Previous |  Up |  Next

Article

Title: On the oscillatory and monotone solutions of ordinary differential equations (English)
Author: Kiguradze, Ivan Tarielovich
Language: English
Journal: Archivum Mathematicum
ISSN: 0044-8753 (print)
ISSN: 1212-5059 (online)
Volume: 14
Issue: 1
Year: 1978
Pages: 21-44
.
Category: math
.
MSC: 34C10
idZBL: Zbl 0408.34033
idMR: MR512742
.
Date available: 2008-06-06T06:04:26Z
Last updated: 2012-05-09
Stable URL: http://hdl.handle.net/10338.dmlcz/106988
.
Reference: [1a] Aripov M. M.: On the solution of the second order ordinary non-linear equation.(Russian), Dokl. Akad. Nauk Uz. SSR, 1970, N7, 6-8. MR 0379935
Reference: [1b] Aripov M. M.: Method of "standard" equations (VKB-method) for second order non-linear equations.(Russian), Izv. Akad. Nauk Uz. SSR ser. fyz.-mat. nauk, 1970, N4, 3-7. MR 0294793
Reference: [2] Atkinson F. V.: On second-order non-linear oscillations.Pacif. J. Math., 1955, 5, N1, 643-647. Zbl 0065.32001, MR 0072316
Reference: [3] Bellman R.: Stability theory of differential equations.New York, Mc Graw-Hill, 1953, Zbl 0053.24705, MR 0061235
Reference: [4a] Belohorec Š.: Oscilatorické riešenia istej nelineárnej diferenciálnej rovnice druhého rádu.Mat.-Fyz. Časopis SAV, 1961, 11, N4, 250-255.
Reference: [4b] Belohorec Š.: Neoscilatorické riešenia istej nelineárnej diferenciálnej rovnice druhého rádu.Mat.-Fyz. Časopis SAV, 1962, 12, N4, 253-262. MR 0159994
Reference: [4c] Belohorec Š.: On some properties of the equation $y"(x) + f(x) y^{\alpha}(x) = 0, 0 < \alpha < 1$.Mat. Časopis, 1967, 17, N1, 10-19. MR 0214854
Reference: [4d] Belohorec Š.: Two remarks on the properties of solutìons of a nonlinear differential equation.Acta F. R. N. Univ. Cоmen. - Mathematica, 1969, 22, 19-26. MR 0289855
Reference: [4e] Belohorec Š.: A criterion for oscillation and nonoscillation.Аcta F. R. N. Univ. Cоmen. - Mathematica, 1969, 20, 75-79. Zbl 0225.34019, MR 0274855
Reference: [4f] Belohorec Š.: Monotone and oscillatory solutions of a class of nonlinear differential equations.Mat. Časоpis, 1969, 19, NЗ. 169-187. Zbl 0271.34045, MR 0304773
Reference: [5] Bоbrоwski D.: O pewnych wlastnościah rozviazan równania rôzniczkowego nieliniowego trzeciego rzedu.Pоznan, Pоlitechnika Pоznanska Rоzprawy, 1967, N24.
