Boundary Value Problem for a Coupled System of Nonlinear Fractional Differential Equation

Abstract

This paper is concentrated on the following coupled system of the nonlinear fractional differential equation

$$ \left\{ \begin{aligned} &D^{\alpha } u\left( t \right) = f\left( {t,v\left( t \right)} \right) + \int_{0}^{t} {K\left( {s,v\left( s \right)} \right)ds,\quad 5 < \alpha ,\beta \le 6,\;0 < t < 1} \hfill \\ &D^{\beta } v\left( t \right) = g\left( {t,u\left( t \right)} \right) + \int_{0}^{t} {H\left( {s,u\left( s \right)} \right)ds} \hfill \\ &u\left( 1 \right) = \mathop {\lim }\limits_{{t \to 0}} u\left( t \right) \cdot t^{{2 - \alpha }} = v\left( 1 \right) = \mathop {\lim }\limits_{{t \to 0}} v\left( t \right) \cdot t^{{2 - \beta }} = 0. \hfill \\ \end{aligned} \right. $$

where $f,\;K,\;g,\;H:\;\left[ {0,\,1} \right]\, \times \,\Re \, \to \,\left[ {0,\, + \infty } \right)$ are the positive continuous functions. $D^{\alpha }$ and $D^{\beta }$ are the standard Riemann–Liouville fractional derivatives with the order $\alpha ,\, \beta,$ respectively. We give the existence and the uniqueness of the solution by using the Schauder fixed point theorem and the generalized Gronwall inequality.