Abstract
In this paper, we consider the existence of positive solutions to a singular semipositone boundary value problem of nonlinear fractional differential equations. By applying the fixed point index theorem, some new results for the existence of positive solutions are obtained. In addition, an example is presented to demonstrate the application of our main results.
Highlights
In this paper, we discuss the following singular semipositone system of nonlinear fractional differential equations: D0α+u(t) + f(t, u(t), v(t)) = 0, 0 < t < 1, D0α+v(t) + g(t, u(t), v(t)) = 0, 0 < t < 1, (1)u(0) = u(1) = u′(0) = u′(1) = v(0) = v(1) = v′(0) = v′(1) = 0, where 3 < α ≤ 4 is a real number, D0α+ is the standard Riemann-Liouville fractional derivative, and f, g ∶ (0, 1) × [0, +∞) × [0, +∞) → (−∞, +∞) are given continuous functions. f, g may be singular at t = 0 and/or t = 1 and may take negative values
In [7], by using the fixed point index theorem, Liu, Zhang and Wu have studied the existence of positive solutions for a nonlinear singular semipositone system:
In [12], Zhu, Liu and Wu have discussed the existence of positive solutions for the fourthorder singular semipositone system:
Summary
In [6], Henderson and Luca have considered the existence of positive solutions for the system of nonlinear fractional differential equations: D0α+u(t) + λf(t, u(t), v(t)) = 0, t ∈ (0, 1), D0β+v(t) + μg(t, u(t), v(t)) = 0, t ∈ (0, 1), with the coupled integral boundary conditions u(0) = u′(0) = ⋯ = u(n−2)(0) = 0, v(0) = v′(0) = ⋯ = v(n−2)(0) = 0, u′(1) = ∫0 v(s) dH(s), v′(1) = ∫0 u(s) dK(s), where α ∈
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