Maximum and anti-maximum principles and monotone methods for boundary value problems for Riemann-Liouville fractional differential equations in neighborhoods of simple eigenvalues

It has been shown that, under suitable hypotheses, boundary value problems of the form, $Ly+\lambda y=f,$ $BC y =0$ where $L$ is a linear ordinary or partial differential operator and $BC$ denotes a linear boundary operator, then there exists $\Lambda >0$ such that $f\ge 0$ implies $\lambda y \ge...

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Bibliographic Details
Published in:Cubo (Temuco, Chile) Vol. 25; no. 2; pp. 251 - 272
Main Authors: Eloe, Paul W., Neugebauer, Jeffrey T.
Format: Journal Article
Language:English
Published: Universidad de La Frontera. Departamento de Matemática y Estadística 01.08.2023
Universidad de La Frontera
Subjects:
anti-maximum principle
boundary value problem
MATHEMATICS
MATHEMATICS, APPLIED
maximum principle
monotone methods
riemann-liouville fractional differential equation
upper and lower solution
Maximum principle
Riemann-Liouville fractional differential equation
boundary value problem
anti-maximum principle
upper and lower solution
monotone methods
ISSN:0719-0646, 0716-7776, 0719-0646
Online Access:Get full text
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Summary:It has been shown that, under suitable hypotheses, boundary value problems of the form, $Ly+\lambda y=f,$ $BC y =0$ where $L$ is a linear ordinary or partial differential operator and $BC$ denotes a linear boundary operator, then there exists $\Lambda >0$ such that $f\ge 0$ implies $\lambda y \ge 0$ for $\lambda\in [-\Lambda ,\Lambda ]\setminus\{0\},$ where $y$ is the unique solution of $Ly+\lambda y=f,$ $BC y =0$. So, the boundary value problem satisfies a maximum principle for $\lambda\in [-\Lambda ,0)$ and the boundary value problem satisfies an anti-maximum principle for $\lambda\in (0, \Lambda ]$. In an abstract result, we shall provide suitable hypotheses such that boundary value problems of the form, $D_{0}^{\alpha}y+\beta D_{0}^{\alpha -1}y=f,$ $BC y =0$ where $D_{0}^{\alpha}$ is a Riemann-Liouville fractional differentiable operator of order $\alpha$, $1<\alpha \le 2$, and $BC$ denotes a linear boundary operator, then there exists $\mathcal{B} >0$ such that $f\ge 0$ implies $\beta D_{0}^{\alpha -1}y \ge 0$ for $\beta \in [-\mathcal{B} ,\mathcal{B} ]\setminus\{0\},$ where $y$ is the unique solution of $D_{0}^{\alpha}y+\beta D_{0}^{\alpha -1}y =f,$ $BC y =0$. Two examples are provided in which the hypotheses of the abstract theorem are satisfied to obtain the sign property of $\beta D_{0}^{\alpha -1}y.$ The boundary conditions are chosen so that with further analysis a sign property of $\beta y$ is also obtained. One application of monotone methods is developed to illustrate the utility of the abstract result.
ISSN:0719-0646
0716-7776
0719-0646
DOI:10.56754/0719-0646.2502.251