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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 280970, 8 pages
Asymptotics of the Eigenvalues of a Self-Adjoint Fourth Order Boundary Value Problem with Four Eigenvalue Parameter Dependent Boundary Conditions
The John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the Witwatersrand (Wits), Private Bag 3, Johannesburg 2050, South Africa
Received 9 April 2013; Revised 24 May 2013; Accepted 27 May 2013
Academic Editor: Janusz Matkowski
Copyright © 2013 Manfred Möller and Bertin Zinsou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Considered is a regular fourth order ordinary differential equation which depends quadratically on the eigenvalue parameter and which has separable boundary conditions depending linearly on . It is shown that the eigenvalues lie in the closed upper half plane or on the imaginary axis and are symmetric with respect to the imaginary axis. The first four terms in the asymptotic expansion of the eigenvalues are provided.
Sturm-Liouville problems have attracted extensive attention due to their intrinsic mathematical challenges and their applications in physics and engineering. Classical Sturm-Liouville problems have been extended to higher-order differential equations and to differential equations with eigenvalue parameter dependent boundary conditions. For example, the generalized Regge problem is realised by a second order differential operator which depends quadratically on the eigenvalue parameter and which has eigenvalue parameter dependent boundary conditions, see . The particular feature of this problem is that the coefficient operators of this pencil are self-adjoint, and it is shown in  that this gives some a priori knowledge about the location of the spectrum. In , this approach has been extended to a fourth order differential equation describing small transversal vibrations of a homogeneous beam compressed or stretched by a force . Again, this problem is represented by a quadratic operator pencil, in a suitably chosen Hilbert space, whose coefficient operators are self-adjoint. In , we have considered this fourth order differential equation with general two-point boundary conditions which depend linearly on the eigenvalue parameter. Necessary and sufficient conditions such that the associated operator pencil consists of self-adjoint operators have been obtained. In [4, 5], we have derived eigenvalue asymptotics associated with particular boundary conditions. In this paper, we are considering the case of separable boundary conditions where all four of these boundary conditions depend on the eigenvalue parameter.
Other recent results on fourth order differential operators whose boundary conditions depend on the eigenvalue parameter but which are represented by linear operator pencils, include spectral asymptotics and basis properties, see [6–8].
In Section 2, we introduce the operator pencil associated with the eigenvalue problem (1), (2), and we derive the boundary conditions such that the operators in the pencil are self-adjoint. In Section 3, we obtain the location of the spectrum and the asymptotic distribution of the eigenvalues for the case . In Section 4, we prove that the boundary value problem under investigation is Birkhoff regular, which implies that the eigenvalues for general are small perturbations of the eigenvalues for . Hence, in Section 5, we derive the first four terms of the asymptotics of the eigenvalues and compare them to those obtained for the boundary conditions considered in .
2. The Quadratic Operator Pencil
On the interval , we consider the boundary value problem where is a real valued function and (2) is separated boundary conditions with the depending linearly on the eigenvalue parameter . The boundary conditions (2) are taken at the endpoint for and at the endpoint for . Further, we assume for simplicity that where for and for , , , and . We recall that the quasi-derivatives associated with (1) are given by see [9, page 26]. In order to have independent boundary conditions, we will also assume that the numbers , , , as well as the numbers , , , are mutually disjoint.
Recall that in applications, using separation of variables, the parameter emanates from derivatives with respect to the time variable in the original partial differential equation, and it is reasonable that the highest space derivative occurs in the term without time derivative. Thus, the most relevant boundary conditions would have for .
Further assumptions on the , , and will be made later and will be justified by the requirements on the operator pencil which we are going to define now.
We denote by the collection of the boundary conditions (2) and define the following operators related to : where is the Sobolev space of order on the interval . We consider the linear operators , , and in the space with domains given by It is easy to check that , , , and . We associate a quadratic operator pencil in the space with the problem (1), (2). We observe that (8) is an operator representation of the eigenvalue problem (1), (2) in the sense that a function satisfies (1), (2) if and only if .
