Hilfer Fractional Differential Equations with Almost Sectorial Operators

  • 40 Accesses


In this article we consider an abstract Cauchy problem with the Hilfer fractional derivative and an almost sectorial operator. We introduce a suitable definition of a mild solution for this evolution equation and establish the existence result for a mild solution. We also give an example to highlight the applicability of theoretical results established.

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

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 99

This is the net price. Taxes to be calculated in checkout.


  1. 1.

    Delbosco, D., Rodino, L.: Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 204, 609–625 (1996)

  2. 2.

    He, J.H.: Some applications of nonlinear fractional differential equations and their approximations. Bull. Sci. Technol. 15(2), 86–90 (1999)

  3. 3.

    He, J.H.: Approximate analytical solutions for seepage flow with fractional derivatives in porous media. Comput. Methods Appl. Mech. Eng. 167, 57–68 (1998)

  4. 4.

    Shin, J.S., Naito, T.: Existence and continuous dependence of mild solutions to semilinear functional differential equations in Banach spaces. Tohoku Math. J. (2) 51(4), 555–583 (1999)

  5. 5.

    Hernandez, E., O’Regan, D., Balachandran, K.: On recent developments in the theory of abstract differential equations with fractional derivatives. Nonlinear Anal. 73, 3462–3471 (2010)

  6. 6.

    Fan, H., Mu, J.: Initial value problem for fractional evolution equations. Adv. Differ. Equ. 2012, 49 (2012)

  7. 7.

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

  8. 8.

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

  9. 9.

    Zhou, Y.: Basic theory of fractional differential equations. World Scientific, Singapore (2014)

  10. 10.

    Pazy, A.: Semigroup of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, Berlin (1983)

  11. 11.

    Von Wahl, W.: Gberochene potenzen eines elliptischen operators und parabolische Differentialgleichungen in Raumen holderstetiger Funktionen. Nacher. Akad. Wiss. Gottingen Math. Phys. Klasse 11, 231–258 (1972)

  12. 12.

    Periago, F., Straub, B.: A functional calculus for almost sectorial operators and applications to abstract evolution equations. J. Evol. Equ. 2, 41–68 (2002)

  13. 13.

    Wang, R.N., Chen, D.H., Xiao, T.J.: Abstract fractional Cauchy problems with almost sectorial operators. J. Differ. Equ. 252(1), 202–235 (2012)

  14. 14.

    Zhang, L., Zhou, Y.: Fractional Cauchy problems with almost sectorial operators. Appl. Math. Comput. 257, 145–157 (2015)

  15. 15.

    Ding, X.L., Ahmad, B.: Analytical solutions to fractional evolution equations with almost sectorial operators. Adv. Differ. Equ. 2016, 203 (2016)

  16. 16.

    Li, F.: Mild solutions for abstract fractional differential equations with almost sectorial operators and infinite delay. Adv. Differ. Equ. 2013, 327 (2013)

  17. 17.

    Carvalho, A.N., Nascimento, M.J.: Semilinear evolution equations with almost sectorial operators. Cadernos De Mathematica 09, 19–44 (2008)

  18. 18.

    Hilfer, R.: Fractional calculus and regular variation in thermodynamics. In: Hilfer, R. (ed.) Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

  19. 19.

    Hilfer, R.: Fractional time evolution. In: Hilfer, R. (ed.) Applications of Fractional Calculus in Physics, pp. 87–130. World Scientific Publishing Company, Singapore (2000)

  20. 20.

    Hilfer, R.: Experimental evidence for fractional time evolution in glass forming materials. Chem. Phys. 284, 399–408 (2002)

  21. 21.

    Ahmed, H.M., Okasha, A.: Nonlocal Hilfer fractional neutral integrodifferential equations. Int. J. Math. Anal. 12(6), 277–288 (2018)

  22. 22.

    Ahmed, H.M., et al.: Impulsive Hilfer fractional differential equations. Adv. Differ. Equ. 2018, 226 (2018)

  23. 23.

    Gu, H., Trujillo, J.J.: Existence of mild solution for evolution equation with Hilfer fractional derivative. Appl. Math. Comput. 257, 344–354 (2015)

  24. 24.

    Harrat, A., Nieto, J.J., Debbouche, A.: Solvability and optimal controls of impulsive Hilfer fractional delay evolution inclusions with Clarke subdifferential. J. Comput. Appl. Math. 344, 725–737 (2018)

  25. 25.

    Guo, D.J., Lakshmikantham, V., Liu, X.Z.: Nonlinear Integral Equations in Abstract Spaces. Kluwer Academic, Dordrecht (1996)

  26. 26.

    Monch, H.: Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces. Nonlinear Anal. TMA 4, 985–999 (1980)

  27. 27.

    Bothe, D.: Multivalued perturbation of \(m\)-accretive differential inclusions. Isr. J. Math. 108, 109–138 (1998)

Download references


The authors are thankful to the anonymous referees for their valuable comments.

Author information

Correspondence to Anjali Jaiswal.

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

Verify currency and authenticity via CrossMark

Cite this article

Jaiswal, A., Bahuguna, D. Hilfer Fractional Differential Equations with Almost Sectorial Operators. Differ Equ Dyn Syst (2020).

Download citation


  • Almost sectorial operator
  • Cauchy problem
  • Hilfer fractional derivative
  • Measure of noncompactness