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

• Paul W. Eloe
• Jeffrey T. Neugebauer

Abstract

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.

Mathematics Subject Classification:

• Department of Mathematics, University of Dayton, Dayton, Ohio 45469, USA.
• Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, Kentucky 40475, USA.
• Pages: 251–272
• Date Published: 2023-08-07
• Vol. 25 No. 2 (2023)

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2023-08-07

How to Cite

[1]
P. W. Eloe and J. T. Neugebauer, “Maximum, anti-maximum principles and monotone methods for boundary value problems for Riemann-Liouville fractional differential equations in neighborhoods of simple eigenvalues”, CUBO, pp. 251–272, Aug. 2023.

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