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((l))(s) + g(s, x(s)) = 0,s is an element of[c, d], satisfying three-point boundary conditions that contain a nonhomogeneous term x(c) = 0,x '(c) = 0,x ''(c) = 0,...,x((l-2))(c) = 0,x((l-2))(d)-beta x((l-2))(eta)=gamma, where l >= 3,0 <= c < eta < d, the constants beta,gamma are real numbers, and g:[c, d]xR -> 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.