Electron. J. Diff. Equ., Vol. 2015 (2015), No. 161, pp. 1-16.

Extremal points for a higher-order fractional boundary-value problem

Aijun Yang, Johnny Henderson, Charles Nelms Jr.

Abstract:
The Krein-Rutman theorem is applied to establish the extremal point, $b_0$, for a higher-order Riemann-Liouville fractional equation, $D_{0+}^{\alpha}y+p(t)y = 0$, $0 <t <b$, $n < \alpha \leq n+1$, $n\geq 2$, under the boundary conditions $y^{(i)}(0)= 0$, $y^{(n-1)}(b) = 0$, $i=0,1,2,\ldots, n-1$. The key argument is that a mapping, which maps a linear, compact operator, depending on $b$ to its spectral radius, is continuous and strictly increasing as a function of b. Furthermore, we also treat a nonlinear problem as an application of the result for the extremal point for the linear case.

Submitted May 19, 2015. Published June 16, 2015.
Math Subject Classifications: 26A33, 34B08, 34B40.
Key Words: u-positive operator; fractional boundary value problem; spectral radius

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Aijun Yang
Zhejiang University of Technology
College of Science
Hangzhou 310023, China
email: yangaij2004@163.com, Aijun_Yang@baylor.edu
Johnny Henderson
Department of Mathematics, Baylor University
Waco, TX 76798-7328, USA
email: Johnny_Henderson@baylor.edu
  Charles Nelms Jr.
Department of Mathematics, Baylor University,
Waco, TX 76798-7328, USA
email: Charles_Nelms@baylor.edu

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