Skip to main content
SpringerLink
Log in

Local and Global Existence and Uniqueness of Solution and Local Well-Posednesss for Abstract Fractional Differential Equations with State-Dependent Delay

  • Published:
Applied Mathematics & Optimization Aims and scope Submit manuscript

Abstract

We study the local and global existence and uniqueness of solution and the local well-possedness for a general class of abstract fractional differential equations with state-dependent delay. Our studies are developed using a general abstract framework which allows us to present comprehensive results and to correct some important errors in the associate literature. Our results on the existence and uniqueness of solution are proved without assuming that the forcing terms are locally Lipschitz. In the last section, we present some examples of fractional in time partial differential equations motivated from the theory of population dynamics.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bajlekova, E. G.: Fractional evolution equations in Banach spaces. Thesis (Dr.)-Technische Universiteit Eindhoven (The Netherlands), p. 113. ProQuest LLC (2001)

  2. Bu, S., Cai, G.: Well-posedness of fractional integro-differential equations in vector-valued functional spaces. Math. Nachr. 292(5), 969–982 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  3. Carvalho Neto, P.: Fractional differential equations: a novel study of local and global solutions in Banach spaces. Tese de Doutorado. Universidade de São Paulo (2013)

  4. Chang, Y.-K., Ponce, R., Rueda, S.: Fractional differential equations of Sobolev type with sectorial operators. Semigroup Forum 99(3), 591–606 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chaudhary, R., Pandey, D.: Existence results for a class of impulsive neutral fractional stochastic integro-differential systems with state dependent delay. Stoch. Anal. Appl. 37(5), 865–892 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  6. Diethelm, K., Freed, A.D.: On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity. In: Keil, F., Mackens, W., Voss, H., Werther, J. (eds.) Scientific Computing in Chemical Engineering II Computational Fluid Dynamics. Reaction Engineering and Molecular Properties, pp. 217–224. Springer-Verlag, Heidelberg (1999)

    Google Scholar 

  7. Dlotko, T.: Semilinear Cauchy problems with almost sectorial operators. Bull. Pol. Acad. Sci. Math. 55(4), 333–346 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Driver, R.D.: A functional-differential system of neutral type arising in a two-body problem of classical electrodynamics. In: LaSalle, J., Lefschtz, S. (eds.) International Symposium on Nonlinear Differential Equations and Nonlinear Mechanics, pp. 474–484. Academic Press, New York (1963)

    Chapter  Google Scholar 

  9. Driver, R.D.: A neutral system with state-dependent delay. J. Differ. Equ. 54, 73–86 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag–Leffler Functions, Related Topics and Applications, vol. 2. Springer, Berlin (2014)

    Book  MATH  Google Scholar 

  11. Hartung, F., Krisztin, T., Walther, H., Wu, J.: Functional differential equations with state-dependent delays: theory and applications. In: Handbook of Differential Equations: Ordinary Differential Equations, vol. III, pp. 435–545. Elsevier, Amsterdam (2006)

    Chapter  Google Scholar 

  12. Hernández, E., Fernandes, D., Wu, J.: Existence and uniqueness of solutions, well-posedness and global attractor for abstract differential equations with state-dependent delay. J. Differ. Equ. 302(25), 753–806 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hernández, E., Wu, J., Chadha, A.: Existence, uniqueness and approximate controllability of abstract differential equations with state-dependent delay. J. Differ. Equ. 269(10), 8701–8735 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hernández, E., Pierri, M., Wu, J.: \( C^{1+\alpha }\)-strict solutions and wellposedness of abstract differential equations with state dependent delay. J. Differ. Equ. 261(12), 6856–6882 (2016)

    Article  MATH  Google Scholar 

  15. Hernández, E., Wu, J.: Existence and uniqueness of \({ C}^{1+\alpha }\)-strict solutions for integro-differential equations with state-dependent delay. Differ. Integral Equ. 32(5–6), 291–322 (2019)

    MATH  Google Scholar 

  16. Hernández, 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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hilfer, R.: Applications of fractional calculus in physics. World Scientific Publishing Co. Inc, River Edge, NJ (2000)

    Book  MATH  Google Scholar 

  18. Hutchinson, G.E.: Circular causal systems in ecology. Ann. NY Acad. Sci. 50(4), 221–246 (1948)

    Article  Google Scholar 

  19. Jothimani, K., Alliammal, N., Ravichandran, C.: Existence result for a neutral fractional integro-differential equation with state dependent delay. J. Appl. Nonlinear Dyn. 7(4), 371–381 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kalamani, P., Baleanu, D., Selvarasu, S., Arjunan, M.: On existence results for impulsive fractional neutral stochastic integro-differential equations with nonlocal and state-dependent delay conditions. Adv. Differ. Equ. 2016, 163 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kilbas, A.A., Srivastava, H.M., Trujillo, J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier Science B.V., Amsterdam (2006)

