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Zbl 1236.34006
Jiang, Weihua
The existence of solutions to boundary value problems of fractional differential equations at resonance.
(English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 5, 1987-1994 (2011). ISSN 0362-546X

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.
[Nickolai Kosmatov (Little Rock)]
MSC 2000:
*34A08
34B10 Multipoint boundary value problems
34B15 Nonlinear boundary value problems of ODE
47N20 Appl. of operator theory to differential and integral equations

Keywords: resonance; Fredholm operator; fractional integral; fractional derivative

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