Ambrosetti–Prodi Type Results for Multiparameter Neumann Systems with Mean Curvature Operator in Minkowski Space

Based on the monotone iteration method and Leray–Schauder degree theory, we obtain the Ambrosetti-Prodi type results for multiparameter Neumann systems with mean curvature operator in Minkowski space \( \left\{ \begin{array}{ll} -(\frac{u'}{\sqrt{1-u'^{2}}})'=f(x,u,v)+r+h(x), & x\in (0,1), \\ -(\frac{v'}{\sqrt{1-v'^{2}}})'=g(x,u,v)+s+l(x), & x\in (0,1), \\ u'(0)=u'(1)=0,\ v'(0)=v'(1)=0, \end{array} \right. \) $$\left\{\begin{array}{cc}-{(\frac{u^{\prime }}{\sqrt{1-u^{\prime 2}}})}^{\prime }=f(x,u,v)+r+h(x),\hfill & x\in (0,1),\hfill \\ -{(\frac{v^{\prime }}{\sqrt{1-v^{\prime 2}}})}^{\prime }=g(x,u,v)+s+l(x),\hfill & x\in (0,1),\hfill \\ u^{\prime }(0)=u^{\prime }(1)=0,v^{\prime }(0)=v^{\prime }(1)=0,\hfill \end{array}\right)$$ where  \(f,g\in C([0,1]\times \mathbb {R}\times \mathbb {R}),\) $f,g\in C([0,1]\times \mathbb{R}\times \mathbb{R}),$ \(r,s\in \mathbb {R}\) $r,s\in \mathbb{R}$  are parameters, \(h,l\in C[0,1],\) $h,l\in C[0,1],$  and   \(\int _{0}^{1}h(x){\text {d}}x=0,\int _{0}^{1}l(x){\text {d}}x=0.\) ${\int }_0^{1}h(x)\text{d}x=0,{\int }_0^{1}l(x)\text{d}x=0.$