Proposition 1. The differential operator associated with (1), (2) is self-adjoint if and only if for ; the numbers , , and satisfy the following conditions: , if is even in case or odd in case , and , otherwise.
Proposition 2. The operator pencil is a Fredholm valued operator function with index . The spectrum of the Fredholm operator consists of discrete eigenvalues of finite multiplicities, and all eigenvalues of , , lie in the closed upper half-plane and on the imaginary axis and are symmetric with respect to the imaginary axis.
Proof. As in [2, Section 3], we can argue that for all , is a relatively compact perturbation of , where is well known to be a Fredholm operator. The statement on the location of the spectrum follows as in [2, Lemma 3.1].
3. Asymptotics of Eigenvalues for
In this section, we investigate the boundary value problem (1), (2) with . We count all eigenvalues with their proper multiplicities and develop a formula for the asymptotic distribution of the eigenvalues, which we will use to obtain the corresponding formula for general . Observe that for , the quasi-derivatives coincide with the standard derivatives . We take the canonical fundamental system , , of (1) with for . It is well known that the functions are analytic on with respect to . Putting the eigenvalues of the boundary value problem (1), (2) are the eigenvalues of the analytic matrix function , where the corresponding geometric and algebraic multiplicities coincide, see [10, Theorem ].
Setting and it is easy to see that
The first and the second rows of have exactly two nonzero entries (for ), and these nonzero entries are: and while and . Therefore, An expansion of gives where with In view of (11), (12), we get that Observing that the are given by (14) and (15) a straightforward calculation leads to It follows that the term with the highest -power in comes from and is a nonzero multiple of
The following result on the zeros of , with proper counting, is obvious.
Lemma 3. has a zero of multiplicity at , simple zeros at simple zeros at , and for , and no other zeros.
Proposition 4. For , there exists a positive integer such that the eigenvalues , , counted with multiplicity, of the problem (1), (9)–(12) can be enumerated in such a way that the eigenvalues are pure imaginary for , and for . For , we can write that , where the have the following asymptotic representation as : In particular, the number of pure imaginary eigenvalues is even.
Proof. It follows from (17) and (21) that
Up to the constant factor , the first term equals . Considering that, therefore,
for , that is, and . It follows that
Fix , and for , let be the squares determined by the vertices , . These squares do not intersect due to . There exists such that for all on the squares , . By periodicity, there are numbers and such that and for all on the boundary of the squares for all . Since , it follows that there is such that for all on the squares , where , the estimate holds. By Lemma 3, has exactly one simple zero inside . Hence, it follows from Rouché’s theorem that there is exactly one (simple) zero of in each for , which proves the existence of zeros of with as .
Let Then, , and for all , Observing that is an even function, it follows altogether that has zeros , for with .
We still have to show that the stated counting and asymptotic behaviour describes all zeros of . To this end, we are going to estimate on the squares , , whose vertices are . For , , and , we have uniformly in as . Hence, there are and such that for all , and , Note that for and , Hence, we have shown that for and , Furthermore, we will make use of the estimates which hold for all and all . Therefore, it follows from (31), (33)-(34), and the symmetry of , see (29), that there is such that for all with . Again from the definition of in (27) and Rouché’s theorem, we conclude that the functions and have the same number of zeros in the square , for with . Since has zeros inside and thus zeros inside , it follows that all such zeros of are accounted for by for and sufficiently large .
Finally, account for all eigenvalues of the problem (1)-(2) since each of these eigenvalues gives rise to two zeros of , counted with multiplicity. By Proposition 2, all eigenvalues with nonzero real part occur in pairs , , which shows that we can index all such eigenvalues as . Since there is an even number of remaining indices, the number of pure imaginary eigenvalues must be even.
4. Birkhoff Regularity
We refer to [10, Definition ] for the definition of Birkhoff regularity.
