Nontrivial solutions of discrete Kirchhoff-type problem via bifurcation theory

<jats:p>In this paper, we show that the bifurcation points for a discrete Kirchhoff-type problem with only local conditions, and we investigate the existence of positive and negative solutions for the problem when the nonlinear term \(f\) is asymptotically linear at zero and is asymptotically 3-linear at infinity. By using bifurcation techniques and the idea of taking limits of connected branches, under the assumption that \(f\) has some non-zero zeros, some results are also obtained.</jats:p>