On calculation of eigenvalues and eigenfunctions of a Sturm-Liouville type problem with retarded argument which contains a spectral parameter in the boundary condition
© Şen and Bayramov; licensee Springer. 2011
Received: 7 June 2011
Accepted: 17 November 2011
Published: 17 November 2011
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.
Keywordsdifferential 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.
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 |γ i | + |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 .
The conditions (6) define a unique solution of Equation 1 on [2, p. 12].
The conditions (7) are defined as a unique solution of Equation 1 on .
is a such solution of Equation 1 on ; which satisfies one of the boundary conditions and both transmission conditions.
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.
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]).
From the latter discussions of , it follows that identically on . But this contradicts (6), thus completing the proof.
2 An existance theorem
By Theorem 1.1, the set of eigenvalues of boundary-value problem (1)-(5) coincides with the set of real roots of Equation 13. Let and .
- (2)Let . Then, for the solution w2 (x, λ) of Equation 9, the following inequality holds:(15)
Hence, if , we get (15).
Theorem 2. The problem (1)-(5) has an infinite set of positive eigenvalues.
Obviously, for large s, Equation 20 has an infinite set of roots. Thus, the theorem is proved.
3 Asymptotic formulas for eigenvalues and eigenfunctions
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.
The derivatives q'(x) and Δ″(x) exist and are bounded in and have finite limits and , respectively.
Δ'(x) ≤ 1 in , Δ(0) = 0 and .
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 → ∞.
Thus, we have proven the following theorem.
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.
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