Research article

On Dirac operator with boundary and transmission conditions depending Herglotz-Nevanlinna type function

  • Received: 27 May 2020 Accepted: 11 January 2021 Published: 25 January 2021
  • MSC : 34A55, 34B24, 34L05

  • In this paper, an inverse problem is considered for Dirac equations with boundary and transmission conditions eigenvalue depending as rational function of Herglotz-Nevanlinna. We give some spectral properties of the problem and also it is shown that the coefficients of the problem are uniquely determined by Weyl function and by classical spectral data made up of eigenvalues and norming constants.

    Citation: Yalçın Güldü, Ebru Mişe. On Dirac operator with boundary and transmission conditions depending Herglotz-Nevanlinna type function[J]. AIMS Mathematics, 2021, 6(4): 3686-3702. doi: 10.3934/math.2021219

    Related Papers:

    [1] Ricardo Almeida . Variational problems of variable fractional order involving arbitrary kernels. AIMS Mathematics, 2022, 7(10): 18690-18707. doi: 10.3934/math.20221028
    [2] Xiaojing Du, Xiaotong Liang, Yonghong Xie . Integral expressions of solutions to higher order $ \lambda $-weighted Dirac equations valued in the parameter dependent Clifford algebra. AIMS Mathematics, 2025, 10(1): 1043-1060. doi: 10.3934/math.2025050
    [3] Yongjian Hu, Huifeng Hao, Xuzhou Zhan . On the solvability of the indefinite Hamburger moment problem. AIMS Mathematics, 2023, 8(12): 30023-30037. doi: 10.3934/math.20231535
    [4] Hong Li, Keyu Zhang, Hongyan Xu . Solutions for systems of complex Fermat type partial differential-difference equations with two complex variables. AIMS Mathematics, 2021, 6(11): 11796-11814. doi: 10.3934/math.2021685
    [5] Kun Li, Peng Wang . Properties for fourth order discontinuous differential operators with eigenparameter dependent boundary conditions. AIMS Mathematics, 2022, 7(6): 11487-11508. doi: 10.3934/math.2022640
    [6] Tuba Gulsen, Emrah Yilmaz, Ayse Çiğdem Yar . Proportional fractional Dirac dynamic system. AIMS Mathematics, 2024, 9(4): 9951-9968. doi: 10.3934/math.2024487
    [7] Valérie Gauthier-Umaña, Henryk Gzyl, Enrique ter Horst . Decoding as a linear ill-posed problem: The entropy minimization approach. AIMS Mathematics, 2025, 10(2): 4139-4152. doi: 10.3934/math.2025192
    [8] Yong Liu, Chaofeng Gao, Shuai Jiang . On meromorphic solutions of certain differential-difference equations. AIMS Mathematics, 2021, 6(9): 10343-10354. doi: 10.3934/math.2021599
    [9] Clara Burgos, Juan Carlos Cortés, Elena López-Navarro, Rafael Jacinto Villanueva . Probabilistic analysis of linear-quadratic logistic-type models with hybrid uncertainties via probability density functions. AIMS Mathematics, 2021, 6(5): 4938-4957. doi: 10.3934/math.2021290
    [10] Noureddine Bahri, Abderrahmane Beniani, Abdelkader Braik, Svetlin G. Georgiev, Zayd Hajjej, Khaled Zennir . Global existence and energy decay for a transmission problem under a boundary fractional derivative type. AIMS Mathematics, 2023, 8(11): 27605-27625. doi: 10.3934/math.20231412
  • In this paper, an inverse problem is considered for Dirac equations with boundary and transmission conditions eigenvalue depending as rational function of Herglotz-Nevanlinna. We give some spectral properties of the problem and also it is shown that the coefficients of the problem are uniquely determined by Weyl function and by classical spectral data made up of eigenvalues and norming constants.



    We consider the system of Dirac equations

    y(x):=By(x)+Q(x)y(x)=λy(x)x[a,b], (1)

    where B=(0110), Q(x)=(p(x)q(x)q(x)p(x)),  y(x)=(y1(x)y2(x)), p(x), q(x) are real valued functions in L2(a,b) and λ is a spectral parameter, with boundary conditions

    U(y):=y2(a)+f1(λ)y1(a)=0 (2)
    V(y):=y2(b)+f2(λ)y1(b)=0 (3)

    and with transmission conditions

    {y1(wi+0)=αiy1(wi0)y2(wi+0)=α1iy2(wi0)+hi(λ)y1(wi0)(i=1,2) (4)

    where fi(λ), hi(λ)(i=1,2) are rational functions of Herglotz-Nevanlinna type such that

    fi(λ)=aiλ+biNik=1fikλgik (5)
    hi(λ)=miλ+niPik=1uikλtik (i=1,2) (6)

    ai, bi, fik, gik,mi,ni,uik and tik are real numbers, a1<0, f1k<0, a2>0, f2k>0,mi>0, uik>0 and gi1<gi2<...<giNi, ti1<ti2<...<tiPi, αi>0 and a<w1<w2<b. In special case, when fi(λ)=, conditions (2) and (3) turn to Dirichlet conditions y1(a)=y1(b)=0 respectively. Moreover, when hi(λ)=, conditions (4) turn to y1(w2+0)=α2y1(w20), y2(w2+0)=α12y2(w20)+h2(λ)y1(w20) and y1(w1+0)=α1y1(w10), y2(w1+0)=α11y2(w10)+h1(λ)y1(w10) according to order i=1,2.

    Inverse problems of spectral analysis compose of recovering operators from their spectral data. Such problems arise in mathematics, physics, geophysics, mechanics, electronics, meteorology and other branches of natural sciences. Inverse problems also play important role in solving many equations in mathematical physics.

    R1(λ)y1(a)+R2(λ)y2(a)=0 is a boundary condition depending spectral parameter where R1(λ) and R2(λ) are polynomials. When degR1(λ)=degR2(λ)=1, this equality depends on spectral parameter as linearly. On the other hand, it is more difficult to search for higher orders of polynomials R1(λ) and R2(λ). When R1(λ)R2(λ) is rational function of Herglotz-Nevanlinna type such that f(λ)=aλ+bNk=1fkλgk in boundary conditions, direct and inverse problems for Sturm-Liouville operator have been studied [1,2,3,4,5,6,7,8,9,10,11]. In this paper, direct and inverse spectral problem is studied for the system of Dirac equations with rational function of Herglotz-Nevanlinna in boundary and transmission conditions.

