Existence and multiplicity of positive solutions for a new class of singular higher-order fractional differential equations with Riemann\u2013Stieltjes integral boundary value conditions

Lishan Liu,Yonghong Wu,Dandan Min

doi:10.1186/s13662-020-02892-7

Abstract

In this work, the aim is to discuss a new class of singular nonlinear higher-order fractional boundary value problems involving multiple Riemann–Liouville fractional derivatives. The boundary conditions are constituted by Riemann–Stieltjes integral boundary conditions. The existence and multiplicity of positive solutions are derived via employing the Guo–Krasnosel’skii fixed point theorem. In addition, the main results are demonstrated by some examples to show their validity.

Highlights

We consider the existence and multiplicity of positive solutions for the following nonlinear singular higher-order fractional differential equations:⎧ ⎪⎪⎨Dα0+ u(t) + f (t, u(t), Dα0+1 u(t), Dα0+2 u(t), . . . , Dα0+n–2 u(t)) = 0,⎪⎪⎩uD(β00+1)u=(1D) γ0=1+ u(0) = Dγ02+η 0 h(s)Dβ0+2 u(0) u(s) =··· dA(s) = +Dγ0n+–2 u(0) = 0,1 0 a(s)Dβ0+3 u(s) dA(s), t ∈ (0, 1), (1.1)where Dα0+ u, Dα0+k u, Dγ0k+ u (k = 1, 2, . . . , n – 2), Dβ0+i u (i = 1, 2, 3) are the standard Riemann
In [6], by means of the fixed point index theory, Zhang et al investigated the existence of positive solutions for the fractional differential equation with integral boundary conditions:
In [7], via employing the topological degree theory, Zhang et al investigated the existence of positive nontrivial solutions for the following nonlinear fractional differential equations with integral boundary conditions:

Summary

Introduction

We consider the existence and multiplicity of positive solutions for the following nonlinear singular higher-order fractional differential equations:. N – 2), Dβ0+i u (i = 1, 2, 3) are the standard Riemann–. N – 1 < α ≤ n (n ≥ 3), k – 1 < αk, γk ≤ k N – 2), 1 < α – αn–2 – 1 ≤ 2, γn–2 – αn–2 ≥ 0, β1 ≥ β2, β1 ≥. Β3, α –βi ≥ 1, βi –αn–2 –1 ≥ 0 (i = 1, 2, 3), β1 ≤ n–1, f : (0, 1)×Rn+–1 → R1+ = [0, +∞) is continuous and a, h ∈ C((0, 1), R1+), A is a function of bounded variation, η 0 h(s)Dβ0+2 u(s) dA(s), dA(s) denote the Riemann–Stieltjes integrals with respect to

Liu et al Advances in Difference Equations

The unique solution of

By using

We denote

There exist

For any