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

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
Full Text: