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

Published online by Cambridge University Press:  23 December 2020

Christopher Goodrich
Affiliation:
School of Mathematics and Statistics, UNSW Australia, Sydney, NSW, 2052, Australia (c.goodrich@unsw.edu.au)
Carlos Lizama
Affiliation:
Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Las Sophoras 173, 9160000, Santiago, Chile (carlos.lizama@usach.cl)

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.

Type
Research Article
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press.

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