Rocky Mountain Journal of Mathematics

Homoclinic orbits of nonlinear functional difference equations with Jacobi operators

Abstract

By using the critical point theory, the existence of a nontrivial homoclinic orbit which decays exponentially at infinity for difference equations containing both advance and retardation is obtained. The proof is based on the mountain pass lemma in combination with periodic approximations. Our results extend the results of 2007.

Article information

Source
Rocky Mountain J. Math., Volume 43, Number 6 (2013), 1991-2008.

Dates
First available in Project Euclid: 25 February 2014

https://projecteuclid.org/euclid.rmjm/1393336665

Digital Object Identifier
doi:10.1216/RMJ-2013-43-6-1991

Mathematical Reviews number (MathSciNet)
MR3178452

Zentralblatt MATH identifier
1302.37017

Citation

Ren, Zhiguo; Zhang, Yuanbiao; Zheng, Bo; Shi, Haiping. Homoclinic orbits of nonlinear functional difference equations with Jacobi operators. Rocky Mountain J. Math. 43 (2013), no. 6, 1991--2008. doi:10.1216/RMJ-2013-43-6-1991. https://projecteuclid.org/euclid.rmjm/1393336665

References

• R.P. Agarwal, Difference equations and inequalities: Theory, methods and applications, Marcel Dekker, New York, 1992.
• R.P. Agarwal, K. Perera and D. O'Regan, Multiple positive solutions of singular and nonsingular discrete problems via variational methods, Nonlin. Anal. 58 (2004), 69-73.
• –––, Multiple positive solutions of singular discrete $p$-Laplacian problems via variational methods, Adv. Differ. Eq. 2005 (2005), 93-99.
• A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381.
• P. Chen and H. Fang, Existence of periodic and subharmonic solutions for second-order $p$-Laplacian difference equations, Adv. Differ. Eq. 2007 (2007), 1-9.
• Y. Ding and M. Girardi, Infinitely many homoclinic orbits of a Hamiltonian system with symmetry, Nonlin. Anal. 38 (1999), 391-415.
• S.N. Elaydi, An introduction to difference equations, Springer-Verlag, New York, 1999.
• S.N. Elaydi and S. Zhang, Stability and periodicity of difference equations with finite delay, Funk. Ekvac. 37 (1994), 401-413.
• R.P. Feynman and A.R. Hibbs, Quantum mechanics and path integrals, McGraw-Hill, New York, 1965.
• Z.M. Guo and Y.T. Xu, Existence of periodic solutions to a class of second-order neutral differential difference equations, Acta Anal. Funct. Appl. 5 (2003), 13-19.
• Z.M. Guo and J.S. Yu, Applications of critical point theory to difference equations, Fields Inst. Comm. 42 (2004), 187-200.
• Z.M. Guo and J.S. Yu, The existence of periodic and subharmonic solutions for second-order superlinear difference equations, Sci. China 46 (2003), 506-515.
• –––, The existence of periodic and subharmonic solutions of subquadratic second order difference equations, J. Lond. Math. Soc. 68 (2003), 419-430.
• H. Hofer and K. Wysocki, First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems, Math. Ann. 288 (1990), 483-503.
• J.L. Kaplan and J.A. Yorke, On the nonlinear differential delay equation $x'(t)=-f(x(t),x(t-1))$, J. Differ. Equat. 23 (1977), 293-314.
• V.L. Kocic and G. Ladas, Global behavior of nonlinear difference equations of high order with applications, Kluwer Academic Publishers, Boston, 1993.
• L.D. Landau and E.M. Lifshitz, Quantum mechanics, Pergamon, New York, 1979.
• J.B. Li and X.Z. He, Proof and generalization of Kaplan-Yorke's conjecture on periodic solution of differential delay equations, Sci. China 42 (1999), 957-964.
• M.J. Ma and Z.M. Guo, Homoclinic orbits and subharmonics for nonlinear second order difference equations, Nonlin. Anal. 67 (2007), 1737-1745.
• H. Matsunaga, T. Hara and S. Sakata, Global attractivity for a nonlinear difference equation with variable delay, Comp. Math. Appl. 41 (2001), 543-551.
• J. Moser, Stable and random motions in dynamical systems, Princeton University Press, Princeton, 1973.
• R.D. Nussbaum, Circulant matrices and differential delay equations, J. Differ. Equat. 60 (1985), 201-217.
• W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Diff. Int. Eq. 5 (1992), 1115-1120.
• A. Pankov and N. Zakharchenko, On some discrete variational problems, Acta Appl. Math. 65 (2001), 295-303.
• H. Poincaré, Les méthodes nouvelles de la mécanique céleste, Gauthier-Villars, Paris, 1899.
• P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, American Mathematical Society, Providence, RI, New York, 1986.
• C.K. Raju, Classical time-symmetric electrodynamics, J. Phys. Math. Gen. 13 (1980), 3303-3317.
• L.S. Schulman, Some differential-difference equations containing both advance and retardation, J. Math. Phys. 15 (1974), 295-298.
• D. Smets and M. Willem, Solitary waves with prescribed speed on infinite lattices, J. Funct. Anal. 149 (1997), 266-275.
• A. Szulkin and W. Zou, Homoclinic orbits for asymptotically linear Hamiltonian systems, J. Funct. Anal. 187 (2001), 25-41.
• G. Teschl, Jacobi operators and completely integrable nonlinear lattices, American Mathematical Society, Providence, RI, New York, 2000.
• J.A. Wheeler and R.P. Feynman, Classical electrodynamics in terms of direct interparticle action, Rev. Mod. Phys. 21 (1949), 425-433.
• J.S. Yu, Y.H. Long and Z.M. Guo, Subharmonic solutions with prescribed minimal period of a discrete forced pendulum equation, J. Dynam. Diff. Eq. 16 (2004), 575-586.
• Z. Zhou and Q. Zhang, Uniform stability of nonlinear difference systems, J. Math. Anal. Appl. 225 (1998), 486-500. \noindentstyle