Advertisement

An Extension of the Fractional Gronwall Inequality

  • Ricardo Almeida
  • Agnieszka B. Malinowska
  • Tatiana OdzijewiczEmail author
Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 559)

Abstract

In this work, we prove a generalization of the Gronwall type inequality. This relation can be used in the qualitative analysis of the solutions to fractional differential equations with the \(\psi \)-fractional derivatives.

Keywords

Fractional operators Gronwall type inequality 

Notes

Acknowledgments

R. Almeida is supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), within project UID/MAT/04106/2019. A. B. Malinowska is supported by the Bialystok University of Technology grant S/WI/1/2016 and T. Odzijewicz by the Warsaw School of Economics grant KAE/S18/08/18.

References

  1. 1.
    Adda, F.B., Cresson, J.: Fractional differential equations and the Schrödinger equation. Appl. Math. Comput. 161, 323–345 (2005)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Almeida, R.: A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul. 44, 460–481 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Almeida, R., Malinowska, A.B., Monteiro, M.T.T.: Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications. Math. Methods Appl. Sci. 41, 336–352 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Jarad, F., Abdeljawad, T., Baleanu, D.: Caputo-type modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2012, 142 (2012).  https://doi.org/10.1186/1687-1847-2012-142MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bagley, R.L., Torvik, J.: Fractional calculus - a different approach to the analysis of viscoelastically damped structures. AIAA J. 21(5), 741–748 (1983)CrossRefGoogle Scholar
  6. 6.
    Bellman, R.: Stability Theory of Differential Equations. McGraw-Hill, New York (1953)zbMATHGoogle Scholar
  7. 7.
    Djordjević, V.D., Jarić, J., Fabry, B., Fredberg, J.J., Stamenović, D.: Fractional derivatives embody essential features of cell rheological behavior. Ann. Biomed. Eng. 31, 692–699 (2003)CrossRefGoogle Scholar
  8. 8.
    Douglas, J.F.: Some applications of fractional calculus to polymer science. In: Prigogine, I., Rice, S.A. (eds.) Advances in Chemical Physics (2007)CrossRefGoogle Scholar
  9. 9.
    Grzesikiewicz, W., Wakulicz, A., Zbiciak, A.: Nonlinear problems of fractional calculus in modeling of mechanical systems. Int. J. Mech. Sci. 70, 89–90 (2013)CrossRefGoogle Scholar
  10. 10.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier Science B.V., Amsterdam (2006)CrossRefGoogle Scholar
  11. 11.
    Koh, C.G., Kelly, J.M.: Application of fractional derivatives to seismic analysis of base-isolated models. Earthq. Eng. Struct. Dyn. 19, 229–241 (1990)CrossRefGoogle Scholar
  12. 12.
    Luchko, Y., Trujillo, J.J.: Caputo-type modification of the Erdélyi-Kober fractional derivative. Fract. Calc. Appl. Anal. 10(3), 249–267 (2007)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Makris, N., Constantinou, M.C.: Fractional-derivative Maxwell model for viscous dampers. J. Struct. Eng. 117(9), 2708–2724 (1991)CrossRefGoogle Scholar
  14. 14.
    Qian, D., Gong, Z., Li, C.: A generalized Gronwall inequality and its application to fractional differential equations with Hadamard derivatives. In: Proceedings of the 3rd Conference on Nonlinear Science and Complexity, Cankaya University (2010)Google Scholar
  15. 15.
    Sousa, J.V.C., Oliveira, E.C.: A Gronwall inequality and the Cauchy-type problem by means of \(\psi \)-Hilfer operator. Differ. Equ. Appl. 11(1), 87–106 (2019)MathSciNetGoogle Scholar
  16. 16.
    Ye, H., Gao, J., Ding, Y.: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328(2), 1075–1081 (2007)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Zhang, Z., Wei, Z.: A generalized Gronwall inequality and its application to fractional neutral evolution inclusions. J. Inequal. Appl. 2016, 45 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Center for Research and Development in Mathematics and Applications (CIDMA), Department of MathematicsUniversity of AveiroAveiroPortugal
  2. 2.Faculty of Computer ScienceBialystok University of TechnologyBiałystokPoland
  3. 3.Department of Mathematics and Mathematical EconomicsWarsaw School of EconomicsWarsawPoland

Personalised recommendations