Skip to main content
SpringerLink
Log in

Weakly Singular Integral Inequalities and Global Solutions for Fractional Differential Equations of Riemann–Liouville Type

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we obtain some new results about weakly singular integral inequalities. These inequalities are used to establish the global existence and uniqueness results for fractional differential equations of Riemann–Liouville type. Some examples are provided to illustrate the applicability of our main results.

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. Abdulla, A.B., Al-Refai, M., Al-Rawashdeh, A.: On the existence and uniqueness of solutions for a class of non-linear fractional boundary value problems. J. King Saud Univ. Sci. 28, 103–110 (2016)

    Article  Google Scholar 

  2. Al-Refai, M.: Maximum principles for nonlinear fractional differential equations in reliable space. Prog. Fract. Differ. Appl. 6(2), 95–99 (2020)

    Article  Google Scholar 

  3. Bainov, D., Simeonov, P.: Integral Inequalities and Applications, vol. 57. Kluwer Academic Publishers, Dordrecht (1992)

    Book  Google Scholar 

  4. Becker, L.C., Burton, T.A., Purnaras, I.K.: Complementary equations: a fractional differential equation and a Volterra integral equation. Electron. J. Qual. Theory. Differ. Equ. 12, 1–24 (2015)

  5. Denton, Z., Vatsala, A.S.: Fractional integral inequalities and applications. Comput. Math. Appl. 59, 1087–1094 (2010)

    Article  MathSciNet  Google Scholar 

  6. Diethelm, K.: The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type, Lecture Notes in Mathematics, vol. 2004. Springer, Berlin (2010)

  7. Haraux, A.: Nonlinear Evolution Equations-Global Behavior of Solutions, Lecture Notes in Mathematics, vol. 841. Springer, Berlin (1981)

  8. Henry, D.: Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840. Springer, Berlin (1981)

  9. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)

  10. Kong, Q.X., Ding, X.L.: A new fractional integral inequality with singularity and its application. Abstr. Appl. Anal., 937908 (2012)

  11. Medveď, M.: A new approach to an analysis of Henry type integral inequalities and their Bihari type versions. J. Math. Anal. Appl. 214, 349–366 (1997)

    Article  MathSciNet  Google Scholar 

  12. Medveď, M.: Integral inequalities and global solutions of semilinear evolution equations. J. Math. Anal. Appl. 267, 643–650 (2002)

    Article  MathSciNet  Google Scholar 

  13. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  14. Pazy, A.: SemiGroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44. Springer, New York (1983)

    MATH  Google Scholar 

  15. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  16. Schaefer, H.: Über die methods der priori Schranhen. Math. Ann. 129, 415–416 (1955)

    Article  MathSciNet  Google Scholar 

  17. Trif, T.: Existence of solutions to initial value problems for nonlinear fractional differential equations on the semi-axis. Frac. Calc. Appl. Anal. 16, 595–612 (2013)

    Article  MathSciNet  Google Scholar 

  18. Webb, J.R.L.: Weakly singular Gronwall inequalities and applications to fractional differential equations. J. Math. Anal. Appl. 471, 692–711 (2019)

    Article  MathSciNet  Google Scholar 

  19. Zhu, T.: Fractional integral inequalities and global solutions of fractional differential equations. Electron. J. Qual. Theory. Differ. Equ. 5, 1–16 (2020)

Download references

Acknowledgements

The research was supported by Scientific Research Foundation of Nanjing Institute of Technology (No: CKJB201508). I would like to express my sincere gratitude to the anonymous referee for his or her comments and suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics and Physics, Nanjing Institute of Technology, Nanjing, 211100, People’s Republic of China

    Tao Zhu

Authors
  1. Tao Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tao Zhu.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhu, T. Weakly Singular Integral Inequalities and Global Solutions for Fractional Differential Equations of Riemann–Liouville Type. Mediterr. J. Math. 18, 184 (2021). https://doi.org/10.1007/s00009-021-01824-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-021-01824-3

Keywords

Mathematics Subject Classification

Search

Navigation