Reference: [6] Butler G. J.: Oscillation theorems for a non-linear analogue of Hill's equation.Quart. J. Math., 1976, 27, N106, 159-171. MR 0409976
Reference: [7] Cоffman C. V., Ullrich D. F.: On the continuation of solutions of a certain non-linear differential equation.Mоnatshefte für Math., 1967, 71, 5, 385-392. MR 0227494
Reference: [8] Cоffman C. V., Wоng J. S. W.: On a second order nonlinear oscillation problem.Тrans. Аmer. Math. Sоc., 1970, 147, N2, 357-366. MR 0257473
Reference: [9] Erbe L. H., Muldowney J. S.: On the existence of oscillatory solutions to nonlinear differential equations.Ann. mat. pura ed appl, 1976, N109, 23-38. Zbl 0345.34022, MR 0481254
Reference: [10] Fite W. B.: Concerning the zeros of the solutions of certain differential equations.Trans. Amer. Math. Soc., 1918, 19, 341-352. MR 1501107
Reference: [11] Galbraith A. S., McShane E. J., Parrish G. B.: On the solutions of linear second-order differential equations.Proc. Nat. Acad. Sci. U.S.A., 1965, 53, N2, 247-249. Zbl 0133.34103, MR 0174817
Reference: [12] Hartman P.: The existence of large or small solutions of linear differential equations.Duke Math. J., 1961, 28, N3, 421-429. Zbl 0102.30301, MR 0130432
Reference: [13] Hastings S. P.: Boundary value problems in one differential equation with a discontinuity.J. Different. Equat., 1965, 1, N2, 346-369. Zbl 0142.06303, MR 0180723
Reference: [14a] Heidel J. W.: Uniqueness, continuation, and nonoscillation for a second order nonlinear differential equation.Pacif. J. Math., 1970, 32, N3, 715-721. Zbl 0188.14301, MR 0259244
Reference: [14b] Heidel J. W.: The existence of oscillatory solutions for a nonlinear odd order differential equation.Czechoslov. Math. J., 1970, 20, N1, 93-97. Zbl 0183.36902, MR 0257468
Reference: [15] Heidel J. W., Kiguradze I. T.: Oscillatory solutions for a generalized sublinear second order differential equation.Proc. Amer. Math. Soc., 1973, 38, N1, 80-82. Zbl 0242.34035, MR 0310339
Reference: [16] Hinton D.: Some stability conditions for a nonlinear differential equation.Trans. Amer. Math. Soc., 1969, 139, 349-358. Zbl 0182.11902, MR 0241767
Reference: [17a] Izjumova D. V.: On the conditions of oscillation and non-oscillation of second order non-linear differential equations.(Russian), Differenciaľnye Uravnenija, 1966, 2, N12, 1572-1585. MR 0209579
Reference: [17b] Izjumova D. V.: A note on oscillation of solutions of second order non-linear differential equations.Soobšč. Akad. Nauk Gr. SSR, 1967, 57, N1, 19-24. MR 0219809
Reference: [17c] Izjumova D. V.: On asymptotic behaviour of solutions of some second order ordinary non-linear differential equations.(Russian), Trudy Tbilis. Gos. Univ., ser. mek.-mat. nauk, 1968, 129, 157-178. MR 0255923
Reference: [17d] Izjumova D. V.: On oscillatory solutions of second-order non-linear differential inequalities.Studies of certain equations of mathematical physics (Russian), Tbilisi, Tbilisi Univ. Press, 1972, 17-22. MR 0361276
Reference: [17e] Izjumova D. V.: Concerning the oscillation of solutions of second order non-linear differential inequalities and equations.Studies of certain equations of mathematical physycs (Russian), Tbilisi, Tbilisi Univ. Press, 1976, 63-70. MR 0594801
Reference: [18] Izjumova D. V., Kiguradze I. T.: Some notes on solutions of the equation $u" + a(t) f(u) = 0$.(Russian), Differenciaľnye Uravnenija, 1968, 4, N4, 589-605. MR 0227544
Reference: [19] Jasny M.: On the existence of an oscillatory solution of the nonlinear differential equation of the second order $y" + f(x) y^{2n-1} = 0, f(x) > 0$.Časopis Pěst. mat., 1960, 85, N1, 78-83. MR 0142840
Reference: [20] Kamenev I. V.: On oscillation of solutions of second order non-linear equation with coefficient with alternating signs.Differentiaľnye Uravnenija, 1970, 6, N9, 1718-1721. MR 0277817
Reference: [21] Kartsatos A. G.: On n-th order differential inequalities.J. Math. Anal. Appl., 1975, 52, N1, 1-9. Zbl 0327.34012, MR 0390369
Reference: [22a] Kiguradze I. T.: On the conditions of oscillation of solutions of the equation $u" + + a(t) | u |^n \sign u = 0$.(Russian), Časopis Pěst. Mat., 1962, 87, N3, 492-495. MR 0181800
Reference: [22b] Kiguradze I. T.: On oscillation of solutions of some ordinary differential equations.(Russian), Dokl. Akad. Nauk SSSR, 1962, 144, N1, 33-36.
