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A unified characterization of convolution coefficients in nonlocal differential equations

Part of: Nonlinear operators and their properties Integral transforms, operational calculus Real functions Elementary classical functions Boundary value problems Harmonic analysis in one variable

Published online by Cambridge University Press:  18 September 2024

Christopher S. Goodrich Open the ORCID record for Christopher S. Goodrich [Opens in a new window]
Christopher S. Goodrich*
Affiliation:
School of Mathematics and Statistics, UNSW Sydney, Sydney, NSW 2052 Australia ([email protected])
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*Corresponding author
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Abstract

In loving memory of my beloved miniature dachshund Maddie (16 March 2002 – 16 March 2020). We consider nonlocal differential equations with convolution coefficients of the form

\[{-}M\Big(\big(a*(g\circ |u|)\big)(1)\Big)u''(t)=\lambda f\big(t,u(t)\big),\quad t\in(0,1), \]
in the case in which $g$ can satisfy very generalized growth conditions; in addition, $M$ is allowed to be both sign-changing and vanishing. Existence of at least one positive solution to this equation equipped with boundary data is considered. We demonstrate that the nonlocal coefficient $M$ allows the forcing term $f$ to be free of almost all assumptions other than continuity.

Keywords

Nonlocal differential equationpositive solutionconvolutionHarnack inequalitytopological fixed point theory

MSC classification

Primary: 33B15: Gamma, beta and polygamma functions 34B10: Nonlocal and multipoint boundary value problems 34B18: Positive solutions of nonlinear boundary value problems 42A85: Convolution, factorization
Secondary: 44A35: Convolution 26A33: Fractional derivatives and integrals 47H30: Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 19
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Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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