The authors study the existence and uniqueness of solutions for a boundary value problem consisting of the higher-order differential equation \[ x^{(l)}(s)+g(s,x(s))=0,\ s\in[c,d], \] subject to the three-point nonhomogeneous conditions \[ x(c)=0,\ x'(c)=0,\ x''(c)=0,\ \dots,\ x^{(l-2)}(c)=0,\] \[x^{(l-2)}(d)-\beta\ x^{(l-2)}(\eta)=\gamma. \] The main tools in the proofs are the Banach fixed point theorem and the Rus fixed point theorem.
First, the authors rewrite the main problem as an equivalent integral equation. Then, the authors define an operator in a suitable set of functions and prove that the operator has a unique fixed point.
In fact, the main results state that if the nonlinearity in the equation satisfies a Lipschitz condition and the length of the interval in the boundary conditions is not large, then the problem has a unique solution.
Examples, which illustrate the applicability of the obtained results, are provided at the end of the article.
The obtained results continue studies of the subject and may be useful for researchers in the field. The list of references contains 26 items.