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Ukrainian Mathematical Journal

, Volume 70, Issue 11, pp 1661–1676 | Cite as

Global Existence Results for Neutral Functional Differential Inclusions with State-Dependent Delay

  • E. Alaidarous
  • M. Benchohra
  • I. Medjadj
Article
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We study the existence of global solutions for a class of neutral functional differential inclusions with state-dependent delay. The proof of the main result is based on the semigroup theory and the Bohnenblust–Karlin fixed-point theorem.

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References

  1. 1.
    N. Abada, R. P. Agarwal, M. Benchohra, and H. Hammouche, “Existence results for nondensely defined impulsive semilinear functional differential equations with state-dependent delay,” Asian-Eur. J. Math., 1, 449–468 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    M. Adimy and K. Ezzinbi, “A class of linear partial neutral functional-differential equations with nondense domain,” J. Different. Equat., 147, 285–332 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    W. G. Aiello, H. I. Freedman, and J.Wu, “Analysis of a model representing stage-structured population growth with state-dependent time delay,” SIAM J. Appl. Math., 52, 855–869 (1992).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    E. Ait Dads and K. Ezzinbi, “Boundedness and almost periodicity for some state-dependent delay differential equations,” Electron. J. Different. Equat., 2002, No. 67, 1–13 (2002).MathSciNetzbMATHGoogle Scholar
  5. 5.
    A. Anguraj, M. M. Arjunan, and E. M. Hernàndez, “Existence results for an impulsive neutral functional differential equation with state-dependent delay,” Appl. Anal., 86, 861–872 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    O. Arino, K. Boushaba, and A. Boussouar, “A mathematical model of the dynamics of the phytoplankton-nutrient system: spatial heterogeneity in ecological models (Alcala de Henares, 1998),” Nonlinear Anal. Real World Appl., 1, 69–87 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston (1990).zbMATHGoogle Scholar
  8. 8.
    S. Baghli and M. Benchohra, “Existence results for semilinear neutral functional differential equations involving evolution operators in Fréchet spaces,” Georgian Math. J., 17, 423–436 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    S. Baghli and M. Benchohra, “Global uniqueness results for partial functional and neutral functional evolution equations with infinite delay,” Different. Integr. Equat., 23, 31–50 (2010).MathSciNetzbMATHGoogle Scholar
  10. 10.
    J. Belair, “Population models with state-dependent delays,” Lect. Notes Pure Appl. Math., 131, 156–176 (1990).Google Scholar
  11. 11.
    M. Benchohra, E. Gatsori, and S. K. Ntouyas, “Existence results for functional and neutral functional integrodifferential inclusions with lower semicontinous right-hand side,” J. Math. Anal. Appl., 281, 525–538 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    M. Benchohra and I. Medjadj, “Global existence results for functional differential equations with delay,” Comm. Appl. Anal., 17, 213–220 (2013).MathSciNetzbMATHGoogle Scholar
  13. 13.
    M. Benchohra, I. Medjadj, J. J. Nieto, and P. Prakash, “Global existence for functional differential equations with state-dependent delay,” J. Funct. Spaces Appl., 2013, Article ID 863561 (2013), 7 p.Google Scholar
  14. 14.
    H. F. Bohnenblust and S. Karlin, “On a theorem of Ville,” in: Contribution to the Theory of Games (AM-24), Vol. 1, Princeton Univ. Press, Princeton (1950), pp. 155–160.Google Scholar
  15. 15.
    Y. Cao, J. Fan, and T. C. Card, “The effects of state-dependent time delay on a stage-structured population growth model,” Nonlin. Anal., 19, 95–105 (1992).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    F. Chen, D. Sun, and J. Shi, “Periodicity in a food-limited population model with toxicants and state-dependent delays,” J. Math. Anal. Appl., 288, 136–146 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    C. Corduneanu, Integral Equations and Stability of Feedback Systems, Academic Press, New York (1973).zbMATHGoogle Scholar
  18. 18.
    K. Deimling, Multivalued Differential Equations, W. de Gruyter, Berlin–New York (1992).Google Scholar
  19. 19.
