Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 141, pp. 1-12.
Title: Existence of solutions to fractional boundary-value
problems with a parameter
Authors: Ya-Ning Li (Lanzhou Univ., Gansu, China)
Hong-Rui Sun (Lanzhou Univ., Gansu, China)
Quan-Guo Zhang (Lanzhou Univ., Gansu, China)
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.