Integral averaging technique for oscillation of damped half-linear oscillators

Yukihide Enaka; Masakazu Onitsuka

Czechoslovak Mathematical Journal (2018)

  • Volume: 68, Issue: 3, page 755-770
  • ISSN: 0011-4642

Abstract

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This paper is concerned with the oscillatory behavior of the damped half-linear oscillator ( a ( t ) φ p ( x ' ) ) ' + b ( t ) φ p ( x ' ) + c ( t ) φ p ( x ) = 0 , where φ p ( x ) = | x | p - 1 sgn x for x ∈ ℝ and p > 1 . A sufficient condition is established for oscillation of all nontrivial solutions of the damped half-linear oscillator under the integral averaging conditions. The main result can be given by using a generalized Young’s inequality and the Riccati type technique. Some examples are included to illustrate the result. Especially, an example which asserts that all nontrivial solutions are oscillatory if and only if p ≠ 2 is presented.

How to cite

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Enaka, Yukihide, and Onitsuka, Masakazu. "Integral averaging technique for oscillation of damped half-linear oscillators." Czechoslovak Mathematical Journal 68.3 (2018): 755-770. <http://eudml.org/doc/294404>.

@article{Enaka2018,
abstract = {This paper is concerned with the oscillatory behavior of the damped half-linear oscillator $(a(t)\phi _p(x^\{\prime \}))^\{\prime \}+b(t)\phi _p(x^\{\prime \})+c(t)\phi _p(x) = 0$, where $\phi _p(x) = |x|^\{p-1\}\mathop \{\rm sgn\} x$ for $x \in \mathbb \{R\}$ and $p > 1$. A sufficient condition is established for oscillation of all nontrivial solutions of the damped half-linear oscillator under the integral averaging conditions. The main result can be given by using a generalized Young’s inequality and the Riccati type technique. Some examples are included to illustrate the result. Especially, an example which asserts that all nontrivial solutions are oscillatory if and only if $p \ne 2$ is presented.},
author = {Enaka, Yukihide, Onitsuka, Masakazu},
journal = {Czechoslovak Mathematical Journal},
keywords = {damped half-linear oscillator; integral averaging technique; Riccati technique; generalized Young inequality; oscillatory solution},
language = {eng},
number = {3},
pages = {755-770},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Integral averaging technique for oscillation of damped half-linear oscillators},
url = {http://eudml.org/doc/294404},
volume = {68},
year = {2018},
}

TY - JOUR
AU - Enaka, Yukihide
AU - Onitsuka, Masakazu
TI - Integral averaging technique for oscillation of damped half-linear oscillators
JO - Czechoslovak Mathematical Journal
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 3
SP - 755
EP - 770
AB - This paper is concerned with the oscillatory behavior of the damped half-linear oscillator $(a(t)\phi _p(x^{\prime }))^{\prime }+b(t)\phi _p(x^{\prime })+c(t)\phi _p(x) = 0$, where $\phi _p(x) = |x|^{p-1}\mathop {\rm sgn} x$ for $x \in \mathbb {R}$ and $p > 1$. A sufficient condition is established for oscillation of all nontrivial solutions of the damped half-linear oscillator under the integral averaging conditions. The main result can be given by using a generalized Young’s inequality and the Riccati type technique. Some examples are included to illustrate the result. Especially, an example which asserts that all nontrivial solutions are oscillatory if and only if $p \ne 2$ is presented.
LA - eng
KW - damped half-linear oscillator; integral averaging technique; Riccati technique; generalized Young inequality; oscillatory solution
UR - http://eudml.org/doc/294404
ER -

