Eur. Phys. J. Special Topics 226, 3445–3456 (2017)
https://doi.org/10.1140/epjst/e2018-00006-0

Regular Article

Extremal points for fractional boundary value problems

¹ Baylor University, Department of Mathematics, Waco, TX 76798-7328, USA
² Wayland Baptist University, School of Math and Science, MSB 124, Plainview, TX 79072, USA
³ Zhejiang University of Technology, College of Science, Hangzhou 310023, P.R. China

^a e-mail: Johnny_Henderson@baylor.edu

Received: 7 April 2017
Received in final form: 31 May 2017
Published online: 25 July 2018

Abstract

This article is concerned with characterizing the first extremal point, b₀, for a Riemann–Liouville fractional boundary value problem, D^α₀₊y + p(t)y = 0, 0 < t < b, y(0) = y′(0) = y″(b) = 0, 2 < α ≤ 3, by applying the theory of u₀-positive operators with respect to a suitable cone in a Banach space. 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, an application to a nonlinear case is given.