R E S E A R C H
Open Access
Self-adjoint fourth order differential
operators with eigenvalue parameter
dependent and periodic boundary conditions
Boitumelo Moletsane and Bertin Zinsou
**Correspondence:
[email protected] School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa
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 polynomialL(λ
) =λ
2M–iαλ
K–A, whereα
> 0,MandKare self-adjoint operators. Necessary and sufficient conditions are given such thatAis self-adjoint.MSC: Primary 34B07; secondary 34B08; 47E05
Keywords: fourth order differential operators; eigenvalue dependent boundary conditions; periodic boundary conditions; quadratic operator pencil; self-adjoint operators
1 Introduction
Higher order linear differential equations occur in applications with or without the eigen-value parameter in the boundary conditions. Such problems are realized as operator poly-nomials, also called operator pencils. Higher order eigenvalue problems are experienc-ing slow but steady developments. Some recent developments of higher order problems whose eigenvalue boundary conditions may depend on the eigenvalue parameter, includ-ing asymptotics of the eigenvalues can be found in [–].
The generalized Regge problem and the small transversal vibration of a homogeneous beam compressed or stretched problems have boundary conditions with partial first derivatives with respect to the time variablet. The self-adjoint sixth order problem [], the self-adjoint higher order problems [] and the fourth order Birkhoff regular problems [] also have the same type of boundary conditions described above. The mathematical model of these problems leads to eigenvalue problems with the eigenvalue parameterλ
occurring linearly in the boundary conditions. Such problems have an operator represen-tation of the form
L(λ) =λM–iλK–A (.)
in the Hilbert spaceH=L(I)⊕Ck, whereIis an interval,kis the number of eigenvalue dependent boundary conditions,M,K, andAare coefficient operators.
A classification of separated eigenvalue boundary conditions of nth order problems for which all the coefficient operators of the operator pencil (.) are self-adjoint is given in [], Theorem ., while an equivalent classification for fourth order problems is given in []. The boundary conditions investigated in [, ] are all separated. Möller and Pivovarchik [] 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 nth order differential equation.
In this paper we extend the work of [], using the same fourth order differential equation (.), to a class of boundary conditions, where two boundary conditions are periodic or anti-periodic at the end points, the remaining two boundary conditions are separated, one of them depends linearly on the eigenvalue parameterλ. The genesis of this problem is the problem studied by Möller and Pivovarchik []. In that problem the boundary condition that has dependence on the eigenvalue parameter has a specific meaning: that the hinge connection at the right end is subjected to a viscous frictionα> in the hinge. In keeping with the pattern of the boundary conditions of the operator studied in [], we confine our boundary conditions such that the two terms of the boundary condition that depends on the eigenvalue parameter are at one end point of the interval and the order of the derivative of the eigenvalue dependent part of this boundary condition is one less than the order of the eigenvalue independent part. We have associated to the problem under consideration the operator polynomial
L(λ) =λM–iαλK–A, (.)
whereα> , whileMandK are self-adjoint operators. We give necessary and sufficient conditions for the operatorAto be self-adjoint.
We give basic definitions and properties needed to conduct the study under inves-tigation in Section . In Section 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 [] and the Möller and Zinsou characterization for sepa-rated boundary conditions [, ]. In Section we present, for the fourth order eigenvalue problems investigated in this paper, the two different characterizations of self-adjoint op-erators used in Section as matrix equations. The Möller and Pivovarchik characteri-zation is given byU(N(U)) =R(U∗), while the Möller and Zinsou characterization is
W(N(U)) =R(U∗), whereUis a × matrix,U is a × matrix,Uis a × ma-trix andW a × matrix ofrank. Finally, in Section 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 nec-essary and sufficient conditions for which the coefficient operatorAis self-adjoint and we provide the structure of the boundary conditions using singular value decomposition.
2 Preliminaries
A Sobolev space is defined as
Wm (,a) :=
g∈L(,a) :∀j∈ {, . . . ,m}g(j)∈L(,a)
,
Letn= kwherek∈N. We consider annth order differential expressionof the form
y= k
m=
gmy(m)
(m)
(.)
on an interval [,a],a> , wheregm∈Wm(,a),m= , . . . ,k, are real valued functions and|gk(x)|>εfor someε> andx∈[,a]. The differential expressionyis well defined fory∈Wn
(,a) in which casey∈L(,a). The operatorLdefined by
D(L) =Wn(,a), Ly=y, y∈Wn(,a), (.)
is called the maximal operator associated with the differential expressionon [,a]. Definition . Lety∈Wn(,a). Forj= , . . . ,nthejth quasi-derivative ofy, denotedy[j], is recursively defined by
y[j]=y(j) forj= , . . . ,k– ,
y[k]=gky(k),
y[j]=y[j–]+gn–jy(n–j) forj=k+ , . . . ,n.
