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On the Luxemburg Functional in the Concave Regime with an Application to Nonlocal Differential Equations
Abstract
For p:[0,1]→(0,1) a continuous function, we consider the Luxemburg functional Λp(·)(u):=inf{δ>0:∫01|u(t)/δ|p(t) dt≤ 1}. We develop a suitable functional analytic framework for Λp(·) so that it can be incorporated into a positive order cone, which can then be used in the analysis of nonlocal differential equations having the general form −A(∫01 b(1−s)(u(s))p(s) ds)u''(t) =λf(t,u(t)), t∈(0,1). By means of a specific numerical example, we demonstrate that, as opposed to a methodology relying only on the supremum norm, our methodology via the Luxemburg functional yields an appreciably sharper localisation of any solutions of the nonlocal equation.
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BibTeX
@misc{goodrich2026luxemburg,
title = {{On the Luxemburg Functional in the Concave Regime with an Application to Nonlocal Differential Equations}},
author = {Christopher S. Goodrich and Gabriel Nakhl},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35878},
eprint = {2610.35878},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35878},
}
RIS
TY - UNPB
TI - On the Luxemburg Functional in the Concave Regime with an Application to Nonlocal Differential Equations
AU - Christopher S. Goodrich
AU - Gabriel Nakhl
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.35878
UR - https://arxiv.org/abs/2610.35878
AB - For p:[0,1]→(0,1) a continuous function, we consider the Luxemburg functional Λ_p(·)(u):=inf{δ>0:∫_0^1|u(t)/δ|^p(t) dt≤ 1}. We develop a suitable functional analytic framework for Λ_p(·) so that it can be incorporated into a positive order cone, which can then be used in the analysis of nonlocal differential equations having the general form −A(∫_0^1 b(1−s)(u(s))^p(s) ds)u''(t) =λf(t,u(t)), t∈(0,1). By means of a specific numerical example, we demonstrate that, as opposed to a methodology relying only on the supremum norm, our methodology via the Luxemburg functional yields an appreciably sharper localisation of any solutions of the nonlocal equation.
ER -