Lower and Upper Solutions for System of Differential Equations Involving Homeomorphism and Nonlinear Boundary Conditions

We study the existence of solution to the system of differential equations \((\phi (u'))'=f(t,u,u')\) ${(\varphi (u^{\prime }))}^{\prime }=f(t,u,u^{\prime })$ with nonlinear boundary conditions \(\begin{aligned} g(u(0),u,u')=0, \quad h(u'(1),u,u')=0, \end{aligned}\) $$\begin{array}{c}\hfill g(u(0),u,u^{\prime })=0,h(u^{\prime }(1),u,u^{\prime })=0,\end{array}$$ where \(f:[0,1]\times \mathbb {R}^{n}\times \mathbb {R}^{n}\rightarrow \mathbb {R}^{n}\) $f:[0,1]\times {\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ , \(g,h:\mathbb {R}^{n}\times C([0,1],\mathbb {R}^{n})\times C([0,1],\mathbb {R}^{n})\rightarrow \mathbb {R}^{n}\) $g,h:{\mathbb{R}}^n\times C([0,1],{\mathbb{R}}^n)\times C([0,1],{\mathbb{R}}^n)\to {\mathbb{R}}^n$ are continuous, \(\phi :\prod _{i=1}^{n}(-a_i,a_i) \rightarrow \mathbb {R}^{n}\) $\varphi :{\prod }_{i=1}^n(-a_i,a_i)\to {\mathbb{R}}^n$ , \(0<a_i\le +\infty \) $0<a_i\le +\infty$ , \(\phi (s)=\left( \phi _1(s_1),\dots ,\phi _n(s_n)\right) \) $\varphi (s)=\left({\varphi }_1,(s_1),,,\cdots ,,,{\varphi }_n,(s_n)\right)$ and \(\phi _i:(-a_i,a_i)\rightarrow \mathbb {R}\) ${\varphi }_i:(-a_i,a_i)\to \mathbb{R}$ is a one dimensional regular or singular homeomorphism. Our proofs are based on the concept of the lower and upper solutions.