Periodic solutions for n -th order delay differential equations with damping terms

Lijun Pan

Archivum Mathematicum (2011)

  • Volume: 047, Issue: 4, page 263-278
  • ISSN: 0044-8753

Abstract

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By using the coincidence degree theory of Mawhin, we study the existence of periodic solutions for n th order delay differential equations with damping terms x ( n ) ( t ) = ∑ i = 1 s b i [ x ( i ) ( t ) ] 2 k - 1 + f ( x ( t - τ ( t ) ) ) + p ( t ) . Some new results on the existence of periodic solutions of the investigated equation are obtained.

How to cite

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Pan, Lijun. "Periodic solutions for $n$-th order delay differential equations with damping terms." Archivum Mathematicum 047.4 (2011): 263-278. <http://eudml.org/doc/247173>.

@article{Pan2011,
abstract = {By using the coincidence degree theory of Mawhin, we study the existence of periodic solutions for $n$ th order delay differential equations with damping terms $x^\{(n)\}(t)=\sum \limits ^\{s\}_\{i=1\}b_\{i\}[x^\{(i)\}(t)]^\{2k-1\}+ f(x(t-\tau (t)))+p(t)$. Some new results on the existence of periodic solutions of the investigated equation are obtained.},
author = {Pan, Lijun},
journal = {Archivum Mathematicum},
keywords = {delay differential equations; periodic solution; coincidence degree; delay differential equation; periodic solution; coincidence degree},
language = {eng},
number = {4},
pages = {263-278},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Periodic solutions for $n$-th order delay differential equations with damping terms},
url = {http://eudml.org/doc/247173},
volume = {047},
year = {2011},
}

TY - JOUR
AU - Pan, Lijun
TI - Periodic solutions for $n$-th order delay differential equations with damping terms
JO - Archivum Mathematicum
PY - 2011
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 047
IS - 4
SP - 263
EP - 278
AB - By using the coincidence degree theory of Mawhin, we study the existence of periodic solutions for $n$ th order delay differential equations with damping terms $x^{(n)}(t)=\sum \limits ^{s}_{i=1}b_{i}[x^{(i)}(t)]^{2k-1}+ f(x(t-\tau (t)))+p(t)$. Some new results on the existence of periodic solutions of the investigated equation are obtained.
LA - eng
KW - delay differential equations; periodic solution; coincidence degree; delay differential equation; periodic solution; coincidence degree
UR - http://eudml.org/doc/247173
ER -