    On the other hand, inverse problem firstly was studied by Ambarzumian in 1929 [12]. After that, G. Borg was proved the most important uniqueness theorem in 1946 [13]. In the light of these studies, we note that for the classical Sturm-Liouville operator and Dirac operator, the inverse problem has been studied fairly (see [14,15,16,17,18,19,20], where further references and links to applications can be found). Then, results in these studies have been extended to other inverse problems with boundary conditions depending spectral parameter and with transmission conditions. Therefore, spectral problems for differential operator with transmission conditions inside an interval and with eigenvalue dependent boundary and transmission conditions as linearly and non-linearly have been studied in so many problems of mathematics as well as in applications (see [21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43] and other works, and see [44,45,46,47,48,49,50,51,52,53,54] and other works cited therein respectively).

    The aim of this article is to get some uniqueness theorems for mentioned above Dirac problem with eigenvalue dependent as rational function of Herglotz-Nevanlinna type in both of the boundary conditions and also transmission conditions at two different points. We take into account inverse problem for reconstruction of considered boundary value problem by Weyl function and by spectral data {λn,ρn}nZ and {λn,μn}nZ. Although the boundary and transmission conditions of the problem are not linearly dependent on the spectral parameter, this allows the eigenvalues to be real and to define normalizing numbers.

    Consider the space H:=L2(a,b)L2(a,b)CN1+1CN2+1CP1+1 CP2+1 and element Y in H is in the form of Y=(y1(x),y2(x),τ,η,β,γ), such that τ=(Y1,Y2,,YN1,YN1+1), η=(L1,L2,,LN2,LN2+1), β=(R1,R2,,RP1,RP1+1), γ=(V1,V2,,VP2,VP2+1). H is a Hilbert space with the inner product defined by

    <Y,Z>:=ba(y1(x)¯z1(x)+y2(x)¯z2(x))dxYN1+1¯YN1+1a1+LN2+1¯LN2+1a2+α1m1RP1+1¯RP1+1+α2m2VP2+1¯VP2+1+N1k=1Yk¯Yk(1f1k)+N2k=1Lk¯Lkf2k+P1k=1α1Rr¯Rru1k+P2k=1α2Vr¯Vru2k (7)

    for Y=(y1(x),y2(x),τ,η,β,γ) ve Z=(z1(x),z2(x),τ,η,β,γ) in H. Define the operator T on the domain

    D(T)={YH:y1(x),y2(x)AC(a,b),

    lyL2(a,b), y1(w+i)=αiy1(wi),i=1,2

    YN1+1:=a1y1(a), LN2+1:=a2y1(b),

    RP1+1:=m1y1(w1),VP2+1:=m2y1(w2)}

    such that

    TY:=(ly,Tτ,Tη,Tβ,Tγ) (8)

    where

    Tτ=TYi={g1iYif1iy1(a)i=¯1,N1y2(a)+b1y1(a)+N1k=1Yki=N1+1 (9)
    Tη=TLi={g2iLif2iy1(b)i=¯1,N2y2(b)+b2y1(b)+N2k=1Lki=N2+1 (10)
    Tβ=TRi={t1iRiu1iy1(w1)i=¯1,P1y2(w+1)+α11y2(w1)+n1y1(w1)+P1k=1Rki=P1+1 (11)
    Tγ=TVi={t2iViu2iy1(w2)i=¯1,P2y2(w+2)+α12y2(w2)+n2y1(w2)+k=1P2Vki=P2+1. (12)

    Accordingly, equality TY=λY corresponds to problem (1)-(4) under the domain D(T)H.

    Theorem 1. The eigenvalues of the operator T and the problem (1)-(4) coincide.

    Proof. Assume that λ is an eigenvalue of T and Y(x)=(y1(x),y2(x),τ,η,β,γ)H is the eigenvector corresponding to λ. Since YD(T), it is obvious that the condition y1(wi+0)αiy1(wi0)=0 and Eq (1) hold. On the other hand, boundary conditions (2)-(3) and the second condition of (4) are satisfied by the following equalities;

    Tτ=TYi=g1iYif1iy1(a)=λYi, i=¯1,N1

    TYN1+1=y2(a)+b1y1(a)+N1k=1Yk=a1y1(a)λ

    Tη=TLi=g2iLif2iy1(b)=λLi, i=¯1,N2

    TLN2+1=y2(b)+b2y1(b)+N2k=1Lk=a2y1(b)λ

    Tβ=TRi=t1iRiu1iy1(w1), i=¯1,P1

    TRP1+1=y2(w+1)+α11y2(w1)+n1y1(w1)+P1k=1Rk=m1y1(w1)λ

    Tγ=TVi=t2iViu2iy1(w2), i=¯1,P2

    TVP2+1=y2(w+2)+α12y2(w2)+n2y1(w2)+k=1P2Vk=m2y1(w2)λ.

    If λ=gik(i=1,2 and k={1,2,Ni}) are eigenvalues of operator T, then, from above equalities and the domain of T, equalities (1), y1(a,g1k)=0, y1(b,g2k)=0 and (4) are satisfied.

    Moreover, If λ=tik(i=1,2 and k={1,2,Pi}) are eigenvalues of operator T, from above equalities and the domain of T, Eqs (1)-(3) and y1(wi,tik)=0=y1(w+i,tik) are valid. In that case, λ is also an eigenvalue of L.

    Conversely, let λ be an eigenvalue of L and (y1(x)y2(x)) be an eigenfunction corresponding to λ. If λgik(i=1,2k={1,2,Ni}) and λtik(i=1,2k={1,2,Pi}) then, it is clear that λ is an eigenvalue of T and the vector

    Y=(y1(x),y2(x),f11g11λy1(a),f12g12λy1(a),,f1N1g1N1λy1(a),a1y1(a),

    f21g21λy1(b),f22g22λy1(b),,f2N2g2N2λy1(b),a2y1(b),

    u11t11λy1(w1),u12t12λy1(w1),,u1P1t1P1λy1(w1),m1y1(w1),

    u21t21λy1(w2),u22t22λy1(w2),,u2P2t2P2λy1(w2),m2y1(w2)) is the eigenvector corresponding to λ.

    If λ=g1k(k={1,2,N1}), then,

    Y=(y1(x),y2(x),Y1,Y2,,YN1,0,L1,L2,,LN2,LN2+1,R1,R2,,RP1,RP1+1, V1,V2,,VP2,VP2+1),

    Yi={0,      iky2(a),i=k,i=1,2,,N1 is the eigenvector of T corresponding to g1k.

