A focal boundary value problem for difference equations (Q1204354)

Consider the eigenvalue problem for difference equations (*) \((-1)^{n- k} \Delta^ n y(t)=\lambda\displaystyle{\sum_{i=0}^{k-1}}p_ i(t)\Delta^ i y(t)\), \(0\leq t\leq T\), \(\Delta^ i y(0)=0\), \(0\leq i\leq k-1\), \(k^{k+i} y(T+1)=0\), \(0\leq i\leq n-k-1\). The authors show that under suitable conditions on the coefficients \(p_ i\), the smallest positive eigenvalue is a decreasing function of \(T\). As a consequence they obtain results concerning the first focal point for the boundary value problem (*) with \(\lambda=1\). Most of the results are motivated from the works of \textit{M. S. Keener} and \textit{C. C. Travis} [Trans. Am. Math. Soc. 237, 331-351 (1978; Zbl 0404.35017)] and \textit{E. C. Tomastik} [Rocky Mt. J. Math. 18, No. 1, 1-11 (1988; Zbl 0667.34046)].