Skip to main content
SpringerLink
Log in

The Choice of the Regularization Parameter for Solving Linear Volterra-Stieltjes Integral Equations of the Third Kind

  • Conference paper
  • First Online:
Scientific and Technical Revolution: Yesterday, Today and Tomorrow (ISC 2019)

Part of the book series: Lecture Notes in Networks and Systems ((LNNS,volume 129))

Included in the following conference series:

  • 1472 Accesses

Abstract

For solving one class of Volterra linear integral equations of the third kind, the regularization parameter for the regularizing operator was chosen by Lavrentiev. In general case, the Volterra-Stieltjes integral equations do not always reduce to Volterra integral equations, since the Stieltjes integral equation does not always reduce to the Riemann integral or Lebesgue integral. Therefore, a Volterra-Stieltjes integral equation was of independent interest. The aim of the study was to construct a regularizing operator and prove the uniqueness theorem. In this study we used the notion of a derivative with respect to an increasing function, regularization method according to M.M. Lavrentiev, methods of functional analysis, methods for transforming equations, methods of integral and differential equations. Proposed methods can be used to study integral and integro-differential equations of the Volterra-Stieltjes type of high orders, as well as in qualitative study of some processes in fields of applied physics, ecology, medicine, and the theory of complex systems control.

Furthermore, they can be used in development of the integral equations theory in classes of inaccurate problems, for the numerical solution of Volterra-Stieltjes integral equations of the third kind and in solving specific applied problems that lead to equations of the third kind.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Asanov, A., Hazar, E., Mustafa, E., Matanova, K., Abdyldaeva, E.: Approximate solution of Volterra-Stieltjes linear integral equations of the second kind with the generalized trapezoid rule. Adv. Math. Phys. (2016). https://doi.org/10.1155/2016/1798050. ID1798050:6

  2. Ashirbayev, N.K., BanaÅ›, J., Bekmoldayeva, R.: A unified approach to some classes of nonlinear integral equations. J. Funct. Spaces (2014). https://doi.org/10.1155/2014/306231. ID306231:9

  3. Yang, Z.W.: Second-kind linear Volterra Integral equations with noncompact operators. Numer. Funct. Anal. Optim. 36(1), 104–131 (2015). https://doi.org/10.1080/01630563.2014.951769

    Article  MathSciNet  MATH  Google Scholar 

  4. Bharucha-Reid, A.T.: Chapter 4 random linear integral equations. Math. Sci. Eng. 96, 134–173 (1972). https://doi.org/10.1016/S0076-5392(08)60807-3. ISBN 9780120957507

    Article  Google Scholar 

  5. Banaś, J., Rzepka, B.: The technique of Volterra-Stieltjes integral equations in the application to infinite systems of nonlinear integral equations of fractional orders. Comput. Math Appl. 64(10), 3108–3116 (2012). https://doi.org/10.1016/j.camwa.2012.03.006

    Article  MathSciNet  MATH  Google Scholar 

  6. Banaś, J., Zaja̧c, T.: A new approach to the theory of functional integral equations of fractional order. J. Math. Anal. Appl. 375(2), 375–387 (2011). https://doi.org/10.1016/j.jmaa.2010.09.004

    Article  MathSciNet  Google Scholar 

  7. Nachbin, L.: Volterra Stieltjes-Integral Equations. Functional Analytic Methods, Linear Constraints, 1st edn, vol. 16, p. 156. North Holland. eBook ISBN: 9780080871271

    Google Scholar 

  8. Polyanin, A., Manzhirov, A.: Handbook of Integral Equations. CRC Press, Boca Raton (1998). ISBN 978-0849328763

    Book  Google Scholar 

  9. Darwish, M.A., BanaÅ›, J.: Existence and characterization of solutions of nonlinear Volterra-Stieltjes integral equations in two variables. Abstr. Appl. Anal. (2014). https://doi.org/10.1155/2014/618434. ID 618434:11

  10. Banaś, J., Kishin, S.: Solvability of Volterra-Stieltjes operator-integral equations and their applications. Comput. Math Appl. 41(12), 1535–1544 (2001). https://doi.org/10.1016/S0898-1221(01)00118-3

    Article  MathSciNet  MATH  Google Scholar 

  11. Banaś, J., Dronka, J.: Integral operators of Volterra-Stieltjes type, their properties and applications. Math. Comput. Model. 32(11–13), 1321–1331 (2000). https://doi.org/10.1016/S0895-7177(00)00207-7

    Article  MathSciNet  MATH  Google Scholar 

  12. Lavrentiev, M.M.: On integral equations of the first kind. SSSR 127(1), 31–38 (1959)

    MathSciNet  Google Scholar 

  13. Asanov, A.: Volterra-Stieltjes integral equations of the second and first kind. J. Nat. Sci. KTUM. 2, 79–95 (2002)

    Google Scholar 

  14. Asanov, A.: Derivative function with increasing function. J. Nat. Sci. KTUM 1, 18–64 (2001)

    Google Scholar 

  15. Shao, J., Meng, F.: Gronwall-Bellman type inequalities and their applications to fractional differential equations. Abstr. Appl. Anal. (2013). https://doi.org/10.1155/2013/217641. ID217641:7

Download references

Acknowledgements

Authors are thankful to professor A. Asanov for discussions and advices during solving the equations.

Author information

Authors and Affiliations

  1. Osh State University, Osh, Kyrgyzstan

    Nurgul Bedelova

Authors
  1. Nurgul Bedelova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nurgul Bedelova .

Editor information

Editors and Affiliations

  1. Moscow State Institute of International Relations, Plekhanov Russian University of Economic, Moscow, Russia

    Elena G. Popkova

  2. University of Messina, Messina, Italy

    Bruno S. Sergi

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Bedelova, N. (2020). The Choice of the Regularization Parameter for Solving Linear Volterra-Stieltjes Integral Equations of the Third Kind. In: Popkova, E., Sergi, B. (eds) Scientific and Technical Revolution: Yesterday, Today and Tomorrow. ISC 2019. Lecture Notes in Networks and Systems, vol 129. Springer, Cham. https://doi.org/10.1007/978-3-030-47945-9_36

Download citation

Publish with us

Policies and ethics

Search

Navigation