Abstract
In this study, we obtain asymptotic formulas for eigenvalues and eigenfunctions of the one-dimensional Sturm–Liouville equation with one classical-type Dirichlet boundary condition and integral-type nonlocal boundary condition. We investigate solutions of special initial value problem and find asymptotic formulas of arbitrary order. We analyze the characteristic equation of the boundary value problem for eigenvalues and derive asymptotic formulas of arbitrary order. We apply the obtained results to the problem with integral-type nonlocal boundary condition.
Highlights
Consider the following one-dimensional Sturm–Liouville equation:− u′′(t) + q(t)u(t) = λu(t), t ∈ (0, 1), (1)where the real-valued function q ∈ C[0, 1]; λ = s2 is a complex spectral parameter, and s = x + ıy; x, y ∈ R.Remark 1
Asymptotic analysis of Sturm–Liouville problem with Nonlocal Boundary Condition (NBC) are valid for eigenvalues and eigenfunctions, respectively, for the SLP (1)–(2), (6)
The considered problem differs from the classical one-dimensional SLP with Boundary Condition (BC) in that it contains a NBC in two cases
Summary
Eigenvalues and eigenfunctions of BVPs with integral-type NBCs and discrete case have been investigated in [1, 6, 8, 10, 11, 16, 19]. Asymptotic analysis of Sturm–Liouville problem with NBCs are valid for eigenvalues and eigenfunctions, respectively, for the SLP (1)–(2), (6). Together with the BCs (2), (6), where the real-valued function q(t) ∈ C[0, 1]; the realvalued function ∆(t) ⩾ 0 is continuous on [0, 1], λ = s2 is a complex spectral parameter They calculate the asymptotics of eigenvalues and eigenfunctions. CE points are the first-order poles of CF Eigenvalues corresponding to these points are positive and simple. Since CF has zeros at points πk, k ∈ N, we have |xk − πk| < π
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