    If λ=g2k(k={1,2,N2}), then,

    Y=(y1(x),y2(x),Y1,Y2,,YN1,YN1+1,L1,L2,,LN2,0,R1,R2,,RP1,RP1+1,V1,V2,,VP2,VP2+1), Li={0,      iky2(b),i=k,i=1,2,,N2 is the eigenvector of T corresponding to g2k.

    Furthermore, if λ=t1k(k={1,2,P1}), then,

    Y=(y1(x),y2(x),Y1,Y2,,YN1,YN1,L1,L2,,LN2,LN2+1,R1,R2,,RP1,0,V1,V2,,VP2,VP2+1), Ri={0,      iky2(w+1)α11y2(w1),i=k,i=1,2,,P1 is the eigenvector corresponding to t1k.

    If λ=t2k(k={1,2,P2}), then, Y=(y1(x),y2(x),Y1,Y2,,YN1,YN1,L1,L2,,LN2,LN2+1,R1,R2,,RP1,RP1+1, V1,V2,,VP2,0), Vi={0,                           iky2(w+2)α12y2(w2),i=k,i=1,2,,P2 is the eigenvector corresponding to t2k.

    It is possible to write fi(λ) as follows:

    fi(λ)=ai(λ)bi(λ), i=1,2 where

    ai(λ)=(aiλ+bi)Nik=1(λgik)Nik=1Nij=1(jk)fik(λgij)

    bi(λ)=Nik=1(λgik).

    Assume that a2(λ) and b2(λ) do not have common zeros.

    Let functions φ(x,λ) and ψ(x,λ) be the solutions of (1) under the initial conditions

    φ(a,λ)=(b1(λ)a1(λ)),ψ(b,λ)=(b2(λ)a2(λ)) (13)

    as well as the transmission conditions (4) respectively such that

    φ(x,λ)={φ1(x,λ)x<w1φ2(x,λ)w1<x<w2φ3(x,λ)w2<x<b and ψ(x,λ)={ψ3(x,λ)x<w1ψ2(x,λ)w1<x<w2ψ1(x,λ)w2<x<b.

    Then it can be easily proven that φi(x,λ) and ψi(x,λ), i=¯1,3 are the solutions of the following integral equations;

    φi+1,1(x,λ)=αiφi1(wi,λ)cosλ(xwi)

    [α1iφi2(wi,λ)+hi(λ)φi1(wi,λ)]sinλ(xwi)

    +xwi[p(t)sinλ(xt)+q(t)cosλ(xt)]φi+1,1(t,λ)dt

    +xwi[q(t)sinλ(xt)p(t)cosλ(xt)]φi+1,2(t,λ)dt,

    φi+1,2(x,λ)=αiφi1(wi,λ)sinλ(xwi)+[α1iφi2(wi,λ)+hi(λ)φi1(wi,λ)]cosλ(xwi)+xwi[p(t)cosλ(xt)+q(t)sinλ(xt)]φi+1,1(t,λ)dt+xwi[q(t)cosλ(xt)p(t)sinλ(xt)]φi+1,2(t,λ)dt,for  i=1,2

    and

    ψi1(x,λ)=α1iψi+1,1(wi,λ)cosλ(xwi)+(αiψi+1,2(wi,λ)+hi(λ)ψi+1,1(wi,λ))sinλ(xwi)wix[p(t)sinλ(xt)+q(t)cosλ(xt)]ψi1(t,λ)dt+wix[q(t)sinλ(xt)+p(t)cosλ(xt)]ψi2(t,λ)dtψi2(x,λ)=α1iψi+1,1(wi,λ)sinλ(xwi)+(αiψi+1,2(wi,λ)hi(λ)ψi+1,1(wi,λ))cosλ(xwi)+wix[p(t)cosλ(xt)q(t)sinλ(xt)]ψi1(t,λ)dt+w2x[q(t)cosλ(xt)+p(t)sinλ(xt)]ψi2(t,λ)dt,for  i=2,1

    Lemma 1. For the solutions φi(x,λ) and ψi(x,λ), i=¯1,3 as |λ|, the following asymptotic estimates hold;

    φ11(x,λ)={a1λN1+1sinλ(xa)+o(|λ|N1+1exp|Imλ|[(xa)]),

    φ12(x,λ)={a1λN1+1cosλ(xa)+o(|λ|N1+1exp|Imλ|[(xa)]),

    φ21(x,λ)={a1m1λL1+N1+2sinλ(w1a)sinλ(xw1)+o(|λ|L1+N1+2exp|Imλ|[(w1a)+(xw1)])

    φ22(x,λ)={a1m1λL1+N1+2sinλ(w1a)cosλ(xw1)+o(|λ|L1+N1+2exp|Imλ|[(w1a)+(xw1)])

    φ31(x,λ)={m2m1a1λL1+L2+N1+3sinλ(w1a)sinλ(w2w1)sinλ(xw2)+o(|λ|L1+L2+N1+3exp|Imλ|[(w1a)+(w2w1)+(xw2)])

    φ32(x,λ)={m2m1a1λL1+L2+N1+3sinλ(w1a)sinλ(w2w1)cosλ(xw2)+o(|λ|L1+L2+N1+3exp|Imλ|[(w1a)+(w2w1)+(xw2)])

    ψ11(x,λ)={a2λN2+1sinλ(xb)+o(|λ|N2+1exp|Imλ|[(xb)])

    ψ12(x,λ)={a2λN2+1cosλ(xb)+o(|λ|N2+1exp|Imλ|[(xb)])

    ψ21(x,λ)={m2a2λN2+L2+2sinλ(w2b)sinλ(xw2)+o(|λ|N2+L2+2exp|Imλ|[(w2b)+(xw2)])

    ψ22(x,λ)={m2a2λN2+L2+2sinλ(w2b)cosλ(xw2)+o(|λ|N2+L2+2exp|Imλ|[(w2b)+(xw2)])

    ψ31(x,λ)={m1m2a2λN2+L1+L2+3sinλ(w2b)sinλ(w1w2)sinλ(xw1)+o(|λ|N2+L1+L2+3exp|Imλ|[(w2b)+(w1w2)+(xw2)])

    ψ32(x,λ)={m1m2a2λN2+L1+L2+3sinλ(w2b)sinλ(w1w2)cosλ(xw1)+o(|λ|N2+L1+L2+3exp|Imλ|[(w2b)+(w1w2)+(xw1)])

    Theorem 2. The eigenvalues {λn}nZ of problem L are real numbers.

