In this study, a discontinuous boundary-value problem with retarded argument which contains a spectral parameter in the boundary condition and with transmission conditions at the point of discontinuity is investigated. We obtained asymptotic formulas for the eigenvalues and eigenfunctions.
MSC (2010): 34L20; 35R10.
Keywords:differential equation with retarded argument; transmission conditions; asymptotics of eigenvalues and eigenfunctions
The asymptotic formulas for the eigenvalues and eigenfunctions of boundary problem of Sturm-Liouville type for second order differential equation with retarded argument were obtained in .
The asymptotic formulas for the eigenvalues and eigenfunctions of Sturm-Liouville problem with the spectral parameter in the boundary condition were obtained in .
In the articles [7-9], the asymptotic formulas for the eigenvalues and eigenfunctions of discontinuous Sturm-Liouville problem with transmission conditions and with the boundary conditions which include spectral parameter were obtained.
In this article, we study the eigenvalues and eigenfunctions of discontinuous boundary-value problem with retarded argument and a spectral parameter in the boundary condition. Namely, we consider the boundary-value problem for the differential equation
and transmission conditions
where if and if , the real-valued function q(x) is continuous in and has a finite limit , the real-valued function Δ(x) ≥ 0 continuous in and has a finite limit , if ; if ; λ is a real spectral parameter; p1, p2, γ1, γ2, δ1, δ2 are arbitrary real numbers and |γi| + |δi| ≠ 0 for i = 1, 2. Also, γ1δ2p1 = γ2δ1p2 holds.
It must be noted that some problems with transmission conditions which arise in mechanics (thermal condition problem for a thin laminated plate) were studied in .
Lemma 1. Let w (x, λ) be a solution of Equation 1 and λ > 0. Then, the following integral equations hold:
Theorem 1. The problem (1)-(5) can have only simple eigenvalues.
be a corresponding eigenfunction. Then, from (2) and (6), it follows that the determinant
and by Theorem 2.2.2 in , the functions and are linearly dependent on . We can also prove that the functions and are linearly dependent on . Hence,
for some K1 ≠ 0 and K2 ≠ 0. We must show that K1 = K2. Suppose that K1 ≠ K2. From the equalities (4) and (10), we have
Since δ1 (K1 - K2) ≠ 0, it follows that
By the same procedure from equality (5), we can derive that
By using we may also find
2 An existance theorem
The function ω(x, λ) defined in Section 1 is a nontrivial solution of Equation 1 satisfying conditions (2), (4) and (5). Putting ω(x, λ) into (3), we get the characteristic equation
If s ≥ 2q1, we get (14). Differentiating (8) with respect to x, we have
From (16) and (14), it follows that, for s ≥ 2q1, the following inequality holds:
Theorem 2. The problem (1)-(5) has an infinite set of positive eigenvalues.
Proof. Differentiating (9) with respect to x, we get
From (8), (9), (13), (16) and (18), we get
Let λ be sufficiently large. Then, by (14) and (15), Equation 19 may be rewritten in the form
Obviously, for large s, Equation 20 has an infinite set of roots. Thus, the theorem is proved.
3 Asymptotic formulas for eigenvalues and eigenfunctions
Now, we begin to study asymptotic properties of eigenvalues and eigenfunctions. In the following, we shall assume that s is sufficiently large. From (8) and (14), we get
From (9) and (15), we get
Proof. We consider the expression which is denoted by O(1) in Equation 20. If formulas (21)-(23) are taken into consideration, it can be shown by differentiation with respect to s that for large s this expression has bounded derivative. It is obvious that for large s the roots of Equation 20 are situated close to entire numbers. We shall show that, for large n, only one root (20) lies near to each . We consider the function . Its derivative, which has the form , does not vanish for s close to n for sufficiently large n. Thus, our assertion follows by Rolle's Theorem.
The formula (24) makes it possible to obtain asymptotic expressions for eigenfunction of the problem (1)-(5). From (8), (16) and (21), we get
From (9), (22), (25) and (26), we get
By putting (24) in (25) and (27), we derive that
Hence, the eigenfunctions un(x) have the following asymptotic representation:
Under some additional conditions, the more exact asymptotic formulas which depend upon the retardation may be obtained. Let us assume that the following conditions are fulfilled:
Using (b), we have
From (25), (27) and (28), we have
Under the conditions (a) and (b), the following formulas
can be proved by the same technique in Lemma 3.3.3 in [?]. Putting these expressions into (19), we have
and using γ1δ2p1 = γ2δ1p2 we get
Thus, we have proven the following theorem.
We now may obtain a sharper asymptotic formula for the eigenfunctions. From (8) and (29),
Replacing s by sn and using (32), we have
From (16) and (29), we have
From (9), (30), (31), (33) and (35), we have
Now, replacing s by sn and using (32), we have
Thus, we have proven the following theorem.
Theorem 5. If conditions (a) and (b) are satisfied, then the eigenfunctions un(x) of the problem (1)-(5) have the following asymptotic representation for n → ∞:
where u1n(x) and u2n(x) defined as in (34) and (36), respectively.
In this study, first, we obtain asymptotic formulas for eigenvalues and eigenfunctions for discontinuous boundary-value problem with retarded argument which contains a spectral parameter in the boundary condition. Then, under additional conditions (a) and (b) the more exact asymptotic formulas, which depend upon the retardation obtained.
5 Competing interests
The authors declare that they have no completing interests.
6 Authors' contributions
Establishment of the problem belongs to AB (advisor). ES obtained the asymptotic formulas for eigenvalues and eigenfunctions. All authors read and approved the final manuscript.
Demidenko, GV, Likhoshvai, VA: On differential equations with retarded argument. Sib Mat Zh. 46, 417–430 (2005). Publisher Full Text
Bayramov, A, Calıṣkan, S, Uslu, S: Computation of eigenvalues and eigen-functions of a discontinuous boundary value problem with retarded argument. Appl Math Comput. 191, 592–600 (2007). Publisher Full Text
Mukhtarov, OSH, Kadakal, M, Muhtarov, FŞ: Eigenvalues and normalized eigenfunctions of discontinuous Sturm-Liouville problem with transmission conditions. Rep Math Phys. 54(1), 41–56 (2004). Publisher Full Text
Altinisik, N, Kadakal, M, Mukhtarov, OSH: Eigenvalues and eigenfunctions of discontinuous Sturm-Liouville problems with eigenparameter-dependent boundary conditions. Acta Math Hungar. 102(1-2), 159–175 (2004)
Akdoğan, Z, Demirci, M, Mukhtarov, OSH: Discontinuous Sturm-Liouville problems with eigenparameter-dependent boundary and transmission conditions. Acta Appl Math. 86, 329–344 (2005). Publisher Full Text
Titeux, I, Yakubov, Y: Completeness of root functions for thermal conduction in a strip with piecewise continuous coefficients. Math Models Methods Appl Sci. 7(7), 1035–1050 (1997). Publisher Full Text