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Differential Equations and Dynamical Systems

, Volume 27, Issue 1–3, pp 313–325 | Cite as

Semilinear Conformable Fractional Differential Equations in Banach Spaces

  • Anjali JaiswalEmail author
  • D. Bahuguna
Original Research
  • 273 Downloads

Abstract

We introduce the concept of a mild solution of conformable fractional abstract initial value problem. We establish the existence and uniqueness theorem using the contraction principle. As a regularity result for a linear problem, we show that the mild solution is in fact a strong solution. We give an example to demonstrate the applicability of the established theoretical results.

Keywords

Conformable fractional derivative Fractional-order differential equation Banach fixed point theorem 

Mathematics Subject Classification

MSC 34G10 MSC 34G20 

Notes

Acknowledgements

The authors would like to thanks the referees for their valuable comments and suggestions which have considerably improved our original manuscript.

References

  1. 1.
    Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abu-Hammad, M., Khalil, R.: Conformable fractional heat differential equation. Int. J. Pure Appl. Math. 94, 215–221 (2014)CrossRefzbMATHGoogle Scholar
  3. 3.
    Abu-Hammad, M., Khalil, R.: Fractional Fourier series and applications. Am. J. Comput. Appl. Math. 4(6), 187–191 (2014)zbMATHGoogle Scholar
  4. 4.
    Bayor, B., Torres, D.F.M.: Existence of solution to a local fractional differential equation. J. Comput. Appl. Math. 312, 127–133 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bryant, V.W.: A remark on a fixed point theorem for iterated mappings. Am. Math. Mon 75, 399–400 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Engel, K.J., Nagel, R.: One parameter semigroups for linear evolution equations. Graduate Texts in Math. Springer, Berlin (2000)Google Scholar
  7. 7.
    Goldstein, Jerome, A.: Semigroups of linear operators and applications. Oxford University Press (1985)Google Scholar
  8. 8.
    Khaldi, R., Guezane-Lakoud, A.: Lyapunov inequality for a boundary value problem involving conformable derivative. Progr. Fract. Differ. Appl. 3(4), 323–329 (2017)CrossRefzbMATHGoogle Scholar
  9. 9.
    Khalil, R., Abu-Shaab, H.: Solution of some conformable fractional differential equations. Int. J. Pure Appl. Math. 103(4), 667–673 (2015)CrossRefGoogle Scholar
  10. 10.
    Khalil, R., Al Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Springer, New York (1983)CrossRefzbMATHGoogle Scholar
  12. 12.
    Pinchover, Y., Rubinstein, J.: An introduction to partial differential equations. Cambridge University Press (2005)Google Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsIndian Institute of Technology KanpurKanpurIndia

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