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The existence of solutions to boundary value problems of fractional differential equations at resonance. (English) Zbl 1236.34006

The author obtains a solution of the Riemann-Liouville fractional differential equation \[ D_{0+}^{\alpha}u(t) = f(t,u(t), D_{0+}^{\alpha-1}u(t)) \quad \mathrm{a. \, e.} \quad t \in (0,1) \] satisfying the non-local conditions \[ u(0) = 0, \quad D_{0+}^{\alpha-1}u(0) = \sum_{i=1}^m a_i D_{0+}^{\alpha-1}u(\xi_i), \quad D_{0+}^{\alpha-2}u(1) = \sum_{i=1}^n b_i D_{0+}^{\alpha-2}u(\eta_i). \] It is assumed that \(2 < \alpha < 3\), \(0 < \xi_1 < \dots < \xi_m < 1\), \(0 < \eta_1 < \dots < \eta_n < 1\), \( \sum_{i=1}^m a_i = 1\), and \(\sum_{i=1}^n b_i \eta_i =1\). The existence of a solution at resonance follows from the coincidence degree theorem of Mawhin.

MSC:

34A08 Fractional ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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