摘要：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 ${}^{c}D^{\alpha}u(t)$ equals to $f_{1} ( t, u(t), {}^{c}D^{ \beta_{1}} u(t), I^{\gamma_{1}} u(t) )$ and $f_{2} ( t, u(t), {}^{c} D^{\beta_{2}} u(t), I^{\gamma_{2}} u(t) )$ for each t belonging to $[0, t_{0}]$ and $[t_{0}, 1]$ , respectively, with simultaneous Dirichlet boundary conditions, where ${}^{c}D^{\alpha}$ and $I^{\alpha}$ denote the Caputo fractional derivative and Riemann–Liouville fractional integral of order α, respectively. Some models are thrown to illustrate our results, too.
