Electronic Journal of Differential Equations (Apr 2015)
An extension of the Lax-Milgram theorem and its application to fractional differential equations
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.
