Application of Schauder fixed point theorem to a coupled system of differential equations of fractional order :

Hao, Mengru¹ ; Zhai, Chengbo¹

¹ School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, P.R. China

Journal of nonlinear sciences and its applications, Tome 7 (2014) no. 2, p. 131-137

Résumé

In this paper, by using Schauder fixed point theorem, we study the existence of at least one positive solution to a coupled system of fractional boundary value problems given by

$ \begin{cases} -D^{\nu_1}_{0^+}y_1(t) = \lambda_1a_1(t)f(t, y_1(t), y_2(t)) + e_1(t),\\ -D^{\nu_2}_{0^+}y_2(t) = \lambda_2a_2(t)g(t, y_1(t), y_2(t)) + e_2(t), \end{cases} $

where $\nu_1,\nu_2\in (n - 1; n]$ for $n > 3$ and $n \in N$, subject to the boundary conditions $y^(i)_1 (0) = 0 = y^(i)_2 (0)$, for $0 \leq i \leq n - 2$, and $[D^{\alpha}_{0^+}y_1(t)]_{t=1}=0=[D^{\alpha}_{0^+}y_2(t)]_{t=1}$, for $1 \leq\alpha\leq n - 2$.

Détail

DOI : 10.22436/jnsa.007.02.07

Classification : 47H10, 34A08, 34B18.
Keywords: Fractional differential equation, Schauder fixed point theorem, Positive solution.

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