American Institute of Mathematical Sciences

Advanced
April  2013, 33(4): 1603-1614. doi: 10.3934/dcds.2013.33.1603

Existence of periodic solutions with nonconstant sign in a class of generalized Abel equations

Josep M. Olm 1, and  Xavier Ros-Oton 2,

1. 

Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, Av. Esteve Terradas 5, 08860 Castelldefels, Spain
2. 

Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain

Received  October 2011 Revised  May 2012 Published  October 2012

This article provides sufficient conditions for the existence of periodic solutions with nonconstant sign in a family of polynomial, non-auto-nomous, first-order differential equations that arise as a generalization of the Abel equation of the second kind.
Keywords: Hilbert's 16th problem., periodic solutions, polynomial differential equations, limit cycles, Abel differential equations.
Mathematics Subject Classification: Primary: 34C07; Secondary: 34C2.
Citation: Josep M. Olm, Xavier Ros-Oton. Existence of periodic solutions with nonconstant sign in a class of generalized Abel equations. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1603-1614. doi: 10.3934/dcds.2013.33.1603
References:
[1]

J. Math. Anal. Applic., 328 (2007), 1108-1116. doi: 10.1016/j.jmaa.2006.05.078.  Google Scholar

[2]

J. Math. Anal. Applic., 329 (2007), 1161-1169. doi: 10.1016/j.jmaa.2006.07.039.  Google Scholar

[3]

$2^{nd}$ edition, Chapman & Hall/CRC, Boca Raton, 2003.  Google Scholar

[4]

Math. Intelligencer, 20 (1998), 7-15. doi: 10.1007/BF03025291.  Google Scholar

[5]

SIAM J. Math. Anal., 21 (1990), 1235-1244. doi: 10.1137/0521068.  Google Scholar

[6]

Nonlinearity, 13 (2000), 1337-1342. doi: 10.1088/0951-7715/13/4/319.  Google Scholar

[7]

J. Differential Equations, 234 (2007), 161-176. doi: 10.1016/j.jde.2006.11.004.  Google Scholar

[8]

J. Math. Anal. Applic, 342 (2008), 931-942. doi: 10.1016/j.jmaa.2007.12.060.  Google Scholar

[9]

Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), 3869-3876. doi: 10.1142/S0218127409025195.  Google Scholar

[10]

Discrete Continuous Dynam. Systems - A, 25 (2009), 1129-1141. doi: 10.3934/dcds.2009.25.1129.  Google Scholar

[11]

Discrete Continuous Dynam. Systems - A, 31 (2011), 25-34. doi: 10.3934/dcds.2011.31.25.  Google Scholar

[12]

J. Math. Anal. Appl., 381 (2011), 582-589. doi: 10.1016/j.jmaa.2011.02.084.  Google Scholar

[13]

Discrete Continuous Dynam. Systems - B, 7 (2007), 53-76.  Google Scholar

[14]

Discrete Continuous Dynam. Systems - B, 15 (2011), 197-215. doi: 10.3934/dcdsb.2011.15.197.  Google Scholar

[15]

J. Differential Equations, 185 (2002), 54-73. doi: 10.1006/jdeq.2002.4172.  Google Scholar

[16]

J. Differential Equations, 3 (1967), 546-570.  Google Scholar

[17]

$2^{nd}$ edition, Springer-Verlag, New York, 1985.  Google Scholar

show all references

References:
[1]

J. Math. Anal. Applic., 328 (2007), 1108-1116. doi: 10.1016/j.jmaa.2006.05.078.  Google Scholar

[2]

J. Math. Anal. Applic., 329 (2007), 1161-1169. doi: 10.1016/j.jmaa.2006.07.039.  Google Scholar

[3]

$2^{nd}$ edition, Chapman & Hall/CRC, Boca Raton, 2003.  Google Scholar

[4]

Math. Intelligencer, 20 (1998), 7-15. doi: 10.1007/BF03025291.  Google Scholar

[5]

SIAM J. Math. Anal., 21 (1990), 1235-1244. doi: 10.1137/0521068.  Google Scholar

[6]

Nonlinearity, 13 (2000), 1337-1342. doi: 10.1088/0951-7715/13/4/319.  Google Scholar

[7]

J. Differential Equations, 234 (2007), 161-176. doi: 10.1016/j.jde.2006.11.004.  Google Scholar

[8]

J. Math. Anal. Applic, 342 (2008), 931-942. doi: 10.1016/j.jmaa.2007.12.060.  Google Scholar

[9]

Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), 3869-3876. doi: 10.1142/S0218127409025195.  Google Scholar

[10]

Discrete Continuous Dynam. Systems - A, 25 (2009), 1129-1141. doi: 10.3934/dcds.2009.25.1129.  Google Scholar

[11]

Discrete Continuous Dynam. Systems - A, 31 (2011), 25-34. doi: 10.3934/dcds.2011.31.25.  Google Scholar

[12]

J. Math. Anal. Appl., 381 (2011), 582-589. doi: 10.1016/j.jmaa.2011.02.084.  Google Scholar

[13]

Discrete Continuous Dynam. Systems - B, 7 (2007), 53-76.  Google Scholar

[14]

Discrete Continuous Dynam. Systems - B, 15 (2011), 197-215. doi: 10.3934/dcdsb.2011.15.197.  Google Scholar

[15]

J. Differential Equations, 185 (2002), 54-73. doi: 10.1006/jdeq.2002.4172.  Google Scholar

[16]

J. Differential Equations, 3 (1967), 546-570.  Google Scholar

[17]

$2^{nd}$ edition, Springer-Verlag, New York, 1985.  Google Scholar

[1]

Jaume Llibre, Claudia Valls. Rational limit cycles of Abel equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1077-1089. doi: 10.3934/cpaa.2021007
[2]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277
[3]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[4]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027
[5]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25
[6]

Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209
[7]

Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203
[8]

Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192
[9]

Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021020
[10]

Pengyu Chen. Periodic solutions to non-autonomous evolution equations with multi-delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2921-2939. doi: 10.3934/dcdsb.2020211
[11]

Shuang Wang, Dingbian Qian. Periodic solutions of p-Laplacian equations via rotation numbers. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021060
[12]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175
[13]

Dariusz Idczak. A Gronwall lemma for functions of two variables and its application to partial differential equations of fractional order. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021019
[14]

Ankit Kumar, Kamal Jeet, Ramesh Kumar Vats. Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021016
[15]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056
[16]

Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199
[17]

Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3063-3092. doi: 10.3934/dcds.2020398
[18]

Wensheng Yin, Jinde Cao, Guoqiang Zheng. Further results on stabilization of stochastic differential equations with delayed feedback control under $ G $-expectation framework. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021072
[19]

Pengyu Chen, Xuping Zhang, Zhitao Zhang. Asymptotic behavior of time periodic solutions for extended Fisher-Kolmogorov equations with delays. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021103
[20]

Nikolaos Roidos. Expanding solutions of quasilinear parabolic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021026

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (42)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences