Abstract
We investigate the existence of positive solutions of a Riemann-Liouville fractional differential equation with sequential derivatives, a positive parameter and a nonnegative singular nonlinearity, supplemented with integral-multipoint boundary conditions which contain fractional derivatives of various orders and Riemann-Stieltjes integrals. Our general boundary conditions cover some symmetry cases for the unknown function. In the proof of our main existence result, we use an application of the Krein-Rutman theorem and two theorems from the fixed point index theory.
Highlights
IntroductionWe consider the ordinary fractional differential equation with sequential derivatives
In this paper, we consider the ordinary fractional differential equation with sequential derivativesD0α+ q(t)D0β+v(t) = λr(t)g(t, v(t)), t ∈ (0, 1), (1)subject to the integral-multipoint boundary conditionsAcademic Editor: Calogero VetroReceived: 17 June 2021 Accepted: 11 August 2021 Published: 13 August 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.v(k)(0) = 0, k = 0, . . . , n − 2, D0β+v(0) = 0, p q(1)D0β+v(1) = aq(ξ)D0β+v(ξ), D0γ+0 v(1) = ∑ j=1 D0γ+j v(t) dH j (t)
Where γ ∈ R, γ ∈ (n − 1, n], n ∈ N, n ≥ 3, m ∈ N, ζj ∈ R for all j = 0, . . . , m, 0 ≤ ζ1 < ζ2 < · · · < ζm ≤ ζ0 < γ − 1, ζ0 ≥ 1, Kj, j = 1, . . . , m are bounded variation functions, μ is a positive parameter, the function g(t, w) is nonnegative and it may have singularity at w = 0 and the function h(t) is nonnegative and it may be singular at the points t = 0 and/or t = 1
Summary
We consider the ordinary fractional differential equation with sequential derivatives. In the proof of our main theorem we use some results from the fixed point index theory Positive solutions for such fractional problems are of great practical importance for describing nonlocal processes with memory, which determines the relevance of the chosen research topic. They present various assumptions for the nonsingular/singular functions φ and ψ, and intervals for parameters μ and ν such that problem (5), (6) has at least one or two positive solutions. They applied the nonlinear alternative of Leray-Schauder type and the GuoKrasnosel’skii fixed point theorem in the proofs of the main existence results.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
