Abstract
This paper focuses on a singular boundary value (SBV) problem of nonlinear fractional differential (NFD) equation defined as follows: D 0 + β υ τ + f τ , υ τ = 0 , τ ∈ 0,1 , υ 0 = υ ′ 0 = υ ″ 0 = υ ″ 1 = 0 , where 3 < β ≤ 4 , D 0 + β is the standard Riemann–Liouville fractional (RLF) derivative. The nonlinear function f τ , υ τ might be singular on the spatial and temporal variables. This paper proves that a positive solution to the SBV problem exists and is unique, taking advantage of Green’s function through a fixed-point (FP) theory on cones and mixed monotone operators.
Highlights
According to Lemma 7, the complete continuity of T is valid from K∖Br into K for any r > 0. en, the existence of two fixed points μ1 and μ2 with 0 < ‖μ1‖ < 2λr < ‖μ2‖ is proved here
We firstly show that Tξ: Qς × Qς ⟶ Qς
Eorem 2 is validated, indicating the presence of a unique positive solution μ∗
Summary
Significant progress has been achieved in solving NFD equations led by both the advance of fractional calculus and its applications to physics, mechanics, chemistry, economics, engineering, and other fields [1,2,3,4,5,6,7,8].e initial value problem and the boundary problem were studied in [9,10,11,12,13,14,15,16,17,18,19,20,21], respectively. e SBV problem was recently studied for NFD [22,23,24,25,26,27,28,29,30,31,32,33], mostly focused on investigating the positive solution. ey were mainly based on nonlinear analysis techniques such as Leray–Schuader theory, FP, topological theories, and mixed monotone method [34,35,36,37,38,39].In [22], the presence of positive solutions for the SVB problem for NFD equation was investigated, which is defined as (1)where 2 < α ≤ 3, CDα0+ is the standard Caputo’s fractional derivative, and f: (0, 1] × [0, +∞) ⟶ [0, +∞) with limτ⟶0+f(τ, ·) +∞. In [22], the presence of positive solutions for the SVB problem for NFD equation was investigated, which is defined as Xu [28] investigated the following SBV problem for NFD: E presence of multiple positive solutions for f(τ, Mathematical Problems in Engineering υ) ≤ a(τ)g(υ) + h(υ) was proved through the Guo–Krasnoselskii FP theory.
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