On the Hadamard and Riemann–Liouville fractional neutral functional integrodifferential equations with finite delay

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Abstract

This paper is concerned with the existence and uniqueness of solutions for Hadamard and Riemann–Liouville fractional neutral functional integrodifferential equations with finite delay. The existence of solutions is derived from Leray–Schauders alternative, whereas the uniqueness of solution is established by Banachs contraction principle. An illustrative example is also included.

Keywords

Hadamard fractional derivative Riemann–Liouville fractional integral Neutral fractional differential equations Fixed point theorems 

Mathematics Subject Classification

34A08 34K37 34K40 

References

  1. 1.
    Abbas, M.I.: Existence and uniqueness of solution for a boundary value problem of fractional order involving two Caputos fractional derivatives. Adv. Differ. Equ. 2015, 252 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Abbas, M.I.: Ulam stability of fractional impulsive differential equations with Riemann–Liouville integral boundary conditions. J. Contemp. Math. Anal. 50(5), 209–219 (2015)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Agarwal, R., Zhou, Y., He, Y.: Existence of fractional neutral functional differential equations. Comput. Math. Appl. 59, 1095–1100 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ahmad, B., Ntouyas, S.K., Tariboon, J.: A nonlocal hybrid boundary value problem of caputo fractional integro-differential equations. Acta Math. Sci. 36 B(6), 1631–1640 (2016)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Ahmad, B., Ntouyas, S.K.: Initial value problems for functional and neutral functional Hadamard type fractional differential inclusions. Miskolc Math. Notes 17(1), 15–27 (2016)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Ahmad, B., Ntouyas, S.K.: Eexistence and uniqueness of solutions for Caputo–Hadamard sequential fractional order neutral functional differential equations. EJDE 2017(36), 1–11 (2017)Google Scholar
  7. 7.
    Ahmad, B., Ntouyas, S.K., Tariboonc, J.: A study of mixed Hadamard and Riemann–Liouville fractional integro-differential inclusions via endpoint theory. Appl. Math. Lett. 52, 9–14 (2016)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Agarwal, P., Chand, M., Singh, G.: Certain fractional kinetic equations involving the product of generalized k-Bessel function. Alexandria Eng. J. 55, 3053–3059 (2016)CrossRefGoogle Scholar
  9. 9.
    Agarwal, P., Choib, J., Paris, R.B.: Extended Riemann–Liouville fractional derivative operator and its applications. J. Nonlinear Sci. Appl. 8, 451–466 (2015)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Agarwal, P.: Some inequalities involving Hadamard-type k-fractional integral operators. Math. Method Appl. Sci. 40(11), 3882–3891 (2017)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Butzer, P.L., Kilbas, A.A., Trujillo, J.J.: Compositions of Hadamard-type fractional integration operators and the semigroup property. J. Math. Anal. Appl. 269, 387–400 (2002)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Butzer, P.L., Kilbas, A.A., Trujillo, J.J.: Fractional calculus in the Mellin setting and Hadamard-type fractional integrals. J. Math. Anal. Appl. 269, 1–27 (2002)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Butzer, P.L., Kilbas, A.A., Trujillo, J.J.: Mellin transform analysis and integration by parts for Hadamard-type fractional integrals. J. Math. Anal. Appl. 270, 1–15 (2002)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York (2003)CrossRefMATHGoogle Scholar
  15. 15.
    Hadamard, J.: Essai sur l’etude des fonctions donnees par leur developpment de Taylor. J. Mater. Pure Appl. Ser. 8, 101–186 (1892)MATHGoogle Scholar
  16. 16.
    Hale, J., Verduyn Lunel, S.: Introduction to Functional Differential Equations, Series Applied Mathematical Sciences, vol. 99. Springer, New York (1993)Google Scholar
  17. 17.
    Kilbas, A.A., Srivastava, Hari M., Juan Trujillo, J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)Google Scholar
  18. 18.
    Kolmanovskii, V., Myshkis, A.: Introduction to the Theory and Applications of Functional Differential Equations, Series Mathematics and Its Applications, vol. 463. Kluwer, Dordrecht (1999)Google Scholar
  19. 19.
    Ma, Q., Wang, R., Wang, J., Ma, Y.: Qualitative analysis for solutions of a certain more generalized two-dimensional fractional differential system with Hadamard derivative. Appl. Math. Comput. 257, 436–45 (2015)MathSciNetMATHGoogle Scholar
  20. 20.
    Ruzhansky, M., Cho, Y.J., Agarwal, P., Area, I.: Advances in Real and Complex Analysis with Applications. Springer, Berlin (2017)CrossRefMATHGoogle Scholar
  21. 21.
    Wang, J., Zhang, Y.: On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives. Appl. Math. Lett. 39, 85–90 (2015)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Computer Science, Faculty of ScienceAlexandria UniversityAlexandriaEgypt

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