Decaying Regularly Varying Solutions of Third-order Differential Equations with a Singular Nonlinearity
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2014)
- Volume: 53, Issue: 1, page 91-105
- ISSN: 0231-9721
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topKučerová, Ivana. "Decaying Regularly Varying Solutions of Third-order Differential Equations with a Singular Nonlinearity." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 53.1 (2014): 91-105. <http://eudml.org/doc/261959>.
@article{Kučerová2014,
abstract = {This paper is concerned with asymptotic analysis of strongly decaying solutions of the third-order singular differential equation $x^\{\prime \prime \prime \}+q(t)x^\{-\gamma \}=0$, by means of regularly varying functions, where $\gamma $ is a positive constant and $q$ is a positive continuous function on $[a,\infty )$. It is shown that if $q$ is a regularly varying function, then it is possible to establish necessary and sufficient conditions for the existence of slowly varying solutions and regularly varying solutions of (A) which decrease to $0$ as $t\rightarrow \infty $ and to acquire precise information about the asymptotic behavior at infinity of these solutions. The main tool is the Schauder–Tychonoff fixed point theorem combined with the basic theory of regular variation.},
author = {Kučerová, Ivana},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {third order nonlinear differential equation; singular nonlinearity; positive solution; decaying solution; asymptotic behavior; regularly varying functions; third-order nonlinear differential equation; singular nonlinearity; positive solution; decaying solution; asymptotic behavior; regularly varying functions},
language = {eng},
number = {1},
pages = {91-105},
publisher = {Palacký University Olomouc},
title = {Decaying Regularly Varying Solutions of Third-order Differential Equations with a Singular Nonlinearity},
url = {http://eudml.org/doc/261959},
volume = {53},
year = {2014},
}
TY - JOUR
AU - Kučerová, Ivana
TI - Decaying Regularly Varying Solutions of Third-order Differential Equations with a Singular Nonlinearity
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2014
PB - Palacký University Olomouc
VL - 53
IS - 1
SP - 91
EP - 105
AB - This paper is concerned with asymptotic analysis of strongly decaying solutions of the third-order singular differential equation $x^{\prime \prime \prime }+q(t)x^{-\gamma }=0$, by means of regularly varying functions, where $\gamma $ is a positive constant and $q$ is a positive continuous function on $[a,\infty )$. It is shown that if $q$ is a regularly varying function, then it is possible to establish necessary and sufficient conditions for the existence of slowly varying solutions and regularly varying solutions of (A) which decrease to $0$ as $t\rightarrow \infty $ and to acquire precise information about the asymptotic behavior at infinity of these solutions. The main tool is the Schauder–Tychonoff fixed point theorem combined with the basic theory of regular variation.
LA - eng
KW - third order nonlinear differential equation; singular nonlinearity; positive solution; decaying solution; asymptotic behavior; regularly varying functions; third-order nonlinear differential equation; singular nonlinearity; positive solution; decaying solution; asymptotic behavior; regularly varying functions
UR - http://eudml.org/doc/261959
ER -
References
top- Bingham, N. H., Goldie, C. M., Teugels, J. L., Regular Variation, Encyklopedia of Mathematics and its Applications 27, Cambridge University Press, Cambridge, 1987. (1987) Zbl0617.26001MR0898871
- Evtukhov, V. M., Samoilenko, A. M., 10.1134/S001226611105003X, Differ. Uravn. 47 (2011), 628–650, (in Russian); translation in Differ. Equ. 47 (2011), 627–649. (2011) Zbl1242.34092MR2918280DOI10.1134/S001226611105003X
- Jaroš, J., Kusano, T., Tanigawa, T., 10.1002/mana.201100296, Math. Nachr. 286 (2013), 205–223. (2013) Zbl1269.34054MR3021476DOI10.1002/mana.201100296
- Jaroš, J., Kusano, T., Tanigawa, T., 10.1515/gmj-2013-0027, Georgian Math. J. 20 (2013), 493–531. (2013) MR3100967DOI10.1515/gmj-2013-0027
- Kamo, K., Usami, H., 10.1016/j.na.2006.12.045, Nonlinear Anal. 68 (2008), 1627–1639. (2008) Zbl1140.34023MR2388837DOI10.1016/j.na.2006.12.045
- Kusano, T., Manojlović, J., 10.14232/ejqtde.2012.1.45, Electron. J. Qual. Theory Differ. Equ. 45 (2012), 1–23. (2012) MR2943099DOI10.14232/ejqtde.2012.1.45
- Kusano, T., Manojlović, J., 10.1016/j.camwa.2011.05.019, Comput. Math. Appl. 62 (2011), 551–565. (2011) Zbl1228.34072MR2817892DOI10.1016/j.camwa.2011.05.019
- Kusano, T., Tanigawa, T., Positive solutions to a class of second order differential equations with singular nonlinearities, Appl. Anal. 69 (1998), 315–331. (1998) Zbl0923.34032MR1706534
- Marić, V., Regular Variation and Differential Equations, Lecture notes in Mathematics 1726, Springer-Verlag, Berlin–Heidelberg, 2000. (2000) MR1753584
- Tanigawa, T., 10.1023/A:1022965616534, Georgian Math. J. 6 (1999), 347–362. (1999) Zbl0933.34028MR1693224DOI10.1023/A:1022965616534
- Taylor, A. E., L’Hospital’s rule, Amer. Math. Monthly 59 (1952), 20–24. (1952) Zbl0046.06202MR0044602
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