Journal of Mathematical Physics, Analysis, Geometry 2020, vol. 16, No 1, pp. 27-45    ( to contents , go back )

### Fractional Boundary Value Problem on the Half-Line

Bilel Khamessi

Department of Mathematics, College of Science, Taibah University, Al-Madinah Al- Munawwarah, Saudi-Arabia

Université Tunis El Manar, Faculté des sciences de Tunis, LR18ES09 Modélisation mathématique, analyse harmonique et théorie du potentiel, 2092 Tunis, Tunisia
E-mail: bilel.khamassi@yahoo.com

Received May 7, 2019, revised October 14, 2019.

Abstract

We consider the semilinear fractional boundary value problem \begin{equation*} D^{\beta}\left(\frac{1}{b(x)}D^{\alpha}u\right)=a(x)u^{\sigma} \quad\text{in } (0,\infty) \end{equation*} with the conditions $\lim_{x\rightarrow 0} x^{2-\beta} \frac{1}{b(x)}D^{\alpha}u(x) =\lim_{x\rightarrow \infty} x^{1-\beta}\frac{1}{b(x)}D^{\alpha}u(x)=0$ and $\lim_{x\rightarrow 0} x^{2-\alpha}u(x)= \lim_{x\rightarrow \infty} x^{1-\alpha}u(x)=0$, where $\beta,\alpha \in (1,2)$, $\sigma\in(-1,1)$ and $D^{\beta}, D^{\alpha}$ stand for the standard Riemann--Liouville fractional derivatives. The functions $a,b : (0,\infty)\rightarrow \mathbb{R}$ are nonnegative continuous functions satisfying some appropriate conditions. The existence and the uniqueness of a positive solution are established. Also, a description of the global behavior of this solution is given.

Mathematics Subject Classification 2000: 34A08, 35B09, 47H10.
Key words: fractional differential equation, positive solution, Schauder fixed point theorem.