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First extremal point comparison for a fractional boundary value problem with a fractional boundary condition


Authors: Johnny Henderson and Jeffrey T. Neugebauer
Journal: Proc. Amer. Math. Soc. 147 (2019), 5323-5327
MSC (2010): Primary 26A33, 34A08; Secondary 34A40, 26D20
DOI: https://doi.org/10.1090/proc/14648
Published electronically: June 14, 2019
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Abstract: Let $ n \geq 2$ be a natural number, and let $ n-1<\alpha \le n$ and $ 0<\gamma \le \alpha -1$ be real numbers. Let $ \beta >0$ and $ b\in (0,\beta ]$. We compare first extremal points of the differential equations $ D_{0+}^\alpha u+p(t)u=0$, $ D_{0+}^\alpha u+q(t)u=0$, $ t\in (0,\beta )$, each of which satisfies the boundary conditions $ u^{(i)}(0)=0$, $ i=0,1,\dots ,n-2$, $ \quad D_{0^+}^\gamma u(b)=0$. While it is assumed that $ q$ is nonnegative, no sign restrictions are put on $ p$. The fact that the associated Green's function $ G(b;t,s)$ is nonnegative and increasing with respect to $ b$ plays an important role in the analysis.


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Additional Information

Johnny Henderson
Affiliation: Department of Mathematics, Baylor University, Waco, Texas 76798-7328
Email: johnny_henderson@baylor.edu

Jeffrey T. Neugebauer
Affiliation: Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, Kentucky 40475
Email: jeffrey.neugebauer@eku.edu

DOI: https://doi.org/10.1090/proc/14648
Keywords: Fractional boundary value problem, extremal point
Received by editor(s): January 26, 2019
Received by editor(s) in revised form: March 13, 2019, and March 28, 2019
Published electronically: June 14, 2019
Communicated by: Wenxian Shen
Article copyright: © Copyright 2019 American Mathematical Society