Geodesic
Periodic solutions for $n$-th order delay differential equations with damping terms
Archivum mathematicum, Tome 47 (2011) no. 4, pp. 263-278
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 34C25
Keywords: delay differential equations; periodic solution; coincidence degree 
@article{ARM_2011_47_4_a3,
     author = {Pan, Lijun},
     title = {Periodic solutions for $n$-th order delay differential equations with damping terms},
     journal = {Archivum mathematicum},
     pages = {263--278},
     year = {2011},
     volume = {47},
     number = {4},
     mrnumber = {2876949},
     zbl = {1249.34206},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2011_47_4_a3/}
}
TY  - JOUR
AU  - Pan, Lijun
TI  - Periodic solutions for $n$-th order delay differential equations with damping terms
JO  - Archivum mathematicum
PY  - 2011
SP  - 263
EP  - 278
VL  - 47
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/ARM_2011_47_4_a3/
LA  - en
ID  - ARM_2011_47_4_a3
ER  - 
%0 Journal Article
%A Pan, Lijun
%T Periodic solutions for $n$-th order delay differential equations with damping terms
%J Archivum mathematicum
%D 2011
%P 263-278
%V 47
%N 4
%U http://geodesic.mathdoc.fr/item/ARM_2011_47_4_a3/
%G en
%F ARM_2011_47_4_a3
Pan, Lijun. Periodic solutions for $n$-th order delay differential equations with damping terms. Archivum mathematicum, Tome 47 (2011) no. 4, pp. 263-278. http://geodesic.mathdoc.fr/item/ARM_2011_47_4_a3/

[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. | MR | Zbl

[2] Cong, F.: Periodic solutions for $2k$th order ordinary differential equations with resonance. Nonlinear Anal. 32 (1998), 787–793. | DOI | MR

[3] Cong, F.: Existence of $2k+1$th order ordinary differential equations. Appl. Math. Lett. 17 (2004), 727–732. | DOI | MR

[4] Cong, F., Huang, Q. D., Shi, S. Y.: Existence and uniqueness of periodic solutions for $2n+1$th order ordinary differential equations. J. Math. Anal. Appl. 241 (2000), 1–9. | DOI | MR

[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. | MR | Zbl

[6] Fabry, C., Mawhin, J., Nkashama, M. N.: A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations. Bull. London Math. Soc. 10 (1986), 173–180. | DOI | MR | Zbl

[7] Gaines, R. E., Mawhin, J.: Concidence degree and nonlinear differential equations. Lecture Notes in Math. 568 (1977), Berlin, New York, Springer–Verlag. | MR

[8] Jiao, G.: Periodic solutions of $2n+1$th order ordinary differential equations. J. Math. Anal. Appl. 272 (2004), 691–699. | MR

[9] Kiguradze, I. T., Půža, B.: On periodic solutions of system of differential equations with deviating arguments. Nonlinear Anal. 42 (2000), 229–242. | DOI | MR

[10] Li, J. W., Wang, G. Q.: Sharp inequalities for periodic functions. Appl. Math. E-Notes 5 (2005), 75–83. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[13] Liu, Y. J., Yang, P. H., Ge, W. G.: Periodic solutions of high–order delay differential equations. Nonlinear Anal. 63 (2005), 136–152. | DOI | MR

[14] Liu, Z. L.: Periodic solution for nonlinear $n$th order ordinary differential equation. J. Math. Anal. Appl. 204 (1996), 46–64. | DOI

[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.

[16] Lu, S., Ge, W.: Sufficient conditions for the existence of periodic solutions to some second order differential equation with a deviating argument. J. Math. Anal. Appl. 308 (2005), 393–419. | DOI | MR

[17] Mawhin, J.: Degré topologique et solutions périodiques des systémes différentiels nonlineaires. Bull. Soc. Roy. Sci. Liége 38 (1969), 308–398. | MR

[18] Mawhin, J.: $L^{2}$–estimates and periodic solutions of some nonlinear differential equations. Boll. Unione Mat. Ital. 10 (1974), 343–354. | MR

[19] Omari, P., Villari, G., Zanolin, F.: Periodic solutions of Lienard equation with one–side growth restrictions. J. Differential Equations 67 (1987), 278–293. | DOI | MR

[20] Pan, L. J.: Periodic solutions for higher order differential equations with deviating argument. J. Math. Anal. Appl. 343 (2008), 904–918. | DOI | MR | Zbl

[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. | MR | Zbl

[22] Reissig, R.: Periodic solutions of third–order nonlinear differential equation. Ann. Mat. Pura Appl. 92 (1972), 193–198. | DOI | MR

[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. | MR

[24] Srzednicki, S.: On periodic solutions of certain $n$th differential equations. J. Math. Anal. Appl. 196 (1995), 666–675. | DOI | MR

[25] Wang, G. Q.: Existence theorems of periodic solutions for a delay nonlinear differential equation with piecewise constant argument. J. Math. Anal. Appl. 298 (2004), 298–307. | DOI | MR

[26] Zhang, Z. Q., Wang, Z. C.: Periodic solution of the third order functional differential equation. J. Math. Anal. Appl. 292 (2004), 115–134. | DOI | MR