Homoclinic solutions in periodic difference equations with saturable nonlinearity (Q547329)

This paper is mainly concerned with the homoclinic solutions of the following difference equation \[ L u_n-\omega u_n=\frac{\sigma\chi_n u^3_n}{1+c_nu^2_n}, \] where \[ Lu_n=a_nu_{n+1}+a_{n-1}u_{n-1}+b_nu_n, \] and \(\{a_n\}\), \(\{b_n\}\), \(\{c_n\}\) and \(\{\chi_n\}\) are positive real valued \(T\)-periodic sequences. By using the linking theorem in combination with periodic approximations, the authors establish sufficient conditions on the nonexistence and on the existence of homoclinic solutions for the above difference equation. Their results solve an open problem proposed by \textit{A. Pankov} [Nonlinearity 19, No. 1, 27--40 (2006; Zbl 1220.35163)]. This paper is interesting and a good contribution in this area.