    Proof. It is enough to prove that eigenvalues of operator T are real. By using inner product (7), for Y in D(T), we compute that

    TY,Y=balyˉydx1a1TYN1+1¯YN1+1+1a2TLN2+1¯LN2+1+α1m1TRP1+1¯RP1+1+α2m2TVP2+1¯VP2+1N1k=1TYk¯Yk(1f1k)+N2k=1TLk¯Lk(1f2k)+P1k=1α1TRk¯Rk(1u1k)+P2k=1α2TVk¯Vk(1u2k).

    If necessary arrangements are made, we get

    TY,Y=bap(x)(|y1|2|y2|2)dx+baq(x)2Re(y2¯y1)dx+b1|y1(a)|+N1k=12Re(Yk¯y1(a))b2|y1(b)|2N2k=12Re(Lk¯y1(b))a1n1|y1(w1)|2P1k=1a12Re(Rky1(w1))a2n2|y1(w2)|2P2k=1a22Re(Vky1(w2))N1k=1g1k|Yk|21f1k+N2k=1g2kf2k|Lk|2+P1k=1a1t1ku1k|Rk|2+P2k=1a2t2ku2k|Vk|2ba2Re(y2¯y1)dx.

    Accordingly, since TY,Y is real for each Y in D(T), λR is obtained.

    Lemma 2. The equality Yn2=ρn is valid such that Yn is eigenvector corresponding to eigenvalue λn of T.

    Proof. Let λngik. When λn=gik, following proof is done with minor changes. By using the structure of D(T) and the Eqs (8)-(12), we get

    Yn2=ba(φ21(x,λn)+φ22(x,λn))dx|YN1+1|2a1+|LN2+1|2a2+α1m1|RP1+1|2+α2m2|VP2+1|2N1k=1|Yk|2f1k+N2k=1|Lk|2f2k+P1k=1α1u1k|Rk|2+P2k=1α2u2k|Vk|2 (14)
    =ba(φ21(x,λn)+φ22(x,λn))dxa1φ21(a,λn)+a2φ21(b,λn)+m1α1φ21(w10,λn) +m2α2φ21(w20,λn)N1k=1f1kφ21(a,λn)(λng1k)2+N2k=1f2kφ21(b,λn)(λng2k)2 +P1k=1α1u1kφ21(w10,λn)(λnt1k)2+P2k=1α2u2kφ21(w20,λn)(λnt2k)2 =ba(φ21(x,λn)+φ22(x,λn))dxφ21(a,λn)(a1+N1k=1f1k(λng1k)2) +φ21(b,λn)(a2+N2k=1f2k(λng2k)2)+α1φ21(w10,λn)(m1+P1k=1u1k(λnt1k)2) +α2φ21(w20,λn)(m2+P2k=1u2k(λnt2k)2) =ba(φ21(x,λn)+φ22(x,λn))dxφ21(a,λn)f1(λn)+φ21(b,λn)f2(λn) +α1φ21(w1,λn)h1(λn)+α2φ21(w2,λn)h2(λn)=ρn.

    On the other hand, the expression

    W(φ,ψ)=φ1(x,λ)ψ2(x,λ)φ2(x,λ)ψ1(x,λ)

    is called characteristic function of problem (1)-(4). Moreover, since solutions φ(x,λ) and ψ(x,λ) satisfy the problem L,

    for x[a,b]

    xW(φ,ψ) =φ1(x,λ)ψ2(x,λ)+ψ2(x,λ)φ1(x,λ)φ2(x,λ)ψ1(x,λ)ψ1(x,λ)φ2(x,λ) =[q(x)φ1(x,λ)p(x)φ2(x,λ)λφ2(x,λ)]ψ2(x,λ) +[p(x)ψ1(x,λ)q(x)ψ2(x,λ)+λψ1(x,λ)]φ1(x,λ) [p(x)φ1(x,λ)q(x)φ2(x,λ)+λφ1(x,λ)]ψ1(x,λ) [q(x)ψ1(x,λ)p(x)ψ2(x,λ)λψ2(x,λ)]φ2(x,λ)=0

    is obtained. Furthermore, since solutions φ(x,λ) and ψ(x,λ) also satisfy transmission conditions (4), we get

    W(wi+0)=φ1(wi+0,λ)ψ2(wi+0,λ)φ2(wi+0,λ)ψ1(wi+0,λ)=αiφ1(wi0,λ)[α1iψ2(wi0,λ)+hi(λ)ψ1(wi0,λ)][α1iφ2(wi0,λ)+hi(λ)φ1(wi0,λ)]αiψ1(wi0,λ)=φ1(wi0,λ)ψ2(wi0,λ)φ2(wi0,λ)ψ1(wi0,λ)=W(wi0).

    Therefore, since characteristic function W(φ,ψ) is independent from x,

    W{φ,ψ}:=Δ(λ)=φ1(x,λ)ψ2(x,λ)φ2(x,λ)ψ1(x,λ)=a2(λ)φ1(b,λ)+b2(λ)φ2(b,λ)=b1(λ)ψ2(a,λ)a1(λ)ψ1(a,λ)

    can be written.

    It is clear that Δ(λ) is an entire function and its zeros namely {λn}nZcoincide with the eigenvalues of the problem L.

    Accordingly, for each eigenvalue λn equality ψ(x,λn)=snφ(x,λn) is valid where sn=ψ1(a,λn)b1(λn)=ψ2(a,λn)a1(λn).

    On the other hand, since ai(gik)0 ve bi(gik)=0 for i{1,2} and k={1,2,,Ni}, gik is an eigenvalue if and only if φ1(b,g2k)=0, φ1(a,g1k)=0 i.e., Δ(gik)=0.

    At the same time, tik is an eigenvalue if and only if φ1(wi,tik)=0=φ1(w+i,tik) i.e., Δ(tik)=0 such that i=1,2 and k={1,2,Pi}.

    Theorem 3. Eigenvalues of problem L are simple.

    Proof. Let λngik and φ(x,λn) be eigenfunction corresponds to the eigenvalue λn. In that case, the Eq (1) can be written for ψ(x,λ) and φ(x,λn) as follows;

    Bψ(x,λ)+Q(x)ψ(x,λ)=λψ(x,λ)Bφ(x,λn)+Q(x)φ(x,λn)=λnφ(x,λn).