The quasi-derivatives depend on the differential expression (.). They are convenient for the formulation of the Lagrange identity when dealing with differential operators which have fairly general coefficients. Let
Y=
y c
, Z=
z d
, W=
w e
(.)
be elements of the Hilbert spaceL(,a)⊕C,y,z,w∈Wn (,a).
A formulation of the Lagrange identity and Green’s formula is quoted below from [], Theorem ...
Theorem . For a differential expressionand y,z∈Wn
(,a),the Lagrange identity (y)z–y(z) = d
dx[y,z] (.)
holds on[,a]almost everywhere,where
[y,z] = k
j=
(–)jy[j–]z[n–j]–y[n–j]z[j–] (.)
and Green’s formula
(y,z) – (y,z) = [y,z](a) – [y,z]() (.)
Letr,q∈N,U ar×nmatrix,U aq×nmatrix, V aq×nmatrix. Then the operatorAin the Hilbert spaceL(,a)⊕Cis defined by
D(A) =Y∈Wn(,a)⊕C,UYˆ = ,c=UYˆ
, (.)
AY=
y VYˆ
, (.)
where
ˆ
Y=y(), . . . ,y[n–](),y(a), . . . ,y[n–](a). (.) Form∈Ndefine
⎧ ⎨ ⎩
Jm,= ((–)s–δs,m+–t)ms,t=, Jm,=
Jm, –Jm∗,
,
Jm=
–Jm, Jm,
.
(.)
Finally define
U=
⎛ ⎜ ⎝
J
V
–U
⎞ ⎟
⎠, (.)
U=
⎛ ⎜ ⎝
U
U –I
V –I
⎞ ⎟
⎠. (.)
Before stipulating a criterion of self-adjointness, we give a proposition which states con-ditions under whichZ∈D(A∗), quoted from [], Proposition ...
Proposition . Assume that rankUU=r+q. Then Z ∈D(A∗) if and only if Z ∈ Wn
(,a)⊕Cand there is e∈Csuch that
[y,z](a) – [y,z]() +d∗VYˆ–e∗UYˆ = (.)
for allYˆ∈N(U).For Z∈D(A∗),e is unique and
A∗Z=
z
e
.
A criterion of self-adjointness as given by [], Theorem .., is quoted below.
Theorem . Assume that
rank
U
U
=r+q.
Then A is self-adjoint if and only if U
In addition to determining ifAis self-adjoint, we use [], Theorem .., quoted below to conclude thatAis bounded below.
Theorem . Assume that A is self-adjoint.Then A has a compact resolvent.Assume ad-ditionally that
(i) (–)kg k> ,
(ii) each component ofUYˆ either contains only quasi-derivativesy[m]withm<kor
contains only quasi-derivativesm≥k,
(iii) each component ofUYˆ either contains only quasi-derivativesy[m]withm<kor
contains only quasi-derivativesm≥k,
(iv) for each component ofUYˆ which only contains quasi-derivativesy[m]withm≥k,
the corresponding component ofVYˆ only contains quasi-derivativesy[m]withm<k.
Then A is bounded below.
Anym×nmatrix can be decomposed into a diagonal matrix of its singular values and orthogonal matrices of ordermandnas stated in [], Theorem ., quoted below as Theorem . Any m×n real matrix,with m≥n,can be factorized as
=
, (.)
where∈Rm×mand∈Rn×nare orthogonal,and ∈Rn×nis diagonal,
=diag(σ,σ, . . . ,σn),
whereσ≥σ≥ · · · ≥σn≥.
3 A particular problem
The boundary value problem with a fourth order differential equation
y()(λ,x) –gy(λ,x) =λy(λ,x), (.)
together with the following boundary conditions:
y(λ, ) –y(λ,a) = , (.)
y[](λ, ) –y[](λ,a) = , (.)
y(λ, ) = , (.)
y(λ,a) +iαλy(λ,a) = , (.)
K, andM, which are coefficients of the operator polynomial in the eigenvalue parameter; see (.) below. Then the eigenfunctions of the operator polynomialLgiven by
L(λ) =λM–iαλK–A (.)
correspond to the non-trivial solutions of (.)-(.), where the operatorsA,K, andMare defined by
D(A) =
Y=
y c
:y∈W(,a),y(λ, ) –y(λ,a) =y(λ, ) = ,
y[](λ, ) –y[](λ,a) = ,c=y(λ,a)
,
D(K) =D(M) =L(,a)⊕C, and
A
y c
=
y()– (gy)
y(a)
, K=
and M=
I
.
