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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

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.
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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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