The multiplicity of solutions for a system of second-order differential equations

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

doi:10.2140/involve.2017.10.77

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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

DOI: 10.2140/involve.2017.10.77

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 $- v^{''} (t) = λ f (t, u (t), u^{'} (t), v (t), v^{'} (t))$ for $t \in (0, 1)$ with right focal boundary conditions $u (0) = v (0) = 0$ , $u^{'} (1) = a$ , and $v^{'} (1) = b$ , where $f, g : [0, 1] \times [0, \infty)$ $^{4} \to [0, \infty)$ are continuous, $a, b, λ \geq 0$ , 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

Vol. 10, No. 1, 2017