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Keywords:
systems of nonlinear differential equations; positive solutions; asymptotic behavior; regularly varying functions
systems of nonlinear differential equations; positive solutions; asymptotic behavior; regularly varying functions
Summary:
The system of nonlinear differential equations \begin{equation*} x^{\prime } + p_1(t)x^{\alpha _1} + q_1(t)y^{\beta _1} = 0\,, \qquad y^{\prime } + p_2(t)x^{\alpha _2} + q_2(t)y^{\beta _2} = 0\,, A \end{equation*} is under consideration, where $\alpha _i$ and $\beta _i$ are positive constants and $p_i(t)$ and $q_i(t)$ are positive continuous functions on $[a,\infty )$. There are three types of different asymptotic behavior at infinity of positive solutions $(x(t),y(t))$ of (). The aim of this paper is to establish criteria for the existence of solutions of these three types by means of fixed point techniques. Special emphasis is placed on those solutions with both components decreasing to zero as $t \rightarrow \infty $, which can be analyzed in detail in the framework of regular variation.
References:
[1] Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Encyclopedia Math. Appl., Cambridge University Press, 1987. MR 0898871 | Zbl 0617.26001
[2] Coppel, W.A.: Stability and Asymptotic Behavior of Differential Equations. D.C. Heath and Company, Boston, 1965. MR 0190463 | Zbl 0154.09301
[3] Evtukhov, V.M., Vladova, E.S.: Asymptotic representations of solutions of essentially nonlinear cyclic systems of ordinary differential equations. Differential Equations 48 (2012), 630–646. DOI 10.1134/S0012266112050023 | MR 3177196 | Zbl 1271.34055
[4] Haupt, O., Aumann, G.: Differential- und Integralrechnung. Walter de Gruyter & Co., Berlin, 1938.
[5] Jaroš, J., Kusano, T.: Existence and precise asymptotic behavior of strongly monotone solutions of systems of nonlinear differential equations. Differ. Equ. Appl. 5 (2013), 185–204. MR 3099987
[6] Jaroš, J., Kusano, T.: Slowly varying solutions of a class of first order systems of nonlinear differentialequations. Acta Math. Univ. Comenian. 82 (2013), 265–284. MR 3106802
[7] Jaroš, J., Kusano, T., Tanigawa, T.: 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. DOI 10.1002/mana.201100296 | MR 3021476 | Zbl 1269.34054
[8] Kusano, T., Manojlović, J.: Asymptotic behavior of positive solutions of sublinear differentialequations of Emden-Fowler type. Comput. Math. Appl. 62 (2011), 551–565. DOI 10.1016/j.camwa.2011.05.019 | MR 2817892
[9] Kusano, T., Manojlović, J.: Precise asymptotic behavior of solutions of the sublinear Emden-Fowlerdifferential equation. Appl. Math. Comput. 217 (2011), 4382–4396. DOI 10.1016/j.amc.2010.09.061 | MR 2745121
[10] Kusano, T., Manojlović, J.: Positive solutions of fourth order Emden-Fowler type differential equationsin the framework of regular variation. Appl. Math. Comput. 218 (2012), 6684–6701. DOI 10.1016/j.amc.2011.12.029 | MR 2880324
[11] Kusano, T., Manojlović, J.: Positive solutions of fourth order Thomas-Fermi type differential equationsin the framework of regular variation. Acta Appl. Math. 121 (2012), 81–103. DOI 10.1007/s10440-012-9691-5 | MR 2966967
[12] Kusano, T., Manojlović, J.: Complete asymptotic analysis of positive solutions of odd-order nonlinear differential equations. Lithuanian Math. J. 53 (2013), 40–62. DOI 10.1007/s10986-013-9192-x | MR 3038148
[13] Kusano, T., Marić, V., Tanigawa, T.: An asymptotic analysis of positive solutions of generalized Thomas-Fermi differential equations – The sub-half-linear case. Nonlinear Anal. 75 (2012), 2474–2485. MR 2870933 | Zbl 1248.34075
[14] Marić, V.: Regular Variation and Differential Equations. Lecture Notes in Math., vol. 1726, Springer-Verlag, Berlin, 2000. DOI 10.1007/BFb0103952 | MR 1753584
[15] Mirzov, J.D.: Asymptotic Properties of Solutions of Systems of Nonlinear Nonautonomous Ordinary Differential Equations. Folia Fac. Sci. Natur. Univ. Masaryk. Brun. Math., Mathematica, vol. 14, Masaryk University, Brno, 2004. MR 2144761 | Zbl 1154.34300
[16] Seneta, E.: Regularly Varying Functions. Lecture Notes in Math., vol. 508, Springer Verlag, Berlin-Heidelberg, 1976. MR 0453936 | Zbl 0324.26002
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