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Existence and monotonicity of nonlocal boundary value problems: the one-dimensional case

Part of: General theory in ordinary differential equations Boundary value problems Integral transforms, operational calculus Partial differential equations Qualitative theory Real functions Harmonic analysis in one variable

Published online by Cambridge University Press:  23 December 2020

Christopher Goodrich  and
Carlos Lizama
Christopher Goodrich
Affiliation:
School of Mathematics and Statistics, UNSW Australia, Sydney, NSW, 2052, Australia ([email protected])
Carlos Lizama
Affiliation:
Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Las Sophoras 173, 9160000, Santiago, Chile ([email protected])
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Abstract

We consider nonlocal equations of the general form

\begin{equation} \left(a*u''\right)(\cdot)+\lambda f\big(\cdot,u(\cdot)\big)=0.\nonumber \end{equation}
By developing a Green's function representation for the solution of the boundary value problem we study existence, uniqueness, and qualitative properties (e.g., positivity or monotonicity) of solutions to these problems. We apply our methods to fractional order differential equations. We also demonstrate an application of our methodology both to convolution equations with nonlocal boundary conditions as well as those with a nonlocal term in the convolution equation itself.

Keywords

Convolutionmonotonicitynonlocal differential equationfractional differential equation

MSC classification

Primary: 34B10: Nonlocal and multipoint boundary value problems 34C11: Growth, boundedness 35B09: Positive solutions 42A85: Convolution, factorization 44A35: Convolution
Secondary: 26A33: Fractional derivatives and integrals 26A48: Monotonic functions, generalizations 34A08: Fractional differential equations 34B27: Green functions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 1 , February 2022 , pp. 1 - 27
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press.

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