Reference: [22c] Kiguradze I. T.: On asymptotic properties of solutions of the equation $u" + a(t) | u |^n \sign u = 0$.(Russian), Soobšč. Akad. Nauk Gr. SSR, 1963, 30, N2, 129-136. MR 0150385
Reference: [22d] Kiguradze I. T.: On non-oscillatory solutions of the equation $u" + a(t) | u |^n \sign u = 0$.(Russian), Soobšč. Akad. Nauk Gr. SSR, 1964, 35, N1, 15-22. MR 0166446
Reference: [22e] Kiguradze I. T.: Concerning the oscillation of solutions of the equation $\frac{d^m u}{dt^m} + a(t) | u |^n \sign u = 0$.(Russian), Mat. Sb., 1964, 65, N2, 172-187. Zbl 0135.14302
Reference: [22f] Kiguradze I. T.: The problem of oscillation of solutions of non-linear differential equations.(Russian), Differentiaľnye Uravnenija, 1965, 1, N8, 995-1006. MR 0196185
Reference: [22g] Kiguradze I. T.: A note on boundedness of solutions of differential equations.(Russian), Trudy Tbilissk. Univ., 1965, 110, 103-108. MR 0208089
Reference: [22h] Kiguradze I. T.: Asymptotic properties of a certain non-linear differential equation of Emden-Fowler type.(Russian), Izv. Akad. Nauk SSSR, ser. mat., 1965, 29, N5, 965-986. MR 0190459
Reference: [22i] Kiguradze I. T.: A note on oscillation of solutions of the equation $u" + a(t) | u |^n \sign u = 0$.(Russian), Časopis Pěst. Mat., 1967, 92, N3, 343-350. MR 0221012
Reference: [22j] Kiguradze I. T.: On monotone solutions of non-linear ordinary differential equations of order n.(Russian), Izv. Akad. Nauk SSSR, ser. mat., 1969, 33, N , 1373-1398. MR 0259212
Reference: [22k] Kiguradze I. T.: On conditions of oscillation of solutions of non-lineary differential equations.(Russian), I, Differenciaľnye Uravnenija 1974, 10, N8, 1387-1399, II ibidum 1974, 10, N9, 1586-1594. MR 0364755
Reference: [22l] Kiguradze I. T.: Some singular boundary value problems for ordinary differential equations.Tbilisi, Tbilisi Univ. Press, 1975. MR 0499402
Reference: [23] Kiguradze I. T., Chanturia T. A.: A note on asymptotic behaviour of solutions of the equation $u" + a(t)u = 0$.(Russian), Differentiaľnye Uravnenija, 1970, 6, N6, 1115-1117. MR 0271481
Reference: [24] Klebanov L. B.: Local behaviour of solutions of ordinary differential equations.Differentiaľnye Uravnenija, 1971, 7, N8, 1393-1397. MR 0288362
Reference: [25] Klokov I. A.: Some theorems on boundedness of solutions of ordinary differential equations.Uspechi Mat. Nauk, 1958, 80, N2, 189-194. MR 0096029
Reference: [26] Kneser A.: Untersuchungen über die reellen Nultstellen der Integrate linearer Differentialgleichungen.Math. Ann., 1893, 42, N3, 409-435. MR 1510784
Reference: [27a] Kondratjev V. A.: On oscillation of solutions of linear third and fourth order differential equations.(Russian), Trudy Mosk. Mat. Obšč., 1959, 8, 259-281. MR 0104877
Reference: [27b] Kondratjev V. A.: On oscillation of solutions of the equation $y" + p(x) y = 0$.(Russian), Trudy Mosk. Mat. Obšč., 1961, 10,419-436. MR 0141842
Reference: [28a] Koplatadze R. G.: On oscillation of solutions of first order non-linear differential equations with retarded argument.(Russian), Soobšč. Akad. Nauk Gr. SSR, 1973, 70, N1, 17-20. MR 0361369
Reference: [28b] Koplatadze R. G.: On existence of oscillatory solutions of second order non-linear differential equations with retarded argument.Dokl. Akad. Nauk SSSR, 1973, 210, N2, 260-262. MR 0320481
Reference: [28c] Koplatadze R. G.: A note on oscillation of solutions of high order differential inequalities and equations with retarded argument.(Russian), DifFerenciaľnye Uravnenija, 1974, 10, N8, 1400-1405. MR 0358026
Reference: [28d] Koplatadze R. G.: On some properties of the solutions of non-linear differential inequalities and equations with retarded argument.(Russian), Differentiaľnye Uravnenija, 1976, 12, N11, 1971-1984. MR 0466843
Reference: [29] Koplatadze R. G., Chanturia T. A.: On oscillatory solutions of functional-differential equations.(Russian), Tbilisi, Tbilisi Univ. Press, 1977.
