Electron. J. Diff. Equ., Vol. 2016 (2016), No. 133, pp. 1-13.

Positive solutions for systems of competitive fractional differential equations

Majda Chaieb, Abdelwaheb Dhifli, Malek Zribi

Abstract:
Using potential theory arguments, we study the existence and boundary behavior of positive solutions in the space of weighted continuous functions, for the fractional differential system
$$\displaylines{
 D^{\alpha }u(x)+p(x)u^{a_1}(x)v^{b_1}(x) =0\quad \text{in }(0,1),\quad
 \lim_{x\to 0^{+}}x^{1-\alpha }u(x)=\lambda >0, \cr
 D^{\beta }v(x)+q(x)v^{a_2}(x)u^{b_2}(x) = 0\quad \text{in }(0,1),\quad
 \lim_{x\to 0^{+}}x^{1-\beta }v(x)=\mu >0,
 }$$
where $\alpha,\beta \in (0,1)$, $a_i>1$, $b_i\geq 0$ for $i\in \{1,2\}$ and $p,q$ are positive continuous functions on $(0,1)$ satisfying a suitable condition relying on fractional potential properties.

Submitted February 27, 2016 Published June 7, 2016.
Math Subject Classifications: 26A33, 31B25, 34A12, 34B18.
Key Words: Fractional differential system; positive solution; potential theory; Schauder fixed point theorem.

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Majda Chaieb
Département de Mathématiques
Faculté des Sciences de Tunis
Campus Universitaire, 2092 Tunis, Tunisia
email: majda.chaieb@gmail.com
Abdelwaheb Dhifli
Département de Mathématiques
Faculté des Sciences de Tunis
Campus Universitaire, 2092 Tunis, Tunisia
email: dhifli_waheb@yahoo.fr
Malek Zribi
Département de Mathématiques
Faculté des Sciences de Tunis
Campus Universitaire, 2092 Tunis, Tunisia
email: malek.zribi@insat.rnu.tn

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