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Eigenvalue comparisons for a class of boundary value problems of second order difference equations. (English) Zbl 1113.39021
This paper deals with the structure of eigenvalues of the boundary value problem for the second order difference equation $$\Delta(r_{i-1}\Delta y_{i-1})-b_i y_i+\lambda a_iy_i=0, 1\le i\le n, y_0-\tau y_1=y_{n+1}-\delta y_n=0$$ under the assumption that the $a_i$’s can be negative at some $i$, $1\le i\le n$. Especially, the authors focus on the comparison of all eigenvalues as the coefficients $\{a_i\}_{i=1}^{n}$, $\{b_i\}_{i=1}^{n}$, $\{r_i\}_{i=1}^{n}$ and the parameter $\tau, \delta$ change. The results extend the authors earlier results to a more general setting, allowing some $a_i$’s to be negative [{\it J. Ji, B. Yang}, J. Math. Anal. Appl. 320, No. 2, 964--972 (2006; Zbl 1111.39012)].

39A12Discrete version of topics in analysis
39A10Additive difference equations
34L15Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators
Full Text: DOI
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