Abstract
This paper concerns a new kind of fractional differential equation of arbitrary order by combining a multi-point boundary condition with an integral boundary condition. By solving the equation which is equivalent to the problem we are going to investigate, the Green’s functions are obtained. By defining a continuous operator on a Banach space and taking advantage of the cone theory and some fixed point theorems, the existence of multiple positive solutions for the BVPs is proved based on some properties of Green’s functions and under the circumstance that the continuous functions f satisfy certain hypothesis. Finally, examples are provided to illustrate the results.
Highlights
Fractional calculus has attracted many researchers’ interests because of its wide application in solving practical problems that arise in fields like viscoelasticity, biological science, ecology, aerodynamics, etc
Numerous writings have showed that fractional-order differential equations could provide more methods to deal with complex problems in statistical physics and environmental issues
Lemma 2.5 Assume that h ∈ L1[0, 1], x ∈ ACn[0, 1] and n – 1 < σ ≤ n with n ≥ 3, the solution to the fractional differential equation
Summary
Fractional calculus has attracted many researchers’ interests because of its wide application in solving practical problems that arise in fields like viscoelasticity, biological science, ecology, aerodynamics, etc. The authors explored the fractional-order equation with integral boundary conditions as follows in [9]: 1. the authors obtained the existence and uniqueness of nontrivial solutions due to the application of Leray-Schauder’s nonlinear alternative and Schauder’s fixed point theorem.
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