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Keywords:
asymptotic behavior; nonoscillatory solution; half-linear differential equation; Hardy-type inequality
asymptotic behavior; nonoscillatory solution; half-linear differential equation; Hardy-type inequality
Summary:
We consider the half-linear differential equation of the form \[ (p(t)|x^{\prime }|^{\alpha }\mathrm{sgn}\,x^{\prime })^{\prime } + q(t)|x|^{\alpha }\mathrm{sgn}\,x = 0\,, \quad t \ge t_{0} \,, \] under the assumption that $p(t)^{-1/\alpha }$ is integrable on $[t_{0}, \infty )$. It is shown that if a certain condition is satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as $t \rightarrow \infty $.
References:
[1] Beesack, P.R.: Hardy’s inequality and its extensions. Pacific J. Math. 11 (1961), 39–61. DOI 10.2140/pjm.1961.11.39 | MR 0121449
[2] Došlý, O., Řehák, P.: Half-Linear Differential Equations. North-Holland Mathematics Studies, vol. 202, Elsevier, Amsterdam, 2005. MR 2158903
[3] Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. second ed., Cambridge University Press, Cambridge, 1952. MR 0046395 | Zbl 0047.05302
[4] Hartman, P.: Ordinary Differential Equations. SIAM, Classics in Applied Mathematics, Wiley, 1964. MR 0171038 | Zbl 0125.32102
[5] Jaroš, J., Kusano, T.: Self-adjoint differential equations and generalized Karamata functions. Bull. T. CXXIX de Acad. Serbe Sci. et Arts, Classe Sci. Mat. Nat. Sci. Math. 29 (2004), 25–60. MR 2099615
[6] Jaroš, J., Kusano, T., Tanigawa, T.: Nonoscillation theory for second order half-linear differential equations in the framework of regular variation. Results Math. 43 (2003), 129–149. DOI 10.1007/BF03322729 | MR 1962855
[7] Jaroš, J., Kusano, T., Tanigawa, T.: Nonoscillatory half-linear differential equations and generalized Karamata functions. Nonlinear Anal. 64 (2006), 762–787. DOI 10.1016/j.na.2005.05.045 | MR 2197094
[8] Kusano, T., Manojlović, J.V.: Asymptotic behavior of solutions of half-linear differential equations and generalized Karamata functions. to appear in Georgian Math. J.
[9] Kusano, T., Manojlović, J.V.: Precise asymptotic behavior of regularly varying solutions of second order half-linear differential equations. Electron. J. Qual. Theory Differ. Equ. (2016), 24 pp., paper No. 62. MR 3547438
[10] Manojlović, J.V.: Asymptotic analysis of regularly varying solutions of second-order half-linear differential equations. Kyoto University, RIMS Kokyuroku 2080 (2018), 4–17.
[11] Marić, V.: Regular Variation and Differential Equations. Lecture Notes in Math., vol. 1726, Springer, 2000. MR 1753584
[12] Naito, M.: Asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations. to appear in Arch. Math. (Basel).
[13] Naito, M.: Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, I. Opuscula Math. 41 (2021), 71–94. DOI 10.7494/OpMath.2021.41.1.71
[14] Řehák, P.: Asymptotic formulae for solutions of half-linear differential equations. Appl. Math. Comput. 292 (2017), 165–177. MR 3542549
[15] Řehák, P., Taddei, V.: Solutions of half-linear differential equations in the classes gamma and pi. Differ. Integral Equ. 29 (2016), 683–714. MR 3498873
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