Abstract
In this paper, we investigate the second order self-adjoint discrete Hamiltonian system $\Delta[p(n)\Delta u(n-1)]-L(n)u(n)+\lambda a(n)\nabla G(u(n))+\mu b(n)\nabla F(u(n))=0$ , where $p,L:\mathbb{Z}\rightarrow\mathbb{R}^{N\times N}$ are both positive definite for all $n\in\mathbb{Z}$ , and no symmetric condition on G and F is needed. We establish two new criteria to guarantee that the above system has at least two nontrivial homoclinic solutions or infinitely many homoclinic solutions via critical point theory.
Highlights
In this paper, we consider the following second order self-adjoint discrete Hamiltonian system: p(n) u(n – ) – L(n)u(n) + λa(n)∇G u(n) + μb(n)∇F u(n) =, ∀n ∈ Z, ( . )where u ∈ RN, u(n) = u(n + ) – u(n) is the forward difference, p, L : Z → RN×N
We investigate the second order self-adjoint discrete Hamiltonian system [p(n) u(n – 1)] – L(n)u(n) + λa(n)∇G(u(n)) + μb(n)∇F(u(n)) = 0, where p, L : Z → RN×N are both positive definite for all n ∈ Z, and no symmetric condition on G and F is needed
The discrete Hamiltonian system has found a great deal of interest last years because it is important in applications but it provides a good model for developing mathematical methods
Summary
When no symmetric condition on the nonlinear term is assumed, as far as the authors are aware, there is no research about the existence of multiple homoclinic solutions for system We obtain the existence of two homoclinic solutions. ([ ], Theorem ) Let X be a separable and reflexive real Banach space; let : X → R be a coercive, sequentially weakly lower semicontinuous C functional, belonging to X , bounded on each bounded subset of X and whose derivative admits a continuous inverse on X∗; J : X → R a C functional with compact derivative. By ([ ], Theorem .A(d)), we see that admits a continuous inverse on X∗ Proof It follows from (F ), (F ), and (F ) that, for any > , there exists T such that.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
