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Existence and uniqueness of global solutions of caputo-type fractional differential equations

  • Chung-Sik Sin EMAIL logo and Liancun Zheng
Published/Copyright: June 23, 2016
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 19 Issue 3

Abstract

In this paper we consider initial value problems for fractional differential equations involving Caputo differential operators. By establishing a new property of the Mittag-Leffler function and using the Schauder fixed point theorem, we obtain new sufficient conditions for the existence and uniqueness of global solutions of initial value problems.

Key Words and Phrases: Caputo derivative, existence and uniqueness of solutions, Mittag-Leffler type function, initial value problem, fractional differential equation
MSC 2010: 34A08, 34A12, 33E12, 26A33

Acknowledgements

The authors would like to thank Prof. Fawang Liu, Prof. Virginia Kiryakova and Prof. Kai Diethelm for their help and advices for the improvement of this article.

The work is supported by the National Natural Science Foundations of China (No. 51276014, 51476191).

References

[1] A. Aghajani, E. Pourhadi, J.J. Trujillo, Application of measure of noncompactness to a Cauchy problem for fractional differential equations in Banach spaces. Fract. Calc. Appl. Anal.16, No 4 (2013), 962–977; DOI: 10.2478/s13540-013-0059-y; http://www.degruyter.com/view/j/fca.2013.16.issue-4/issue-files/fca.2013.16.issue-4.xmlSearch in Google Scholar

[2] K. Diethelm, The Analysis of Fractional Differential Equations. Springer, Berlin, 2010; http://www.springerlink.com/content/978-3-642-14573-510.1007/978-3-642-14574-2Search in Google Scholar

[3] K. Diethelm, The mean value theorems and a Nagumo-type uniqueness theorem for Caputo's fractional calculus. Fract. Calc. Appl. Anal.15, No 2 (2012), 304–313; DOI: 10.2478/s13540-012-0022-3; http://www.degruyter.com/view/j/fca.2012.15.issue-2/issue-files/fca.2012.15.issue-2.xmlSearch in Google Scholar

[4] K. Diethelm, N.J. Ford, Analysis of fractional differential equations. J. Math. Anal. Appl.265 (2002), 229–24810.1007/978-3-642-14574-2Search in Google Scholar

[5] R.A.C. Ferreria, A uniqueness result for a fractional differential equation. Fract. Calc. Appl. Anal.15, No 4 (2012), 611–615; DOI: 10.2478/s13540-012-0042-z; http://www.degruyter.com/view/j/fca.2012.15.issue-4/issue-files/fca.2012.15.issue-4.xmlSearch in Google Scholar

[6] R. Gorenflo, A.A Kilbas, F. Mainardi, S.V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications. Springer, 2014; http://www.springer.com/mathematics/analysis/book/978-3-662-43929-610.1007/978-3-662-43930-2Search in Google Scholar

[7] H.J. Haubold, A.M. Mathai, R.K. Saxena, Mittag-Leffler functions and their applications. J. Appl. Math.2011 (2011), Article ID 298628, 51 pages; DOI: 10.1155/2011/298628Search in Google Scholar

[8] D. Idczak, R. Kamocki, On the existence and uniqueness and formula for the solution of R-L fractional Cauchy problem in Rn. Fract. Calc. Appl. Anal.14, No 4 (2011), 538–553; DOI: 10.2478/s13540-011-0033-5; http://www.degruyter.com/view/j/fca.2011.14.issue-4/issue-files/fca.2011.14.issue-4.xmlSearch in Google Scholar

[9] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, 2006Search in Google Scholar

[10] V. Lakshmikantham, S. Leela, Nagumo-type uniqueness result for fractional differential equations. Nonlinear Anal., Theory Methods Appl.71 (2009), 2886–288910.1016/j.na.2009.01.169Search in Google Scholar

[11] V. Lakshmikantham, S. Leela, A Krasnoselskii-Krein-type uniqueness result for fractional differential equations. Nonlinear Anal., Theory Methods Appl.71 (2009), 3421–342410.1016/j.na.2009.02.008Search in Google Scholar

[12] W. Lin, Global existence theory and chaos control of fractional differential equations. J. Math. Anal. Appl.332 (2007), 709–72610.1016/j.jmaa.2006.10.040Search in Google Scholar

[13] F. Mainardi, An historical perspective on fractional calculus in linear viscoelasticity. Fract. Calc. Appl. Anal.15, No 4 (2012), 712–717; DOI: 10.2478/s13540-012-0048-6; http://www.degruyter.com/view/j/fca.2012.15.issue-4/issue-files/fca.2012.15.issue-4.xmlSearch in Google Scholar

[14] I. Podlubny, Fractional Differential Equations. Academic Press, New York, 1999Search in Google Scholar

Received: 2015-4-17
Revised: 2015-11-21
Published Online: 2016-6-23
Published in Print: 2016-6-1

© 2016 Diogenes Co., Sofia

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  2. Editorial Note
  3. FCAA related news, events and books (FCAA-volume 19-3-2016)
  4. Survey Paper
  5. Fractional integrals and derivatives: mapping properties
  6. Research Paper
  7. Riesz fractional integrals in grand lebesgue spaces on ℝn
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  9. United lattice fractional integro-differentiation
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  11. Integral equations of fractional order in Lebesgue spaces
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  19. Pseudo almost automorphy of semilinear fractional differential equations in Banach Spaces
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https://doi.org/10.1515/fca-2016-0040
Keywords for this article
Caputo derivative, existence and uniqueness of solutions, Mittag-Leffler type function, initial value problem, fractional differential equation

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA-volume 19-3-2016)
  4. Survey Paper
  5. Fractional integrals and derivatives: mapping properties
  6. Research Paper
  7. Riesz fractional integrals in grand lebesgue spaces on ℝn
  8. Survey Paper
  9. United lattice fractional integro-differentiation
  10. Research Paper
  11. Integral equations of fractional order in Lebesgue spaces
  12. Research Paper
  13. General time-fractional diffusion equation: some uniqueness and existence results for the initial-boundary-value problems
  14. Research Paper
  15. Multilinear integral operators in weighted grand Lebesgue spaces
  16. Research Paper
  17. Fractional integration operator on some radial rays and intertwining for the Dunkl operator
  18. Research Paper
  19. Pseudo almost automorphy of semilinear fractional differential equations in Banach Spaces
  20. Research Paper
  21. Existence and uniqueness of global solutions of caputo-type fractional differential equations
  22. Research Paper
  23. Perfect nonlinear observers of fractional descriptor continuous-time nonlinear systems
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