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Comparison Theorems for Weighted Focal Points of Conjoined Bases of Hamiltonian Difference Systems

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Differential and Difference Equations with Applications (ICDDEA 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 164))

Abstract

In this paper we prove comparison theorems for the number of weighted focal points of conjoined bases of Hamiltonian difference systems. The notion of a weighted focal point introduced by O. Došlý and J. Elyseeva (Appl. Math. Lett. (43) 2015, 114–119) plays an important role in spectral theory for discrete Hamiltonian eigenvalue problems with nonlinear dependence on the spectral parameter. We present new relations between the numbers of weighted focal points of conjoined bases of two Hamiltonian systems and derive corollaries to these relations generalizing comparison results for the classical number of focal points. The consideration is based on the comparative index theory for symplectic difference systems.

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Acknowledgements

This research is supported by the Federal Programme of Ministry of Education and Science of the Russian Federation in the framework of the state order [grant number 2014/105].

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Authors and Affiliations

  1. Department of Applied Mathematics, Moscow State University of Technology, Vadkovskii per. 3a, 101472, Moscow, Russia

    Julia Elyseeva

Authors
  1. Julia Elyseeva
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Correspondence to Julia Elyseeva .

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Editors and Affiliations

  1. Departamento de Ciências Exactas e Naturais, Academia Militar, Amadora, Portugal

    Sandra Pinelas

  2. Department of Mathematics and Statistics, Masaryk University, Brno, Czech Republic

    Zuzana Došlá

  3. Department of Mathematics and Statistics, Masaryk University, Brno, Czech Republic

    Ondřej Došlý

  4. School of Mathematics and Stastistics, Huazhong University of Science and Technology, Wuhan, China

    Peter E. Kloeden

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Elyseeva, J. (2016). Comparison Theorems for Weighted Focal Points of Conjoined Bases of Hamiltonian Difference Systems. In: Pinelas, S., Došlá, Z., Došlý, O., Kloeden, P. (eds) Differential and Difference Equations with Applications. ICDDEA 2015. Springer Proceedings in Mathematics & Statistics, vol 164. Springer, Cham. https://doi.org/10.1007/978-3-319-32857-7_21

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