Proposition . The operators A,K,and M are self-adjoint,M and K are bounded,K hasrank,M≥,K≥,M+K,N(M)∩N(A) ={}and A is bounded below and has a compact resolvent.
Proof The statements aboutKandMare obvious. If (y,c)∈N(M)∩N(A) then (y,c)∈
N(M) givesy= , and (y,c)∈D(A) wherec=y(a) leads toc=y(a) = . HenceN(M)∩
N(A) ={}. We are going to use Theorem . to verify thatAis self-adjoint. By the differ-ential expression (.) withn= ,g= ,g= –g∈C[,a] andg= one represents (.) as
y= (gy) +
gy
+gy
=y()–gy=L(λ)y. (.) The quasi-derivatives associated with (.) are
y[]=y, y[]=y, y[]=y, y[]=y()–gy, y[]=y()–gy.
The number of eigenvalue independent boundary conditions as given by (.)-(.) cor-responds withr= , leaving only one boundary condition dependent on the eigenvalue parameter meaning thatq= .Uis a × matrix and bothUandVare × matrices given by
U=
⎛ ⎜ ⎝
–
–
⎞ ⎟
⎠, (.)
U=
, (.)
V=
Then the operatorAcan also be defined in terms of these matrices as
AY=
y VYˆ
,
D(A) =Y∈W(,a)⊕C,UYˆ = ,c=UYˆ
,
similar to (.) and (.). We now specify the matricesJ,U, andUas
J=
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ – – – – ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , (.)
U=
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ – – – – – ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , (.) and U= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ – – – – ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . (.)
Jis a × matrix,Uis a × matrix andUis a × matrix whereIin (.) is a × matrix.
We findN(U) andR(U∗) as
N(U) =span{e+e,e,e+e,e,e} ⊂C (.) and
The coefficient of the highest derivative in the differential component ofAisg= > as required by Theorem .(i). Particular values in assumptions of Theorem .(ii)-(iv) for (.)-(.) are
UYˆ =
⎛ ⎜ ⎝
y() –y(a)
y[]() –y[](a)
y[]()
⎞ ⎟
⎠, (.)
UYˆ =y[](a), (.)
VYˆ =y[](a). (.)
The first and third component ofUYˆ have quasi-derivatives of order zero and one. Hence their order given bymis less thank= , half the order of the differential equation and the second component has order three which ism= ≥k= . The component ofUYˆ has order one which is less thank.Adoes not have components ofUYˆwith quasi-derivatives that are greater thankand the condition onVYˆ is irrelevant. Thus all the conditions of
Theorem . are fulfilled andAis bounded below.
An alternative criterion is used to show that (.)-(.) is self-adjoint. First, define
W=J+U∗V–V∗U. (.)
ThenW(N(U)) andR(U∗) are given by
WN(U)=span{e–e,e, –e+e} ⊂C× (.) and
RU∗=span{e–e,e,e–e} ⊂C×. (.) A comparison ofW(N(U)) andR(U∗) shows thatW(N(U)) =R(U∗), which is a necessary and sufficient condition of self-adjointness, see Theorem . on p..
4 Periodic and a single eigenvalue dependent boundary condition
Consider on the interval [,a], wherea> , the differential equation (.) with boundary conditions
UYˆ = , (.)
(V+iαU)Yˆ= , (.)
where the matricesU,U, andVare of the following form:
U=
ui,j,i=,j=, (.)
U=
ui,j,i=,j=, (.)
Consider a particular case whereUandV contain exactly one non-zero element such that the non-zero element ofUis in a different column to the non-zero element ofVand the non-zero elements ofUare positioned such that the first column of (.) has linearly independent rows. The operatorAin (.) is given by
AY=
y VYˆ
,
D(A) =Y∈W(,a)⊕C,UYˆ = ,c=UYˆ
.
We recall that the dimension of the domain of a linear map between two spaces is given by the sum of the dimension of the null space and the rank of this linear map. In addition, two finite dimensional spaces coincide if one space is contained in the other and their dimensions are equal. A vector spaceCacted upon by these three matricesU
,U, and
V means thatrankUandrankV are given by
–dimN(U)= and –dimN(V)= ,
respectively.
Proposition . Let Uand V contain exactly one zero element such that the
non-zero element in Uis in a different column to the non-zero element in V.Let
W=J+U∗V–V∗U. (.)
Then U∗V and V∗Uare×matrices ofrank,U∗V–V∗Uis an×matrix ofrank and W is an×matrix ofrankat least.
