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Published in:
2013 | OriginalPaper | Chapter
Boundary Value Problem for a Coupled System of Nonlinear Fractional Differential Equation
Authors : Ya-ling Li, Shi-you Lin
Published in: Proceedings of The Eighth International Conference on Bio-Inspired Computing: Theories and Applications (BIC-TA), 2013
Publisher: Springer Berlin Heidelberg
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Abstract
This paper is concentrated on the following coupled system of the nonlinear fractional differential equationwhere \( f,\;K,\;g,\;H:\;\left[ {0,\,1} \right]\, \times \,\Re \, \to \,\left[ {0,\, + \infty } \right) \) 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.
$$ \left\{ \begin{aligned} &D^{\alpha } u\left( t \right) = f\left(
{t,v\left( t \right)} \right) + \int_{0}^{t} {K\left( {s,v\left( s
\right)} \right)ds,\quad 5 < \alpha ,\beta \le 6,\;0 < t < 1}
\hfill \\ &D^{\beta } v\left( t \right) = g\left( {t,u\left( t
\right)} \right) + \int_{0}^{t} {H\left( {s,u\left( s \right)}
\right)ds} \hfill \\ &u\left( 1 \right) = \mathop {\lim
}\limits_{{t \to 0}} u\left( t \right) \cdot t^{{2 - \alpha }} =
v\left( 1 \right) = \mathop {\lim }\limits_{{t \to 0}} v\left( t
\right) \cdot t^{{2 - \beta }} = 0. \hfill \\ \end{aligned}
\right. $$