Positive solutions of fractional differential equation with two pieces in chain interval and simultaneous Dirichlet boundary conditions

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 ᶜDαu(t) equals to f₁ ( t, u(t), ᶜD^ β₁ u(t), I^γ₁ u(t) ) and f₂ ( t, u(t), ᶜ D^β₂ u(t), I^γ₂ u(t) ) for each t belonging to [0, t₀] and [t₀, 1] , respectively, with simultaneous Dirichlet boundary conditions, where ᶜDα and Iα denote the Caputo fractional derivative and Riemann–Liouville fractional integral of order α, respectively. Some models are thrown to illustrate our results, too.