# Article

 Title: On the asymptotic behavior at infinity of solutions to quasi-linear differential equations (English) Author: Astashova, Irina Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 (print) ISSN: 2464-7136 (online) Volume: 135 Issue: 4 Year: 2010 Pages: 373-382 Summary lang: English . Category: math . Summary: Sufficient conditions are formulated for existence of non-oscillatory solutions to the equation $$y^{(n)}+\sum _{j=0}^{n-1}a_j(x)y^{(j)}+p(x)|y|^k \mathop {\rm sgn} y =0$$ with $n\ge 1$, real (not necessarily natural) $k>1$, and continuous functions $p(x)$ and $a_j(x)$ defined in a neighborhood of $+\infty$. For this equation with positive potential $p(x)$ a criterion is formulated for existence of non-oscillatory solutions with non-zero limit at infinity. In the case of even order, a criterion is obtained for all solutions of this equation at infinity to be oscillatory. \endgraf Sufficient conditions are obtained for existence of solution to this equation which is equivalent to a polynomial. (English) Keyword: quasi-linear ordinary differential equation of higher order Keyword: existence of non-oscillatory solution Keyword: oscillatory solution MSC: 34C10 MSC: 34C15 idZBL: Zbl 1224.34098 idMR: MR2681011 DOI: 10.21136/MB.2010.140828 . Date available: 2010-11-24T08:25:33Z Last updated: 2020-07-29 Stable URL: http://hdl.handle.net/10338.dmlcz/140828 . Reference: [1] Atkinson, F. V.: On second order nonlinear oscillations.Pacif. J. Math. 5 (1955), 643-647. MR 0072316, 10.2140/pjm.1955.5.643 Reference: [2] Astashova, I. V.: Application of dynamical systems to the study of asymptotic properties of solutions to nonlinear higher-order differential equations.J. Math. Sci., New York 126 (2005), 1361-1391. Zbl 1093.37005, MR 2157611, 10.1007/PL00021970 Reference: [3] Astashova, I. V.: Uniform estimates to the positive solutions of quasilinear differential equations of even order.J. Math. Sci., New York 135 (2006), 2616-2624. MR 2271904, 10.1007/s10958-006-0133-7 Reference: [4] Astashova, I. V.: On existence of non-oscillatory solutions to quasi-linear differential equations.Georgian Math. J. 14 (2007), 223-238. MR 2341274 Reference: [5] Belohorec, S. A.: A criterion for oscillation and nonoscillation.Acta F. R. N. Univ. Comen. Math. 20 (1969), 75-79. Zbl 0225.34019, MR 0274855 Reference: [6] Kartsatos, A. G.: $N$th order oscillations with middle terms of order $N-2$.Pacific J. Math. 67 (1976), 477-488. MR 0440122, 10.2140/pjm.1976.67.477 Reference: [7] Kiguradze, I. T.: On conditions for oscillation of solutions of the equation $u''+a(t) |u|^n \*\mathop sgnu=0$.Čas. Pěst. Mat. 87 (1962), 492-495 Russian. Zbl 0138.33504, MR 0181800 Reference: [8] Kiguradze, I. T.: On the oscillation of solution of the equation $d^m/ d t^m+a(t)|u|^n \*\mathop sign u=0$.Mat. Sbornik 65 (1964), 172-187 Russian. Zbl 0135.14302, MR 0173060 Reference: [9] Kiguradze, I. T., Chanturiya, T. A.: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations.Kluver Academic Publishers, Dordrecht (1993). Zbl 0782.34002, MR 1220223 Reference: [10] Kiguradze, I. T.: On the oscillation criteria for one class of ordinary differential equations.Diff. Uravnenija 28 (1992), 207-219 Russian. Reference: [11] Kondratiev, V. A., Samovol, V. S.: On some asymptotic properties of solutions for the Emden-Fowler type equations.Diff. Uravnenija 17 (1981), 749-750 Russian. Reference: [12] Kusano, T., Naito, M.: Nonlinear oscillation of fourth-order differential equations.Canad. J. Math. 28 (1976), 840-852. Zbl 0432.34022, MR 0430415, 10.4153/CJM-1976-081-0 Reference: [13] Levin, A. Yu.: Nonoscillation of solutions of the equation $x^{(n)}+p_1(t)x^{(n-1)}+\dots+p_n(t)\* x=0$.Usp. Mat. Nauk. 24 (1969), 43-96 Russian. MR 0254328 Reference: [14] Lovelady, D. L.: On the oscillatory behavior of bounded solutions of higher order differential equations.J. Diff. Equations 19 (1975), 167-175. Zbl 0333.34030, MR 0382781, 10.1016/0022-0396(75)90026-1 Reference: [15] Lovelady, D. L.: An oscillation criterion for a fourth-order integrally superlinear differential equation.Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Natur. 8 (1975), 531-536. Zbl 0348.34026, MR 0422766 Reference: [16] Masci, J. W., Wong, J. S. W.: Oscillation of solutions to second-order nonlinear differential equations.Pacif. J. Math. 24 (1968), 111-117. MR 0224908, 10.2140/pjm.1968.24.111 Reference: [17] Pólya, G.: On the mean-value theorem corresponding to a given linear homogeneous differential equation.Trans. Amer. Math. Soc. 24 (1924), 312-324. MR 1501228, 10.2307/1988819 Reference: [18] Sobol, I. M.: On asymptotical behavior of solutions to linear differential equations.Doklady Akad. Nauk SSSR 61 (1948), 219-222 Russian. MR 0025650 Reference: [19] Taylor, W. E. Jr.: Oscillation criteria for certain nonlinear fourth order equations.Internat. J. Math. 6 (1983), 551-557. Zbl 0539.34021, MR 0712574, 10.1155/S0161171283000502 Reference: [20] Vallée-Poussin, Ch. I. de la: Sur l'équation différentielle linéaire du second ordre. Détermination d'une intégrale par deux valeurs assignées. Extension aux équations d'ordre $n$.J. Math. Pures Appl. 9 (1929), 125-144. Reference: [21] Waltman, P.: Some properties of solutions of $u''+a(t) f(u)=0$.Monatsh. Math. 67 (1963), 50-54. Zbl 0116.29401, MR 0147700, 10.1007/BF01300681 Reference: [22] Wong, J. S. W.: On second-order nonlinear oscillation.Funkcialaj Ekvacioj 11 (1968), 207-234. Zbl 0157.14802, MR 0245915 .

## Files

Files Size Format View
MathBohem_135-2010-4_4.pdf 246.4Kb application/pdf View/Open

Partner of