References

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  1. Chen, X. R., Pan, L. J., Existence of periodic solutions for n th order differential equations with deviating argument, Int. J. Pure. Appl. Math. 55 (2009), 319–333. (2009) Zbl1185.34092MR2561457
  2. Cong, F., 10.1016/S0362-546X(97)00517-8, Nonlinear Anal. 32 (1998), 787–793. (1998) MR1612146DOI10.1016/S0362-546X(97)00517-8
  3. Cong, F., 10.1016/S0893-9659(04)90112-7, Appl. Math. Lett. 17 (2004), 727–732. (2004) MR2064187DOI10.1016/S0893-9659(04)90112-7
  4. Cong, F., Huang, Q. D., Shi, S. Y., 10.1006/jmaa.1999.6471, J. Math. Anal. Appl. 241 (2000), 1–9. (2000) MR1738328DOI10.1006/jmaa.1999.6471
  5. Ezeilo, J. O. C., On the existence of periodic solutions of a certain third–order differential equation, Proc. Cambridge Philos. Soc. 56 (1960), 381–389. (1960) Zbl0097.29404MR0121539
  6. Fabry, C., Mawhin, J., Nkashama, M. N., 10.1112/blms/18.2.173, Bull. London Math. Soc. 10 (1986), 173–180. (1986) Zbl0586.34038MR0818822DOI10.1112/blms/18.2.173
  7. Gaines, R. E., Mawhin, J., Concidence degree and nonlinear differential equations, Lecture Notes in Math. 568 (1977), Berlin, New York, Springer–Verlag. (1977) MR0637067
  8. Jiao, G., Periodic solutions of 2 n + 1 th order ordinary differential equations, J. Math. Anal. Appl. 272 (2004), 691–699. (2004) MR1930716
  9. Kiguradze, I. T., Půža, B., 10.1016/S0362-546X(98)00342-3, Nonlinear Anal. 42 (2000), 229–242. (2000) MR1773980DOI10.1016/S0362-546X(98)00342-3
  10. Li, J. W., Wang, G. Q., Sharp inequalities for periodic functions, Appl. Math. E-Notes 5 (2005), 75–83. (2005) Zbl1076.26011MR2112160
  11. Liu, B. W., Huang, L. H., Existence of periodic solutions for nonlinear n th order ordinary differential equations, Acta Math. Sinica 47 (2004), 1133–1140, in Chinese. (2004) Zbl1124.34334MR2128079
  12. Liu, W. B., Li, Y., The existence of periodic solutions for high order duffing equations, Acta Math. Sinica 46 (2003), 49–56, in Chinese. (2003) Zbl1036.34052MR1971712
  13. Liu, Y. J., Yang, P. H., Ge, W. G., 10.1016/j.na.2005.04.038, Nonlinear Anal. 63 (2005), 136–152. (2005) MR2167321DOI10.1016/j.na.2005.04.038
  14. Liu, Z. L., 10.1006/jmaa.1996.0423, J. Math. Anal. Appl. 204 (1996), 46–64. (1996) DOI10.1006/jmaa.1996.0423
  15. Lu, S., Ge, W., On the existence of periodic solutions for Lienard equation with a deviating argument, J. Math. Anal. Appl. 289 (2004), 241–243. (2004) 
  16. Lu, S., Ge, W., 10.1016/j.jmaa.2004.09.010, J. Math. Anal. Appl. 308 (2005), 393–419. (2005) MR2150099DOI10.1016/j.jmaa.2004.09.010
  17. Mawhin, J., Degré topologique et solutions périodiques des systémes différentiels nonlineaires, Bull. Soc. Roy. Sci. Liége 38 (1969), 308–398. (1969) MR0594965
  18. Mawhin, J., L 2 –estimates and periodic solutions of some nonlinear differential equations, Boll. Unione Mat. Ital. 10 (1974), 343–354. (1974) MR0369823
  19. Omari, P., Villari, G., Zanolin, F., 10.1016/0022-0396(87)90151-3, J. Differential Equations 67 (1987), 278–293. (1987) MR0879698DOI10.1016/0022-0396(87)90151-3
  20. Pan, L. J., 10.1016/j.jmaa.2008.01.096, J. Math. Anal. Appl. 343 (2008), 904–918. (2008) Zbl1160.34065MR2417111DOI10.1016/j.jmaa.2008.01.096
  21. Pan, L. J., Chen, X. R., Periodic solutions for n th order functional differential equations, Bull. Belg. Math. Soc. Simon Stevin 17 (2010), 109–126. (2010) Zbl1202.34121MR2656675
  22. Reissig, R., 10.1007/BF02417946, Ann. Mat. Pura Appl. 92 (1972), 193–198. (1972) MR0316827DOI10.1007/BF02417946
  23. Ren, J. L., Ge, W. G., On the existence of periodic solutions for the second order functional differential equation, Acta Math. Sinica 47 (2004), 569–578, in Chinese. (2004) MR2083276
  24. Srzednicki, S., 10.1006/jmaa.1995.1433, J. Math. Anal. Appl. 196 (1995), 666–675. (1995) MR1362714DOI10.1006/jmaa.1995.1433
  25. Wang, G. Q., 10.1016/j.jmaa.2004.05.016, J. Math. Anal. Appl. 298 (2004), 298–307. (2004) MR2086548DOI10.1016/j.jmaa.2004.05.016
  26. Zhang, Z. Q., Wang, Z. C., 10.1016/j.jmaa.2003.11.059, J. Math. Anal. Appl. 292 (2004), 115–134. (2004) MR2050220DOI10.1016/j.jmaa.2003.11.059

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