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Zbl 1234.34005
Bai, Chuanzhi
Impulsive periodic boundary value problems for fractional differential equation involving Riemann-Liouville sequential fractional derivative. (English)
[J] J. Math. Anal. Appl. 384, No. 2, 211-231 (2011). ISSN 0022-247X

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.
[K. Rajendra Prasad (Visakhapatnam)]

MSC 2000:: *34A08
34B37 Boundary value problems with impulses
34A45 Theoretical approximation of solutions of ODE

Keywords: fractional order; impulsive boundary value problems; periodic boundary conditions; upper and lower solutions; iterative method

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