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Anti-periodic Boundary Value Problems of φ-Laplacian Impulsive Differential Equations

Received: 3 May 2016     Accepted: 13 May 2016     Published: 30 May 2016
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Abstract

This paper concerned with the existence of solutions of anti-periodic boundary value problems for impulsive differential equations with φ-Laplacian operator. Firstly, the definition a pair of coupled lower and upper solutions of the problem is introduced. Then, under the approach of coupled upper and lower solutions together with Nagumo condition, we prove that there exists at least one solution of anti-periodic boundary value problems for impulsive differential equations with φ-Laplacian operator.

Published in Applied and Computational Mathematics (Volume 5, Issue 2)
DOI 10.11648/j.acm.20160502.19
Page(s) 91-96
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Anti-periodic Boundary Value Problems, Impulsive Differential Equations, φ-Laplacian Operator, Coupled Lower and Upper Solutions

References
[1] D. Bainov, V. Covachev, Impulsive Differential Equations With a Small Parameter, World Scientific, Singapore, 1994.
[2] M. Benchohra, J. Henderson, S. K. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publishing Corparation, New York, 2006.
[3] V. Lakshmikantham, D. D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
[4] A. Cabada, J. Tomecek, Extremal solutions for nonlinear functional φ-Laplacian impulsive equations, Nonlinear Anal. 67 (2007) 827-841.
[5] W. Ding, M. A. Han, J. Mi, Periodic boundary value problem for the second order impulsive functional equations, Comput. Math. Appl. 50 (2005) 491-507.
[6] A. Cabada, B. Thompson, Nonlinear second-order equations with functional implicit impulses and nonlinear functional boundary conditions, Nonlinear Anal. 74 (2011) 7198-7209.
[7] E. N. Dancer. On the Dirichlet problem for weakly non-linear elliptic partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A, 76 (1977) 283-300.
[8] A. Cabada, An overview of the lower and upper solutions method with nonlinear boundary value conditions, Bound. Value Probl. Art. ID 893753 (2011) 1-18.
[9] H. Kleinert, A. Chervyakov, Functional determinants from Wronski Green function, J. Math. Phys. 40 (1999) 6044-6051.
[10] M. Wang, A. Cabada, J. J. Nieto, Monotone method for nonlinear second order periodic boundary value problems with Carath′eodory functions, Ann. Polon. Math. 58(3) (1993) 221-235.
[11] A. Cabada, D. R. Vivero, Existence and uniqueness of solutions of higher-order antiperiodic dynamic equations, Adv. Difference Equ. 4 (2004) 291-310.
[12] X. Guo, L. Lu, Z. Liu, BVPs for higher-order integro-differential equations with Φ-Laplacian and functional boundary conditions, Adv. Differ. Equa. 2014:285 (2014) 1-13.
[13] P. Jebelean, C. Şerban, Boundary value problems for discontinuous perturbations of singular Φ-Laplacian operator. J. Math. Anal. Appl. 431(1) (2015) 662-681.
[14] H. Okochi, On the existence of periodic solutions to nonlinear abstract parabolic equations, J. Math. Soc. Japan 40(3) (1988) 541-553.
[15] F. Delvos, L. Knoche, Lacunary interpolation by anti-periodic trigonometric polynomials, BIT 39 (1999) 439-450.
[16] P. Wang, W. Wang, Anti-periodic boundary value problem for first order impulsive delay difference equations, Adv. Differ. Equ. 2015(1) (2015) 1-13.
[17] T. Zhang, Y. Li, Global exponential stability and existence of anti-periodic solutions to impulsive Cohen-Grossberg neural networks on time scales, Topol. Methods Nonlinear Anal. 45(2) (2015) 363-384.
Cite This Article
  • APA Style

    Xiufeng Guo, Yuan Gu. (2016). Anti-periodic Boundary Value Problems of φ-Laplacian Impulsive Differential Equations. Applied and Computational Mathematics, 5(2), 91-96. https://doi.org/10.11648/j.acm.20160502.19

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    ACS Style

    Xiufeng Guo; Yuan Gu. Anti-periodic Boundary Value Problems of φ-Laplacian Impulsive Differential Equations. Appl. Comput. Math. 2016, 5(2), 91-96. doi: 10.11648/j.acm.20160502.19

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    AMA Style

    Xiufeng Guo, Yuan Gu. Anti-periodic Boundary Value Problems of φ-Laplacian Impulsive Differential Equations. Appl Comput Math. 2016;5(2):91-96. doi: 10.11648/j.acm.20160502.19

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  • @article{10.11648/j.acm.20160502.19,
      author = {Xiufeng Guo and Yuan Gu},
      title = {Anti-periodic Boundary Value Problems of φ-Laplacian Impulsive Differential Equations},
      journal = {Applied and Computational Mathematics},
      volume = {5},
      number = {2},
      pages = {91-96},
      doi = {10.11648/j.acm.20160502.19},
      url = {https://doi.org/10.11648/j.acm.20160502.19},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20160502.19},
      abstract = {This paper concerned with the existence of solutions of anti-periodic boundary value problems for impulsive differential equations with φ-Laplacian operator. Firstly, the definition a pair of coupled lower and upper solutions of the problem is introduced. Then, under the approach of coupled upper and lower solutions together with Nagumo condition, we prove that there exists at least one solution of anti-periodic boundary value problems for impulsive differential equations with φ-Laplacian operator.},
     year = {2016}
    }
    

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    T1  - Anti-periodic Boundary Value Problems of φ-Laplacian Impulsive Differential Equations
    AU  - Xiufeng Guo
    AU  - Yuan Gu
    Y1  - 2016/05/30
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    N1  - https://doi.org/10.11648/j.acm.20160502.19
    DO  - 10.11648/j.acm.20160502.19
    T2  - Applied and Computational Mathematics
    JF  - Applied and Computational Mathematics
    JO  - Applied and Computational Mathematics
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    PB  - Science Publishing Group
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    AB  - This paper concerned with the existence of solutions of anti-periodic boundary value problems for impulsive differential equations with φ-Laplacian operator. Firstly, the definition a pair of coupled lower and upper solutions of the problem is introduced. Then, under the approach of coupled upper and lower solutions together with Nagumo condition, we prove that there exists at least one solution of anti-periodic boundary value problems for impulsive differential equations with φ-Laplacian operator.
    VL  - 5
    IS  - 2
    ER  - 

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Author Information
  • College of Sciences, Hezhou University, Hezhou, Guangxi, China

  • College of Sciences, Hezhou University, Hezhou, Guangxi, China

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