Abstract
It is proved that if a nonlinear system possesses some group-symmetry, then under certain transversality it admits solutions with the corresponding symmetry. The method is due to Mawhin’s guiding function one.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Buică, A., Ortega, R. Persistence of equilibria as periodic solutions of forced systems. J. Differential Equations, 252(3): 2210–2221 (2012)
Chen, Y., Nieto, J.J., O’Regan, D. Anti-periodic solutions for evolution equations associated with maximal monotone mappings. Appl. Math. Lett., 24(3): 302–307 (2011)
Chen, Y., O’Regan, D., Agarwal, Ravi P. Anti-periodic solutions for semilinear evolution equations in Banach spaces. J. Appl. Math. Comput., 38(1–2): 63–70 (2012)
Dilna, N., Fečkan, M. On symmetric and periodic solutions of parametric weakly nonlinear ODE with time-reversal symmetries. Bull. Belg. Math. Soc. Simon Stevin, 18(5): 896–923 (2011)
Fonda, A., Toader, R. Periodic solutions of radially symmetric perturbations of Newtonian systems. Proc. Amer. Math. Soc., 140(4): 1331–1341 (2012)
Graef, J.R., Kong, L. Periodic solutions of first order functional differential equations. Appl. Math. Lett., 24(12): 1981–1985 (2011)
Gurevich, P., Tikhomirov, S. Symmetric periodic solutions of parabolic problems with discontinuous hysteresis. J. Dynam. Differential Equations, 23(4): 923–960 (2011)
Julka, K.M. Periodic solutions of nonlinear differential equations. Adv. Dyn. Syst. Appl., 7(1): 89–93 (2012)
Krasnosel’skii, A.M., Krasnosel’skii, M.A., Mawhin, J., Pokrovskii, A. Generalized guiding functions in a problem on high frequency forced oscillations. Nonlinear Anal., 22(11): 1357–1371 (1994)
Krasnosel’skii, A.M., Mawhin, J. Periodic solutions of equations with oscillating nonlinearities. Nonlinear operator theory. Math. Comput. Modelling, 32(11–13): 1445–1455 (2000)
Liu, Y. Anti-periodic solutions of nonlinear first order impulsive functional differential equations. Math. Slovaca, 62(4): 695–720 (2012)
Liu, A., Feng, C. Anti-periodic solutions for a kind of high order differential equations with multi-delay. Commun. Math. Anal., 11(1): 137–150 (2011)
Liu, L., Li, Y. Existence and uniqueness of anti-periodic solutions for a class of nonlinear n-th order functional differential equations. Opuscula Math., 31(1): 61–74 (2011)
Liu, J., Liu, Z. On the existence of anti-periodic solutions for implicit differential equations. Acta Math. Hungar., 132(3): 294–305 (2011)
Liang, J., Liu, J.H., Xiao, T.J. Periodic solutions of delay impulsive differential equations. Nonlinear Anal., 74(17): 6835–6842 (2011)
Mawhin, J. Reduction and continuation theorems for Brouwer degree and applications to nonlinear difference equations. Opuscula Math., 28(4): 541–560 (2008)
Mawhin, J. Leray-Schauder degree: a half century of extensions and applications. Topol. Methods Nonlinear Anal., 14(2): 195–228 (1999)
Mawhin, J. Continuation theorems and periodic solutions of ordinary differential equations. Topological methods in differential equations and inclusions (Montreal, PQ, 1994), 291–375, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 472, Kluwer Acad. Publ., Dordrecht, 1995
Mawhin, J., Thompson, H.B. Periodic or bounded solutions of Carath odory systems of ordinary differential equations. Special issue dedicated to Victor A. Pliss on the occasion of his 70th birthday. J. Dynam. Differential Equations, 15(2–3): 327–334 (2003)
Mawhin, J., Ward, J.R. Jr. Guiding-like functions for periodic or bounded solutions of ordinary differential equations. Discrete Contin. Dyn. Syst., 8(1): 39–54 (2002)
Mawhin, J., Rebelo, C., Zanolin, F. Continuation theorems for Ambrosetti-Prodi type periodic problems. Commun. Contemp. Math., 2(1): 87–126 (2000)
Pan, L., Cao, J. Anti-periodic solution for delayed cellular neural networks with impulsive effects. Nonlinear Anal. Real World Appl., 12(6): 3014–3027 (2011)
Tian, Y., Henderson, J. Three anti-periodic solutions for second-order impulsive differential inclusions via nonsmooth critical point theory. Nonlinear Anal., 75(18): 6496–6505 (2012)
Wu, R., Cong, F., Li, Y. Anti-periodic solutions for second order differential equations. Appl. Math. Lett., 24(6): 860–863 (2011)
Zhao, L., Li, Y. Existence and exponential stability of anti-periodic solutions of high-order Hopfield neural networks with delays on time scales. Differ. Equ. Dyn. Syst., 19(1–2): 13–26 (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by National Basic Research Program of China (grant No. 2013CB834100), National Natural Science Foundation of China (grant No. 11171132), and National Natural Science Foundation of China (grant No. 11201173).
Rights and permissions
About this article
Cite this article
Wang, Hr., Yang, X. & Yang, X. Rotating-symmetric solutions for nonlinear systems with symmetry. Acta Math. Appl. Sin. Engl. Ser. 31, 307–312 (2015). https://doi.org/10.1007/s10255-015-0484-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10255-015-0484-2