Electronic Journal of Differential Equations,
Vol. 2015 (2015), No. 95, pp. 1-9.
Title: An extension of the Lax-Milgram theorem and its
application to fractional differential equations
Authors: Nemat Nyamoradi (Razi Univ., Kermanshah, Iran)
Mohammad Rassol Hamidi (Razi Univ., Kermanshah, Iran)
Abstract:
In this article, using an iterative technique, we introduce an
extension of the Lax-Milgram theorem which can be used for proving
the existence of solutions to boundary-value problems.
Also, we apply of the obtained result to the
fractional differential equation
$$\displaylines{
{}_t D_T^{\alpha}{}_0 D_t^{\alpha}u(t)+u(t)
=\lambda f (t, u(t)) \quad t \in (0,T),\cr
u(0)=u(T)=0,
}$$
where ${}_tD_T^\alpha$ and ${}_0D_t^\alpha$ are the right and
left Riemann-Liouville fractional derivative of order
$\frac{1}{2}< \alpha \leq 1$ respectively, $\lambda$ is a
parameter and $f:[0,T]\times\mathbb{R}\to\mathbb{R}$
is a continuous function. Applying a regularity argument to this
equation, we show that every weak solution is a classical solution.
Submitted February 1, 2015. Published April 13, 2015.
Math Subject Classifications: 34A08, 35A15, 35B38.
Key Words: Lax-Milgram theorem; fractional differential equation.