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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.

34A08Fractional differential equations
34B37Boundary value problems for ODE with impulses
34A45Theoretical approximation of solutions of ODE
Full Text: DOI
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