Abstract
In this study, a discontinuous boundaryvalue 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 eigenfunctions1 Introduction
Boundaryvalue problems for differential equations of the second order with retarded argument were studied in [15], and various physical applications of such problems can be found in [2].
The asymptotic formulas for the eigenvalues and eigenfunctions of boundary problem of SturmLiouville type for second order differential equation with retarded argument were obtained in [5].
The asymptotic formulas for the eigenvalues and eigenfunctions of SturmLiouville problem with the spectral parameter in the boundary condition were obtained in [6].
In the articles [79], the asymptotic formulas for the eigenvalues and eigenfunctions of discontinuous SturmLiouville 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 boundaryvalue problem with retarded argument and a spectral parameter in the boundary condition. Namely, we consider the boundaryvalue problem for the differential equation
and transmission conditions
where if and if , the realvalued function q(x) is continuous in and has a finite limit , the realvalued function Δ(x) ≥ 0 continuous in and has a finite limit , if ; if ; λ is a real spectral parameter; p_{1}, p_{2}, γ_{1}, γ_{2}, δ_{1}, δ_{2 }are arbitrary real numbers and γ_{i} + δi ≠ 0 for i = 1, 2. Also, γ_{1}δ_{2}p_{1 }= γ_{2}δ_{1}p_{2 }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 [10].
Let w_{1}(x, λ) be a solution of Equation 1 on , satisfying the initial conditions
The conditions (6) define a unique solution of Equation 1 on [2, p. 12].
After defining above solution, we shall define the solution w_{2 }(x, λ) of Equation 1 on by means of the solution w_{1}(x, λ) by the initial conditions
The conditions (7) are defined as a unique solution of Equation 1 on .
Consequently, the function w (x, λ) is defined on by the equality
is a such solution of Equation 1 on ; which satisfies one of the boundary conditions and both transmission conditions.
Lemma 1. Let w (x, λ) be a solution of Equation 1 and λ > 0. Then, the following integral equations hold:
Proof. To prove this, it is enough to substitute and instead of and in the integrals in (8) and (9), respectively, and integrate by parts twice.
Theorem 1. The problem (1)(5) can have only simple eigenvalues.
Proof. Let be an eigenvalue of the problem (1)(5) and
be a corresponding eigenfunction. Then, from (2) and (6), it follows that the determinant
and by Theorem 2.2.2 in [2], the functions and are linearly dependent on . We can also prove that the functions and are linearly dependent on . Hence,
for some K_{1 }≠ 0 and K_{2 }≠ 0. We must show that K_{1 }= K_{2}. Suppose that K_{1 }≠ K_{2}. From the equalities (4) and (10), we have
Since δ_{1 }(K_{1 } K_{2}) ≠ 0, it follows that
By the same procedure from equality (5), we can derive that
From the fact that is a solution of the differential equation (1) on and satisfies the initial conditions (11) and (12) it follows that identically on (cf. [2, p. 12, Theorem 1.2.1]).
By using we may also find
From the latter discussions of , it follows that identically on . But this contradicts (6), thus completing the proof.
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
By Theorem 1.1, the set of eigenvalues of boundaryvalue problem (1)(5) coincides with the set of real roots of Equation 13. Let and .
Lemma 2. (1) Let . Then, for the solution w_{1}(x, λ) of Equation 8, the following inequality holds:
(2) Let . Then, for the solution w_{2 }(x, λ) of Equation 9, the following inequality holds:
Proof. Let . Then, from (8), it follows that, for every λ > 0, the following inequality holds:
If s ≥ 2q_{1}, we get (14). Differentiating (8) with respect to x, we have
From (16) and (14), it follows that, for s ≥ 2q_{1}, the following inequality holds:
Hence,
Let . Then, from (9), (14) and (17), it follows that, for s ≥ 2q_{1}, the following inequalities 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
The existence and continuity of the derivatives for , and for , follows from Theorem 1.4.1 in [?].
Theorem 3. Let n be a natural number. For each sufficiently large n, there is exactly one eigenvalue of the problem (1)(5) near .
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.
Let n be sufficiently large. In what follows, we shall denote by the eigenvalue of the problem (1)(5) situated near . We set . From (20), it follows that . Consequently
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 u_{n}(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:
(a) The derivatives q'(x) and Δ″(x) exist and are bounded in and have finite limits and , respectively.
(b) Δ'(x) ≤ 1 in , Δ(0) = 0 and .
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 γ_{1}δ_{2}p_{1 }= γ_{2}δ_{1}p_{2 }we get
Dividing by s and using , we have
Hence,
and finally
Thus, we have proven the following theorem.
Theorem 4. If conditions (a) and (b) are satisfied, then the positive eigenvalues of the problem (1)(5) have the (32) asymptotic representation for n → ∞.
We now may obtain a sharper asymptotic formula for the eigenfunctions. From (8) and (29),
Replacing s by s_{n }and using (32), we have
From (16) and (29), we have
From (9), (30), (31), (33) and (35), we have
Now, replacing s by s_{n }and using (32), we have
Thus, we have proven the following theorem.
Theorem 5. If conditions (a) and (b) are satisfied, then the eigenfunctions u_{n}(x) of the problem (1)(5) have the following asymptotic representation for n → ∞:
where u_{1n}(x) and u_{2n}(x) defined as in (34) and (36), respectively.
4 Conclusion
In this study, first, we obtain asymptotic formulas for eigenvalues and eigenfunctions for discontinuous boundaryvalue 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.
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