Reference: [30] Kostin A. V.: On asymptotic behaviour of continued solutions of the equation of Emden-Fowler type.(Russian), Dokl. Akad. Nauk SSSR, 1971, 200, N1, 28-31. d2Y MR 0296437
Reference: [31] Kuo-liang Chiou: The existence of oscillatory solutions for the equation $\frac{d^2 y}{dt^2} a(t) y^{\tau} =0, 0< \tau< 1$.Proc. Arner. Math. Soc., 1972, 35, 120-122. MR 0301292
Reference: [32a] Kurzweil J.: Sur l'équation $\ddot{x} + f(t) x = 0$.Časopis Pěst. Mat., 1957, 82, N2, 218-226. MR 0094506
Reference: [32b] Kurzweil J.: A note on oscillatory solutions of the equation $y" + f(x) y^{2n-1} = 0$.(Russian), Časop. pro Pěstov. Mat., 1960, 85, N3, 357-358. MR 0126025
Reference: [33] Kusano T., Onose H.: Oscillatory and asymptotic behaviour of sublinear retarded differential equations.Hiroshima Math. J., 1974, 4, N2, 343-353. MR 0350154
Reference: [34] Kvinikadze G. G.: Some notes on solutions of Kneser's problem.(Russian), Differenciaľnye Uravnenija, (to appear).
Reference: [35] Ladas G., Ladde G., Papadakis J. S.: Oscillations of functional-differential equations generated by delays.J. Different. Equat. 1972, 12, N2, 385-395. Zbl 0227.34053, MR 0333399
Reference: [36] Ladas G., Lakshmikantham V., Papadakis J. S.: Oscillations of higher-order retarded differential equations generated by the retarded argument.Delay and Functional Different Equat. and their Appl., New York, Acad. Press, 1972, 219-231. Zbl 0273.34052, MR 0387776
Reference: [37] Ličko I., Švec M.: Le caractère oscillatoire des solutions de l'équation $y^{(n)} + f(x) y^{\alpha} = 0$, n> 1.Czechoslov. Math. J., 1963, 13, N4, 481-491. MR 0161001
Reference: [38] Marušiak P.: On oscillation and monotonicity of solutions of non-linear differential equations with retarded argument.(Russian), Math. Slov., 1976, 26, N2, 139-152. MR 0466867
Reference: [39] Mikusinski J. G.: On Fite's oscillation theorems.Coiloq. Math., 1949, 2, N1, 34-39. Zbl 0039.31302, MR 0041307
Reference: [40] Milloux H.: Sur l'équation différentielle $x" + A(t) x = 0$.Prace Mat. Fizy., 1934, 41, 39-54
Reference: [41a] Mirzov J. D.: Concerning oscillation of solutions of a system of non-linear differential equations.(Russian), Mat. Zametki, 1974, 16, N4, 571-576. MR 0374562
Reference: [41b] Mirzov J. D.: On certain asymptotic properties of solutions of two-dimensional differential systems.Soobšč. Akad. Nauk Gr. SSR, 1975, 80, N3, 545-548. MR 0407378
Reference: [41c] Mirzov J. D.: On the analogue of Kurzweil-Jasny's theorem for the system of two differential equations.(Russian), Časopis Pěst. Mat., 1976, 101, N1, 45-52. MR 0450679
Reference: [41d] Mirzov J. D.: Some notes on asymptotic behaviour of solutions of two-dimensional non-linear differential systems.Studies of certain equations of mathematical physics (Russian) Tbilisi, Tbilisi Univ. Press, 1976, 131-153.
Reference: [42] Mitropolsky I. A., Shevelo V. N.: On the progress of the theory of oscillation of solutions of differential equations with retarded argument.(Russian), Ukr. Mat. Ž., 1977, 29, N3, 313-323.