    If we multiply these equations by φ(x,λn) and ψ(x,λ) respectively and add side by side, we get the following equality;

    (ψ2(x,λ)φ1(x,λn)ψ1(x,λ)φ2(x,λn))=(λλn)(ψ1(x,λ)φ1(x,λn)+ψ2(x,λ)φ2(x,λn)).

    Then if last equality is integrated over the interval [a,b] and the initial conditions (13) and transmission conditions (4) are used to get

    ba(ψ1(x,λ)φ1(x,λn)+ψ2(x,λ)φ2(x,λn))dx+α2φ1(w2,λn)ψ1(w2,λ)h2(λ)h2(λn)λλn+α1ψ1(w1,λ)φ1(w1,λn)h1(λ)h1(λn)λλn+ψ1(b,λn)φ1(b,λn)f2(λ)f2(λn)λλnφ1(a,λn)ψ1(a,λ)f1(λ)f1(λn)λλn=(Δ(λ)Δ(λn)(λλn))

    Then, considering that ψ(x,λn)=snφ(x,λn)

    if the limit is passed when λλnsnρn=˙Δ(λn) is obtained.

    If g1k and g2k are non-simple eigenvalues then φ1(a,g1k)=0, φ1(b,g2k)=0 and so ba(φ21(x,λn)+φ22(x,λn))dx=[α1φ21(w1,λn)h1(λn)+α2φ21(w2,λn)h2(λn)] is obtained. Since α1, α2 and for all λn, h1(λn), h2(λn) are positive, we have a contradiction. Therefore, eigenvalues gik are also simple.

    Using expressions a2(λ), b2(λ) and asymptotic behaviour of solution φ(x,λ), we obtain the following asymptotic of characteristic function Δ(λ) as |λ|; Δ(λ)=a1a2m1m2λN1+N2+L1+L2+4sinλ(w1a)sinλ(w2w1)sinλ(bw2)+o(|λ|N1+N2+L1+L2+4e|Imλ|(ba)).

    Let Φ(x,λ):=(Φ1(x,λ)Φ2(x,λ)) be the solution of Eq (1) under the conditions U(Φ)=1, V(Φ)=0 as well as the transmission conditions (4).

    Since V(Φ)=0=V(ψ), it can be supposed that Φ(x,λ)=kψ(x,λ) (k0) where k is a constant.

    W(φ,Φ)=φ1(x,λ)Φ2(x,λ)φ2(x,λ)Φ1(x,λ)|x=a=b1(λ)Φ2(a,λ)a1(λ)Φ1(a,λ)=U(Φ)=1.

    By the relation U(Φ)=1, we get k[b1(λ)ψ2(a,λ)+a1(λ)ψ1(a,λ)]=1. Since U(ψ)=Δ(λ), we obtain Φ(x,λ)=kψ(x,λ)=ψ(x,λ)Δ(λ) for λλn.

    Let S(x,λ)=(S1(x,λ)S2(x,λ)) and C(x,λ)=(C1(x,λ)C2(x,λ)) be solutions of (1) satify the conditions S(a,λ)=(01), C(a,λ)=(10) and transmission conditions (4).

    Accordingly, the following equalities are obtained:

    φ1(x,λ)=b1(λ)C(x,λ)+a1(λ)S(x,λ) (15)
    Φ(x,λ)=1b1(λ)(S(x,λ)Φ1(a,λ)φ(x,λ)). (16)

    The function Φ(x,λ) is called Weyl solution and the function M(λ)=Φ1(a,λ) is called Weyl function of problem L. Therefore, since Φ(x,λ)=ψ(x,λ)Δ(λ), we set M(λ):=ψ1(a,λ)Δ(λ).

    Consider the boundary value problem ˜L in the same form with L but different coefficients. Here, the expressions related to the L problem are shown with s and the ones related to ˜L are shown with ˜s. According to this statement, we set the problem ˜L as follows:

    ˜[y(x)]:=By(x)+˜Q(x)y(x)=λy(x)x[a,b]
    ˜U(y):=y2(a)+˜f1(λ)y1(a)=0˜V(y):=y2(b)+˜f2(λ)y1(b)=0y1(wi+0)=˜αiy1(wi0)y2(wi+0)=˜α1iy2(wi0)+˜hi(λ)y1(wi0)

    where ˜Q(x)=(˜p(x)q(x)q(x)˜p(x)).

    Theorem 4. If M(λ)=˜M(λ), f1(λ)=˜f1(λ), then Q(x)=˜Q(x) almost everywhere in (a,b), f2(λ)=˜f2(λ), hi(λ)=˜hi(λ),and αi(λ)=˜αi(λ) (i=1,2).

    Proof. Introduce a matrix P(x,λ)=[Pij(x,λ)]i,j=1,2 by the equality as follows;

    (P11P12P21P22)(˜φ1˜Φ1˜φ2˜Φ2)=(φ1Φ1φ2Φ2).

    According to this, we get

    P11(x,λ)=φ1(x,λ)˜Φ2(x,λ)+Φ1(x,λ)˜φ2(x,λ)P12(x,λ)=˜φ1(x,λ)Φ1(x,λ)+φ1(x,λ)˜Φ1(x,λ)P21(x,λ)=φ2(x,λ)˜Φ2(x,λ)+Φ2(x,λ)˜φ2(x,λ)P22(x,λ)=˜φ1(x,λ)Φ2(x,λ)+φ2(x,λ)˜Φ1(x,λ) (17)

    or by using the relation Φ(x,λ)=ψ(x,λ)Δ(λ),

    we obtain

    P11(x,λ)=φ1(x,λ)˜ψ2(x,λ)˜Δ(λ)˜ψ2(x,λ)ψ1(x,λ)Δ(λ)P12(x,λ)=φ1(x,λ)˜ψ1(x,λ)˜Δ(λ)+˜φ1(x,λ)ψ1(x,λ)Δ(λ)P21(x,λ)=φ2(x,λ)˜ψ2(x,λ)˜Δ(λ)˜φ2(x,λ)ψ2(x,λ)Δ(λ)P22(x,λ)=˜φ1(x,λ)ψ2(x,λ)Δ(λ)φ2(x,λ)˜ψ1(x,λ)˜Δ(λ). (18)

    Taking into account the Eqs (15) and (16) and M(λ)=˜M(λ), we can easily get

    P11(x,λ)=C1(x,λ)˜S2(x,λ)S1(x,λ)˜C2(x,λ)P12(x,λ)=˜C1(x,λ)S1(x,λ)C1(x,λ)˜S1(x,λ)P21(x,λ)=C2(x,λ)˜S2(x,λ)S2(x,λ)˜C2(x,λ)P22(x,λ)=˜C1(x,λ)S2(x,λ)C2(x,λ)˜S1(x,λ).

