Cyclic sums of comparative indices and oscillation theory of symplectic difference systems
Volume 38, Issue 4, pp 479--501
https://dx.doi.org/10.22436/jmcs.038.04.05
Publication Date: February 11, 2025
Submission Date: August 08, 2024
Revision Date: October 26, 2024
Accteptance Date: January 08, 2025
Authors
J. Elyseeva
- Department of Applied Mathematics, Moscow State University of Technology, Vadkovskii per. 3a, 101472, Moscow, Russia.
Abstract
In this paper, we generalize the notion of the comparative index which has fundamental applications in oscillation theory of symplectic difference systems and linear differential Hamiltonian systems. We introduce cyclic sums \(\mu_c(Y_1,Y_2,\ldots,Y_m)\) and \(\mu_c^{*}(Y_1,Y_2,\ldots,Y_m),\,m\ge 2\) of the comparative indices for the set of \(n\)-dimensional Lagrangian subspaces. We formulate and prove main properties of the cyclic sums, in particular, we connect the cyclic sums of the comparative indices with the number of positive and negative eigenvalues of \(mn\times mn\) symmetric matrices defined in terms of the Wronskians \(Y_i^T J Y_j\), \(i,j=1,\ldots,m.\) We present first applications of the cyclic sums in the oscillation theory of the discrete symplectic systems connecting the number of focal points of their conjoined bases with the positive and negative inertia of symmetric matrices.
Share and Cite
ISRP Style
J. Elyseeva, Cyclic sums of comparative indices and oscillation theory of symplectic difference systems, Journal of Mathematics and Computer Science, 38 (2025), no. 4, 479--501
AMA Style
Elyseeva J., Cyclic sums of comparative indices and oscillation theory of symplectic difference systems. J Math Comput SCI-JM. (2025); 38(4):479--501
Chicago/Turabian Style
Elyseeva, J.. "Cyclic sums of comparative indices and oscillation theory of symplectic difference systems." Journal of Mathematics and Computer Science, 38, no. 4 (2025): 479--501
Keywords
- Symplectic difference systems
- focal points
- comparative index
- symmetric eigenvalue problems
MSC
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