Proof Let the non-zero element ofVbe atj=pand that ofUbe atj=s,s=p. Then
U∗V=u ij
, i=,j=
(vij),i=,j==
u jvi
, j=,i=,
has exactly one non-zero element,u
svp, atj=s,i=p. The position of the only non-zero element ofV∗U is in rowpand columns, thusU∗V –V∗U has rank .J in (.) is invertible with rank andU∗V–V∗Uhas rank . Hence, the rank ofW is at least . Remark . WheneverY ∈D(A) thenYˆ ∈N(U), and for every u∈N(U) there is a
Y∈D(A) such thatYˆ =u.
Corollary . If A is self-adjoint thenrankW= and W(N(U)) =R(U∗).
Proof Proposition . states thatZ∈D(A∗) if and only ifZ∈W(,a)⊕Cand there is
e∈Csuch that
for allYˆ ∈N(U). ForZ∈D(A∗),eis unique and
A∗Z=
z
e
.
We use (.) forW,Z∈D(A) =D(A∗) and
[y,z](a) – [y,z]() =Zˆ∗JYˆ
together with values ofeanddas implied by (.) and (.), respectively, which we sub-stitute into (.) to get
= [y,z](a) – [y,z]() +d∗VYˆ –e∗UYˆ
= [y,z](a) – [y,z]() + (UZˆ)∗VYˆ – (VZˆ)∗UYˆ =Zˆ∗JYˆ +Zˆ∗U∗VYˆ–Zˆ∗V∗UYˆ
=Zˆ∗J+U∗V–V∗U
ˆ
Y
=Zˆ∗WYˆ,
where Yˆ and Zˆ are as defined in (.). This means that WYˆ ⊥ ˆZ, i.e. W(N(U))⊂ (N(U))⊥=R(U∗). We use this containment ofW(N(U)) inR(U∗) to compare their di-mensions as
=rankU∗≥dimWN(U)
≥dimN(U)– ( –rankW)
= – +rankW. (.)
HencerankW≤. By Proposition .rankW= , and hence all the inequalities in (.) are equalities anddim(W(N(U))) =dim(R(U∗)) holds. ThusW(N(U)) =R(U∗). Theorem . The following statements are equivalent:
(i) Ais self-adjoint,
(ii) U(N(U)) =R(U∗), (iii) W(N(U)) =R(U∗).
Proof Suppose (i) holds. Then Corollary . implies (iii).
Suppose (iii) holds. Letu∈N(U). Then there isv∈D(U∗) such thatWu=U∗v i.e. U∗v=Wu=J+U∗V–V∗U
u. (.)
Consider
Uu=
⎛ ⎜ ⎝
J
V
–U
⎞ ⎟
⎠u=
⎛ ⎜ ⎝
Ju
Vu
–Uu
⎞ ⎟
Letb= –Vuandc=Uu i.e. =Vu+band =Uu–cand substitute (.) below. Then
⎛ ⎜ ⎝
Ju
Vu
–Uu
⎞ ⎟ ⎠= ⎛ ⎜ ⎝
Ju+U∗(Vu+b) –V∗(Uu–c) –b –c ⎞ ⎟ ⎠ = ⎛ ⎜ ⎝
(J+U∗V–V∗U)u+U∗b+V∗c –b –c ⎞ ⎟ ⎠ = ⎛ ⎜ ⎝
U∗v+U∗b+V∗c
–b –c ⎞ ⎟ ⎠ = ⎛ ⎜ ⎝
U∗ U∗ V∗
–I
–I
⎞ ⎟ ⎠ ⎛ ⎜ ⎝ v b c ⎞ ⎟
⎠=U∗
⎛ ⎜ ⎝ v b c ⎞ ⎟ ⎠.
ThusU(N(U))⊂R(U∗) anddim(U(N(U)))≤rankU∗. The mapU:C→C, in (.), hasdim(N(U)) =dim(C) –rankU= – = as given by the rank nullity theorem. Simi-larlyUwith the first column given by (.)-(.) hasrankU= thusdim(N(U)) =rankU∗. We then conclude thatU(N(U)) =R(U∗) by showing thatUis injectivei.e. is the only element inN(U). SupposeUu= . Then
=Uu=
⎛ ⎜ ⎝
Ju
Vu
–Uu
⎞ ⎟
⎠, (.)
andJu= impliesu= sinceJis invertible. Hence (ii) follows.
Suppose that (ii) holds. Then by Theorem . we have (i).
