Application of Schauder fixed point theorem to a coupled system of differential equations of fractional order. (English) Zbl 1477.34012

Hao, Mengru; Zhai, Chengbo

J. Nonlinear Sci. Appl. 7, No. 2, 131-137 (2014).

Summary: 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 \[ \begin{cases} -D^{\nu_1}_{0^+}y_1(t) = \lambda_1a_1(t)f (t, y_1(t), y_2(t)) + e_1(t),\\ -D^{\nu_2}_{0^+}y_2(t) = \lambda_2a_2(t)f (t, y_1(t), y_2(t)) + e_2(t), \end{cases} \] 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 \(1 \leq \alpha \leq n - 2\).

MSC:

34A08	Fractional ordinary differential equations
34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
47H10	Fixed-point theorems

Keywords:

fractional differential equation; Schauder fixed point theorem; positive solution