Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Existence results for φ-Laplacian impulsive differential equations with periodic conditions

1 Department of Mathematics, Baylor University Waco, Texas 76798-7328 USA
2 Laboratory of Mathematics, Sidi-Bel-Abbès University, P. O. Box 89, 22000 Sidi-Bel-Abbès, Algeria
3 Department of Mathematics and Computer Science University of Ahmed Draia Adrar National Road No. 06, 01000, Adrar, Algeria
4 Ecole Superieur en Informatique, Sidi Bel-Abbes, Algeria

Special Issues: Initial and Boundary Value Problems for Differential Equations

Based on a Manasevich and Mawhin continuation theorem and some analysis skills we obtain sufficient conditions for existence results for φ-Laplacian nonlinear impulsive differential equations with periodic boundary conditions: \begin{gather*} (\phi(y'))' = f(t,y(t),y'(t)), \quad\text{a.e. } t\in [0,b],\\ y(t^+_{k})-y(t^-_k)=I_{k}(y(t_{k}^{-})), \quad k=1,\dots,m,\\ y'(t^+_{k})-y'(t^-_k)=\overline{I}_{k}(y(t_{k}^{-})), \quad k=1,\dots,m,\\ y(0)=y(b),\quad y'(0)=y'(b), \end{gather*} where $0<t_{1}<t_{2}<\cdots<t_{m}<b$, $f: [0,b]\times \mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ is a Carathéodory function, $I_{k},\bar I_{k}\in C(\mathbb{R}^{n},\mathbb{R}^{n})$ and $\phi: \mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is a suitable monotone homeomorphism.
  Figure/Table
  Supplementary
  Article Metrics

Keywords φ-Laplacian; impulsive problem; quasi-linear operator; M-compact mapping; topological degree

Citation: Johnny Henderson, Abdelghani Ouahab, Samia Youcefi. Existence results for φ-Laplacian impulsive differential equations with periodic conditions. AIMS Mathematics, 2019, 4(6): 1610-1633. doi: 10.3934/math.2019.6.1610

