Vol. 10, No. 1, 2017

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ISSN: 1944-4184 (e-only)
ISSN: 1944-4176 (print)
The multiplicity of solutions for a system of second-order differential equations

Olivia Bennett, Daniel Brumley, Britney Hopkins, Kristi Karber and Thomas Milligan

Vol. 10 (2017), No. 1, 77–87
Abstract

Making use of the Guo–Krasnosel’skiĭ fixed point theorem multiple times, we establish the existence of at least three positive solutions for the system of second-order differential equations u(t) = g(t,u(t),u(t),v(t),v(t)) and v(t) = λf(t,u(t),u(t),v(t),v(t)) for t (0,1) with right focal boundary conditions u(0) = v(0) = 0, u(1) = a, and v(1) = b, where f,g : [0,1] × [0,)4 [0,) are continuous, a,b,λ 0, and a + b > 0. Our technique involves transforming the system of differential equations to a new system with homogeneous boundary conditions prior to applying the aforementioned fixed point theorem.

Keywords
differential equations, boundary value problem, multiple solutions, positive solutions
Mathematical Subject Classification 2010
Primary: 34B18
Milestones
Received: 31 August 2015
Accepted: 14 January 2016
Published: 11 October 2016

Communicated by John Baxley
Authors
Olivia Bennett
Department of Mathematics and Statistics
University of Central Oklahoma
100 N. University Drive
Edmond, OK 73034
United States
Daniel Brumley
Department of Mathematics and Statistics
University of Central Oklahoma
100 N. University Drive
Edmond, OK 73034
United States
Britney Hopkins
Department of Mathematics and Statistics
University of Central Oklahoma
100 N. University Drive
Edmond, OK 73034
United States
Kristi Karber
Department of Mathematics and Statistics
University of Central Oklahoma
100 N. University Drive
Edmond, OK 73034
United States
Thomas Milligan
Department of Mathematics and Statistics
University of Central Oklahoma
100 N. University Drive
Edmond, OK 73034
United States