Solvability of fractional three-point boundary value problems with nonlinear growth. (English) Zbl 1235.34007

Summary: We consider the following fractional boundary value problem: \[ \begin{gathered} D^\alpha_{0+}u(t)=f(t,u(t),D^{\alpha-1}_{0+}u(t)),\;0<t<1,\\ u(0)=0,\quad u(1)=\sigma u(\eta),\end{gathered} \] where \(1<\alpha\leq 2\) is a real number, \(D^\alpha_{0+}\) is the standard Riemann-Liouville derivative, \(f:[0,1]\times\mathbb{R}^2\to\mathbb{R}\) is continuous and \(\sigma\in(0,\infty)\), \(\eta\in(0,1)\) are given constants such that \(\sigma\eta^{\alpha-1}=1\).
By using the coincidence degree theory, we present an existence result at the resonance case.


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