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

 Download this article For screen For printing  Recent Issues  The Journal About the Journal Editorial Board Subscriptions Editors’ Interests Scientific Advantages Submission Guidelines Submission Form Ethics Statement Editorial Login ISSN: 1944-4184 (e-only) ISSN: 1944-4176 (print) Author Index Coming Soon Other MSP Journals
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 