References

  • 1. Z. Agur, L. Cojocaru, G. Mazaur, et al. Pulse mass measles vaccination across age cohorts, Proc. Nat. Acad. Sci. USA, 90 (1993), 11698-11702.    
  • 2. D. D. Bainov, P. S. Simeonov, Systems with Impulse Effect: Stability, Theory and Applications, New York: Halsted Press, 1989.
  • 3. M. Benchohra, J. Henderson, S. K. Ntouyas, Impulsive Differential Equations and Inclusions, New York: Hindawi Publishing Corporation, 2006.
  • 4. A. Benmezaï, S. Djebali, T. Moussaoui, Multiple positive solutions for φ-Laplacian BVPs, Panamer. Math. J., 17 (2007), 53-73.
  • 5. C. Bereanu, J. Mawhin, Non-homogeneous boundary value problems for some nonlinear equations with singular φ-Laplacian, J. Math. Anal. Appl., 352 (2009), 218-233.    
  • 6. C. Bereanu, J. Mawhin, Periodic solutions of nonlinear perturbations of φ-Laplacians with possibly bounded φ, Nonlinear Anal. Theor., 68 (2008), 1668-1681.    
  • 7. A. Capietto, J. Mawhin, F. Zanolin, Continuation theorems for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc., 329 (1992), 41-72.    
  • 8. S. Djebali, L. Gorniewicz, A. Ouahab, Existence and Structure of Solution Sets for Impulsive Differential Inclusions, Lecture Notes, Nicolaus Copernicus University, 13 (2012).
  • 9. S. Djebali, L. Gorniewicz, A. Ouahab, Solutions Sets for Differential Equations and Inclusions, Berlin: Walter de Gruyter, 2013.
  • 10. P. Fitzpatrick, M. Martelli, J. Mawhin, et al. Topological Methods for Ordinary Differential Equations, Springer-Verlag, 1991.
  • 11. R. E. Gaines, J. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Berlin: Springer-Verlag, 1977.
  • 12. W. Ge, J. Ren, An extension of Mawhin's continuation theorem and its application to boundary value problems with a p-Laplacian, Nonlinear Anal. Theor., 58 (2004), 477-488.    
  • 13. J. R. Graef, J. Henderson, A. Ouahab, Impulsive Differential Inclusions: A Fixed Pont Approach, Berlin: Walter de Gruyter, 2013.
  • 14. A. Halanay, D. Wexler, Teoria Calitativa a Systeme cu Impulduri, Editura Republicii Socialiste Romania, Bucharest, 1968.
  • 15. J. Henderson, A. Ouahab, S. Youcefi, Existence and topological structure of solution sets for φ-Laplacian impulsive differential equations, Electron. J. Differ. Eq., 56 (2012), 1-16.
  • 16. V. Lakshmikantham, D. Bainov, P. S. Simenov, Theory of Impulsive Differential Equations, Singapore: World Scientific, 1989.
  • 17. J. Mawhin, Periodic solutions of nonlinear functional differential equations, J. Differ. Eq., 10 (1971), 240-261.    
  • 18. R. Manasevich, J. Mawhin, Periodic solutions for nonlinear systems with p-Laplacian-like operators, J. Differ. Eq., 145 (1998), 367-393.    
  • 19. V. D. Milman, A. A. Myshkis, On the stability of motion in the presence of impulses (in Russian), Sib. Math. J., 1 (1960), 233-237.
  • 20. J. J. Nieto, D. O'Regan, Variational approach to impulsive differential equations, Nonlinear Anal. Real., 10 (2009), 680-690.    
  • 21. D. O'Regan, Y. J. Cho, Y. Q. Chen, Topological Degree Theory and Applications, Chapman and Hall, 2006.
  • 22. L. Pan, Existence of periodic solutions for second order delay differential equations with impulses, Electron. J. Differ. Eq., 37 (2011), 1-12.
  • 23. D. Qian, X. Li, Periodic solutions for ordinary differential equations with sublinear impulsive effects, J. Math. Anal. Appl., 303 (2005), 288-303.    
  • 24. I. Rachunkova, J. Stryja, Dirichlet problem with φ-Laplacian and mixed singularities, Nonlinear Oscil., 11 (2008), 80-96.    
  • 25. I. Rachunkova, M. Tvrdy, Second order periodic problem with φ-Laplacian and impulses, Nonlinear Anal. Theor., 63 (2005), 257-266.    
  • 26. I. Rachunkova, M. Tvrdy, Periodic problems with φ-Laplacian involving non-ordered lower and upper functions, Fixed Point Theory, 6 (2005), 99-112.
  • 27. I. Rachunkova, M. Tverdy, Existence result for impulsive second order periodic problems, Nonlinear Anal. Theor., 59 (2004), 133-146.    
  • 28. M. Samoilenko, N. Perestyuk, Impulsive Differential Equations, Singapore: World Scientific, 1995.
  • 29. N. A. Perestyuk, V. A. Plotnikov, A. M. Samoilenko, et al. Differential Equations with Impulse Effects. Multivalued Right-hand Sides with Discontinuities, Berlin: Walter de Gruyter, 2011.
  • 30. J. Sun, H. Chen, L. Yang, Existence and multiplicity of solutions for impulsive differential equation with two parameters via variational method, Nonlinear Anal. Theor., 73 (2010), 440-449.    
  • 31. J. Tomeček, Dirichlet boundary value problem for differential equation with φ-Laplacian and state-dependent impulses, Math. Slovaca, 67 (2017), 483-500.
  • 32. J. Zhen, M. Zhien, H. Maoan, The existence of periodic solutions of the n-species Lotka-Volterra competition systems with impulsive, Chaos, Solitons, Fractals, 22 (2004), 181-188.    
  • 33. Z. Zhitao, Existence of solutions for second order impulsive differential equations, Appl. Math. JCU, 12 (1997), 307-320.

 

Reader Comments

your name: *   your email: *  

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved