Geodesic
Fractional boundary value problem on the half-line
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 16 (2020) no. 1, pp. 27-45 Cet article a éte moissonné depuis la source Math-Net.Ru

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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} \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. 
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Bilel Khamessi. Fractional boundary value problem on the half-line. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 16 (2020) no. 1, pp. 27-45. http://geodesic.mathdoc.fr/item/JMAG_2020_16_1_a1/

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