    R. D. Driver, “A neutral system with state-dependent delays,” J. Different. Equat., 54, No. 1, 73–86 (1984).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    J. K. Hale, “Partial neutral functional-differential equations,” Rev. Roumaine Math. Pures Appl., 39, No. 4, 339–344 (1994).MathSciNetzbMATHGoogle Scholar
  21. 21.
    J. Hale and J. Kato, “Phase space for retarded equations with infinite delay,” Funkcial. Ekvac., 21, 11–41 (1978).MathSciNetzbMATHGoogle Scholar
  22. 22.
    J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York (1993).CrossRefzbMATHGoogle Scholar
  23. 23.
    F. Hartung, T. Krisztin, H. O. Walther, and J. Wu, “Chapter 5. Functional differential equations with state-dependent delays: theory and applications,” in: Handbook of Differential Equations: Ordinary Differential Equations, Vol. 3 (2006), pp. 435–545.Google Scholar
  24. 24.
    E. Hernandez and H. Henriquez, “Existence results for partial neutral functional differential equations with unbounded delay,” J. Math. Anal. Appl., 221, 452–475 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    E. Hernandez and H. Henriquez, “Existence of periodic solutions of partial neutral functional differential equations with unbounded delay,” J. Math. Anal. Appl., 221, 499–522 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    E. Hernandez, A. Prokopezyk, and L. Ladeira, “A note on partial functional differential equation with state-dependent delay,” Nonlin. Anal. Real World Appl., 7, 511–519 (2006).MathSciNetCrossRefGoogle Scholar
  27. 27.
    Y. Hino, S. Murakami, and T. Naito, Functional Differential Equations with Unbounded Delay, Springer-Verlag, Berlin (1991).CrossRefzbMATHGoogle Scholar
  28. 28.
    Sh. Hu and N. Papageorgiou, Handbook of Multivalued Analysis. Vol. 1: Theory, Kluwer AP, Dordrecht (1997).Google Scholar
  29. 29.
    A. Lasota and Z. Opial, “An application of the Kakutani–Ky Fan theorem in the theory of ordinary differential equations,” Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 13, 781–786 (1965).MathSciNetzbMATHGoogle Scholar
  30. 30.
    W. S. Li, Y. K. Chang, and J. J. Nieto, “Solvability of impulsive neutral evolution differential inclusions with state-dependent delay,” Math. Comput. Model.: Int. J., 49, 1920–1927 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    K. Nandakumar and M. Wiercigroch, “Galerkin projections for state-dependent delay differential equations with applications to drilling,” Appl. Math. Model., 37, 1705–1722 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    S. K. Ntouyas, “Global existence for neutral functional integrodifferential equations,” Nonlin. Anal., 30, 2133–2142 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    S. K. Ntouyas, Y. G. Sficas, and C. P. Tsamatos, “Existence results for initial value problems for neutral functional-differential equations,” J. Different. Equat., 114, 527–537 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    L. Górniewicz, “Topological fixed point theory of multivalued mappings,” in: Mathematics and Its Applications, 495, Springer Netherlands (1999).Google Scholar
  35. 35.
    Y. Kuang and H. L. Smith, “Slowly oscillating periodic solutions of autonomous state-dependent delay equations,” Nonlin. Anal., 19, 855–1318 (1992).MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York (1983).CrossRefzbMATHGoogle Scholar
  37. 37.
    N. A. Perestyuk, V. A. Plotnikov, A. M. Samoilenko, and N. V. Skripnik, Differential Equations with Impulse Effects. Multivalued Right-Hand Sides with Discontinuities, W. de Gruyter, Berlin–Boston (2011).Google Scholar
  38. 38.
    J. Wu and H. Xia, “Self-sustained oscillations in a ring array of coupled lossless transmission lines,” J. Different. Equat., 124, 247–278 (1996).MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    J. Wu, Theory and Applications of Partial Functional-Differential Equations, Springer, New York (1996).CrossRefzbMATHGoogle Scholar
  40. 40.
    Z. Yang and J. Cao, “Existence of periodic solutions in neutral state-dependent delays equations and models,” J. Comput. Appl. Math., 174, 179–199 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    K. Yosida, Functional Analysis, 6th ed., Springer-Verlag, Berlin (1980).zbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • E. Alaidarous
    • 1
  • M. Benchohra
    • 2
  • I. Medjadj
    • 2
  1. 1.King Abdulaziz UniversityJeddahSaudi Arabia
  2. 2.Djillali Liabes UniversitySidi Bel-AbbèsAlgeria

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