*(English)*Zbl 1005.39014

The authors provide sufficient conditions for the existence of (at least) three positive solutions of the discrete two-point boundary value problem

where ${\Delta}{y}_{k}:={y}_{k+1}-{y}_{k}$ is the usual forward difference operator (note a slightly modified notation used by the authors) and where $f:\{1,2,\cdots ,n\}\times {\mathbb{R}}^{2}\to \mathbb{R}$ is continuous. The proof is based on the existence of pairs of discrete lower solutions ${\alpha}_{1}$, ${\alpha}_{2}$ and discrete upper solutions ${\beta}_{1}$, ${\beta}_{2}$ that satisfy the inequalities ${\alpha}_{1}\le {\alpha}_{2}$ and ${\beta}_{1}\le {\beta}_{2}$. This method is a discrete analog of the authorsâ€™ continuous-time results [J. Differ Equations 166, No. 2, 443-454 (2000; Zbl 1013.34017)]. However, in this paper the assumptions do not require ${\alpha}_{2}\le {\beta}_{1}$ and allow $f$ being dependent on ${\Delta}{y}_{k-1}$. If $f$ is a function of its second variable ${y}_{k}$ only, then the results of this paper are sharp and generalize those of *R. J. Avery* and *A. C. Peterson* [Panam. Math. J. 8, No. 3, 1-12 (1998; Zbl 0959.39006)]. This paper will be useful for researchers interested in two-point boundary value problems and/or their positive solutions.

##### MSC:

39A11 | Stability of difference equations (MSC2000) |

39A12 | Discrete version of topics in analysis |

34B15 | Nonlinear boundary value problems for ODE |