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

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


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.

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