Abstract
We study the three-point boundary value problem of higher-order fractional differential equations of the formDc0+ρut+ft, ut=0,0<t<1,2⩽n-1<ρ<n,u′(0)=u′′(0)=⋯=un-1(0)=0,u(1)+pu′(1)=qu′(ξ), where cD0+ρis the Caputo fractional derivative of orderρ, and the functionf:[0,1]×[0,∞)↦[0,+∞)is continuously differentiable. Here,0⩽q⩽p,0<ξ<1,2⩽n-1<ρ<n. By virtue of some fixed point theorems, some sufficient criteria for the existence and multiplicity results of positive solutions are established and the obtained results also guarantee that the positive solutions discussed are monotone and concave.
Highlights
Applications of fractional differential equations can be found in various areas, including engineering, physics, and chemistry [1,2,3,4]
If a function y in C[0, 1] is a solution of the equation u(t) = I0ρ+g(t, u(t)), y ∈ C(n−1)[0, 1] and y(n) ∈ C(0, 1] ∩ L(0, 1], and the relation cD0ρ+y(t) = g(t, y(t)) holds for each t in [0, 1]
By Lemma 6, we present an integral representation of the solution of the linearized problem corresponding to BVP (2)
Summary
Applications of fractional differential equations can be found in various areas, including engineering, physics, and chemistry [1,2,3,4]. It is worth pointing out that Wang et al [7] obtained the existence and multiplicity results of the positive, monotone, and concave solutions to the following problem: cD0ρ+u (t) + f (t, u (t)) = 0, 0 < t < 1, n − 1 < ρ < n, u (0) = u (0) = ⋅ ⋅ ⋅ = u(n−1) (0) = 0, u (1) = 0, (1).
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