Electron. J. Diff. Equ., Vol. 2013 (2013), No. 141, pp. 1-12.

Existence of solutions to fractional boundary-value problems with a parameter

Ya-Ning Li, Hong-Rui Sun, Quan-Guo Zhang

Abstract:
This article concerns the existence of solutions to the fractional boundary-value problem
$$\displaylines{ -\frac{d}{dt} \big(\frac{1}{2} {}_0D_t^{-\beta}+ \frac{1}{2}{}_tD_{T}^{-\beta}\big)u'(t)=\lambda u(t)+\nabla F(t,u(t)),\quad \hbox{a.e. } t\in[0,T], \cr u(0)=0,\quad u(T)=0. }$$
First for the eigenvalue problem associated with it, we prove that there is a sequence of positive and increasing real eigenvalues; a characterization of the first eigenvalue is also given. Then under different assumptions on the nonlinearity F(t,u), we show the existence of weak solutions of the problem when $\lambda$ lies in various intervals. Our main tools are variational methods and critical point theorems.

Submitted January 27, 2013. Published June 21, 2013.
Math Subject Classifications: 34A08, 34B09.
Key Words: Fractional differential equation; eigenvalue; critical point theory; boundary value problem.

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Ya-Ning Li
School of Mathematics and Statistics
Lanzhou University
Lanzhou, Gansu 730000, China
email: liyn08@lzu.edu.cn
Hong-Rui Sun
School of Mathematics and Statistics
Lanzhou University
Lanzhou, Gansu 730000, China
email: hrsun@lzu.edu.cn
Quan-Guo Zhang
School of Mathematics and Statistics
Lanzhou University
Lanzhou, Gansu 730000, China
email: zhangqg07@lzu.edu.cn

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