摘要：Based upon the well-known coincidence degree theory of Mawhin, we obtain some new existence results for a class of nonlocal fractional boundary value problems at resonance given by { D 0 + α u ( t ) = f ( t , u ( t ) , D 0 + α − 1 u ( t ) , D 0 + α − 2 u ( t ) ) , t ∈ ( 0 , 1 ) , I 0 + 3 − α u ( 0 ) = u ′ ( 0 ) = 0 , D 0 + β u ( 1 ) = ∫ 0 1 D 0 + β u ( t ) d A ( t ) , $$ \textstyle\begin{cases} D_{0+}^{\alpha}u(t)=f(t,u(t),D_{0+}^{\alpha-1}u(t),D_{0+}^{\alpha-2}u(t)),\quad t\in(0,1), \\ I_{0^{+}}^{3-\alpha}u ( 0 ) =u' ( 0 ) =0,\quad\quad D_{0+} ^{\beta}u(1)=\int_(^)D_{0+}^{\beta}u(t)\,dA(t), \end{cases} $$ where α, β are real numbers with 2 < α ≤ 3 $2<\alpha\leq3$ , 0 < β ≤ 1 $0<\beta\leq1$ , D 0 + α $D_{0+}^{\alpha}$ and I 0 + α $I_{0+}^{\alpha}$ respectively denote Riemann-Liouville derivative and integral of order α, f : [ 0 , 1 ] × R 3 → R $f:[0,1]\times\mathbb{R}^"\rightarrow\mathbb{R}$ satisfies the Carathéodory conditions, ∫ 0 1 D 0 + β u ( t ) d A ( t ) $\int_(^)D_{0+}^{\beta}u(t)\,dA(t)$ is a Riemann-Stieltjes integral with ∫ 0 1 t α − β − 1 d A ( t ) = 1 $\int_(^)t^{\alpha-\beta-1}\,dA(t)=1$ . We also present an example to demonstrate the application of the main results.
关键词：fractional differential equation ; resonance ; Riemann-Stieltjes integral ; coincidence degree theory
