Skip to main content
SpringerLink
Log in

On implicit abstract neutral nonlinear differential equations

  • Published:
Applied Mathematics & Optimization Submit manuscript

Abstract

In this paper we continue our developments in Hernández and O’Regan (J Funct Anal 261:3457–3481, 2011) on the existence of solutions for abstract neutral differential equations. In particular we extend the results in Hernández and O’Regan (J Funct Anal 261:3457–3481, 2011) for the case of implicit nonlinear neutral equations and we focus on applications to partial “nonlinear” neutral differential equations. Some applications involving partial neutral differential equations are presented.

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. Hernández, E., O’Regan, D.: On a new class of abstract neutral differential equations. J. Funct. Anal. 261(12), 3457–3481 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Gurtin, M.E., Pipkin, A.C.: A general theory of heat conduction with finite wave speed. Arch. Ration. Mech. Anal. 31, 113–126 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  3. Nunziato, J.W.: On heat conduction in materials with memory. Q. Appl. Math. 29, 187–204 (1971)

    MathSciNet  MATH  Google Scholar 

  4. Cannarsa, P., Sforza, D.: Global solutions of abstract semilinear parabolic equations with memory terms. NoDEA Nonlinear Diff. Equ. Appl. 10(4), 399–430 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Clément, Ph, Nohel, J.A.: Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels. SIAM J. Math. Anal. 12(4), 514–535 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  6. Lunardi, A.: Laplace transform methods in integro-differential equations. Integro-differential evolution equations and application. J. Integral Equ. 10(1—-3), 185–211 (1985)

    MathSciNet  MATH  Google Scholar 

  7. Hernández, E., O’Regan, D.: Existence results for abstract partial neutral differential equations. Proc. Am. Math. Soc. 137(10), 3309–3318 (2009)

    Article  MATH  Google Scholar 

  8. Favini, A., Yagi, A.: Degenerate differential equations in Banach spaces. Monographs and textbooks in pure and applied mathematics, vol. 215. Marcel Dekker Inc., New York (1999)

    MATH  Google Scholar 

  9. Favini, A., Marinoschi, G.: Degenerate nonlinear diffusion equations. Lecture Notes in Mathematics, vol. 2049. Springer, Heidelberg (2012)

    MATH  Google Scholar 

  10. Hale, J., Verduyn Lunel, S.M.: Introduction to functional-differential equations. Applied mathematical sciences, vol. 99. Springer, New York (1993)

    Book  MATH  Google Scholar 

  11. Hartung, F.: On differentiability of solutions with respect to parameters in neutral differential equations with state-dependent delays. Ann. Mat. Pura Appl. 192, 17–47 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kuang, Y.: Qualitative analysis of one-or two-species neutral delay population models. SIAM J. Math. Anal. 23(1), 181–200 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Rabah, R., Sklyar, G.M., Rezounenko, A.V.: On strong regular stabilizability for linear neutral type systems. J. Diff. Equ. 245(3), 569–593 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hale, J.: Partial neutral functional-differential equations. Rev. Roum. Math. Pure Appl. 39(4), 339–344 (1994)

    MathSciNet  MATH  Google Scholar 

  15. Wu, J.: Theory and applications of partial functional-differential equations. Applied mathematical sciences, vol. 119. Springer, New York (1996)

