Advertisement

Existence result for impulsive coupled systems on the half-line

  • Feliz Minhós
  • Robert de Sousa
Original Paper

Abstract

This work considers a second order impulsive coupled system of differential equations with generalized jump conditions in half-line, which can depend on the impulses of the unknown functions and their first derivatives. The arguments apply the fixed point theory, Green’s functions technique, \(L^{1}\)-Carathéodory functions and sequences and Schauder’s fixed point theorem. The method is based on Carathéodory concept of functions and sequences, together with the equiconvergence on infinity and on each impulsive moment, and it allows to consider coupled fully nonlinearities and very general impulsive functions.

Keywords

Coupled systems \(L^{1}\)-Carathéodory functions Green’s functions Equiconvergence at infinity and at the impulsive points Schauder’s fixed-point theorem Problems on the half-line 

Mathematics Subject Classification

34B15 34B27 34L30 92B05 

References

  1. 1.
    Bai, C., Fang, J.: On positive solutions of boundary value problems for second-order functional differentia equations on infinite intervals. J. Math. Anal. Appl. 282, 711–731 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Dishliev, A., Dishlieva, K., Nenov, S.: Specific Asymptotic Properties of the Solutions of Impulsive Differential Equations. Methods and Applications. Academic Publications Ltd, USA (2012)Google Scholar
  3. 3.
    Eloe, P., Kaufmann, E., Tisdell, C.: Multiple solutions of a boundary value problem on an unbounded domain. Dyn. Syst. Appl. 15(1), 53–63 (2006)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Guo, D.: Boundary value problems for impulsive integro-differential equation on unbounded domains in a Banach space. Appl. Math. Comput. 99, 1–15 (1999)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Guo, D.: A class of second-order impulsive integro-differential equations on unbounded domain in Banach space. Appl. Math. Comput. 125, 59–77 (2002)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Hao, Z., Liang, J., Xiao, T.: Positive solutions of operator equations on half-line. J. Math. Anal. Appl. 314, 423–435 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kaufmann, E., Kosmatov, N., Raffoul, Y.: A second-order boundary value problem with impulsive effects on an unbounded domain. Nonlinear Anal. 69, 2924–2929 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lee, E., Lee, Y.-H.: Multiple positive solutions of a singular gelfand type problem for second-order impulsive differential systems. Math. Comput. Modell. 40, 307–328 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lee, E., Lee, Y.-H.: Multiple positive solutions of singular two point boundary value problems for second order impulsive differential equations. Appl. Math. Comput. 158, 745–759 (2004)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Lee, Y.-H., Liu, X.: Study of singular boundary value problems for second order impulsive differential equations. J. Math. Anal. Appl. 331, 159–176 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lian, H., Ge, W.: Solvability for second-order three-point boundary value problems on a half-line. Appl. Math. Lett. 19, 1000–1006 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Liu, Y.: Existence of solutions of boundary value problems for coupled singular differential equations on whole line with impulses. Mediterr. J. Math. 12, 697–716 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Liu, L., Hao, X., Wu, Y.: Unbounded solutions of second-order multipoint boundary value problem on the half- line. Bound. Value Probl. 2010, 15 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Meehan, M., O’Regan, D.: Multiple nonnegative solutions of nonlinear integral equations on compact and semi- infinite intervals. Appl. Anal. 74, 413–427 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Minhós, F.: Impulsive problems on the half-line with infinite impulse moments. Lith. Math. J. 57, 69 (2017).  https://doi.org/10.1007/s10986-017-9344-5 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Minhós, F., Carapinha, R.: Half-linear impulsives problems for classical singular \(\phi \)-Laplacian with generalized impulsive conditions (to appear)Google Scholar
  17. 17.
    Minhós, F., Sousa, R.: Solvability of second order coupled systems on the half-line (to appear)Google Scholar
  18. 18.
    Nagle, R., Saff, E., Snider, A.: Fundamentals of Differential Equations, 8th edn. Pearson Education Limited, London (2014)zbMATHGoogle Scholar
  19. 19.
    Palamides, P., Galanis, G.: Positive, unbounded and monotone solutions of the singular second Painlev equation on the half-line. Nonlinear Anal. 57, 401–419 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Pang, H., Lu, M., Cai, C.: The method of upper and lower solutions to impulsive differential equations with integral boundary conditions. Ad. Differ. Equ. 2014, 183 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Wang, Y., Liu, L., Wu, Y.: Positive solutions of singular boundary value problems on the half-line. Appl. Math. Comput. 197, 789–796 (2008)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Wang, W.-X., Zhang, L., Liang, Z.: Initial value problems for nonlinear impulsive integro-differential equations in Banach space. J. Math. Anal. Appl. 320, 510–527 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Zeidler, E.: Nonlinear Functional Analysis and Its Applications: Fixed-Point Theorems. Springer, New York (1986)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Matemática, Escola de Ciências e Tecnologia, Centro de Investigação em Matemática e Aplicaçõ es (CIMA), Instituto de Investigação e Formação AvançadaUniversidade de ÉvoraÉvoraPortugal
  2. 2.Faculdade de Ciências e Tecnologia, Núcleo de Matemática e Aplicações (NUMAT)Universidade de Cabo VerdePraiaCabo Verde

Personalised recommendations