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Abstract
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In the current study, by using some fixed point technique such as Banach contraction principle and fixed point theorem of Krasnoselskii, we look into the positive solutions for fractional differential equation ${}^cD^\alpha u(t)$ is equals to $f_1\left( t, u(t), {}^cD^{\beta_1} u(t), I^{\gamma_1} u(t) \right)$ and $f_2 \left( t, u(t), {}^c D^{\beta_2} u(t), I^{\gamma_2} u(t) \right)$, for each $t$ belongs to $[0, t_0]$ and $[t_0, 1]$, respectively, with simultaneous Dirichlet boundary conditions, where ${}^cD^\alpha$ and $I^\alpha$ denote the Caputo fractional derivative and Riemann--Liouville fractional integral of order $\alpha$, respectively. Some models are thrown to illustrate our results, too.
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