    Google Scholar 

  22. Krisztin, T., Rezounenkob, A.: Parabolic partial differential equations with discrete state-dependent delay: classical solutions and solution manifold. J. Differ. Equ. 260(5), 4454–4472 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kosovalic, N., Magpantay, F.M.G., Chen, Y., Wu, J.: Abstract algebraic-delay differential systems and age structured population dynamics. J. Differ. Equ. 255(3), 593–609 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kosovalic, N., Chen, Y., Wu, J.: Algebraic-delay differential systems:\(C^{0}\)-extendable submanifolds and linearization. Trans. Am. Math. Soc. 369(5), 3387–3419 (2017)

    Article  MATH  Google Scholar 

  25. Lv, Y., Pei, Y., Yuan, R.: Principle of linearized stability and instability for parabolic partial differential equations with state-dependent delay. J. Differ. Equ. 267(3), 1671–1704 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  26. Li, K., Peng, J., Jia, J.: Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives. J. Funct. Anal. 263(2), 476–510 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics. Fractals and fractional calculus in continuum mechanics (Udine,: CISM Courses and Lect, vol. 378, pp. 291–348. Springer, Vienna (1997)

  28. Mallika, D., Suganya, S., Baleanu, D., Mallika Arjunan, M.: A note on Sobolev form fractional integro-differential equation with state-dependent delay via resolvent operators. Nonlinear Stud. 24(3), 553–573 (2017)

    MathSciNet  MATH  Google Scholar 

  29. Periago, F., Straub, B.: A functional calculus for almost sectorial operators and applications to abstract evolution equations. J. Evol. Equ. 2(1), 41–68 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  30. Podlubny, I. Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Mathematics in Science and Engineering, p. 198

  31. Ponce, R.: On the well-posedness of degenerate fractional differential equations in vector valued function spaces. Isr. J. Math. 219(2), 727–755 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ponce, R.: Hölder continuous solutions for fractional differential equations and maximal regularity. J. Differ. Equ. 255(10), 3284–3304 (2013)

    Article  MATH  Google Scholar 

  33. Ravichandran, C., Valliammal, N., Nieto, J.: New results on exact controllability of a class of fractional neutral integro-differential systems with state-dependent delay in Banach spaces. J. Franklin Inst. 356(3), 1535–1565 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  34. Rezounenko, A.: A condition on delay for differential equations with discrete state-dependent delay. J. Math. Anal. Appl. 385(1), 506–516 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  35. Sousa, J.C., Capelas De Oliveira, E.: On the \(\psi \)-Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 60, 72–91 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  36. Suganya, S., Baleanu, D., Kalamani, P. Mallika., Arjunan, M.: On fractional neutral integro-differential systems with state-dependent delay and non-instantaneous impulses. Adv. Differ. Equ. 2015, 372 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  37. Suganya, S., Mallika Arjunan, M., Trujillo, J.J.: Existence results for an impulsive fractional integro-differential equation with state-dependent delay. Appl. Math. Comput. 266, 54–69 (2015)

    MathSciNet  MATH  Google Scholar 

  38. Yan, Z., Lu, F.: Existence and controllability of fractional stochastic neutral functional integro-differential systems with state-dependent delay in Fréchet spaces. J. Nonlinear Sci. Appl. 9, 603–616 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  39. Yan, Z., Jia, X.: On a fractional impulsive partial stochastic integro-differential equation with state-dependent delay and optimal controls. Stochastics 88(8), 1115–1146 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  40. 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)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zhang, X., Zhu, C., Yuan, C.: Approximate controllability of impulsive fractional stochastic differential equations with state-dependent delay. Adv. Differ. Equ. 2015, 91 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhou, Y., Suganya, S., Mallika Arjunan, M.: Approximate controllability of impulsive fractional integro-differential equation with state-dependent delay in Hilbert spaces. IMA J. Math. Control Inform. 36, 603–622 (2019)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors wish to thank the referees and the editor responsible for this paper, for their valuable comments and suggestions.

Author information

Authors and Affiliations

  1. Departamento de Computação e Matemática, Faculdade de Filosofia Ciências e Letras de Ribeirão Preto, Universidade de São Paulo, Ribeirão Preto, SP, CEP 14040-901, Brazil

    Eduardo Hernandez & Laura R. Gambera

  2. Department of Mathematics, Federal University of Alfenas, Alfenas, CEP 37130-000, MG, Brazil

    José Paulo Carvalho dos Santos

Authors
  1. Eduardo Hernandez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Laura R. Gambera
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. José Paulo Carvalho dos Santos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eduardo Hernandez.

Ethics declarations

Conflict of interest:

The authors have not disclosed any competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hernandez, E., Gambera, L.R. & Santos, J.P.C.d. Local and Global Existence and Uniqueness of Solution and Local Well-Posednesss for Abstract Fractional Differential Equations with State-Dependent Delay. Appl Math Optim 87, 41 (2023). https://doi.org/10.1007/s00245-022-09955-z

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00245-022-09955-z

Keywords

Mathematics Subject Classification

Search

Navigation