Electron. J. Diff. Equ., Vol. 2015 (2015), No. 95, pp. 1-9.

An extension of the Lax-Milgram theorem and its application to fractional differential equations

Nemat Nyamoradi, Mohammad Rassol Hamidi

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.

Show me the PDF file (216 KB), TEX file, and other files for this article.

Nemat Nyamoradi
Department of Mathematics, Faculty of Sciences
Razi University, 67149 Kermanshah, Iran
email: nyamoradi@razi.ac.ir, neamat80@yahoo.com
Mohammad Rassol Hamidi
Department of Mathematics, Faculty of Sciences
Razi University, 67149 Kermanshah, Iran
email: mohammadrassol.hamidi@yahoo.com

Return to the EJDE web page