Proof. The characteristic function of (1) as defined in [10, ] is , and its zeros are , . We can choose according to [10, Theorem ]. The boundary condition (9)–(12) can be written in the form Thus, the boundary matrices defined in [10, ] are given by where Choosing , it follows that , where The Birkhoff matrices are where , are the diagonal matrices with 2 consecutive ones and 2 consecutive zeros in the diagonal in a cyclic arrangement, see [10, Definition and Proposition ]. It is easy to see that after a permutation of columns, the matrices (40) are block diagonal matrices consisting of blocks taken from two consecutive columns (in the sense of cyclic arrangement) of the first two rows of and the last two rows of , respectively. Hence, the determinants of the Birkhoff matrices (40) are Thus, the problem (1), (9)–(12) is Birkhoff regular for .
5. Asymptotic Expansions of Eigenvalues
With , defines a characteristic determinant of the problem (1), (9)–(12) with respect to the fundamental system , considered in Section 3. Observe that is the corresponding characteristic determinant for . Due to the Birkhoff regularity, only influences lower order terms in . Together with the estimates in Section 3, it can be inferred that outside the interior of the small squares , , , around the zeros of , if is sufficiently large. Since the fundamental system , , depends analytically on , also depends analytically on . Hence, applying Rouché’s theorem both to the large squares and to the small squares which are sufficiently far away from the origin, it follows that the eigenvalues of the boundary value problem for general have the same asymptotic distribution as for . Hence Proposition 4 leads to the following.
Proposition 6. For , there exists a positive integer such that the eigenvalues , counted with multiplicity, of the problem (1), (9)–(12), where can be enumerated in such a way that the eigenvalues are pure imaginary for , and for . For , we can write , where the have the following asymptotic representation as : In particular, the number of pure imaginary eigenvalues is even.
In the remainder of the section, we establish more precise asymptotic expansions of the eigenvalues. For this, it is more convenient to replace with the asymptotic fundamental system obtained in [10, Theorem ], (1), which can be written as where and means that we omit those terms of the Leibniz expansion which contain a function with . Since the coefficient of in (1) is zero, we have , see [10, ].
We will now determine the functions and . In this regard, observe that and in the notation of [10, and ], see [10, Theorem ]. From [10, ], we know that where is the th unit vector in , , and are matrices given by [10, , , and ], that is, , where , , and . Letting , a lengthy but straightforward calculation gives and thus for , where means that the estimate is uniform in .
In view of (44), the characteristic determinant of (1), (9)–(12) with respect to the fundamental system is where An expansion of leads to where , , , , , and each of the functions has asymptotic representations of the form .
It follows from (51) that where , , , . If , we have for and the terms for , can be absorbed by as they are of the form for any integer . Hence, for , where A straightforward calculation gives that Hence,
We know by Proposition 6 that the zeros of satisfy the asymptotic representations as . In order to improve on these asymptotic representations, write Because of the symmetry of the eigenvalues, we will only need to find the asymptotic expansions as . We know from Proposition 6 that , and it is our aim to find and . To this end, we will substitute (58) into , and we will then compare the coefficients of , , and .
Theorem 7. For , there exists a positive integer such that the eigenvalues , , counted with multiplicity, of the problem (1), (9)–(12), where can be enumerated in such a way that the eigenvalues are pure imaginary for , and for , where and the have the asymptotic representations
and the numbers , , are as follows:
In particular, the number of pure imaginary eigenvalues is even.
Remark 8. In , we have considered the differential equation (1) with the same boundary conditions , at as in this paper but only one -dependent boundary conditions at . We observe that the first two terms in the eigenvalue expansion coincide with those in Case of , which differs from the present case that the -term is absent in the boundary condition (10). However, the third and fourth terms are similar but different.
Various of the above calculations have been verified with Sage. This work is based upon research supported by the National Research Foundation of South Africa under Grant number 80956. Any opinion, findings and conclusions or recommendations expressed in this material are those of the authors and therefore the NRF do not accept any liability in regard thereto.
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