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Spectral theory of left definite difference operators. (English) Zbl 1127.39046

This paper is concerned with the spectral theory for the second-order left definite difference boundary value problems. Existence of eigenvalues of boundary value problems is proved, the numbers of their eigenvalues are calculated and fundamental spectral results are obtained.

MSC:

39A70 Difference operators
39A12 Discrete version of topics in analysis
34L05 General spectral theory of ordinary differential operators
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[1] Atkinson, F.V., Discrete and continuous boundary problems, (1964), Academic Press, Inc. New York · Zbl 0117.05806
[2] Jirari, A., Second-order sturm – liouville difference equations and orthogonal polynomials, Mem. amer. math. soc., 113, (1995) · Zbl 0817.39004
[3] Ahlbrandt, C.D.; Peterson, A., Discrete linear Hamiltonian systems. difference equations, continued fractions, and Riccati equations, (1996), Kluwer Academic Press Dordrecht · Zbl 0860.39001
[4] Bohner, M., Discrete linear Hamiltonian eigenvalue problems, Comput. math. appl., 36, 179-192, (1998) · Zbl 0933.39033
[5] Shi, Y.; Chen, S., Spectral theory of second-order vector difference equations, J. math. anal. appl., 239, 195-212, (1999) · Zbl 0934.39002
[6] Agarwal, R.P.; Bohner, M.; Wong, P.J.Y., Sturm – liouville eigenvalue problems on time scales, Appl. math. comput., 99, 153-166, (1999) · Zbl 0938.34015
[7] Bohner, M.; Hilscher, R., An eigenvalue problem for linear Hamiltonian dynamic systems, Fasc. math., 35, 35-49, (2005) · Zbl 1096.39017
[8] Bohner, M.; Došlý, O.; Kratz, W., An oscillation theorem for discrete eigenvalue problems, Rocky mountain J. math., 33, 1233-1260, (2003) · Zbl 1060.39003
[9] Erbe, L.; Peterson, A.; Saker, S.H., Oscillation criteria for second-order nonlinear dynamic equations on time scales, J. London math. soc., 67, 701-714, (2003) · Zbl 1050.34042
[10] Kratz, W., Sturm – liouville difference equations and banded matrices, Arch. math. (Brno), 36, 499-505, (2000) · Zbl 1072.39500
[11] Chen, S.; Erbe, L., Oscillation and nonoscillation for systems of selfadjoint second-order difference equations, SIAM J. math. anal., 20, 939-949, (1989) · Zbl 0687.39001
[12] Krall, A.M., Left-definite theory for second order differential operators with mixed boundary conditions, J. differential equations, 118, 153-165, (1995) · Zbl 0827.34073
[13] Krall, A.M., Left-definite regular Hamiltonian systems, Math. nachr., 174, 203-217, (1995) · Zbl 0832.34010
[14] Kong, Q.; Wu, H.; Zettl, A., Left-definite sturm – liouville problems, J. differential equations, 177, 1-26, (2001) · Zbl 1002.34016
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