Fixed point analysis for non-oscillatory solutions of quasi linear ordinary differential equations
Luisa Malaguti; Valentina Taddei
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2005)
- Volume: 44, Issue: 1, page 97-113
- ISSN: 0231-9721
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topMalaguti, Luisa, and Taddei, Valentina. "Fixed point analysis for non-oscillatory solutions of quasi linear ordinary differential equations." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 44.1 (2005): 97-113. <http://eudml.org/doc/32454>.
@article{Malaguti2005,
abstract = {The paper deals with the quasi-linear ordinary differential equation $(r(t)\varphi (u^\{\prime \}))^\{\prime \}+g(t,u)=0$ with $t \in [0, \infty )$. We treat the case when $g$ is not necessarily monotone in its second argument and assume usual conditions on $r(t)$ and $\varphi (u)$. We find necessary and sufficient conditions for the existence of unbounded non-oscillatory solutions. By means of a fixed point technique we investigate their growth, proving the coexistence of solutions with different asymptotic behaviors. The results generalize previous ones due to Elbert–Kusano, [Acta Math. Hung. 1990]. In some special cases we are able to show the exact asymptotic growth of these solutions. We apply previous analysis for studying the non-oscillatory problem associated to the equation when $\varphi (u)=u$. Several examples are included.},
author = {Malaguti, Luisa, Taddei, Valentina},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {quasi-linear second order equations; unbounded; oscillatory and non-oscillatory solutions; fixed-point techniques; quasi-linear differential equation; oscillatory solution},
language = {eng},
number = {1},
pages = {97-113},
publisher = {Palacký University Olomouc},
title = {Fixed point analysis for non-oscillatory solutions of quasi linear ordinary differential equations},
url = {http://eudml.org/doc/32454},
volume = {44},
year = {2005},
}
TY - JOUR
AU - Malaguti, Luisa
AU - Taddei, Valentina
TI - Fixed point analysis for non-oscillatory solutions of quasi linear ordinary differential equations
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2005
PB - Palacký University Olomouc
VL - 44
IS - 1
SP - 97
EP - 113
AB - The paper deals with the quasi-linear ordinary differential equation $(r(t)\varphi (u^{\prime }))^{\prime }+g(t,u)=0$ with $t \in [0, \infty )$. We treat the case when $g$ is not necessarily monotone in its second argument and assume usual conditions on $r(t)$ and $\varphi (u)$. We find necessary and sufficient conditions for the existence of unbounded non-oscillatory solutions. By means of a fixed point technique we investigate their growth, proving the coexistence of solutions with different asymptotic behaviors. The results generalize previous ones due to Elbert–Kusano, [Acta Math. Hung. 1990]. In some special cases we are able to show the exact asymptotic growth of these solutions. We apply previous analysis for studying the non-oscillatory problem associated to the equation when $\varphi (u)=u$. Several examples are included.
LA - eng
KW - quasi-linear second order equations; unbounded; oscillatory and non-oscillatory solutions; fixed-point techniques; quasi-linear differential equation; oscillatory solution
UR - http://eudml.org/doc/32454
ER -
References
top- Cecchi M., Marini M., Villari G., On some classes of continuable solutions of a nonlinear differential equation, J. Diff. Equat. 118 (1995), 403–419. (1995) Zbl0827.34020
- Cecchi M., Marini M., Villari G., Topological and variational approaches for nonlinear oscillation: an extension of a Bhatia result, Proc. First World Congress Nonlinear Analysts, Walter de Gruyter, Berlin, 1996, 1505–1514. (1996) Zbl0846.34027
- Cecchi M., Marini M., Villari G., Comparison results for oscillation of nonlinear differential equations, Nonlin. Diff. Equat. Appl. 6 (1999), 173–190. (1999) Zbl0927.34023
- Coffman C. V., Wong J. S. W., Oscillation and nonoscillation of solutions of generalized Emden–Fowler equations, Trans. Amer. Math. Soc. 167 (1972), 399–434. (1972) Zbl0278.34026
- Došlá Z., Vrkoč I., On an extension of the Fubini theorem and its applications in ODEs, Nonlinear Anal. 57 (2004), 531–548. Zbl1053.34033
- Elbert A., Kusano T., Oscillation and non-oscillation theorems for a class of second order quasilinear differential equations, Acta Math. Hung. 56 (1990), 325–336. (1990) MR1111319
- Kiyomura J., Kusano T., Naito M., Positive solutions of second order quasilinear ordinary differential equations with general nonlinearities, St. Sc. Math. Hung. 35 (1999), 39–51. (1999) MR1690272
- Kusano T., Norio Y., Nonoscillation theorems for a class of quasilinear differential equations of second order, J. Math. An. Appl. 189 (1995), 115–127. (1995) Zbl0823.34039MR1312033
- Tanigawa T., Existence and asymptotic behaviour of positive solutions of second order quasilinear differential equations, Adv. Math. Sc. Appl. 9, 2 (1999), 907–938. (1999) MR1725693
- Wang J., On second order quasilinear oscillations, Funk. Ekv. 41 (1998), 25–54. (1998) Zbl1140.34356MR1627369
- Wong J. S. W., On the generalized Emden–Fowler equation, SIAM Review 17 (1975), 339–360. (1975) Zbl0295.34026MR0367368
- Wong J. S. W., A nonoscillation theorem for Emden–Fowler equations, J. Math. Anal. Appl. 274 (2002), 746–754. Zbl1036.34039MR1936728
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