Homoclinic orbits for second order self-adjoint difference equations

Abstract In this paper we discuss how to use variational methods to study the existence of nontrivial homoclinic orbits of the following nonlinear difference equations Δ [ p ( t ) Δ u ( t − 1 ) ] + q ( t ) u ( t ) = f ( t , u ( t ) ) , t ∈ Z , without any periodicity assumptions on p ( t ) , q ( t ) and f , providing that f ( t , x ) grows superlinearly both at origin and at infinity or is an odd function with respect to x ∈ R , and satisfies some additional assumptions.

[1]  H. Poincaré,et al.  Les méthodes nouvelles de la mécanique céleste , 1899 .

[2]  Alexander Pankov,et al.  On Some Discrete Variational Problems , 2001 .

[3]  P. Rabinowitz,et al.  Dual variational methods in critical point theory and applications , 1973 .

[4]  É. Séré.  Existence of infinitely many homoclinic orbits in Hamiltonian systems , 1992 .

[5]  Ravi P. Agarwal,et al.  Difference equations and inequalities , 1992 .

[6]  Jianshe Yu,et al.  The Existence of Periodic and Subharmonic Solutions of Subquadratic Second Order Difference Equations , 2003 .

[7]  M. Willem,et al.  Homoclinic orbits for a class of Hamiltonian systems , 1992, Differential and Integral Equations.

[8]  Donal O'Regan,et al.  Multiple positive solutions of singular discrete p-Laplacian problems via variational methods , 2005 .

[9]  Yanheng Ding,et al.  Homoclinic Orbits for First Order Hamiltonian Systems , 1995 .

[10]  J. Mawhin,et al.  Critical Point Theory and Hamiltonian Systems , 1989 .

[11]  L. M. Berkovich.  The Generalized Emden-Fowler Equation , 1997 .

[12]  P. Rabinowitz.  Minimax methods in critical point theory with applications to differential equations , 1986 .

[13]  H. Hofer,et al.  First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems , 1990 .

[14]  Kazunaga Tanaka.  Homoclinic orbits in a first order superquadratic Hamiltonian system , 1991 .

[15]  X. Zou,et al.  Periodic Solutions of Second Order Self‐Adjoint Difference Equations , 2005 .

[16]  Vittorio Coti Zelati,et al.  Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials , 1991 .

[17]  C. Ahlbrandt,et al.  Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations , 1996 .

[18]  Jianshe Yu,et al.  Existence of periodic and subharmonic solutions for second-order superlinear difference equations , 2003 .

[19]  Ravi P. Agarwal,et al.  Periodic solutions of first order linear difference equations , 1995 .