The existence of solutions to higher-order differential equations with nonhomogeneous conditions

We prove the existence and uniqueness of solutions to the differential equations of higher order \({x}^{\left(l\right)}\left(s\right)+g\left(s,x\left(s\right)\right)=0,s\in \left[c,d\right],\) satisfying three-point boundary conditions that contain a nonhomogeneous term \(x\left(c\right)=0,{x}{\prime}\left(c\right)=0,{x}^{^{\prime\prime} }\left(c\right)=0,\dots {x}^{\left(l-2\right)}\left(c\right)=0,{x}^{\left(l-2\right)}\left(d\right)-{\beta x}^{\left(l-2\right)}\left(\eta \right)=\upgamma ,\) where \(l\ge \mathrm{3,0}\le c<\eta <d,\) the constants \(\beta ,\upgamma \) are real numbers, and \(g:\left[c,d\right]\times {\mathbb{R}}\to {\mathbb{R}}\) is a continuous function. By using finer bounds on the integral of kernel, the Banach and Rus fixed point theorems on metric spaces are utilized to prove the existence and uniqueness of a solution to the problem.