    Book  MATH  Google Scholar 

  16. Wu, J., Xia, H.: Rotating waves in neutral partial functional-differential equations. J. Dyn. Diff. Equ. 11(2), 209–238 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wu, J., Xia, H.: Self-sustained oscillations in a ring array of coupled lossless transmission lines. J. Diff. Equ. 124(1), 247–278 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  18. Adimy, M., Ezzinbi, K.: A class of linear partial neutral functional-differential equations with nondense domain. J. Diff. Equ. 147(2), 285–332 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. Adimy, M., Ezzinbi, K.: Existence and stability in the a-norm for partial functional differential equations of neutral type. Ann. Mat. Pura Appl. 185(3), 437–460 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Agarwal, R.P., Baghli, S., Benchohra, M.: Controllability for semilinear functional and neutral functional evolution equations with infinite delay in Fréchet spaces. Appl. Math. Optim. 60(2), 253–274 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Radhakrishnan, B., Balachandran, K.: Controllability of neutral evolution integrodifferential systems with state dependent delay. J. Optim. Theory Appl. 153(1), 85–97 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chang, Y.K., Li, W.S.: Solvability for impulsive neutral integro-differential equations with state-dependent delay via fractional operators. J. Optim. Theory Appl. 144(3), 445–459 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Datko, R.: Linear autonomous neutral differential equations in a Banach space. J. Diff. Equ. 25(2), 258–274 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hadd, S.: Singular functional differential equations of neutral type in Banach spaces. J. Funct. Anal. 254(8), 2069–2091 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Henríquez, H.R., Cuevas, C.: Almost automorphy for abstract neutral differential equations via control theory. Ann. Mat. Pura Appl. 192(3), 393–405 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hernández, E., Dos Santos, J.P., Henríquez, H.: Existence results for neutral integro-differential equations with unbounded delay. J. Integr. Equ. Appl. 23(2), 289–330 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Hernández, E., Henríquez, H.: Existence results for partial neutral functional differential equation with unbounded delay. J. Math. Anal. Appl. 221(2), 452–475 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hernández, E.: Existence results for partial neutral integro-differential equations with unbounded delay. J. Math. Anal. Appl. 292(1), 194–210 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hernández, E., O’Regan, D.: Existence of solutions for abstract neutral integro-differential equations with unbounded delay. Czechoslov. Math. J. 61(3), 691–706 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Hernández, E., Henríquez, H.: Existence of periodic solution of partial neutral functional differential equation with unbounded delay. J. Math. Anal. Appl. 221(2), 499–522 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  31. Liu, B.: Controllability of nonlinear neutral evolution integrodifferential systems with infinite delay. J. Optim. Theory Appl. 122(1), 87–109 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Liu, B.: Controllability of neutral functional differential and integrodifferential inclusions with infinite delay. J. Optim. Theory Appl. 123(3), 573–593 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  33. Ntouyas, S.K., O’Regan, D.: Controllability for semilinear neutral functional differential inclusions via analytic semigroups. J. Optim. Theory Appl. 135(3), 491–513 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. Hernández, E., O’Regan, D.: Existence of solutions for abstract non-autonomous neutral differential equations. Can. Math. Bull. 55(4), 736–751 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  35. Hernández, E., O’Regan, D.: \(C^\alpha \)-Hölder classical solutions for non-autonomous neutral differential equations. Discret. Contin. Dyn. Syst. 29(1), 241–260 (2011)

    MATH  Google Scholar 

  36. Lunardi, A.: Analytic semigroups and optimal regularity in parabolic problems, PNLDE. Birkhäauser, Basel (1995)

    Book  MATH  Google Scholar 

  37. Hino, Y., Murakami, S., Naito, T.: Functional differential equations with infinite delay. Lecture Notes in Mathematics. Springer, Berlin (1991)

    MATH  Google Scholar 

  38. Hale, J.K., Kato, J.: Phase space for retarded equations with infinite delay. Funkc. Ekvac. 21(1), 11–41 (1978)

    MathSciNet  MATH  Google Scholar 

  39. Henríquez, H.R.: Periodic solutions of quasi-linear partial functional-differential equations with unbounded delay. Funkc. Ekvac. 37(2), 329–343 (1994)

    MathSciNet  MATH  Google Scholar 

  40. Hernández, E., O’Regan, D.: Existence results for nonlinear abstract neutral differential equations. Submitted for publication

  41. Pierri, M., O’Regan, D.: On non-autonomous abstract nonlinear fractional differential equations. Appl. Anal. 94(5), 879–890 (2015)

Download references

Acknowledgments

The author wishes to thank to the referees for their valuable comments and suggestions. The work of Eduardo Hernández was supported by Fapesp, Grant 2011/17832-3.

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 Prêto, SP, CEP 14040-901, Brazil

    Eduardo Hernández

  2. School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

    Donal O’Regan

Authors
  1. Eduardo Hernández
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Donal O’Regan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eduardo Hernández.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hernández, E., O’Regan, D. On implicit abstract neutral nonlinear differential equations. Appl Math Optim 73, 329–347 (2016). https://doi.org/10.1007/s00245-015-9305-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00245-015-9305-5

Keywords

Mathematics Subject Classification

Search

Navigation