Positive Solutions of a Fractional BVP and the Reach of the Omitted Ray Fixed Point Theorem
- 1Department of Mathematics, Vignan’s Institute of Information Technology (Autonomous), Visakhapatnam 530049, Andhra Pradesh, India
- 2School of Sciences, Woxsen University, Hyderabad 500033, Telangana, India
- 3Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
Abstract
In this paper, we study the problem D0+αu+f(t,u)=0 on (0,1), u(0)=0, D0+α−1u(1)=∫01u dA, where 1<α≤2 and the measure dA may change sign. For a measure made of two point masses of opposite sign, we show that the function KA(s)=∫01G(τ,s) dA(τ), through which dA enters the Green function, is nonnegative exactly when the positive weight is at least θ∗ times the size of the negative one, where θ∗ is given in closed form. Together with a simple condition Δ>0, this makes the Green function nonnegative. The threshold θ∗ decreases in α and tends to infinity as α→1+. For α<2 the Green function has no uniform Harnack estimate, so we work in a cone weighted by Ψ(t)=∫01G(t,s) ds, where Schauder’s theorem gives a positive solution with explicit bounds whenever f lies between two positive constants. For the omitted ray fixed point theorem of Avery, Henderson and Liu, we describe exactly when its key inequality holds for the two-point functional used here, and we show that the hypotheses under which we can verify it already imply the classical inward condition. Examples with α=3/2 and α=1.08 illustrate the results.
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BibTeX
@article{khuddush2026positive,
title = {{Positive Solutions of a Fractional BVP and the Reach of the Omitted Ray Fixed Point Theorem}},
author = {Mahammad Khuddush and Saleh S. Almuthaybiri},
journal = {Fractal and Fractional},
year = {2026},
month = oct,
doi = {10.3390/fractalfract10100716},
url = {https://www.mdpi.com/2504-3110/10/10/716},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Positive Solutions of a Fractional BVP and the Reach of the Omitted Ray Fixed Point Theorem
AU - Mahammad Khuddush
AU - Saleh S. Almuthaybiri
PY - 2026
DA - 2026/10/13
JO - Fractal and Fractional
DO - 10.3390/fractalfract10100716
UR - https://www.mdpi.com/2504-3110/10/10/716
AB - In this paper, we study the problem D0+αu+f(t,u)=0 on (0,1), u(0)=0, D0+α−1u(1)=∫01u dA, where 1<α≤2 and the measure dA may change sign. For a measure made of two point masses of opposite sign, we show that the function KA(s)=∫01G(τ,s) dA(τ), through which dA enters the Green function, is nonnegative exactly when the positive weight is at least θ∗ times the size of the negative one, where θ∗ is given in closed form. Together with a simple condition Δ>0, this makes the Green function nonnegative. The threshold θ∗ decreases in α and tends to infinity as α→1+. For α<2 the Green function has no uniform Harnack estimate, so we work in a cone weighted by Ψ(t)=∫01G(t,s) ds, where Schauder’s theorem gives a positive solution with explicit bounds whenever f lies between two positive constants. For the omitted ray fixed point theorem of Avery, Henderson and Liu, we describe exactly when its key inequality holds for the two-point functional used here, and we show that the hypotheses under which we can verify it already imply the classical inward condition. Examples with α=3/2 and α=1.08 illustrate the results.
SN - 2504-3110
ER -