Multiplicity results for a class of fractional boundary value problems

Nemat Nyamoradi¹

¹ Department of Mathematics Faculty of Sciences Razi University 67149 Kermanshah, Iran

Annales Polonici Mathematici, Tome 109 (2013) no. 1, pp. 59-73.

Résumé

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].

Détail

DOI : 10.4064/ap109-1-5

Keywords: prove existence least three solutions following fractional boundary value problem begin cases frac frac sigma frac sigma right lambda beta gamma quad textrm end cases where sigma sigma right riemann liouville fractional integrals order leq sigma respectively approach based recent three critical points theorem ricceri ricceri further refinement three critical points theorem nonlinear anal