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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.

MSC:

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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