    Hence, the functions Pij(x,λ) are entire in λ. Denote

    Gδ:={λ:|λλn|δ, n=0,±1,±2,},δ>0 and

    ˜Gδ:={λ:|λ˜λn|δ, n=0,±1,±2,} where δ>0 is sufficiently small and fixed.

    Clearly, for λGδ˜Gδ, |sinλx|Cδe|Imλ|x, |λ|.

    Therefore, |Δ(λ)|CδλN1+N2+L1+L2+4e|Imλ|(ba), \ λGδ˜Gδ, |λ|λ for sufficiently large λ=λ(δ) and from (18) we see that Pij(x,λ) are bounded with respect to λ where λGδ˜Gδ and |λ| sufficiently large. From Liouville's theorem, it is obtained that these functions do not depend on λ.

    On the other hand, from (18)

    P11(x,λ)1=φ1(x,λ)(˜ψ2(x,λ)˜Δ(λ)ψ2(x,λ)Δ(λ))ψ1(x,λ)Δ(λ)(˜φ2(x,λ)φ2(x,λ))

    P12(x,λ)=˜φ1(x,λ)(ψ1(x,λ)Δ(λ)˜ψ1(x,λ)˜Δ(λ))˜ψ1(x,λ)˜Δ(λ)(φ1(x,λ)˜φ1(x,λ))

    P21(x,λ)=φ2(x,λ)(˜ψ2(x,λ)˜Δ(λ)ψ2(x,λ)Δ(λ))ψ2(x,λ)Δ(λ)(˜φ2(x,λ)φ2(x,λ))

    P22(x,λ)1=ψ2(x,λ)Δ(λ)(˜φ1(x,λ)φ1(x,λ))φ2(x,λ)(˜ψ1(x,λ)˜Δ(λ)ψ1(x,λ)Δ(λ)).

    If it is considered that Pij(x,λ) do not depend on λ and asymptotic formulas of solutions φ(x,λ) and ψ(x,λ), we obtain

    limλφ1(x,λ)(˜ψ2(x,λ)˜Δ(λ)ψ2(x,λ)Δ(λ))=0,

    limλψ1(x,λ)Δ(λ)(˜φ2(x,λ)φ2(x,λ))=0

    for all x in [a,b]. Hence, limλ[P11(x,λ)1]=0.

    Thus, P11(x,λ)=1 and similarly, P22(x,λ)=1 and P12(x,λ)=P21(x,λ)=0.

    Substitute these relations in (17), to obtain

    φ1(x,λ)=˜φ1(x,λ), ψ1(x,λ)Δ(λ)=˜ψ1(x,λ)˜Δ(λ)

    φ2(x,λ)=˜φ2(x,λ), ψ2(x,λ)Δ(λ)=˜ψ2(x,λ)˜Δ(λ) for all x and λ.

    Taking into account these results and Eq (1), we have

    (Q(x)˜Q(x))φ(x,λ)=0.

    Therefore, Q(x)=˜Q(x) i.e., p(x)=˜p(x). Moreover, it is considered that

    ψ1(x,λ)Δ(λ)=˜ψ1(x,λ)˜Δ(λ), ψ2(x,λ)Δ(λ)=˜ψ2(x,λ)˜Δ(λ)

    and

    b2(λ)ψ2(x,λ)+a2(λ)ψ1(x,λ)=0

    ˜b2(λ)˜ψ2(x,λ)+˜a2(λ)˜ψ1(x,λ)=0

    we get a2(λ)˜b2(λ)b2(λ)˜a2(λ)=0. As we have said above, a2(λ), b2(λ) as well as ˜a2(λ), ˜b2(λ) do not have common zeros. Hence, a2(λ)=˜a2(λ), b2(λ)=˜b2(λ), i.e., f2(λ)=˜f2(λ).

    On the other hand, substituting φ1 and φ2 into transmission conditions (4), we get

    φ1(w+i,λ)=αiφ1(wi,λ),  ˜φ1(w+i,λ)=˜ai˜φ1(wi,λ)

    φ2(w+i,λ)=α1iφ2(w1,λ)+hi(λ)φ1(wi,λ),

    ˜φ2(w+i,λ)=˜α1i˜φ2(wi,λ)+˜hi(λ)˜φ1(wi,λ), i=1,2.

    Therefore, since φ1(x,λ)=˜φ1(x,λ), φ2(x,λ)=˜φ2(x,λ), these yield that α1=˜α1, α2=˜α2

    and h1(λ)=˜h1(λ), h2(λ)=˜h2(λ).

    Theorem 5. If {λn,ρn}nZ={˜λn,˜ρn}nZ, f1(λ)=˜f1(λ) then Q(x)=˜Q(x) almost everywhere in (a,b), f2(λ)=˜f2(λ), hi(λ)=˜hi(λ),and αi(λ)=˜αi(λ) (i=1,2).

    Proof. Since λn=˜λn, Δ(λ)=c˜Δ(λ). On the other hand, also since snρn=˙Δ(λn) and ρn=˜ρn, we get that sn=c˜sn. Therefore, ψ1(a,λn)=c˜ψ1(a,λn) is obtained.

    Denote H(λ):=ψ1(a,λ)c˜ψ1(a,λ)Δ(λ) which is an entire function in λ. Since lim|λ|H(λ)=0, H(λ)0 and so ψ1(a,λ)=c˜ψ1(a,λ). Hence, M(λ)=˜M(λ). As a result, the proof of theorem is finished by Theorem 4.

    We examine the boundary value problem L1 with the condition y1(a)=0 instead of (2) in problem L. Let {μn}nZ be eigenvalues of the problem L1. It is clear that {μn}nZ are zeros of Δ1(μ):=ψ1(a,μ).

    Theorem 6. If {λn,μn}nZ={˜λn,˜μn}nZ, f1(λ)=˜f1(λ) and K=˜K such that K=a2m1m2, ˜K=˜a2˜m1˜m2 then Q(x)=˜Q(x) almost everywhere in (a,b), f2(λ)=˜f2(λ), hi(λ)=˜hi(λ),and αi(λ)=˜αi(λ) (i=1,2).

