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.

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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

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