## Eigenvalue intervals and double positive solutions of certain discrete boundary value problems.(English)Zbl 0923.39002

The authors study the $$n$$-th order difference equation $\Delta^n y + \lambda Q(k,y,\Delta y, \dots, \Delta^{n-2} y) = \lambda P(k,y,\Delta y,\dots,\Delta^{n-1} y), \quad k \in [0,N], \tag{1}$ with the boundary conditions $\Delta^i y(0) = 0, \quad 0 \leq i \leq n-3, \tag{2}$
$\alpha \Delta^{n-2} y(0) - \beta \Delta^{n-1} y(0) = 0, \tag{3}$
$\gamma \Delta^{n-2} y(N+1) + \delta \Delta^{n-1} y(N+1) = 0, \tag{4}$ where $$\lambda > 0$$, $$\alpha, \beta,\gamma$$ and $$\delta$$ are constants satisfying certain inequalities. Intervals of $$\lambda$$ are established such that the boundary value problem (BVP) (1)–(4) has a positive solution. Criteria for existence of double positive solutions of the BVP (1)–(4) are obtained in the case $$\lambda=1$$. Also, two special difference equations subjected to (3), (4) are considered and lower and upper bounds for the positive solutions of these equations are obtained.
Reviewer: D.Bainov (Sofia)

### MSC:

 39A10 Additive difference equations 39A12 Discrete version of topics in analysis