| Title:
|
Fixed point analysis for non-oscillatory solutions of quasi linear ordinary differential equations (English) |
| Author:
|
Malaguti, Luisa |
| Author:
|
Taddei, Valentina |
| Language:
|
English |
| Journal:
|
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica |
| ISSN:
|
0231-9721 |
| Volume:
|
44 |
| Issue:
|
1 |
| Year:
|
2005 |
| Pages:
|
97-113 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
quasi-linear second order equations |
| Keyword:
|
unbounded |
| Keyword:
|
oscillatory and non-oscillatory solutions |
| Keyword:
|
fixed-point techniques |
| MSC:
|
34C10 |
| MSC:
|
34C11 |
| MSC:
|
47N20 |
| idZBL:
|
Zbl 1098.34025 |
| idMR:
|
MR2218571 |
| . |
| Date available:
|
2009-08-21T06:48:50Z |
| Last updated:
|
2012-05-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/133386 |
| . |
| Reference:
|
[1] Cecchi M., Marini M., Villari G.: On some classes of continuable solutions of a nonlinear differential equation.J. Diff. Equat. 118 (1995), 403–419. Zbl 0827.34020 |
| Reference:
|
[2] 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. Zbl 0846.34027 |
| Reference:
|
[3] Cecchi M., Marini M., Villari G.: Comparison results for oscillation of nonlinear differential equations.Nonlin. Diff. Equat. Appl. 6 (1999), 173–190. Zbl 0927.34023 |
| Reference:
|
[4] 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. Zbl 0278.34026 |
| Reference:
|
[5] Došlá Z., Vrkoč I.: On an extension of the Fubini theorem and its applications in ODEs.Nonlinear Anal. 57 (2004), 531–548. Zbl 1053.34033 |
| Reference:
|
[6] 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. MR 1111319 |
| Reference:
|
[7] 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. MR 1690272 |
| Reference:
|
[8] Kusano T., Norio Y.: Nonoscillation theorems for a class of quasilinear differential equations of second order.J. Math. An. Appl. 189 (1995), 115–127. Zbl 0823.34039, MR 1312033 |
| Reference:
|
[9] Tanigawa T.: Existence and asymptotic behaviour of positive solutions of second order quasilinear differential equations.Adv. Math. Sc. Appl. 9, 2 (1999), 907–938. MR 1725693 |
| Reference:
|
[10] Wang J.: On second order quasilinear oscillations.Funk. Ekv. 41 (1998), 25–54. Zbl 1140.34356, MR 1627369 |
| Reference:
|
[11] Wong J. S. W.: On the generalized Emden–Fowler equation.SIAM Review 17 (1975), 339–360. Zbl 0295.34026, MR 0367368 |
| Reference:
|
[12] Wong J. S. W.: A nonoscillation theorem for Emden–Fowler equations.J. Math. Anal. Appl. 274 (2002), 746–754. Zbl 1036.34039, MR 1936728 |
| . |
