Skip to main content
SpringerLink
Log in

Hyers–Ulam Stability of Differential Operators on Reproducing Kernel Function Spaces

  • Published:
Complex Analysis and Operator Theory Aims and scope Submit manuscript

Abstract

In this paper, we investigate the Hyers–Ulam stability of the differential operators \(T_\lambda \) and D on the weighted Hardy spaces \(H_\beta ^2\) with the reproducing property. We obtain a necessary and sufficient condition in order that D is stable on \(H_\beta ^2\), and construct an example concerning the stability of \(T_\lambda \) on \(H_\beta ^2\). Moreover, we also investigate the Hyers–Ulam stability of the partial differential operators \(D_i\) on the several variables reproducing kernel space \(H_f^2(\mathbb {B}_d)\).

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Brzdȩk, J., Jung, S.-M.: A note on stability of an operator linear equation of the second order. Abstr. Appl. Anal. 2011(2011), 1–15. Article ID 602713

  2. Brzdȩk, J., Popa, D., Xu, B.: On approximate solutions of the linear functional equation of higher order. J. Math. Anal. Appl. 373, 680–689 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chan, K.C., Shapiro, J.H.: The cyclic behavior of translation operators on Hilbert spaces of entire functions. Indiana Univ. Math. J. 40, 1421–1449 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ciepliński, K.: Generalized stability of multi-additive mappings. Appl. Math. Lett. 23, 1291–1294 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ciepliński, K.: Applications of fixed point theorems to the Hyers–Ulam stability of functional equations—a survey. Ann. Funct. Anal. 3, 151–164 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Crismale, V.: Quantum stochastic calculus on interacting Fock spaces: semimartignale estimates and stochastic integral. Commun. Stoch. Anal. 1, 321–341 (2007)

    MathSciNet  MATH  Google Scholar 

  7. Das, B.K., Lindsay, J.M., Tripak, O.: Sesquilinear quantum stochastic analysis in Banach space. J. Math. Anal. Appl. 409, 1032–1051 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Găvruţa, P.: A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  9. Guo, K., Hu, J., Xu, X.: Toeplitz algebras, subnormal tuples and rigidity on reproducing C[\(z_1,\ldots, z_d\)]-modules. J. Funct. Anal. 210, 214–247 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222–224 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hatori, O., Kobayasi, K., Miura, T., Takagi, H., Takahasi, S.E.: On the best constant of Hyers–Ulam stability. J. Nonlinear Convex Anal. 5, 387–393 (2004)

    MathSciNet  MATH  Google Scholar 

  12. Hirasawa, G., Miura, T.: Hyers–Ulam stability of a closed operator in a Hilbert space. Bull. Korean Math. Soc. 43, 107–117 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ji, U.C., Obata, N.: Quantum stochastic integral representations of Fock space operators. Stochastics 81, 367–384 (2009)

    MathSciNet  MATH  Google Scholar 

  14. Miura, T., Hirasawa, G., Takahasi, S.E.: Ger-type and Hyers–Ulam stabilities for the first-order linear differential operators of entire functions. Int. J. Math. Math. Sci. 2004, 1151–1158 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Miura, T., Hirasawa, G., Takahasi, S.E.: Stability of multipliers on Banach algebras. Int. J. Math. Math. Sci. 2004, 2377–2381 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Popa, D., Raşa, I.: The Fréchet functional equation with application to the stability of certain operators. J. Approx. Theory 164, 138–144 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Popa, D., Raşa, I.: On the stability of some classical operators from approximation theory. Expo. Math. 31, 205–214 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Rassias, ThM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  19. Takagi, H., Miura, T., Takahasi, S.E.: Essential norms and stability constants of weighted composition operators on \(C(X)\). Bull. Korean Math. Soc. 40, 583–591 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ulam, S.M.: A Collection of Mathematical Problems. Interscience Publishers, New York (1960)

    MATH  Google Scholar 

  21. Wang, C., Xu, T.Z.: Hyers–Ulam stability of differentiation operator on Hilbert spaces of entire functions. J. Funct. Spaces 2014(2014), 1–6. Article ID 398673

  22. Xu, T.Z.: Approximate multi-Jensen, multi-Euler–Lagrange additive and quadratic mappings in \(n\)-Banach spaces. Abstr. Appl. Anal. 2013(2013), 1–12. Article ID 648709

  23. Xu, T.Z.: On the stability of multi-Jensen mappings in \(\beta \)-normed spaces. Appl. Math. Lett. 25, 1866–1870 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Xu, T.Z., Yang, Z., Rassias, J.M.: Direct and fixed point approaches to the stability of an AQ-functional equation in non- Archimedean normed spaces. J. Comput. Anal. Appl. 17, 697–706 (2014)

    MathSciNet  MATH  Google Scholar 

  25. Xu, T.Z.: On fuzzy approximately cubic type mapping in fuzzy Banach spaces. Inform. Sci. 278, 56–66 (2014)

    Article  MathSciNet  Google Scholar 

  26. Xu, T.Z., Wang, C.: Rassias, Th.M: On the stability of multi-additive mappings in non-Archimedean normed spaces. J. Comput. Anal. Appl. 18, 1102–1110 (2015)

Download references

Acknowledgments

The authors are very grateful to anonymous referees and editors for their valuable and detailed suggestions and insightful comments to improve the original manuscript.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China

    Chun Wang & Tian-Zhou Xu

  2. Department of Mathematics, Changzhi University, Changzhi, 046011, Shanxi, People’s Republic of China

    Chun Wang

Authors
  1. Chun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tian-Zhou Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chun Wang.

Additional information

Communicated by Ding-Xuan Zhou.

This work is supported by the National Natural Science Foundation of China (11171022).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, C., Xu, TZ. Hyers–Ulam Stability of Differential Operators on Reproducing Kernel Function Spaces. Complex Anal. Oper. Theory 10, 795–813 (2016). https://doi.org/10.1007/s11785-015-0486-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11785-015-0486-3

Keywords

Mathematics Subject Classification

Search

Navigation