An Existence Result for Impulsive Multi-point Boundary Value Systems Using a Local Minimization Principle

  • Ghasem A. Afrouzi
  • Martin Bohner
  • Giuseppe Caristi
  • Shapour Heidarkhani
  • Shahin Moradi


In this article, multi-point boundary value systems with impulsive effects are considered. Existence of at least one classical solution is investigated. The basis of the approach is an application of certain variational methods for smooth functionals, which are defined on reflexive Banach spaces. Examples are provided in order to illustrate how the presented results can be applied.


Existence result Multi-point boundary value problems Minimization principle Classical solution Impulsive effects Critical point theory Variational methods 

Mathematics Subject Classification

34B10 34B15 34A37 



The authors wish to thank all involved Editors and Referees.


  1. 1.
    Moshinsky, M.: On one-dimensional boundary value problems of a discontinuous nature. Bol. Soc. Mat. Mexicana 4, 1–25 (1947)MathSciNetGoogle Scholar
  2. 2.
    Timoshenko, S.P., Gere, J.M.: Theory of Elastic Stability, Engineering Societies Monographs, 2nd edn. McGraw-Hill Book Co., Inc., New York (1961)Google Scholar
  3. 3.
    Du, Z., Kong, L.: Existence of three solutions for systems of multi-point boundary value problems. Electron. J. Qual. Theory Differ. Equ. Special Edition I(10), 1–17 (2009).
  4. 4.
    Feng, H.Y., Ge, W.G.: Existence of three positive solutions for \(M\)-point boundary-value problem with one-dimensional \(P\)-Laplacian. Taiwanese J. Math. 14(2), 647–665 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Graef, J.R., Heidarkhani, S., Kong, L.: A critical points approach to multiplicity results for multi-point boundary value problems. Appl. Anal. 90(12), 1909–1925 (2011). MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Graef, J.R., Heidarkhani, S., Kong, L.: Existence of nontrivial solutions to systems of multi-point boundary value problems. Discrete Contin. Dyn. Syst. 2013(Supplement), 273–281 (2013)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Graef, J.R., Heidarkhani, S., Kong, L.: Infinitely many solutions for systems of multi-point boundary value problems using variational methods. Topol. Methods Nonlinear Anal. 42(1), 105–118 (2013)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Ma, D.X., Chen, X.G.: Existence and iteration of positive solution for a multi-point boundary value problem with a \(p\)-Laplacian operator. Port. Math. 65(1), 67–80 (2008). MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ma, R.: Existence of positive solutions for superlinear semipositone \(m\)-point boundary-value problems. Proc. Edinb. Math. Soc. (2) 46(2), 279–292 (2003). MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Zhang, X., Liu, L.: Positive solutions for \(m\)-point boundary-value problems with one-dimensional \(p\)-Laplacian. J. Appl. Math. Comput. 37(1–2), 523–531 (2011). MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bohner, M., Heidarkhani, S., Salari, A., Caristi, G.: Existence of three solutions for impulsive multi-point boundary value problems. Opuscu. Math. 37(3), 353–379 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Baĭnov, D.D., Simeonov, P.S.: Systems with Impulse Effect. Ellis Horwood Series: Mathematics and its Applications (Stability, Theory and Applications). Wiley, New York (1989)Google Scholar
  13. 13.
    Benchohra, M., Henderson, J., Ntouyas, S.: Impulsive Differential Equations and Inclusions, Contemporary Mathematics and Its Applications, vol. 2. Hindawi Publishing Corporation, New York (2006). CrossRefzbMATHGoogle Scholar
  14. 14.
    Lakshmikantham, V., Baĭnov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics, vol. 6. World Scientific Publishing Co., Inc., Teaneck (1989). CrossRefGoogle Scholar
  15. 15.
    Samoĭlenko, A.M., Perestyuk, N.A.: Impulsive Differential Equations, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, vol. 14. World Scientific Publishing Co., Inc., River Edge (1995). (With a preface by Yu. A. Mitropol\(^{\prime }\)skiĭ and a supplement by S. I. Trofimchuk, Translated from the Russian by Y. Chapovsky)
  16. 16.
    Bai, L., Dai, B.: Three solutions for a \(p\)-Laplacian boundary value problem with impulsive effects. Appl. Math. Comput. 217(24), 9895–9904 (2011). MathSciNetzbMATHGoogle Scholar
  17. 17.
