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Dirichlet Problem via Lower and Upper Functions

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State-Dependent Impulses

Part of the book series: Atlantis Briefs in Differential Equations ((ABDE,volume 6))

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Abstract

The chapter continues the study of the second order Dirichlet boundary value problem

$$\begin{aligned} z''(t)&= f(t,z(t)) \quad \text {for a.e. } t \in [0,T]\subset \mathbb {R}, \nonumber \\ z'(t+) - z'(t-)&= M_i(t,z(t)), \quad t = \gamma _i(z(t)), \quad i = 1,\ldots ,p,\nonumber \\&z(0) = 0, \quad z(T) = 0, \nonumber \end{aligned}$$

but now, in contrast to Chap. 6, the problem is subject to p state-dependent impulse conditions, where \(p \in \mathbb {N}\). The solvability of this problem is proved under the assumption that there exists a well-ordered couple of lower and upper functions to the corresponding Dirichlet problem without impulses. The main result is contained in the existence theorem which can be applied to problems that are not covered by the existence theorem of Chap. 6 even in the case \(p=1\).

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References

  1. Bainov, D., Hristova, S., Hu, S., Lakshmikantham, V.: Periodic boundary value problems for systems of first order impulsive differential equations. Differ. Integral Equ. 2(1), 37–43 (1989)

    MATH  MathSciNet  Google Scholar 

  2. Bajo, I., Liz, E.: Periodic boundary value problem for first order differential equations with impulses at variable times. J. Math. Anal. Appl. 204(1), 65–73 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. Benbouziane, Z., Boucherif, A., Bouguima, S.: Existence result for impulsive third order periodic boundary value problems. Appl. Math. Comput. 206, 728–737 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Benchohra, M., Graef, J., Ntouyas, S., Ouahab, A.: Upper and lower solutions method for impulsive differential inclusions with nonlinear boundary conditions and variable times. Dyn. Contin., Discret. Impuls. Syst. Series A: Math. Anal. 12(3–4), 383–396 (2005)

    MATH  MathSciNet  Google Scholar 

  5. Chen, P., Tang, X.: Existence and multiplicity of solutions for second-order impulsive differential equations with Dirichlet problems. Appl. Math. Comput. 218, 11775–11789 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chen, P., Tang, X.: New existence and multiplicity of solutions for some Dirichlet problems with impulsive effects. Math. Comput. Model. 55, 723–739 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Doddaballapur, V., Eloe, P., Zhang, Y.: Quadratic convergence of approximate solutions of two-point boundary value problems with impulse. Electron. J. Differ. Equ. Conf. 1, 81–95 (1997)

    MathSciNet  Google Scholar 

  8. Erbe, L., Liu, X.: Existence results for boundary value problems of second order impulsive differential equations. J. Math. Anal. Appl. 149(1), 56–69 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hu, S., Lakshmikantham, V.: Periodic boundary value problems for second order impulsive differential systems. Nonlinear Anal., Theory Methods Appl. 13(1), 75–85 (1989)

    Google Scholar 

  10. Lin, X., Jiang, D.: Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations. J. Math. Anal. Appl. 321, 501–514 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Liu, Z., Chen, H., Zhou, T.: Variational methods to the second-order impulsive differential equation with Dirichlet boundary value problem. Comput. Math. Appl. 61, 1687–1699 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. Rach\(\mathring{\rm {u}}\)nková, I., Tomeček, J.: Second order BVPs with state dependent impulses via lower and upper functions. Cent. Eur. J. Math. 12(1), 128–140 (2014)

    Google Scholar 

  13. Rach\(\mathring{\rm {u}}\)nková, I., Tvrdý, M.: Nonmonotone impulse effects in second order periodic boundary value problems. Funct. Differ. Equ. 9, 471–498 (2002)

    Google Scholar 

  14. Sun, J., Chen, H.: Multiplicity of solutions for a class of impulsive differential equations with Dirichlet boundary conditions via variant fountain theorems. Nonlinear Anal., Real World Appl. 11, 4062–4071 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Wang, W., Yang, X.: Multiple solutions of boundary-value problems for impulsive differential equations. Math. Meth. Appl. Sci. 34, 1649–1657 (2011)

    Article  MATH  Google Scholar 

  16. Xiao, J., Nieto, J.: Variational approach to some damped Dirichlet nonlinear impulsive differential equations. J. Frankl. Inst. 348(2), 369–377 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  17. Zhang, Z., Yuan, R.: An application of variational methods to Dirichlet boundary value problem with impulses. Nonlinear Anal. 11(1), 155–162 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  18. Zhou, J., Li, Y.: Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects. Nonlinear Anal. 71(7–8), 2856–2865 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Science, Palacký University, 17. listopadu 12, Olomouc, Czech Republic

    Irena Rachůnková & Jan Tomeček

Authors
  1. Irena Rachůnková
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  2. Jan Tomeček
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Correspondence to Irena Rachůnková .

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Rachůnková, I., Tomeček, J. (2015). Dirichlet Problem via Lower and Upper Functions. In: State-Dependent Impulses. Atlantis Briefs in Differential Equations, vol 6. Atlantis Press, Paris. https://doi.org/10.2991/978-94-6239-127-7_7

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