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Impulsive periodic boundary value problems for fractional differential equation involving Riemann-Liouville sequential fractional derivative. (English) Zbl 1234.34005

The author considers nonlinear impulsive fractional differential equations involving Riemann-Liouville fractional derivatives with periodic boundary conditions: \[ \mathcal{D}^{2\alpha}u(t)=f(t,u,\mathcal{D}^{\alpha}u),\;t\in(0,1]\setminus\{t_1,\dots,t_m\},\;0<\alpha\leq 1, \]
\[ \lim_{t\rightarrow 0^+}t^{1-\alpha}u(t)=u(1),\;\lim_{t\rightarrow 0^+}t^{1-\alpha}\mathcal{D}^{\alpha}u(t)=\mathcal{D}^{\alpha}u(1), \]
\[ \lim_{t\rightarrow t_j^+}(t-t_j)^{1-\alpha}(u(t)-u(t_j))=I_j(u(t_j)), \]
\[ \lim_{t\rightarrow t_j^+}(t-t_j)^{1-\alpha}(\mathcal{D}^{\alpha}u(t)-\mathcal{D}^{\alpha}u(t_j))=\overline{I}_j(u(t_j)). \] The author establishes the existence of solutions to the periodic boundary value problem using upper and lower solutions and the monotone iterative method.

MSC:

34A08 Fractional ordinary differential equations
34B37 Boundary value problems with impulses for ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
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