Triple Positive Solutions for a Class of Fractional Boundary Value Problem System (Journal of Nonlinear Modeling and Analysis, 2024)
In this paper, the solvability for the following fractional boundaryvalue 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 $a > 0,$ $−1 < b < 0,$ $2 < σ_1, σ_2 ≤ 3,$ $0 < η < 1,$ $0 < μ_1, μ_2 ≤ 1,$ $0 < θ_2 ≤ θ_1 ≤ 1,$ $f_1, f_2:$ $[0, 1] × \mathbb{R}^+ × \mathbb{R} → \mathbb{R}^+$ are continuous, $^CD^{σ_1} _{0+}v_1(t),$ $^CD^{σ_2}_{0+}v_2(t)$ are the Caputo fractional derivatives, and $D^{μ_1}_{0+}v_2(t),$ $D^{μ_2}_{0+}v_1(t)$ arethe Riemann-Liouville fractional derivatives. The fixed point theorem is usedto prove that there are three positive solutions to problems.