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This article concerns the existence of solutions to the fractional boundary-value problem $$displaylines{ -frac{d}{dt} ig(frac{1}{2} {}_0D_t^{-eta}+ frac{1}{2}{}_tD_{T}^{-eta}ig)u'(t)=lambda u(t)+abla F(t,u(t)),quad hbox{a.e. } tin[0,T], cr u(0)=0,quad u(T)=0. }$$ First for the eigenvalue problem associated with it, we prove that there is a sequence of positive and increasing real eigenvalues; a characterization of the first eigenvalue is also given. Then under different assumptions on the nonlinearity F(t,u), we show the existence of weak solutions of the problem when $lambda$ lies in various intervals. Our main tools are variational methods and critical point theorems.