (1 + o(1)), arg y(x) = arg Y (|x|) (1 + o(1)), x .
(7.4). Re p(x) > p > 0. y(x) (36), (x0-, x0+) , y(x0) = 0, C 2 < |y(x0)|-m, p C > 0, m.
(7.4.1). p(x) 7.4. y(x) (36), [a, b], C m |y(x)| < 2p x [a + , b - ].
(7.4.2). p(x) 7.4. y(x) (36), (-, x0) (x0, +), m |y(x)| < |x - x0|-2/m C/p.
(7.4.3). Re p(x) > qx-r, q > 0, r > 0, y(x) (36), (0, +), x > 0 m |y(x)| < x(r-2)/m C/q.
C m 7.4.
(7.4.4). p(x) 7.4, (36), (-, +), y(x) 0.
.. .
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