| In this paper, using techniques from fractional variational calculus and some critical point theorems,
we prove the existence of weak solution. Then, we deduce the existence of solution for the following fractional
boundary value problem:
\begin{equation*}
\left\{
\begin{array}{l}
_{t}D_{T}^{\alpha }(_{0}D_{t}^{\alpha }k(t))=f(t,~_{0}D_{t}^{\alpha
}k(t)),~~a.e.~~t\in \lbrack 0,T], \k^{(j)}(0)=0,~~j = 0, 1, 2, ... , 2(n - 1),\k^{(l)}(T)=0,~~l = 0, 1, 2, ... , n - 1,%
\end{array}%
\right. \label{a2}
\end{equation*}
where $_{t}D_{T}^{\alpha }$ and $_{0}D_{t}^{\alpha }$ are the right and
left Riemann-Liouville fractional derivatives of order $n-1<\alpha
|
