Electron. J. Diff. Equ., Vol. 2016 (2016), No. 116, pp. 1-11.

Existence of solutions for fractional differential equations with Dirichlet boundary conditions

Khaled Ben Ali, Abdeljabbar Ghanmi, Khaled Kefi

Abstract:
In this article, we apply the Nehari manifold to prove the existence of a solution of the fractional differential equation
$$\displaylines{
 \frac{d}{dt} \Big(\frac12 {\,}_0D_t^{-\beta}(u'(t))
 +\frac12 {\,}_tD_T^{-\beta}(u'(t)))= f(t,u(t))
 + \lambda h(t)|u(t)|^{r-2}u(t), \cr
 \text{a.e } t\in [0,T],\cr
 u(0)=u(T)=0,
 }$$
where $ _0D_t^{-\beta},\; _tD_T^{-\beta}$ are the left and right Riemann-Liouville fractional integrals, respectively, of order $0< \beta < 1$.

Submitted January 28, 2016. Published May 10, 2016.
Math Subject Classifications: 26A33, 58E05, 35J60.
Key Words: Fractional differential equation; left and right fractional derivatives; boundary value problem; Nehari manifold.

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Khaled Ben Ali
Département de Mathématiques
Faculté des Sciences de Tunis
Campus Universitaire, 2092 Tunis, Tunisia
email: benali.khaled@yahoo.fr
Abdeljabbar Ghanmi
Department of Mathematics
Faculty of Sciences and Arts Khulais,
University of Jeddah, Saudi Arabia
email: Abdeljabbar.ghanmi@lamsin.rnu.tn
Khaled Kefi
Département de Mathématiques
Faculté des Sciences de Tunis
Campus Universitaire, 2092 Tunis, Tunisia
email: khaled_kefi@yahoo.fr

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