Homoclinic orbits for second order self-adjoint difference equations

Homoclinic orbits for second order self-adjoint difference equations

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)Δu(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 satisfie...

Homoclinic Orbits of Second-order Nonlinear Difference Equations

We establish existence criteria for homoclinic orbits of secondorder nonlinear difference equations by using the critical point theory in combination with periodic approximations.

Conjugacy of Self-Adjoint Difference Equations of Even Order

and Applied Analysis 3 Let us consider, together with system 2.1 , the matrix linear Hamiltonian system ΔXk AXk 1 BkUk, ΔUk CkXk 1 −AUk, 2.4 where the matrices A,Bk, and Ck are also given by 2.3 . We say that the matrix solution X,U of 2.4 is generated by the solutions y 1 , . . . , y n of 1.1 if and only if its columns are generated by y 1 , . . . , y n , respectively, that is, X,U x y1 , . . ...

Infinitely Many Homoclinic Orbits for 2nth-Order Nonlinear Functional Difference Equations Involving the p-Laplacian

and Applied Analysis 3 F2 F t, xn, . . . , x0 W t, x0 − H t, xn, . . . , x0 , for every t ∈ Z, W,H are continuously differentiable in x0 and xn, . . . , x0, respectively. Moreover, there is a bounded set J ⊂ Z such that H t, xn, . . . , x0 ≥ 0; 2.2 F3 There is a constant μ > p such that 0 < μW t, x0 ≤ W ′ 2 t, x0 x0, ∀ t, x0 ∈ Z × R \ {0} ; 2.3 F4 H t, 0, . . . , 0 ≡ 0, and there is a constant ...

On Self-Adjoint Differential Equations of Second Order.