Skip to main content
SpringerLink
Log in

On Integral Solutions for a Class of Mixed Volterra-Fredholm Integro Differential Equations with Caputo Fractional Derivatives

  • Conference paper
  • First Online:
Mathematical Analysis and Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 381))

  • 371 Accesses

Abstract

This work studies the existence of integral solution for a class of neutral integro-differential equation of mixed type involving Caputo fractional derivative under the assumption that the associated operator A is not dense. Utilizing semigroup theory, fractional calculus, Darbo-Sadovskii’s fixed point theorem and measure of noncompactness, we have established some sufficient conditions which ensure the existence of integral solutions of our problem.

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 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover 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. Baleanu, D., Chen, W., Chen, Y.Q., Sun, H.G., Zhang, Y.: A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simul. 64, 213–231 (2018)

    Article  Google Scholar 

  2. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  3. Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, New York (2010)

    Book  Google Scholar 

  4. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, New York (2006)

    MATH  Google Scholar 

  5. Al-Omari, A., Jaradat, O.K., Momani, S.: Existence of the mild solution for fractional semilinear initial value problems. Nonlinear Anal. 69, 3153–3159 (2008)

    Article  MathSciNet  Google Scholar 

  6. Mophou, G.: Existence and uniqueness of mild solutions to impulsive fractional differential equations. Nonlinear Anal. 72, 1604–1615 (2010)

    Article  MathSciNet  Google Scholar 

  7. Jia, J., Li, K.: Existence and uniqueness of mild solutions for abstract delay fractional differential equations. Comput. Math. Appl. 62, 1398–1404 (2011)

    Article  MathSciNet  Google Scholar 

  8. Jiao, F., Zhou, Y.: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal. Real World Appl. 11, 4465–4475 (2010)

    Article  MathSciNet  Google Scholar 

  9. Shen, X.H., Zhang, L., Zhou, Y.: Existence of mild solutions for fractional evolution equations. J. Int. Equ. Appl. 25, 557–586 (2013)

    MathSciNet  MATH  Google Scholar 

  10. Shu, X.B., Wang, Q.: The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order \(1<\alpha <2\). Comput. Math. Appl. 64, 2100–2110 (2012)

    Article  MathSciNet  Google Scholar 

  11. Jia, J., Li, K., Peng, J.: Cauchy problems for fractional evolution equations with Riemann-Liouville fractional derivatives. J. Funct. Anal. 263, 476–510 (2012)

    Article  MathSciNet  Google Scholar 

  12. Feng, Z., Li, Y.N., Sun, H.R.: Fractional abstract Cauchy problem with order \(\alpha \in (1,2)\). Dyn. Partial Differ. Equ. 13, 155–177 (2016)

    Article  MathSciNet  Google Scholar 

  13. Prato, G.D., Sinestrari, E.: Differential operators with nondense domain. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14, 285–344 (1987)

    Google Scholar 

  14. Ezzinbi, K., Liu, J.H.: Nondensely defined evolution equations with nonlocal conditions. Math. Comput. Model. 36, 1027–1038 (2002)

    Article  MathSciNet  Google Scholar 

  15. Fan, Z.: Existence of nondensely defined evolution equations with nonlocal conditions. Nonlinear Anal. 70, 3829–3836 (2009)

    Article  MathSciNet  Google Scholar 

  16. Guérékata, G.M.N., Mophou, G.M.: On integral solutions of some nonlocal fractional differential equations with nondense domain. Nonlinear Anal. 71, 4668–4675 (2009)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  18. Ahmad, B., Alsaedi, A., Gu, H., Zhou, Y.: Integral solutions of fractional evolution equations with nondense domain. Electron. J. Differ. Equ. 2017, 1–15 (2017)

    Article  MathSciNet  Google Scholar 

  19. Fu, X.: On solutions of neutral nonlocal evolution equations with nondense domain. J. Math. Anal. Appl. 299, 392–410 (2004)

    Article  MathSciNet  Google Scholar 

  20. Benchohra, M., Seba, D.: Impulsive fractional differential equations in Banach spaces. Electron. J. Qual. Theory Differ. Equ. Special Edition I, 1–14 (2009)

    Google Scholar 

  21. Chen, P., Li, Y.: Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions. Results Math. 63, 731–744 (2013)

    Article  MathSciNet  Google Scholar 

  22. Banas, J., Goebel, K.: Measure of Noncompactness in Banach spaces. In: Lecture Notes in Pure and Applied Mathematics. Marcel Dekker, Inc., New York (1980)

    Google Scholar 

  23. Liu, Y.L., Lv, J.Y.: Existence results for Riemann-Liouville fractional neutral evolution equations. Adv. Differ. Equ. 2014, 1–16 (2014)

    Article  MathSciNet  Google Scholar 

  24. Deimling, K.: Nonlinear Functional Analysis. Springer, New York (1985)

    Book  Google Scholar 

  25. Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014)

    Book  Google Scholar 

Download references

Acknowledgements

The first author expresses her gratitude to Indian Institute of Technology Guwahati for providing her fellowship to pursue her PhD.

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, 781039, India

    Bandita Roy & Swaroop Nandan Bora

Authors
  1. Bandita Roy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Swaroop Nandan Bora
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bandita Roy .

Editor information

Editors and Affiliations

  1. Faculty of Economy, Ibn Zohr University, Agadir, Morocco

    Ouayl Chadli

  2. Department of Mathematics, National Institute of Technology Jamshedpur, Jamshedpur, India

    Sourav Das

  3. Mathematics, University of Central Florida, Orlando, FL, USA

    Ram N. Mohapatra

  4. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India

    A. Swaminathan

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Roy, B., Bora, S.N. (2021). On Integral Solutions for a Class of Mixed Volterra-Fredholm Integro Differential Equations with Caputo Fractional Derivatives. In: Chadli, O., Das, S., Mohapatra, R.N., Swaminathan, A. (eds) Mathematical Analysis and Applications. Springer Proceedings in Mathematics & Statistics, vol 381. Springer, Singapore. https://doi.org/10.1007/978-981-16-8177-6_6

Download citation

Publish with us

Policies and ethics

Search

Navigation