References

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  1. Agarwal, R. P., Grace, S. R., O'Regan, D., 10.1007/978-94-017-2515-6, Kluwer Academic Publishers, Dordrecht (2002). (2002) Zbl1073.34002MR2091751DOI10.1007/978-94-017-2515-6
  2. Çakmak, D., 10.1016/j.jmaa.2004.06.046, J. Math. Anal. Appl. 300 (2004), 408-425. (2004) Zbl1082.34030MR2098218DOI10.1016/j.jmaa.2004.06.046
  3. Cecchi, M., Došlá, Z., Došlý, O., Marini, M., 10.14232/ejqtde.2013.1.12, Electron. J. Qual. Theory Differ. Equ. 2013 (2013), Paper No. 12, 14 pages. (2013) Zbl1340.34125MR3019662DOI10.14232/ejqtde.2013.1.12
  4. Coles, W. J., 10.2307/2035563, Proc. Am. Math. Soc. 19 (1968), 507. (1968) Zbl0155.12802MR0223644DOI10.2307/2035563
  5. Coppel, W. A., 10.1007/BFb0058618, Lecture Notes in Mathematics 220, Springer, Berlin (1971). (1971) Zbl0224.34003MR0460785DOI10.1007/BFb0058618
  6. Došlý, O., Half-linear differential equations, Handbook of Differential Equations: Ordinary Differential Equations. Vol. I. Elsevier/North-Holland, Amsterdam (2004), 161-357 A. Cañada et al. (2004) Zbl1090.34027MR2166491
  7. Došlý, O., Řehák, P., 10.1016/S0304-0208(13)72439-3, North-Holland Mathematics Studies 202, Elsevier Science, Amsterdam (2005). (2005) Zbl1090.34001MR2158903DOI10.1016/S0304-0208(13)72439-3
  8. Fišnarová, S., Mařík, R., 10.1155/2011/638271, Abstr. Appl. Anal. 2011 (2011), Article ID 638271, 15 pages. (2011) Zbl1232.34052MR2846243DOI10.1155/2011/638271
  9. Grace, S. R., Lalli, B. S., 10.1007/BF01762792, Ann. Mat. Pura Appl., IV. Ser. 151 (1988), 149-159. (1988) Zbl0648.34039MR0964507DOI10.1007/BF01762792
  10. Grace, S. R., Lalli, B. S., 10.1016/0022-247X(90)90301-U, J. Math. Anal. Appl. 149 (1990), 277-311. (1990) Zbl0697.34040MR1054809DOI10.1016/0022-247X(90)90301-U
  11. Hartman, P., Ordinary Differential Equations, John Wiley and Sons, New York (1964). (1964) Zbl0125.32102MR0171038
  12. Hasil, P., Veselý, M., 10.1556/SScMath.51.2014.3.1283, Stud. Sci. Math. Hung. 51 (2014), 303-321. (2014) Zbl1340.34127MR3254918DOI10.1556/SScMath.51.2014.3.1283
  13. Hasil, P., Veselý, M., 10.1186/s13662-015-0533-4, Adv. Difference Equ. 2015 (2015), Article No. 190, 17 pages. (2015) MR3358100DOI10.1186/s13662-015-0533-4
  14. Hata, S., Sugie, J., 10.1007/s10474-012-0259-7, Acta Math. Hung. 138 (2013), 156-172. (2013) Zbl1299.34193MR3015969DOI10.1007/s10474-012-0259-7
  15. Kamenev, I. V., 10.1007/BF01153154, Math. Notes 23 (1978), 136-138. English. Russian original translation from Mat. Zametki 23 1978 249-251. (1978) Zbl0408.34031MR0486798DOI10.1007/BF01153154
  16. Li, T., Rogovchenko, Y. V., Tang, S., 10.2478/s12175-014-0271-1, Math. Slovaca 64 (2014), 1227-1236. (2014) Zbl1349.34111MR3277849DOI10.2478/s12175-014-0271-1
  17. Mirzov, J. D., Asymptotic Properties of Solutions of Systems of Nonlinear Nonautonomous Ordinary Differential Equations, Folia Facultatis Scientiarum Naturalium Universitatis Masarykianae Brunensis. Mathematica 14, Masaryk University, Brno 14 (2004). (2004) Zbl1154.34300MR2144761
  18. Onitsuka, M., Soeda, T., 10.1186/s13662-015-0494-7, Adv. Difference Equ. 2015 (2015), Article No. 158, 24 pages. (2015) MR3359784DOI10.1186/s13662-015-0494-7
