NTMSCI 6, No. 2, 247-257 (2018) 247
New Trends in Mathematical Sciences
http://dx.doi.org/10.20852/ntmsci.2018.288
Asymptotics of eigenvalues for Sturm-Liouville problem
with eigenparameter-dependent boundary conditions
Elif Baskaya
Department of Mathematics, Karadeniz Technical University, Trabzon, Turkey
Received: 15 December 2017, Accepted: 25 December 2017
Published online: 30 June 2018.
Abstract: In this paper, we obtain asymptotic estimates of eigenvalues for regular Sturm-Liouville problems having the eigenvalue
parameter in all boundary conditions with the potential that is continuous and its differentiation exists and is integrable.
Keywords: Sturm-Liouville Problems, integrable potential, eigenvalues, asymptotics.
1 Introduction
In this paper, we consider the boundary value problem
y
′′
(t )+ {λ − q (t )} y (t )= 0, t ∈ [a, b] , (1)
a
1
y (a)+ a
2
y
′
(a)= λ
a
′
1
y (a)+ a
′
2
y
′
(a)
, (2)
b
1
y (b)+ b
2
y
′
(b)= λ
b
′
1
y (b)+ b
′
2
y
′
(b)
, (3)
where λ is a real parameter; q (t ) is a real-valued function; a
i
, a
′
i
, b
i
, b
′
i
∈ R, i = 0, 1. Also we assume that q (t ) is
continuous, its differentiation exists and is integrable. This problem differs from the usual regular Sturm-Liouville
problem in the sense that eigenvalue parameter λ is contained in the boundary condition at a. Problems of this type arise
from the method of separation of variables applied to mathematical models for certain physical problems including heat
conduction and wave propagation, etc. [8]. It is shown by Walter [15] that this problem is self-adjoint problem. The
purpose of this paper is to obtain asymptotic approximations for the eigenvalues of (1)-(3).
Approximations of this type have been derived before. We mention in particular [7, 8] and [2]. Fulton’s approach in [7] is
based on an iteration of the usual Volterra integral equation, producing an asymptotic expansion of the solution in higher
powers of 1/λ
1/2
as λ → ∞ and in [8] is based on the analysis of [14] for regular Sturm-Liouville problems on a finite
closed interval and involves some operator-theoretical results of [15]. The approach used in [2] is based on an iterative
procedure solving the associated Riccati equation and producing an asymptotic expansion of the solution in the higher
powers of 1/λ
1/2
as λ → ∞ for smooth q (t ) . There is also a vast amount of literature dealing with asymptotic estimates
of eigenvalues for standart Sturm-Liouville problems with regular endpoints [3, 4, 5, 6, 9, 10, 11, 13, 14]. Here we follow
the similar approach in [4, 10, 12]. We assume without loss of generality, that q (t ) has mean value zero. That is
b
a
q (t ) dt = 0.
2 Conclusion
Theorem 1. The eigenvalues λ
n
of (1)-(3) satisfy as λ → ∞,
∗
Corresponding author e-mail: elifbekar@ktu.edu.tr
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