#### 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