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Multiplicity results for a class of fractional boundary value problems

Tom 109 / 2013

Annales Polonici Mathematici 109 (2013), 59-73 MSC: 35A15, 34A08. DOI: 10.4064/ap109-1-5

Streszczenie

We prove the existence of at least three solutions to the following fractional boundary value problem: $$\begin{cases} - \frac{d}{d t}\left(\frac{1}{2} \,{}_0D_t^{- \sigma} (u' (t)) + \frac{1}{2} \,{}_tD_T^{- \sigma} (u' (t))\right) - \lambda \beta (t) f (u (t)) - \mu \gamma (t) g (u (t)) = 0, \quad\textrm{a.e.}\ t \in [0, T],\\ u (0) = u (T) = 0, \end{cases}$$ where ${}_0D_t^{- \sigma}$ and ${}_tD_T^{- \sigma}$ are the left and right Riemann–Liouville fractional integrals of order $0 \leq \sigma < 1$ respectively. The approach is based on a recent three critical points theorem of Ricceri [B. Ricceri, A further refinement of a three critical points theorem, Nonlinear Anal. 74 (2011), 7446–7454].