Boundary value problems of higher order have been examined due to their mathematical importance and applications in diversified applied sciences. The higher-order boundary value problems occur in the study of fluid dynamics, astrophysics, hydrodynamic, hydromagnetic stability, astronomy, beam and long wave theory, induction motors, engineering, and applied physics. Such problems have been studied by many authors. For example, Oderinu [
24] applied weighted residual via partition method to obtain a numerical solution for 10th and 12th order linear and nonlinear boundary value problems. Mohyud-Din and Yildirim [
22] used modified variational iteration method for solving 9th and 10th order boundary value problems. Iqbal et al. [
11] constructed a cubic spline algorithm to approximate 10th order boundary value problems. Noor et al. [
23] applied variational iterative method to solve 10th order boundary value problems. Mai-Duy [
20] have presented Chebyshev spectral collocation method to solve high-order ordinary differential equations. Islam et al. [
13] solved 10th and 12th order linear and nonlinear boundary value problems numerically by the Galerkin weighted residual technique with two point boundary conditions. Akgül et al. [
5] have given some reproducing kernel functions to find approximate solutions of 10th order boundary value problems. Jørgensen et al. [
15] have presented a hierarchical basis of arbitrary order for integral equations solved with the method of moments (MoM). Akgül et al. [
4] implemented reproducing kernel Hilbert space method to obtain approximate solutions to 10th order boundary value problems. Ramadan et al. [
25] used homotopy analysis method to solve 7th, 8th, and 10th order boundary value problems. Ramos and Singh [
26] have presented a two-step hybrid block method for the numerical integration of ordinary differential initial value systems. Siddiqi and Twizell [
30] developed an algorithm to approximate the solutions, and their higher-order derivatives, of differential equations. Siddiqi and Akram [
28] found numerical solutions for 10th-order linear special case boundary value problems using an 11th degree spline. The same authors in [
29] used a nonpolynomial spline to obtain numerical solutions for 10th order linear special case boundary value problems. Twizell et al. [
31] have developed 2nd order finite-difference methods to obtain the numerical solutions for 8th, 10th, and 12th order eigenvalue problems. Wazwaz [
32] proposed an algorithm for solving linear and nonlinear boundary value problems with two-point boundary conditions of 10th and 12th order. Ma [
19] has given existence and uniqueness theorems based on the Leray–Schauder fixed point theorem for some 4th order nonlinear boundary value problems. Zvyagin and Baranovskii [
33] have constructed a topological characteristic to investigate a class of controllable systems. Ahmad and Ntouyas [
3] conferred some existence results based on some standard fixed point theorems and Leray–Schauder degree theory for a higher-order nonlinear differential equation with four-point nonlocal integral boundary conditions. In [
14], a theorem of coupled fixed point on ordered sets has been proved, and its results have been used to obtain the existence and uniqueness of positive solution for a class of boundary value problems for fractional differential equations with singularities. Afsari et al. [
2] introduced some new coupled fixed point theorems which have been used for finding a solution to a fractional differential equation of order
\(\alpha\in(0,1)\). In [
6], generalized
α‐
ψ-contractive mappings have been introduced in metric-like spaces and some fixed point theorems have been proved. Such results are applied to a two-point boundary value problem for 2nd order differential equations. In [
17], a new notion of Berinde type
\((\alpha,\psi)\) contraction and the existence and uniqueness of a fixed point for such mapping have been proved, and an application to nonlinear fractional differential equation was given. Aydi et al. [
7] improved and extended a previous proof for existence and uniqueness for a differential problem with fixed point results by replacing
α-admissibility with orbital
α-admissibility. In the framework of extended b-metric spaces, Abdeljawad et al. [
1] suggested fixed points results to a nonlinear Volterra–Fredholm integral equation and a Caputo fractional derivative differential equation. Karapinar et al. [
16] unified existing fixed point results in the literature to show the existence of solutions for 2nd order nonlinear differential equations and Caputo fractional derivative boundary value problem of order
\(\beta\in[1,2]\). More about the results related to the fixed point theory can be found in [
10] and [
21]. Motivated by these studies, we investigate the generic differential equation of order 2
n and hence seek results on the existence of solutions for 10th order boundary value problems.