  19. Onitsuka, M., Sugie, J., 10.1017/S0308210510000326, Proc. R. Soc. Edinb., Sect. A, Math. 141 (2011), 1083-1101. (2011) Zbl1232.34081MR2838369DOI10.1017/S0308210510000326
  20. Onitsuka, M., Tanaka, S., 10.1007/s10474-017-0722-6, Acta Math. Hung. 152 (2017), 336-364. (2017) Zbl06806136MR3682888DOI10.1007/s10474-017-0722-6
  21. Pašić, M., 10.1155/2013/852180, Abstr. Appl. Anal. 2013 (2013), Article ID 852180, 10 pages. (2013) Zbl1308.34043MR3034985DOI10.1155/2013/852180
  22. Řehák, P., A Riccati technique for proving oscillation of a half-linear equation, Electron. J. Differ. Equ. 2008 (2008), Article No. 105, 8 pages. (2008) Zbl1170.34317MR2430902
  23. Řehák, P., 10.1007/s10474-008-7181-z, Acta Math. Hung. 121 (2008), 93-105. (2008) Zbl1199.34167MR2463252DOI10.1007/s10474-008-7181-z
  24. Řehák, P., 10.1007/s10474-015-0522-9, Acta Math. Hung. 147 (2015), 158-171. (2015) Zbl1374.34102MR3391519DOI10.1007/s10474-015-0522-9
  25. Sugie, J., 10.1016/j.na.2011.07.028, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74 (2011), 7151-7167. (2011) Zbl1243.34073MR2833701DOI10.1016/j.na.2011.07.028
  26. Sugie, J., Matsumura, K., 10.1016/j.amc.2007.10.007, Appl. Math. Comput. 199 (2008), 447-455. (2008) Zbl1217.34056MR2420574DOI10.1016/j.amc.2007.10.007
  27. Sugie, J., Onitsuka, M., Global asymptotic stability for damped half-linear differential equations, Acta Sci. Math. 73 (2007), 613-636. (2007) Zbl1265.34199MR2380068
  28. Sugie, J., Onitsuka, M., 10.1016/j.na.2013.12.005, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 98 (2014), 83-103. (2014) Zbl1291.34096MR3158447DOI10.1016/j.na.2013.12.005
  29. Swanson, C. A., 10.1016/s0076-5392(08)x6131-4, Mathematics in Science and Engineering 48. Academic Press, New York (1968). (1968) Zbl0191.09904MR0463570DOI10.1016/s0076-5392(08)x6131-4
  30. Tiryaki, A., 10.14232/ejqtde.2009.1.61, Electron. J. Qual. Theory Differ. Equ. 2009 (2009), Article No. 61, 11 pages. (2009) Zbl1195.34100MR2558636DOI10.14232/ejqtde.2009.1.61
  31. Tiryaki, A., Çakmak, D., Ayanlar, B., 10.1016/S0022-247X(03)00145-8, J. Math. Anal. Appl. 281 (2003), 565-574. (2003) Zbl1030.34033MR1982674DOI10.1016/S0022-247X(03)00145-8
  32. Tunç, E., Avcı, H., 10.1007/s11253-012-0590-8, Ukr. Math. J. 63 (2012), 1441-1457. English. Russian original translation from Ukr. Mat. Zh. 63 2011 1263-1278. (2012) Zbl1256.34023MR3109665DOI10.1007/s11253-012-0590-8
  33. Wintner, A., 10.1090/qam/28499, Q. Appl. Math. 7 (1949), 115-117. (1949) Zbl0032.34801MR0028499DOI10.1090/qam/28499
  34. Wong, J. S. W., 10.1006/jmaa.2000.7376, J. Math. Anal. Appl. 258 (2001), 244-257. (2001) Zbl0987.34024MR1828103DOI10.1006/jmaa.2000.7376
  35. Yamaoka, N., 10.1016/j.jmaa.2006.02.021, J. Math. Anal. Appl. 325 (2007), 932-948. (2007) Zbl1108.34027MR2270061DOI10.1016/j.jmaa.2006.02.021
  36. Zhang, Q., Wang, L., 10.1007/s10440-009-9483-8, Acta Appl. Math. 110 (2010), 885-893. (2010) Zbl1197.34050MR2610598DOI10.1007/s10440-009-9483-8
  37. Zheng, W., Sugie, J., 10.1007/s00605-014-0695-2, Monatsh. Math. 179 (2016), 149-160. (2016) Zbl1344.34066MR3439277DOI10.1007/s00605-014-0695-2
  38. Zheng, Z., 10.1016/S0022-247X(02)00297-4, J. Math. Anal. Appl. 274 (2002), 466-473. (2002) Zbl1025.34030MR1936709DOI10.1016/S0022-247X(02)00297-4

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