Abstract
In this paper, we study the existence and multiplicity of positive solutions of a class of nonlinear fractional boundary value problems with Dirichlet boundary conditions. By applying the fixed point theory on cones we establish a series of criteria for the existence of one, two, any arbitrary finite number, and an infinite number of positive solutions. A criterion for the nonexistence of positive solutions is also derived. Several examples are given for demonstration.
Highlights
We study the boundary value problem (BVP) consisting of the fractional differential equation
Fractional differential equations have attracted extensive attention as they can be applied in various fields of science and engineering
Our results reveal the fact that under our conditions, the existence of one or more positive solutions is determined by the behavior
Summary
We study the boundary value problem (BVP) consisting of the fractional differential equation. To the best knowledge of the authors, there are no results yet on the existence of an arbitrary number of positive solutions of BVP (1.4), (1.2) This is due to the fact that the Green’s function G(t, s) is positive in the interior of its domain {(t, s) : 0 ≤ t, s ≤ 1}, it becomes zero on the boundary, and does not satisfy the following condition γΦ(s) ≤ G(t, s) ≤ Φ(s), t ∈ [a, b] ⊂ [0, 1], s ∈ [0, 1],. By carefully manipulating the Green’s function G(t, s), we obtain a weaker condition similar to (1.5) so that we are able to apply the fixed point theory on cones to BVP (1.1), (1.2), where the functions w(t) and f (x) satisfy certain conditions. All the proofs of the main results are given in the last section
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