https://doi.org/10.1140/epjst/e2013-01980-2
Regular Article
Infinitely many solutions for a perturbed nonlinear fractional boundary value problems depending on two parameters
1 Department of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah, Iran
2 School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, PR China
a e-mail: nyamoradi@razi.ac.ir; neamat80@yahoo.com
b e-mail: yzhou@xtu.edu.cn
Received: 3 June 2013
Revised: 6 August 2013
Published online: 7 October 2013
In this paper we prove the existence and multiplicity of (weak) solutions for the following fractional boundary value problem:
![\begin{equation*} \left\{\begin{array}{@{}l@{}} -\frac{d}{d t} \Big(\frac{1}{2} {}_0D_t^{\alpha - 1} ({}_0^c D_t^\alpha u (t) ) - \frac{1}{2} {}_tD_T^{\alpha - 1} ({}_t^c D_T^\alpha u (t)) \Big) + A(t) u (t) \\[6pt] \hspace{2cm} = \lambda \nabla F (t, u (t)) + \mu \nabla G (t, u (t)),\\[6pt] u (0) = u (T) = 0, \end{array}\right. \end{equation*}](/articles/epjst/abs/2013/14/epjst2228021/epjst2228021-eq1.jpg)
where α ∊ (½,1], 0Dtα−1 and tDTα−1 are the left and right Riemann-Liouville fractional integrals of order 1 − α respectively, λ,μ ∊ [0,+∞), T > 0, F,G ∊ C([0,T] × RN;R)\{0} and A = (aij(t))N×N is symmetric. Our approach is based on variational methods.
© EDP Sciences, Springer-Verlag, 2013
