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Inverse problems for evolution equations with fractional integrals at boundary-value conditions

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Abstract

The following problem is considered: to find a solution and a term of a first-order differential equation in a Banach space if the initial-value condition and an excessive condition containing the fractional Riemann–Liouville integral are given. We show that the solvability of the considered problem depends on the distributions of zeroes of the Mittag-Leffler function.

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References

  1. W. Arendt, “Vector valued Laplace transforms and Cauchy problems,” Israel J. Math., 59, 327–352 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  2. N. Danford and Dž. Švarc, Linear operators. Part I: General theory [Russian translation], Inostr. Lit., Moscow (1962).

  3. M. M. Dzhrbashyan, Integral Transformations and the Presentation of Functions in a Complex Domain [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  4. Yu. S. Èĭdel’man, “Two-point boundary value problem for a differential equation with a parameter,” Dokl. Akad. Nauk Ukrain. SSR Ser. A, No. 4, 15–18 (1983).

  5. Yu. S. Èĭdel’man, “An inverse problem for an evolution equation,” Math. Notes, 49, No. 5-6, 535–540 (1991).

    MathSciNet  Google Scholar 

  6. A. V. Glushak, “On a relation between the integrated operator cosine function and the operator Bessel function,” Differ. Equ., 42, No. 5, 619–626 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  7. E. Hille and R. Phillips, Functional Analysis and Semigroups [Russian translation], Inostr. Lit., Moscow (1962).

    Google Scholar 

  8. I. V. Mel’nikova and A. I. Filinkov, “Integrated semigroups and C-semigroups. Well-posedness and regularization of operator-differential problems,” Russian Math. Surveys, 49, No. 6, 115–155 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  9. M. Mijatović, S. Pilipović, and F. Vejzović, “α-Times integrated semigroups (α ∈ ℝ+),” J. Math. Anal. Appl., 210, 790–803 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  10. F. Neubrander, “Integrated semigroups and their applications to the abstract Cauchy problem,” Pacific J. Math., 135, 111–155 (1988).

    MATH  MathSciNet  Google Scholar 

  11. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York (1983).

    MATH  Google Scholar 

  12. A. I. Prilepko, D.G. Orlovsky, and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, Basel–New York (2000).

  13. A. M. Sedletskiĭ, “On the zeros of a function of Mittag-Leffler type,” Math. Notes, 68, No. 5-6, 602–613 (2000).

    Article  MathSciNet  Google Scholar 

  14. I. V. Tikhonov and Yu. S. Èĭdel’man, “An inverse problem for a differential equation in a Banach space and the distribution of zeros of an entire function of Mittag-Leffler type,” Differ. Equ., 38, No. 5, 669–677 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  15. I. V. Tikhonov and Yu. S. Èĭdel’man, “A uniqueness criterion in an inverse problem for an abstract differential equation with a nonstationary inhomogeneous term,” Math. Notes, 77, No. 1-2, 246–262 (2005).

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Belgorod State University, Mathematical Analysis Department, Belgorod, Ukraine

    A. V. Glushak

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  1. A. V. Glushak
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Correspondence to A. V. Glushak.

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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 29, Proceedings of KROMSH, 2008.

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Glushak, A.V. Inverse problems for evolution equations with fractional integrals at boundary-value conditions. J Math Sci 164, 518–530 (2010). https://doi.org/10.1007/s10958-010-9760-0

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