Asymptotic analysis of Sturm–Liouville problem with Neumann and nonlocal two-point boundary conditions

In this study, we obtain asymptotic expansions for eigenvalues and eigenfunctions of the one-dimensional Sturm–Liouville equation with one classical Neumann-type boundary condition and a two-point nonlocal boundary condition. We investigate solutions of special initial value problem and find their asymptotic expansions of any order. We analyze the characteristic equation of the boundary value problem for eigenvalues and derive asymptotic expansions of arbitrary order. We apply the obtained results to the problem with two-point nonlocal boundary condition.