SpringerLink
Log in

Abstract degenerate Volterra integro-differential equations: inverse generator problem

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

In this paper, we investigate the inverse generator problem for abstract degenerate Volterra integro-differential equations in locally convex spaces. The classes of degenerate (a, k)-regularized C-resolvent families and degenerate mild (a, k)-regularized \((C_{1},C_{2})\)-existence and uniqueness families play an important role in our analysis. We provide several illustrative applications of the theoretical results obtained primarily to abstract degenerate fractional differential equations with Caputo derivatives.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Canada)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics, vol. 96 (Birkhäuser/Springer Basel AG, Basel, 2001)

    Book  MATH  Google Scholar 

  2. W. Arendt, O. El-Mennaoui, V. Keyantuo, Local integrated semigroups: evolution with jumps of regularity. J. Math. Anal. Appl. 186, 572–595 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. E. Bazhlekova, Fractional evolution equations in Banach spaces. Ph.D. Thesis, Eindhoven University of Technology, Eindhoven (2001)

  4. R. Cross, Multivalued Linear Operators (Marcel Dekker Inc., New York, 1998)

    MATH  Google Scholar 

  5. R. deLaubenfels, Existence Families, Functional Calculi and Evolution Equations, Lect. Notes Math., vol. 1570. Springer, New York (1994)

  6. R. de Laubenfels, Inverses of generators. Proc. Am. Math. Soc. 104, 443–448 (1988)

    Article  MathSciNet  Google Scholar 

  7. R. de Laubenfels, Inverses of generators of nonanalytic semigroups. Stud. Math. 191, 11–38 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. R. de Laubenfels, Inverses of generators of integrated or regularized semigroups. Semigroup Forum 75, 457–463 (2007)

    Article  MathSciNet  Google Scholar 

  9. T. Eisner, H. Zwart, The growth of a \(C_{0}\)-semigroup characterised by its cogenerator. J. Evol. Equ. 8, 749–764 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. S. Fackler, A short counterexample to the inverse generator problem on non-Hilbertian reflexive \(L^p\)-spaces. Arch. Math. 106, 383–389 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Favini, A. Yagi, Degenerate Differential Equations in Banach Spaces. Pure and Applied Mathematics (Chapman and Hall/CRC, New York, 1998)

    Book  Google Scholar 

  12. V.E. Fedorov, D.M. Gordievskikh, M.K. Plekhanova, Equations in Banach spaces with a degenerate operator under a fractional derivative. Differ. Uravn. 51, 1360–1368 (2015)

    MathSciNet  MATH  Google Scholar 

  13. A. Gomilko, Inverses of semigroup generators: a survey and remarks. Banach Center Publ. 112, 107–142 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Gomilko, H. Zwart, Y. Tomilov, On the inverse of the generator of\(C_{0}\)-semigroup, Mat. Sb. 198 (2007), 35–50 (in Russian); English transl.: Sb. Math. 198 (2007), 1095–1110

  15. R. Gorenflo, Y. Luchko, F. Mainardi, Analytical properties and applications of the Wright function. Fract. Calc. Appl. Anal. 2, 383–414 (1999)

    MathSciNet  MATH  Google Scholar 

  16. A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006)

    MATH  Google Scholar 

  17. V. Kiryakova, Generalized Fractional Calculus and Applications, Longman Scientific & Technical, Harlow, 1994, copublished in the United States with John Wiley & Sons, Inc., New York

  18. H. Komatsu, Fractional powers of operators. Pac. J. Math. 19, 285–346 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  19. M. Kostić, Abstract Volterra Integro-Differential Equations (CRC Press, Boca Raton, 2015)

    Book  MATH  Google Scholar 

  20. M. Kostić, Almost Periodic and Almost Automorphic Solutions to Integro-Differential Equations (W. de Gruyter, Berlin, 2019)

    Book  MATH  Google Scholar 

  21. M. Kostić, Abstract Degenerate Volterra Integro-Differential Equations (Mathematical Institute SANU, Belgrade, 2020)

    MATH  Google Scholar 

  22. M. Kostić, Degenerate abstract Volterra equations in locally convex spaces. Filomat 31, 597–619 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Li, J. Pastor, S. Piskarev, Inverses of generators of integrated fractional resolvent operator functions. Fract. Calc. Appl. Anal. 21, 1542–1564 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  24. Y.-N. Li, H.-J. Sun, Z. Feng, Integrated fractional resolvent operator function and fractional abstract Cauchy problem. Abstr. Appl. Anal. 2014, Article ID 430418. https://doi.org/10.1155/2014/430418

  25. Y.-N. Li, H.-J. Sun, Z. Feng, Fractional abstract Cauchy problem with order \(\alpha \in (1,2),\). Dyn. PDEs 13, 155–177 (2016)

    MathSciNet  MATH  Google Scholar 

  26. R. Liu, Is \(A^{-1}\) an infinitesimal generator? J. Appl. Math. Phys. 6, 1979–1987 (2018)

    Article  Google Scholar 

  27. Y. Luchko, Asymptotics of zeros of the Wright function. J. Math. Anal. Appl. 19, 1–12 (2000)

    MathSciNet  MATH  Google Scholar 

  28. O.I. Marichev, S.G. Samko, A.A. Kilbas, Fractional Integrals and Derivates. Theory and Applications (Gordon and Breach Science Publishers, Yverdon, 1993)

    MATH  Google Scholar 

  29. I.V. Melnikova, A.I. Filinkov, Abstract Cauchy Problems: Three Approaches (Chapman and Hall/CRC, Boca Raton, 2001)

    Book  MATH  Google Scholar 

  30. S. Piskarev, H. Zwart, Crank-Nicolson scheme for abstract linear systems. Numer. Funct. Anal. Optim. 28, 717–736 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. J. Prüss, Evolutionary Integral Equations and Applications, Monogr. Math. 87, Birkhäuser, Basel (1993)

  32. G.A. Sviridyuk, V.E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, Inverse and Ill-Posed Problems (Book 42) (VSP, Utrecht, Boston, 2003)

    Book  Google Scholar 

  33. G.N. Watson, Theory of Bessel Functions, 2nd edn. (Cambridge University Press, Cambridge, 1962)

    Google Scholar 

  34. T.-J. **ao, J. Liang, The Cauchy Problem for Higher-Order Abstract Differential Equations (Springer, Berlin, 1998)

    Book  MATH  Google Scholar 

  35. T.-J. **ao, J. Liang, Abstract degenerate Cauchy problems in locally convex spaces. J. Math. Anal. Appl. 259, 398–412 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  36. H. Zwart, Is \(A^{-1}\) an infinitesimal generator? in Perspectives in Operator Theory, Banach Center Publ. 75, Inst. Math., Polish Acad. Sci. (2007), pp. 303–313

  37. H. Zwart, Growth estimates for \(\exp (A^{-1}t)\) on a Hilbert space. Semigroup Forum 74, 487–494 (2007)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovića 6, 21125, Novi Sad, Serbia

    Marko Kostić

Authors
  1. Marko Kostić
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marko Kostić.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The author is partially supported by Grant 174024 of the Ministry of Science and Technological Development, Republic of Serbia.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kostić, M. Abstract degenerate Volterra integro-differential equations: inverse generator problem. Period Math Hung 85, 405–426 (2022). https://doi.org/10.1007/s10998-022-00449-7

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10998-022-00449-7

Keywords

Mathematics Subject Classification

Search

Navigation