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The uniqueness of a solution to the renewal type system of integral equations on the line

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Abstract

We study the uniqueness of a solution to a renewal type system of integral equations z=g+F * z on the line ℝ; here z is the unknown vector function, g is a known vector function, and F is a nonlattice matrix of finite measures on ℝ such that the matrix F(ℝ) is of spectral radius 1 and indecomposable. We show that in a certain class of functions each solution to the corresponding homogeneous system coincides almost everywhere with a right eigenvector of F(ℝ) with eigenvalue 1.

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References

  1. Rudin W., Functional Analysis, McGraw-Hill, New York (1991).

    MATH  Google Scholar 

  2. Sgibnev M. S., “Stone’s decomposition of the renewal measure via Banach-algebraic techniques,“ Proc. Amer. Math. Soc., 130, 2425–2430 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  3. Horn R. A. and Johnson Ch. R., Matrix Analysis, Cambridge Univ. Press, Cambridge (1985).

    MATH  Google Scholar 

  4. Sgibnev M. S., “Systems of renewal equations on the line,“ J. Math. Sci. Univ. Tokyo, 10, 495–517 (2003).

    MATH  MathSciNet  Google Scholar 

  5. Sgibnev M. S., “Systems of renewal-type integral equations on the line,“ Differential Equations, 40, No. 1, 137–147 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  6. Feller W., An Introduction to Probability Theory and Its Applications. Vol. 2, John Wiley and Sons, New York etc. (1971).

    MATH  Google Scholar 

  7. Lukacs E., Characteristic Functions, Griffin, London (1970).

    MATH  Google Scholar 

  8. Crump K. S., “On systems of renewal equations,“ J. Math. Anal. Appl., 30, 425–434 (1970).

    Article  MATH  MathSciNet  Google Scholar 

  9. Sgibnev M. S., “The matrix analog of the Blackwell renewal theorem on the real line,“ Sb.: Math., 197, No. 3, 369–386 (2006).

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    M. S. Sgibnev

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  1. M. S. Sgibnev
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Correspondence to M. S. Sgibnev.

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Original Russian Text Copyright © 2010 Sgibnev M. S.

Novosibirsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 51, No. 1, pp. 204–211, January–February, 2010.

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Sgibnev, M.S. The uniqueness of a solution to the renewal type system of integral equations on the line. Sib Math J 51, 168–173 (2010). https://doi.org/10.1007/s11202-010-0017-4

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  • DOI: https://doi.org/10.1007/s11202-010-0017-4

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