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Existence of solutions of some impulsive fractional integrodifferential equations. (English) Zbl 1252.45004

This paper deals with the existence of solutions of some impulsive fractional integrodifferential equations with finite number of impulses: \[ D^{\alpha}u(t) = f(t, u(t), \int_{t_0}^t a(t -s)g(t, u(t))dt);\quad t \in [t_0, a],\;t \neq t_k,\;k = 1, \dots, m, \] with the initial condition \[ D^{\alpha-1}u(t_0) = u_0;\quad (t - t_0)^{1-\alpha}u(t)|_{t=t_0} = \frac{u_0}{\Gamma(\alpha)}, \] and subject to the impulsive conditions \[ D^{\alpha-1}(u(t_k^+) - u(t_k^-)) = I_k(t);\quad t = t_k,\;k = 1, 2, \dots, m; \]
\[ (t - t_k)^{1-\alpha}u(t^+)|_{t=t_k} = \frac{I_k(t_k)}{\Gamma(\alpha)},\quad k = 1, \dots \] The existence and uniqueness of local solutions and global solutions are established via the Schauder’s fixed point theorem and the Brouwer’s fixed point theorem, respectively.

MSC:

45J05 Integro-ordinary differential equations
45G10 Other nonlinear integral equations
26A33 Fractional derivatives and integrals
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