Abstract
The purpose of this study is to discuss the existence of solutions for a boundary value problem at resonance generated by a nonlinear differential equation involving both right and left Caputo fractional derivatives. The proofs of the existence of solutions are mainly based on Mawhin’s coincidence degree theory. We provide an example to illustrate the main result.
Highlights
Introduction and preliminariesMathematical structures describe the complex systems which involve multiple elements and interact between one another in various forms
The immediate observation would be a system of differential equations
Upon solving such differential equations, the obtained function will have some information that can be used to extract and understand the data at hand and further predict the future information related to the data
Summary
We prove KP(I – Q)N( ) is equicontinuous. – (t2 – s)υ–1I1θ– (I – Q)Nx(s) ds (t1 – s)ν–1 – (t2 – s)υ–1 I1θ– (I – Q)Nx(s) ds (13). + (t2 – s)υ–1 I1θ– (I – Q)Nx(s) ds. It follows that KP(I – Q)N( ) is equicontinuous on [0, 1]. Lemma 10 Let 1 = {x ∈ dom L\ ker L : Lx = λNx for some λ ∈ (0, 1)}. If condition (H1) holds, 1 is bounded. Proof Suppose that x ∈ 1, x = (x – Px) + Px ∈ dom L\ ker L. That is, (I – P)x ∈ dom L ∩ ker P and Px ∈ ker L, i.e., LPx = 0, from Lemma 8, we get (I – P)x
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