Homoclinic solutions in periodic difference equations with mixed nonlinearities (Q2786693)

The paper is devoted to the nonlinear difference equation NEWLINE\[NEWLINELu_n-\omega\,u_n=\sigma\, f_n(u_n),\quad n\in\mathbb{Z},\tag{1}NEWLINE\]NEWLINE where \(\sigma=\pm1\), the function \(f_n(u)\) is continuous in \(u\) with the property \(f_{n+T}(u)=f_n(u)\) for each \(n\in\mathbb{Z}\), and \(L\) is a Jacobi operator NEWLINE\[NEWLINELu_n=a_n\,u_{n+1}+a_{n-1}\,u_{n-1}+b_n\,u_nNEWLINE\]NEWLINE for some real-valued \(T\)-periodic sequences \(\{a_n\}\) and \(\{b_n\}\). The number \(\omega\) belongs to a spectral gap of \(L\). A motivation for the studied problem is given in the introductory section. In the first main result, sufficient conditions for the nonexistence of a nontrivial \(l^2\)-solution of Equation (1) are derived, see Proposition 3.1. In the second main result, sufficient conditions for the existence of a nontrivial \(l^2\)-solution of Equation (1) are established, see Theorem 4.1 and Sections 5--6. In addition, a connection between the latter main result and some results known in the literature is discussed in detail, see Remarks 4.1--12.