    Proof. Since for all nZ, λn=˜λn and μn=˜μn, Δ(λ)˜Δ(λ) and Δ1(μ)˜Δ1(μ) are entire functions in λ and in μ respectively. On the other hand, taking into account the asymptotic behaviours of Δ(λ), Δ1(μ) and K=˜K, we obtain limλΔ(λ)˜Δ(λ)=1 and limμΔ1(μ)˜Δ1(μ)=1. Therefore, since λn=˜λn and μn=˜μn, we get Δ(λ)=˜Δ(λ) and Δ1(μ)=˜Δ1(μ). If we consider the case Δ1(μ)=˜Δ1(μ), then ψ1(a,μ)=˜ψ1(a,μ) is obtained. Furthermore, since M(λ)=ψ1(a,λ)Δ(λ), M(λ)=˜M(λ). Hence, the proof is completed by Theorem 4.

    The purpose of this paper is to state and prove some uniqueness theorems for Dirac equations with boundary and transmission conditions depending rational function of Herglotz-Nevanlinna. Accordingly, it has been proved that while f1(λ) in condition (2) is known, the coefficients of the boundary value problem (1)-(4) can be determined uniquely by each of the following;

    i) The Weyl function M(λ)

    ii) Spectral data {λn,ρn} forming eigenvalues and normalizing constants respectively

    iii) Two given spectra {λn,μn}

    These results are the application of the classical uniqueness theorems of Marchenko, Gelfand, Levitan and Borg to such Dirac equations. Considering this study, similar studies can be made for classical Sturm-Liouville operators, the system of Dirac equations and diffusion operators with finite number of transmission conditions depending spectral parameter as Herglotz-Nevanlinna function.

    There is no conflict of interest.