Reference: [43] Myshkis A. D.: The example of the solution of oscillatory type of the second order differential equation which can't be continued to the whole axis.(Russian), Differenciaľnye Uravnenija, 1969, 5, N12, 2667-2668.
Reference: [44] Naito M.: Oscillation of differential inequalities with retarded arguments.Hiroshima Math. J., 1975, 5, N2, 187-192. MR 0374614
Reference: [45] Nehari Z.: A nonlinear oscillation problem.J. Different. Equat., 1969, 5, N3, 452-460. Zbl 0181.09702, MR 0235203
Reference: [46] Půža B.: On one comparison theorem for the system of two differential equations.(Russian), Differenciaľnye Uravnenija, 1977, 13, N7, 1336-1338. MR 0499444
Reference: [47] Sobol I. M.: On asymptotic behaviour of solutions of linear differential equations.(Russian), Dokl. Akad. Nauk SSSR, 1948, 61, N2, 219-222. MR 0025650
Reference: [48] Švec M.: Sur une propriété des intégrales de l'equation $y^{(n)} + Q(x) y = 0, n = 3, 4$.Czechoslov. Math. J., 1957, 7, 450-462. MR 0095313
Reference: [49] Varekh N. V., Shevelo V. N.: On certain properties of solutions of delay differential equations.(Russian), Ukr. Mat. Ž., 1972, 24, N6, 807-814.
Reference: [50] Waltman P.: An oscillation criterion for a nonlinear second order equation.J. Math. Anal. Appl., 1965, 10, N2, 439-441. Zbl 0131.08902, MR 0173050
Reference: [51a] Čanturia (Chanturia) T. A.: On asymptotic representation of solutions of the equation $u" = a(t) | u |^n \sign u$.(Russian), Differenciaľnye Uravnenija, 1972, 8, N7, 1195-1206. MR 0306638
Reference: [51b] Čanturia (Chanturia) T. A.: On asymptotic representation of oscillatory solutions of the perturbed equation of Emden-Fowler type.Studies of certain equations of mathematical physics (Russian), Tbilisi, Tbilisi Univ. Press, 1972, 5-15.
Reference: [51c] Čanturia (Chanturia) T. A.: On asymptotic behaviour of oscillatory solutions of second order ordinary differential equations.(Russian), Differenciaľnye Uravnenija, 1975, 11, N7, 1232-1245. MR 0377179
Reference: [51d] Čanturia (Chanturia) T. A.: On singular solutions of nonlinear systems of ordinary differential equations.Coll. Math. Soc. János Bolyai, 15. Different. Equat., Keszthely (Hungary), 1975, 107-119. MR 0591720
Reference: [51e] Čanturia (Chanturia) T. A.: On one comparison theorem for linear differential equations.(Russian), Izv. Akad. Nauk SSSR, ser. mat., 1976, 40, N5, 1128-1142. MR 0432973
Reference: [51f] Čanturia (Chanturia) T. A.: On singular solutions of strongly nonlinear systems of ordinary equations.Funct. Theoretic Methods in Different. Equat., London, Pitman Publ., 1976, 196-204. MR 0477225
Reference: [51g] Čanturia (Chanturia) T. A.: Some notes on oscillation of solutions of differential equations with retarded argument.Differenciaľnye Uravnenija (to appear).
Reference: [52a] Ráb M.: Kriterien für die Oszillation der Lösungen der Differentialgleichung $(p(x) y')' + q(x) y = 0$.Časopis Pěst. Mat., 1959, 84, No 3, 335-370. MR 0114964
Reference: [52b] Ráb M.: Asymptotische Formeln für die Lösungen der Differentialgleichung $y" + q(x) y = 0$.Čechoslov. mat. žurnal 1964, 14 (89), 203-221. MR 0165173
Reference: [52c] Ráb M.: Les développements asymptotiques des solutions de l'équation $(py')' + qy = 0$.Arch. Math., 2, (1966), 1-17. MR 0199488
.

Files

Files Size Format View
ArchMath_014-1978-1_3.pdf 2.576Mb application/pdf View/Open
Back to standard record
Partner of
EuDML logo