Discrete Hamiltonian Systems

A Short Note on Simple First Order Linear Difference Equations

“…The `A' term is important because it considers all possible solutions to (2). Now ponder about the particular solution, which can be any solution to the equation (1). Since any solution is allowed, it can be, for example, traced out a solution that sets all the X's equal.…”

“…Wimp [10] discusses the development of this method and related algorithms due to Olver, Clenshaw, Gautschi, Wimp, and others. Further development of the theory of linear difference equations has brought the subject to a state comparable to that of linear differential equations (see Hartman [9], and Ahlbrandt and Peterson [1]). …”

Multiple Periodic Solutions to Nonlinear Discrete Hamiltonian Systems

“…For example, by using the critical point theory, some significant results for the existence and multiplicity of periodic and subharmonic solutions to (1.2) were obtained in [1][2][3][4][5]. Some authors have also contributed to the study of (1.1) for the disconjugacy, boundary value problems, oscillations, and asymptotic behavior, see, for example, [6][7][8][9]. In recent years, existence and multiplicity results of periodic solutions to discrete Hamiltonian systems employing the minimax theory and the geometrical index theory have appeared in the literature.…”

The existence of periodic and subharmonic solutions to subquadratic discrete Hamiltonian systems

“…Since the behaviour of a discrete system is sometimes sharply different to the behaviour of the corresponding continuous system (see, for example, [1,16,17]), many scholars have investigated discrete Hamiltonian systems for the disconjugacy, boundary value problems, oscillations and asymptotic behaviour of (1.1) [3,5,7,8,11]. Only very few papers deal with the existence of periodic solutions even to general difference equations, see for example [2,9,16,17].…”