    [1] P. A. Binding, P. J. Browne, B. A. Watson, Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter I, P. Edinburgh Math. Soc., 45 (2002), 631-645. doi: 10.1017/S0013091501000773
    [2] P. A. Binding, P. J. Browne, B. A. Watson, Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter II, J. Comput. Appl. Math., 148 (2002), 147-168. doi: 10.1016/S0377-0427(02)00579-4
    [3] P. A. Binding, P. J. Browne, B. A. Watson, Equivalence of inverse Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter, J. Math. Anal. Appl., 291 (2004), 246-261. doi: 10.1016/j.jmaa.2003.11.025
    [4] A. Chernozhukova, G. Freiling, A uniqueness theorem for the boundary value problems with non-linear dependence on the spectral parameter in the boundary conditions, Inverse Probl. Sci. Eng., 17 (2009), 777-785. doi: 10.1080/17415970802538550
    [5] G. Freiling, V. A. Yurko, Inverse problems for Sturm- Liouville equations with boundary conditions polynomially dependent on the spectral parameter, Inverse Probl., 26 (2010), 055003. doi: 10.1088/0266-5611/26/5/055003
    [6] R. Mennicken, H. Schmid, A. A. Shkalikov, On the eigenvalue accumulation of Sturm-Liouville problems depending nonlinearly on the spectral parameter, Math. Nachr., 189 (1998), 157-170. doi: 10.1002/mana.19981890110
    [7] A. S. Ozkan, Inverse Sturm-Liouville problems with eigenvalue-dependent boundary and discontinuity conditions, Inverse Probl. Sci. Eng., 20 (2012), 857-868. doi: 10.1080/17415977.2012.658519
    [8] H. Schmid, C. Tretter, Singular Dirac systems and Sturm-Liouville problems nonlinear in the spectral parameter, J. Differ. Equations, 181 (2002), 511-542. doi: 10.1006/jdeq.2001.4082
    [9] V. A. Yurko, Boundary value problems with a parameter in the boundary conditions, Sov. J. Contemp. Math. Anal., Arm. Acad. Sci., 19 (1984), 62-73.
    [10] V. A. Yurko, An inverse problem for pencils of differential operators, Sb. Math., 191 (2000), 137-158. doi: 10.4213/sm520
    [11] A. S. Ozkan, An impulsive Sturm-Liouville problem with boundary conditions containing Herglotz-Nevanlinna type functions, Appl. Math. Inform. Sci., 9 (2015), 205-211.
    [12] V. A. Ambarzumian, Über eine Frage der Eigenwerttheorie, Zs. f. Phys., 53 (1929), 690-695. doi: 10.1007/BF01330827
    [13] G. Borg, Eine umkehrung der Sturm-Liouvilleschen eigenwertaufgabe: bestimmung der differentialgleichung durch die eigenwerte, Acta Math., 78 (1946), 1-96. doi: 10.1007/BF02421600
    [14] B. M. Levitan, Inverse Sturm-Liouville problems, Moscow: Nauka, 1984.
    [15] V. A. Marchenko, Sturm-Liouville operators and their applications, Kiev: Nukova Dumka, 1977.
    [16] A. G. Ramm, Inverse problems, New York: Springer, 2005.
    [17] B. M. Levitan, I. S. Sargsyan, Sturm-Liouville and Dirac operators, Moscow: Nauka, 1988.
    [18] F. Gesztesy, B. Simon, Inverse spectral analysis with partial information on the potential II. The case of discrete spectrum, T. Am. Math. Soc., 352 (2000), 2765-2787.
    [19] M. G. Gasymov, Inverse problem of the scattering theory for Dirac system of order 2n, Tr. Mosk. Mat. Obs., 19 (1968), 41-112.
    [20] M. G. Gasymov, T. T. Dzhabiev, Determination of a system of Dirac differential equations using two spectra, In: Proceedings of School-Seminar on the Spectral Theory of Operators and Representations of Group Theory, 1975, 46-71.
    [21] I. M. Guseinov, On the representation of Jost solutions of a system of Dirac differential equations with discontinuous coefficients, Trans. Acad. Sci. Azerb., Ser. Phys. Tech. Math. Sci., 19 (1999), 42-45.
    [22] Z. Akdogan, M. Demirci, O. Sh. Mukhtarov, Discontinuous Sturm-Liouville problems with eigenparameter-dependent boundary and transmissions conditions, Acta Appl. Math., 86 (2005), 329-344. doi: 10.1007/s10440-004-7466-3
    [23] R. Kh. Amirov, On system of Dirac differential equations with discontinuity conditions inside an interval, Ukrainian Math. J., 57 (2005), 712-727. doi: 10.1007/s11253-005-0222-7
    [24] R. Kh. Amirov, A. S. Ozkan, B. Keskin, Inverse problems for impulsive Sturm-Liouville operator with spectral parameter linearly contained in boundary conditions, Integr. Trans. Spec. Funct., 20 (2009), 607-618. doi: 10.1080/10652460902726443
    [25] R. S. Anderssen, The effect of discontinuities in density and shear velocity on the asymptotic overtone structure of tortional eigenfrequencies of the Earth, Geophys. J. R. Astr. Soc., 50 (1997), 303-309.
    [26] G. Freiling, V. A. Yurko, Inverse Sturm-Liouville problems and their applications, New York: Nova Science, 2001.
    [27] C. T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, P. R. Soc. Edinburgh A., 77 (1977), 293-308. doi: 10.1017/S030821050002521X
    [28] C. T. Fulton, Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions, P. R. Soc. Edinburgh A., 87 (1980), 1-34. doi: 10.1017/S0308210500012312
    [29] O. H. Hald, Discontiuous inverse eigenvalue problems, Commun. Pure Appl. Math., 37 (1984), 539-577. doi: 10.1002/cpa.3160370502
    [30] C. M. McCarthy, W. Rundell, Eigenparameter dependent inverse Sturm-Liouville problems, Numer. Funct. Anal. Optim., 24 (2003), 85-105. doi: 10.1081/NFA-120020248
    [31] Kh. R. Mamedov, O. Akcay, Inverse problem for a class of Dirac operator, Taiwanese J. Math., 18 (2014), 753-772. doi: 10.11650/tjm.18.2014.2768
    [32] C. Van der Mee, V. N. Pivovarchik, A Sturm-Liouville inverse spectral problem withboundary conditions depending on the spectral parameter, Funct. Anal. Appl., 36 (2002), 74-77. doi: 10.1023/A:1014490403525
    [33] A. S. Ozkan, R. Kh. Amirov, Inverse problems for impulsive Dirac operators with eigenvalue dependent boundary condition, J. Adv. Res. Appl. Math., 3 (2011), 33-43. doi: 10.5373/jaram.820.030711
    [34] A. S. Ozkan, Half inverse problem for a class of differential operator with eigenvalue dependent boundary and jump conditions, J. Adv. Res. Appl. Math., 4 (2012), 43-49. doi: 10.5373/jaram.1260.010412
    [35] Y. P. Wang, Uniqueness theorems for Sturm-Liouville operators with boundary conditions polynomially dependent on the eigenparameter from spectral data, Results Math., 63 (2013), 1131-1144. doi: 10.1007/s00025-012-0258-6
    [36] W. Rundell, P. Sacks, Numerical technique for the inverse resonance problem, J. Comput. Appl. Math., 170 (2004), 337-347. doi: 10.1016/j.cam.2004.01.035
    [37] V. A. Yurko, Integral transforms connected with discontinuous boundary value problems, Integr. Trans. Spec. Funct., 10 (2000), 141-164. doi: 10.1080/10652460008819282
    [38] V. A. Yurko, Inverse spectral problems for differential operators and their applications, Amsterdam: Gordon and Breach Science, 2000.
    [39] A. S. Ozkan, B. Keskin, Y. Cakmak, Double discontinuous inverse problems for Sturm-Liouville operator with parameter-dependent conditions, Abstr. Appl. Anal., 2013 (2013), 794262.
    [40] A. S. Ozkan, Half-inverse Sturm-Liouville problem with boundary and discontinuity conditions dependent on the spectral parameter, Inverse Probl. Sci. Eng., 22 (2013), 848-859.
    [41] B. Keskin, A. S. Ozkan, N. Yalçin, Inverse spectral problems for discontinuous Sturm-Liouville operator with eigenparameter dependent boundary conditions, Commun. Fac. Sci. Univ. Ank. Sér. A., 1 (2011), 15-25.
    [42] D. G. Shepelsky, The inverse problem of reconstruction of the medium's conductivity in a class of discontinuous and increasing functions, Adv. Soviet Math., 19 (1994), 209-232.
    [43] V. A. Yurko, On boundary value problems with jump conditions inside the interval, Differ. Uravn., 36 (2000), 1139-1140.
    [44] A. Benedek, R. Panzone, On inverse eigenvalue problems for a second-order differential equations with parameter contained in the boundary conditions, Notas Algebra y Analysis, 1980, 1-13.
    [45] P. A. Binding, P. J. Browne, B. A. Watson, Inverse spectral problems for Sturm-Liouville equations with eigenparameter dependent boundary conditions, J. London Math. Soc., 62 (2000), 161-182. doi: 10.1112/S0024610700008899
    [46] P. J. Browne, B. D. Sleeman, A uniqueness theorem for inverse eigenparameter dependent Sturm-Liouville problems, Inverse Probl., 13 (1997), 1453-1462. doi: 10.1088/0266-5611/13/6/003
    [47] M. V. Chugunova, Inverse spectral problem for the Sturm-Liouville operator with eigenvalue parameter dependent boundary conditions, Oper. Theory, 123 (2001), 187-194.
    [48] B. Keskin, A. S. Ozkan, Inverse spectral problems for Dirac operator with eigenvalue dependent boundary and jump conditions, Acta Math. Hungar., 130 (2011), 309-320. doi: 10.1007/s10474-010-0052-4
    [49] A. S. Ozkan, B. Keskin, Spectral problems for Sturm-Liouville operator with boundary and jump conditions linearly dependent on the eigenparameter, Inverse Probl. Sci. Eng., 20 (2012), 799-808. doi: 10.1080/17415977.2011.652957
    [50] R. Mennicken, M. Möller, Non-self-adjoint boundary eigenvalue problems, North Holland: Elsevier, 2003.
    [51] A. A. Shkalikov, Boundary problems for ordinary problems for differential equations with parameter in the boundary conditions, J. Sov. Math., 33 (1986), 1311-1342. doi: 10.1007/BF01084754
    [52] C. Tretter, Boundary eigenvalue problems with differential equations Nη=λPη with λ-polynomial boundary conditions, J. Differ. Equations., 170 (2001), 408-471. doi: 10.1006/jdeq.2000.3829
    [53] C. F. Yang, Uniqueness theorems for differential pencils with eigenparameter boundary conditions and transmission conditions, J. Differ. Equations, 255 (2013), 2615-2635. doi: 10.1016/j.jde.2013.07.005
    [54] C. F. Yang, Inverse problems for Dirac equations polynomially depending on the spectral parameter, Appl. Anal., 95 (2016), 1280-1306. doi: 10.1080/00036811.2015.1061654
  • This article has been cited by:

    1. Mehmet Kayalar, A uniqueness theorem for singular Sturm-liouville operator, 2023, 34, 1012-9405, 10.1007/s13370-023-01097-x
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2483) PDF downloads(175) Cited by(1)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog