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)


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