Abstract
In this article, by means of fixed point theorem on mixed monotone operator, we establish the uniqueness of positive solution for some nonlocal singular higher-order fractional differential equations involving arbitrary derivatives. We also give iterative schemes for approximating this unique positive solution.
Highlights
We are interested in investigating the existence and iterative schemes of the unique positive solution for the following fractional differential equation (FDE): Dα0+u (t) + f (t, u (t), Dδ0+u (t)) = 0, 0 < t < 1, Dδ0+u (0) = Dδ0++1u (0) = ⋅ ⋅ ⋅ = Dδ0++n−2u (0) = 0, (1)
Under different conjugate type integral conditions such as no parameters, only one or two parameters involved in boundary conditions, [8,9,10,11,12,13,14,15,16, 33, 34] investigate the existence, uniqueness, and multiplicity of positive solutions for FDEs when f is either continuous or singular
X is a positive solution of BVP (8) if and only if x ∈ C[0, 1] is a solution of the following nonlinear integral equation: x (t) = ∫ G (t, s) f (s, I0δ+x (s), x (s)) ds
Summary
We are interested in investigating the existence and iterative schemes of the unique positive solution for the following fractional differential equation (FDE): Dα0+u (t) + f (t, u (t) , Dδ0+u (t)) = 0, 0 < t < 1, Dδ0+u (0) = Dδ0++1u (0) = ⋅ ⋅ ⋅ = Dδ0++n−2u (0) = 0,. Under different conjugate type integral conditions such as no parameters, only one or two parameters involved in boundary conditions, [8,9,10,11,12,13,14,15,16, 33, 34] investigate the existence, uniqueness, and multiplicity of positive solutions for FDEs when f is either continuous or singular. The method used in this paper is different from that in [16]
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