| Title:
|
Oscillation criteria for nonlinear differential equations with $p(t)$-Laplacian (English) |
| Author:
|
Shoukaku, Yutaka |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
141 |
| Issue:
|
1 |
| Year:
|
2016 |
| Pages:
|
71-81 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Recently there has been an increasing interest in studying $p(t)$-Laplacian equations, an example of which is given in the following form $$ (|u'(t)|^{p(t)-2}u'(t))'+c(t)|u(t)|^{q(t)-2}u(t)= 0, \quad t>0. $$ In particular, the first study of sufficient conditions for oscillatory solution of $p(t)$-Laplacian equations was made by Zhang (2007), but to our knowledge, there has not been a paper which gives the oscillatory conditions by utilizing Riccati inequality. Therefore, we establish sufficient conditions for oscillatory solution of nonlinear differential equations with $p(t)$-Laplacian via Riccati method. The results obtained are new and rare, except for a work of Zhang (2007). We present more detailed results than Zhang (2007). (English) |
| Keyword:
|
$p(t)$-Laplacian |
| Keyword:
|
oscillation theory |
| Keyword:
|
Riccati inequality |
| MSC:
|
34K11 |
| MSC:
|
35B05 |
| MSC:
|
35R10 |
| idZBL:
|
Zbl 06562159 |
| idMR:
|
MR3475138 |
| DOI:
|
10.21136/MB.2016.5 |
| . |
| Date available:
|
2016-03-17T19:46:38Z |
| Last updated:
|
2020-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144852 |
| . |
| Reference:
|
[1] Diening, L., Harjulehto, P., H{ä}stö, P., Růžička, M.: Lebesgue and Sobolev Spaces with Variable Exponents.Lecture Notes in Mathematics 2017 Springer, Berlin (2011). Zbl 1222.46002, MR 2790542 |
| Reference:
|
[2] Harjulehto, P., H{ä}stö, P., Lê, Ú. V., Nuortio, M.: Overview of differential equations with non-standard growth.Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72 (2010), 4551-4574. Zbl 1188.35072, MR 2639204, 10.1016/j.na.2010.02.033 |
| Reference:
|
[3] H{ä}st{ö}, P. A.: The {$p(x)$}-{L}aplacian and applications.J. Anal. 15 (2007), 53-62. Zbl 1185.46020, MR 2554092 |
| Reference:
|
[4] {Ş}ahiner, Y., Zafer, A.: Oscillation of nonlinear elliptic inequalities with {$p(x)$}-{L}aplacian.Complex Var. Elliptic Equ. 58 (2013), 537-546. Zbl 1270.35026, MR 3038745, 10.1080/17476933.2012.686493 |
| Reference:
|
[5] {Ş}ahiner, Y., Zafer, A.: Oscillation of {$p(x)$}-{L}aplacian elliptic inequalities with mixed variable exponents.Math. Inequal. Appl. 16 (2013), 947-961. Zbl 1280.35056, MR 3134774 |
| Reference:
|
[6] Usami, H.: Some oscillation theorems for a class of quasilinear elliptic equations.Ann. Mat. Pura Appl. (4) 175 (1998), 277-283. Zbl 0953.35043, MR 1748227 |
| Reference:
|
[7] Yoshida, N.: Picone-type inequality and Sturmian comparison theorems for quasilinear elliptic operators with {$p(x)$}-{L}aplacians.Electron. J. Differ. Equ. (electronic only) 2012 (2012), Article No. 01, 9 pages. Zbl 1239.35057, MR 2889607 |
| Reference:
|
[8] Yoshida, N.: Oscillation criteria for half-linear elliptic inequalities with {$p(x)$}-{L}aplacians via Riccati method.Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74 (2011), 2563-2575. Zbl 1211.35294, MR 2776508, 10.1016/j.na.2010.12.011 |
| Reference:
|
[9] Zhang, Q.: Oscillatory property of solutions for {$p(t)$}-{L}aplacian equations.J. Inequal. Appl. (electronic only) 2007 (2007), Article No. 58548, 8 pages. Zbl 1163.35388, MR 2335972 |
| . |
