#### Vol. 10, No. 1, 2017

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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
##### 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 $-{u}^{\prime \prime }\left(t\right)=g\left(t,u\left(t\right),{u}^{\prime }\left(t\right),v\left(t\right),{v}^{\prime }\left(t\right)\right)$ and $-{v}^{\prime \prime }\left(t\right)=\lambda f\left(t,u\left(t\right),{u}^{\prime }\left(t\right),v\left(t\right),{v}^{\prime }\left(t\right)\right)$ for $t\in \left(0,1\right)$ with right focal boundary conditions $u\left(0\right)=v\left(0\right)=0$, $\phantom{\rule{0.3em}{0ex}}{u}^{\prime }\left(1\right)=a$, and ${v}^{\prime }\left(1\right)=b$, where $f,g:\left[0,1\right]×\left[0,\infty {}^{\right)}4\to \left[0,\infty \right)$ are continuous, $a,b,\lambda \ge 0$, 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
Primary: 34B18
##### Milestones
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