Homoclinic orbits for nonlinear difference equations containing both advance and retardation. (English) Zbl 1160.39311

Summary: We discuss how to use the critical point theory to study the existence of a nontrivial homoclinic orbit for nonlinear difference equations containing both advance and retardation without any periodic assumptions. Moreover, if the nonlinearity is an odd function, the existence of an unbounded sequence of homoclinic orbits is obtained.


39A11 Stability of difference equations (MSC2000)
Full Text: DOI


[1] Agarwal, R. P., Difference Equations and Inequalities: Theory, Methods and Applications (1992), Marcel Dekker: Marcel Dekker New York · Zbl 0925.39001
[2] Agarwal, R. P.; Perera, K.; O’Regan, D., Multiple positive solutions of singular and nonsingular discrete problems via variational methods, Nonlinear Anal., 58, 69-73 (2004) · Zbl 1070.39005
[3] Agarwal, R. P.; Perera, K.; O’Regan, D., Multiple positive solutions of singular discrete \(p\)-Laplacian problems via variational methods, Adv. Difference Equ., 2005, 93-99 (2005) · Zbl 1098.39001
[4] Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349-381 (1973) · Zbl 0273.49063
[5] Chen, P.; Fang, H., Existence of periodic and subharmonic solutions for second-order \(p\)-Laplacian difference equations, Adv. Difference Equ., 2007 (2007), Article ID 42530, doi:10.1155/2007/42530 · Zbl 1148.39002
[6] Ding, Y.; Girardi, M., Infinitely many homoclinic orbits of a Hamiltonian system with symmetry, Nonlinear Anal., 38, 391-415 (1999) · Zbl 0938.37034
[7] Feynman, R. P.; Hibbs, A. R., Quantum Mechanics and Path Integrals (1965), McGraw-Hill: McGraw-Hill New York · Zbl 0176.54902
[8] Guo, Z. M.; Yu, J. S., Applications of critical point theory to difference equations, Fields Inst. Commun., 42, 187-200 (2004) · Zbl 1067.39007
[9] Guo, Z. M.; Yu, J. S., The existence of periodic and subharmonic solutions for second-order superlinear difference equations, Sci. China Ser. A, 46, 506-515 (2003) · Zbl 1215.39001
[10] Guo, Z. M.; Yu, J. S., The existence of periodic and subharmonic solutions of subquadratic second order difference equations, J. London Math. Soc., 68, 419-430 (2003) · Zbl 1046.39005
[11] Guo, Z. M.; Xu, Y. T., Existence of periodic solutions to a class of second-order neutral differential difference equations, Acta Anal. Funct. Appl., 5, 13-19 (2003) · Zbl 1024.34063
[12] Hofer, H.; Wysocki, K., First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems, Math. Ann., 288, 483-503 (1990) · Zbl 0702.34039
[13] Kaplan, J. L.; Yorke, J. A., On the nonlinear differential delay equation \(x^\prime(t) = - f(x(t), x(t - 1))\), J. Differential Equations, 23, 293-314 (1977) · Zbl 0307.34070
[14] Kocic, V. L.; Ladas, G., Global Behavior of Nonlinear Difference Equations of High Order with Applications (1993), Kluwer Academic: Kluwer Academic Boston · Zbl 0787.39001
[15] Landau, L. D.; Lifshitz, E. M., Quantum Mechanics (1979), Pergamon: Pergamon New York · Zbl 0081.22207
[16] Li, J. B.; He, X. Z., Proof and generalization of Kaplan-Yorke’s conjecture on periodic solution of differential delay equations, Sci. China Ser. A, 42, 957-964 (1999) · Zbl 0983.34061
[17] Ma, M. J.; Guo, Z. M., Homoclinic orbits for second order self-adjoint difference equations, J. Math. Anal. Appl., 323, 513-521 (2006) · Zbl 1107.39022
[18] Matsunaga, H.; Hara, T.; Sakata, S., Global attractivity for a nonlinear difference equation with variable delay, Comput. Math. Appl., 41, 543-551 (2001) · Zbl 0985.39009
[19] Moser, J., Stable and Radom Motions in Dynamical Systems (1973), Princeton University Press: Princeton University Press Princeton · Zbl 0271.70009
[20] Nussbaum, R. D., Circulant matrices and differential delay equations, J. Differential Equations, 60, 201-217 (1985) · Zbl 0622.34076
[21] Omana, W.; Willem, M., Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5, 1115-1120 (1992) · Zbl 0759.58018
[22] Pankov, A.; Zakharchenko, N., On some discrete variational problems, Acta Appl. Math., 65, 295-303 (2001) · Zbl 0993.39011
[23] Poincaré, H., Les méthodes nouvelles de la mécanique céleste (1899), Gauthier-Villars: Gauthier-Villars Paris
[24] Rabinowitz, P. H., Minimax Methods in Critical Point Theory with Applications to Differential Equations (1986), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0609.58002
[25] Raju, C. K., Classical time-symmetric electrodynamics, J. Phys. A, 13, 3303-3317 (1980)
[26] Schulman, L. S., Some differential-difference equations containing both advance and retardation, J. Math. Phys., 15, 295-298 (1974) · Zbl 0277.34081
[27] Smets, D.; Willem, M., Solitary waves with prescribed speed on infinite lattices, J. Funct. Anal., 149, 266-275 (1997) · Zbl 0889.34059
[28] Szulkin, A.; Zou, W., Homoclinic orbits for asymptotically linear Hamiltonian systems, J. Funct. Anal., 187, 25-41 (2001) · Zbl 0984.37072
[29] Wheeler, J. A.; Feynman, R. P., Classical electrodynamics in terms of direct interparticle action, Rev. Modern Phys., 21, 425-433 (1949) · Zbl 0034.27801
[30] Yu, J. S.; Long, Y. H.; Guo, Z. M., Subharmonic solutions with prescribed minimal period of a discrete forced pendulum equation, J. Dynam. Differential Equations, 16, 575-586 (2004) · Zbl 1067.39022
[31] Zhou, Z.; Yu, J. S.; Guo, Z. M., Periodic solutions of higher-dimensional discrete systems, Proc. Roy. Soc. Edinburgh Sect. A, 134, 1013-1022 (2004) · Zbl 1073.39010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.