5 Further examples of self-adjoint operators with periodic and a single eigenvalue dependent boundary conditions
Keeping with the pattern of the boundary conditions of the operator studied in [], using the differential equation (.) and Theorem ., we identify the boundary conditions of the self-adjoint operators under investigation as follows:
y[β](λ, ) –
y[β](λ,a) = , (.)
y[β](λ, ) –
y[β](λ,a) = , (.)
δy[β](λ, ) + ( –δ)y[β](λ,a) = , (.) ( –δ)y[β](λ, ) +
iαλy[β](λ, )
=δy[β](λ,a) +
iαλy[β](λ,a)
, (.)
Theorem . The quadratic operator polynomial representing the fourth order differential equation(.)with the boundary conditions(.)-(.)is self-adjoint if and only if these boundary conditions have the following structure:
= , (.)
= – forδ= , (.)
= forδ= , (.)
β= , (.)
β= , (.)
β= , . (.)
Proof Consider the matricesU,U, andV of the form (.)-(.). Let the non-zero ele-ments ofUandVbe atu,andv,, respectively. Using the representation of (.), these corresponds toβ= andβ= . Let= –,β= ,= –,β= ,= – andβ= . Starting with this choice ofUandV, which implies thatδ= ,Ugiven by these param-eters is
U=
⎛ ⎜ ⎝
–
–
⎞ ⎟
⎠. (.)
Then we considerUandVwhere the non-zero elements are atu,andv,, respectively, correspond toβ= ,β= ,δ= and= . A matrixUwith such periodic boundary conditions is given by
U=
⎛ ⎜ ⎝
–
⎞ ⎟
⎠. (.)
The assumption of Theorem . is fulfilled sincerankUU= for both (.) and (.) together with their correspondingU. For eachUwe computeU(N(U)) and the corre-spondingR(U∗). The result is thatU(N(U)) =R(U∗) for each of the two cases and any of the combination of the parameters stated. Thus the operatorAfor each of the cases is self-adjoint. A self-adjoint quadratic operator polynomial representing the fourth order differential equation (.) with boundary conditions that satisfy (.)-(.) satisfies
U
N(U)
=RU∗.
If we represent boundary conditions with
U=
⎛ ⎜ ⎝
a a
b b
c
⎞ ⎟
⎠, (.)
U=
d
V=
e
, (.)
such that (.)-(.) is satisfied. Then, using Matlab, we prove that ifU(N(U)) =R(U∗) then the values of parametersβ,δ, andare as given by (.)-(.).
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 .. The matrixUdefined below was decomposed into its singular values and orthogonal matrices in an effort to find a relationship in all the cases.
Define a matrix
U:=
⎛ ⎜ ⎝
U
U
V
⎞ ⎟
⎠. (.)
All theUthat result from (.)-(.) and satisfy Theorem . are such that eachUcolumn has at most one non-zero element and each of its rows has at least one non-zero element.
Theorem . The self-adjoint quadratic operator polynomial representing the fourth or-der differential equation(.)with boundary conditions(.)-(.)that satisfy Theorem.
has
U=
,
where=I, =diag(
√
,√, , , )and∈R×.
Proof Consider (.)-(.) withβ= ,β= ,β= ,β= ,β= ,δ= and,,= . This choice of parameters results inU,U, andVgiven in (.)-(.). We then compute singular values ofUwith
U=
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
–
–
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
. (.)
Then
UU=
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
The eigenvalues ofUUareσ= with eigenvectors ( ), ( )andσ= with ( ), ( ), and ( ). We construct a matrixCwhose columns are the eigenvectors ofUUand order these eigenvectors by the magnitude of their eigen-valuesi.e. C= (eeeee). Then we implement the Gram-Schmidt orthonormalization process which in this case is=I×. We repeat the process withUUto find. The eigenvalues ofUUare , , and with multiplicities of two, three and three, respectively. We list the eigenvectors ofUUas columns ofD= (di) ordered below in decreasing magnitude of their eigenvalues as
d= –
√
(e–e),
d= –
√
(e–e),
d=e,
d=e,
d=e,
d= –
√
(e+e),
d=e,
d= –
√
(e+e).
Then we project thediand normalize them as before, which gives
= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ √ – √ –√ – √ √ – –√ – √ –√ – ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . (.)
The operators have the sameand withbeing the only distinguishing matrix in the
decompositions of theirU, whereW=J+U∗V–V∗U, andU=
J V –U . Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The subject of this paper is part of the Ph.D. thesis of BM. The subject has been suggested and supervised by Prof Manfred Möller, BZ is the co-supervisor. The initial version of the paper has been written by BM. The submitted version has been verified and discussed by BM, BZ, and MM.
Acknowledgements
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Received: 19 October 2016 Accepted: 6 March 2017 References
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