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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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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


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