Opuscula Math. 35, no. 1 (2015), 47-69
http://dx.doi.org/10.7494/OpMath.2015.35.1.47

 
Opuscula Mathematica

Strongly increasing solutions of cyclic systems of second order differential equations with power-type nonlinearities

Jaroslav Jaroš
Kusano Takaŝi

Abstract. We consider \(n\)-dimensional cyclic systems of second order differential equations \[(p_i(t)|x_{i}'|^{\alpha_i -1}x_{i}')' = q_{i}(t)|x_{i+1}|^{\beta_i-1}x_{i+1},\] \[\quad i = 1,\ldots,n, \quad (x_{n+1} = x_1) \tag{\(\ast\)}\] under the assumption that the positive constants \(\alpha_i\) and \(\beta_i\) satisfy \(\alpha_1{\ldots}\alpha_n \gt \beta_1{\ldots}\beta_n\) and \(p_i(t)\) and \(q_i(t)\) are regularly varying functions, and analyze positive strongly increasing solutions of system (\(\ast\)) in the framework of regular variation. We show that the situation for the existence of regularly varying solutions of positive indices for (\(\ast\)) can be characterized completely, and moreover that the asymptotic behavior of such solutions is governed by the unique formula describing their order of growth precisely. We give examples demonstrating that the main results for (\(\ast\)) can be applied to some classes of partial differential equations with radial symmetry to acquire accurate information about the existence and the asymptotic behavior of their radial positive strongly increasing solutions.

Keywords: systems of differential equations, positive solutions, asymptotic behavior, regularly varying functions.

Mathematics Subject Classification: 34C11, 26A12.

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  1. J.A.D. Appleby, D.D. Patterson, Classification of convergence rates of solutions of perturbed ordinary differential equations with regularly varying nonlinearity, preprint, 2013, arXiv:1303.3345v3.
  2. N.H. Bingham, C.M. Goldie, J.L. Teugels, Regular Variation, Encyclopedia of Mathematics and its Applications, Vol. 27, Cambridge Universiy Press, 1987.
  3. C. Cîrstea, V. Radulescu, Nonlinear problems with boundary blow-up: a Karamata regular variation theory approach, Asymptot. Anal. 46 (2006), 275-298.
  4. V.M. Evtukhov, E.S. Vladova, Asymptotic representations of solutions of essentially nonlinear cyclic systems of ordinary differential equations, Differ. Equ. 48 (2012), 630-646.
  5. O. Haupt, G. Aumann, Differential- Und Integralrechnung, Walter de Gruyter, Berlin, 1938.
  6. J. Jaroš, T. Kusano, Existence and precise asymptotic behavior of strongly monotone solutions of systems of nonlinear differential equations, Differ. Equ. Appl. 5 (2013), 185-204.
  7. J. Jaroš, T. Kusano, Asymptotic behavior of positive solutions of a class of systems of second order nonlinear differential equations, Electron. J. Qual. Theory Differ. Equ. 23 (2013), 1-23.
  8. J. Jaroš, T. Kusano, On strongly decreasing solutions of cyclic systems of second order nonlinear differential equations, Proc. Roy. Soc. Edinburgh Sect. A, to appear.
  9. J. Jaroš, T. Kusano, T. Tanigawa, Asymptotic analysis of positive solutions of a class of third order nonlinear differential equations in the framework of regular variation, Math. Nachr. 286 (2013), 205-223.
  10. T. Kusano, J. Manojlović, Asymptotic behavior of positive solutions of sublinear differential equations of Emden-Fowler type, Comput. Math. Appl. 62 (2011), 551-565.
  11. T. Kusano, J. Manojlović, Positive solutions of fourth order Thomas-Fermi type differential equations in the framework of regular variation, Acta Appl. Math. 121 (2012), 81-103.
  12. T. Kusano, J. Manojlović, Asymptotic behavior of positive solutions of odd order Emden-Fowler type differential equations in the framework of regular variation, Electron. J. Qual. Theory Differ. Equ. 45 (2012), 1-23.
  13. T. Kusano, V. Marić, T. Tanigawa, An asymptotic analysis of positive solutions of generalized Thomas-Fermi differential equations - the sub-half-linear case, Nonlinear Anal. 75 (2012), 2474-2485.
  14. V. Marić, Regular Variation and Differential Equations, Lecture Notes in Mathematics, Vol. 1726, Springer Verlag, Berlin-Heidelberg, 2000.
  15. E. Seneta, Regularly Varying Functions, Lecture Notes in Mathematics, Vol. 508, Springer Verlag, Berlin-Heidelberg, 1976.
  • Jaroslav Jaroš
  • Comenius University, Faculty of Mathematics, Physics and Informatics, Department of Mathematical Analysis and Numerical Mathematics, 842 48 Bratislava, Slovakia
  • Kusano Takaŝi
  • Hiroshima University, Faculty of Science, Department of Mathematics, Higashi-Hiroshima 739-8526, Japan
  • Communicated by Alexander Domoshnitsky.
  • Received: 2014-02-14.
  • Revised: 2014-05-20.
  • Accepted: 2014-05-31.
  • Published online: 2014-11-12.
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Cite this article as:
Jaroslav Jaroš, Kusano Takaŝi, Strongly increasing solutions of cyclic systems of second order differential equations with power-type nonlinearities, Opuscula Math. 35, no. 1 (2015), 47-69, http://dx.doi.org/10.7494/OpMath.2015.35.1.47

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