(untitled) (Journal of Nonlinear Modeling and Analysis, 2024)
In this paper, the solvability for the following fractional boundary value problem system ^CD^σ_1_0+v_1(t) = f_1(t, v_2(t), D^μ_1_0+v_2(t)), 0 < t < 1, ^CD^σ_2_0+v_2(t) = f_2(t, v_1(t), D^μ_2_0+v_1(t)), 0 < t < 1, v_1' (0) = bv_1(0), v_1''(0) = 0, ^CD^θ_1_0+v_1(1) = a · ^C D^θ_2 _0+v_1(η), v_2'(0) = bv_2(0), v_2''(0) = 0, ^CD^θ_1_0+v_2(1) = a · ^CD^θ_2_0+v_2(η), is studied, where a > 0, −1 < b < 0, 2 < σ_1, σ_2 ≤ 3, 0 < η < 1, 0 < μ_1, μ_2 ≤ 1, 0 < θ_2 ≤ θ_1 ≤ 1, f_1, f_2: [0, 1] × R^+ × R → R^+ are continuous, ^CD^σ_1 _0+v_1(t), ^CD^σ_2_0+v_2(t) are the Caputo fractional derivatives, and D^μ_1_0+v_2(t), D^μ_2_0+v_1(t) are the Riemann-Liouville fractional derivatives. The fixed point theorem is used to prove that there are three positive solutions to problems.