Self-adjoint fourth order differential operators with eigenvalue parameter dependent and periodic boundary conditions
- Boitumelo Moletsane^{1} and
- Bertin Zinsou^{1}Email authorView ORCID ID profile
Received: 19 October 2016
Accepted: 6 March 2017
Published: 14 March 2017
Abstract
Fourth order eigenvalue problems with periodic and separated boundary conditions are considered. One of the separated boundary conditions depends linearly on the eigenvalue parameter λ. These problems can be represented by an operator polynomial \(L(\lambda)=\lambda ^{2}M-i\alpha\lambda K-A\), where \(\alpha>0\), M and K are self-adjoint operators. Necessary and sufficient conditions are given such that A is self-adjoint.
Keywords
MSC
1 Introduction
Higher order linear differential equations occur in applications with or without the eigenvalue parameter in the boundary conditions. Such problems are realized as operator polynomials, also called operator pencils. Higher order eigenvalue problems are experiencing slow but steady developments. Some recent developments of higher order problems whose eigenvalue boundary conditions may depend on the eigenvalue parameter, including asymptotics of the eigenvalues can be found in [1–14].
A classification of separated eigenvalue boundary conditions of 2nth order problems for which all the coefficient operators of the operator pencil (1.1) are self-adjoint is given in [9], Theorem 4.7, while an equivalent classification for fourth order problems is given in [4]. The boundary conditions investigated in [4, 9] are all separated. Möller and Pivovarchik [15] give necessary and sufficient conditions for an operator to be self-adjoint in terms of the null and image spaces of matrices defined by any type of boundary conditions for a 2nth order differential equation.
We give basic definitions and properties needed to conduct the study under investigation in Section 2. In Section 3 we prove that a particular fourth order periodic eigenvalue problem is self-adjoint using two different characterizations of self-adjoint operators. These characterizations are the Möller and Pivovarchik characterization for general boundary conditions [15] and the Möller and Zinsou characterization for separated boundary conditions [4, 9]. In Section 4 we present, for the fourth order eigenvalue problems investigated in this paper, the two different characterizations of self-adjoint operators used in Section 3 as matrix equations. The Möller and Pivovarchik characterization is given by \(U_{3}(N(U_{1}))=R(U^{*})\), while the Möller and Zinsou characterization is \(W(N(U_{1}))=R(U_{1}^{*})\), where \(U_{3}\) is a \(8\times10\) matrix, \(U_{1}\) is a \(3\times8\) matrix, U is a \(5\times10\) matrix and W a \(8\times8\) matrix of rank6. Finally, in Section 5 we consider a class of periodic eigenvalue problems consisting of two periodic boundary conditions and two separated boundary conditions, one of them depends on the eigenvalue parameter. We derive necessary and sufficient conditions for which the coefficient operator A is self-adjoint and we provide the structure of the boundary conditions using singular value decomposition.
2 Preliminaries
Definition 2.1
A formulation of the Lagrange identity and Green’s formula is quoted below from [15], Theorem 10.2.3.
Theorem 2.2
Proposition 2.3
A criterion of self-adjointness as given by [15], Theorem 10.3.5, is quoted below.
Theorem 2.4
In addition to determining if A is self-adjoint, we use [15], Theorem 10.3.8, quoted below to conclude that A is bounded below.
Theorem 2.5
- (i)
\((-1)^{k}g_{k}>0\),
- (ii)
each component of \(U_{1}\hat{Y}\) either contains only quasi-derivatives \(y^{[m]}\) with \(m< k\) or contains only quasi-derivatives \(m\geq k\),
- (iii)
each component of \(U_{2}\hat{Y}\) either contains only quasi-derivatives \(y^{[m]}\) with \(m< k\) or contains only quasi-derivatives \(m\geq k\),
- (iv)
for each component of \(U_{2}\hat{Y}\) which only contains quasi-derivatives \(y^{[m]}\) with \(m\geq k\), the corresponding component of VŶ only contains quasi-derivatives \(y^{[m]}\) with \(m< k\).
Any \(m\times n\) matrix can be decomposed into a diagonal matrix of its singular values and orthogonal matrices of order m and n as stated in [16], Theorem 6.1, quoted below as
Theorem 2.6
3 A particular problem
Proposition 3.1
The operators A, K, and M are self-adjoint, M and K are bounded, K has rank 1, \(M\geq0\), \(K\geq0\), \(M+K\gg0\), \(N(M)\cap N(A)= \{0\}\) and A is bounded below and has a compact resolvent.
Proof
4 Periodic and a single eigenvalue dependent boundary condition
Proposition 4.1
Proof
Remark 4.2
Whenever \(Y\in D(A)\) then \(\hat{Y}\in N(U_{1})\), and for every \(u \in N(U_{1})\) there is a \(Y\in D(A)\) such that \(\hat{Y}=u\).
Corollary 4.3
If A is self-adjoint then \(\operatorname {rank}W=6 \) and \(W(N(U_{1}))=R(U_{1}^{*})\).
Proof
Theorem 4.4
- (i)
A is self-adjoint,
- (ii)
\(U_{3}(N(U_{1}))=R(U^{*})\),
- (iii)
\(W(N(U_{1}))=R(U_{1}^{*})\).
Proof
Suppose (i) holds. Then Corollary 4.3 implies (iii).
Suppose that (ii) holds. Then by Theorem 2.4 we have (i). □
5 Further examples of self-adjoint operators with periodic and a single eigenvalue dependent boundary conditions
Theorem 5.1
Proof
We find an unifying structure of boundary conditions that are periodic or anti-periodic at the end points of the interval and have an eigenvalue parameter dependence in one of them as described by Theorem 2.6. The matrix \(U_{4}\) defined below was decomposed into its singular values and orthogonal matrices in an effort to find a relationship in all the cases.
Theorem 5.2
Proof
Declarations
Acknowledgements
This research was supported by a grant from the NRF of South Africa, Grant number 80956. Various of the above calculations have been verified with MathLab.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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