    Heidarkhani, S., Ferrara, M., Salari, A.: Infinitely many periodic solutions for a class of perturbed second-order differential equations with impulses. Acta Appl. Math. 139, 81–94 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Tian, Y., Ge, W.: Applications of variational methods to boundary-value problem for impulsive differential equations. Proc. Edinb. Math. Soc. (2) 51(2), 509–527 (2008). MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Tian, Y., Ge, W.: Variational methods to Sturm–Liouville boundary value problem for impulsive differential equations. Nonlinear Anal. 72(1), 277–287 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Xiao, J., Nieto, J.J., Luo, Z.: Multiplicity of solutions for nonlinear second order impulsive differential equations with linear derivative dependence via variational methods. Commun. Nonlinear Sci. Numer. Simul. 17(1), 426–432 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Feng, M., Pang, H.: A class of three-point boundary-value problems for second-order impulsive integro-differential equations in Banach spaces. Nonlinear Anal. 70(1), 64–82 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Feng, M., Xie, D.: Multiple positive solutions of multi-point boundary value problem for second-order impulsive differential equations. J. Comput. Appl. Math. 223(1), 438–448 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Liu, B., Yu, J.: Existence of solution of \(m\)-point boundary value problems of second-order differential systems with impulses. Appl. Math. Comput. 125(2–3), 155–175 (2002). MathSciNetzbMATHGoogle Scholar
  24. 24.
    Thaiprayoon, C., Samana, D., Tariboon, J.: Multi-point boundary value problem for first order impulsive integro-differential equations with multi-point jump conditions. Bound. Value Probl. 2012, 38 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Breckner, B.E., Varga, C.: Multiple solutions of Dirichlet problems on the Sierpinski gasket. J. Optim. Theory Appl. 167(3), 842–861 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lisei, H., Varga, C.: A multiplicity result for a class of elliptic problems on a compactRiemannian manifold. J. Optim. Theory Appl. 167(3), 912–927 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Papageorgiou, N.S., Rădulescu, V.D., Repovš, D.D.: Nonhomogeneous hemivariational inequalities with indefinite potential and Robin boundary condition. J. Optim. Theory Appl. 175(2), 293–323 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ricceri, B.: A general variational principle and some of its applications. Fixed point theory with applications in nonlinear analysis. J. Comput. Appl. Math. 113(1–2), 401–410 (2000). MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Bonanno, G., Molica Bisci, G.: Infinitely many solutions for a boundary value problem with discontinuous nonlinearities. Bound. Value Probl. pp. Art. ID 670,675, 20 (2009)Google Scholar
  30. 30.
    Afrouzi, G.A., Hadjian, A., Molica Bisci, G.: Some remarks for one-dimensional mean curvature problems through a local minimization principle. Adv. Nonlinear Anal. 2(4), 427–441 (2013)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Galewski, M., Molica Bisci, G.: Existence results for one-dimensional fractional equations. Math. Methods Appl. Sci. 39(6), 1480–1492 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Heidarkhani, S., Afrouzi, G.A., Ferrara, M., Caristi, G., Moradi, S.: Existence results for impulsive damped vibration systems. Bull. Malays. Math. Sci. Soc. (2016).
  33. 33.
    Heidarkhani, S., Afrouzi, G.A., Moradi, S., Caristi, G., Ge, B.: Existence of one weak solution for \(p(x)\)-biharmonic equations with Navier boundary conditions. Z. Angew. Math. Phys. 67(3), 73 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Heidarkhani, S., Zhou, Y., Caristi, G., Afrouzi, G.A., Moradi, S.: Existence results for fractional differential systems through a local minimization principle. Comput. Math. Appl. (2016).
  35. 35.
    Afrouzi, G.A., Heidarkhani, S., Moradi, S.: Existence of weak solutions for three-point boundary-value problems of Kirchhoff-type. Electron. J. Differ. Equ. 2016(234), 1–13 (2016).
  36. 36.
    Zeidler, E.: Nonlinear Functional Analysis and Its Applications. II/A. Springer, New York (1990). (Linear monotone operators, Translated from the German by the author and Leo F. Boron)

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of MazandaranBabolsarIran
  2. 2.Missouri S&TRollaUSA
  3. 3.University of MessinaMessinaItaly
  4. 4.Razi UniversityKermanshahIran
  5. 5.Young Researchers and Elite ClubIslamic Azad UniversityKermanshahIran

Personalised recommendations