Anti-periodic Boundary Value Problems of <i>&phi;</i>-Laplacian Impulsive Differential Equations

Xiufeng Guo; Yuan Gu

doi:doi:10.11648/j.acm.20160502.19

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

Xiufeng Guo, Yuan Gu

Published in Applied and Computational Mathematics (Volume 5, Issue 2)

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.

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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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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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TY  - JOUR
T1  - Anti-periodic Boundary Value Problems of φ-Laplacian Impulsive Differential Equations
AU  - Xiufeng Guo
AU  - Yuan Gu
Y1  - 2016/05/30
PY  - 2016
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DO  - 10.11648/j.acm.20160502.19
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 91
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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

Xiufeng Guo

College of Sciences, Hezhou University, Hezhou, Guangxi, China
Yuan Gu

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

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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

Copy | Download

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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
T1  - Anti-periodic Boundary Value Problems of φ-Laplacian Impulsive Differential Equations
AU  - Xiufeng Guo
AU  - Yuan Gu
Y1  - 2016/05/30
PY  - 2016
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
SP  - 91
EP  - 96
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20160502.19
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  -

Copy | Download