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

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 $u\left(0\right)=v\left(0\right)=0$, $u^{\prime }\left(1\right)=a$, and $v^{\prime }\left(1\right)=b$, where $f,g:\left[0,1\right]\times [0,\infty {}^{)}4\to \left[0,\infty \right)$ 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.