### AIMS Mathematics

2019, Issue 6: 1610-1633. doi: 10.3934/math.2019.6.1610
Research article Special Issues

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

• 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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