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AIMS Mathematics

2019, Volume 4, Issue 6: 1610-1633. doi: 10.3934/math.2019.6.1610
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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
  • Received: 10 July 2019 Accepted: 09 September 2019 Published: 14 October 2019

  • MSC : 34B15, 34B37, 34C25

  • 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:$ (\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), $ where zhongwenzy $< t_{1} < t_{2} < \cdots < t_{m} < b$, $f: [0, b]\times \mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ 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.

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

    Related Papers:

  • 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:$ (\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), $ where zhongwenzy $< t_{1} < t_{2} < \cdots < t_{m} < b$, $f: [0, b]\times \mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ 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.


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    [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.
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