J. Math. Anal. Appl. 409 (2014) 567–575

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Journal of Mathematical Analysis andApplications

journal homepage: www.elsevier.com/locate/jmaa

On the perturbation determinants of a singular dissipativeboundary value problem with finite transmission conditionsEkin UğurluDepartment of Mathematics, Ankara University, 06100 Tandoğan, Ankara, Turkey

a r t i c l e i n f o

Article history:Received 15 September 2012Available online 24 July 2013Submitted by J. Mawhin

Keywords:Completeness theoremLivšic’s theoremDissipative operatorsTransmission conditions

a b s t r a c t

In this paper a singular dissipative boundary value problem with finite transmissionconditions is investigated. Using Livšic’s theorem, it is proved that the system of all eigenand associated functions of this problem is complete in the Hilbert space.

© 2013 Elsevier Inc. All rights reserved.

1. Introduction

Let us consider the differential expression

η(y) =1

w(x)

−(p(x)y′)′ + q(x)y

, x ∈ Λ :=

n+1k=1

Λk,

where Λk = (ck−1, ck) and −∞ < c0 < c1 < · · · < cn+1 ≤ ∞. In this paper the following properties are assumed to besatisfied:

(i) the points c0, c1, . . . , cn are regular and cn+1 is singular for the differential expression η,(ii) p, q andw are real-valued, Lebesgue measurable functions onΛ,(iii) p−1, q, w ∈ L1loc(Λk), k = 1, 2, . . . , n + 1, and(iv) w(x) > 0 for almost all x onΛ.

Let L2w(Λ) denote the Hilbert space consisting of all complex valued functions y such thatΛw(x) |y(x)|2 dx < ∞ with

the inner product

(y, χ) =

Λ

w(x)y(x)χ(x)dx.

Let D denote the set of all functions y ∈ L2w(Λ) such that y, py′ are locally absolutely continuous functions on allΛk, k = 1, 2, . . . , n + 1, and η(y) ∈ L2w(Λ). For arbitrary y, χ ∈ D, Green’s formula is

Λ

w(x)η(y)χ(x)dx −

Λ

w(x)y(x)η(χ)dx =

n+1k=1

[y, χ]ck−ck−1+

,

E-mail address: ekinugurlu@yahoo.com.

0022-247X/$ – see front matter© 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.jmaa.2013.07.040

568 E. Uğurlu / J. Math. Anal. Appl. 409 (2014) 567–575

where [y, χ]ck−ck−1+

= [y, χ]ck− −[y, χ]ck−1+, [y, χ]x := y(x)χ [1](x)− y[1](x)χ(x) and y[1] denotes py′. This equation impliesthat at singular point cn+1 for all y, χ ∈ D, the limit [y, χ]cn+1 := [y, χ]cn+1− = limx→cn+1−[y, χ]x exists and is finite.

In this paper it is assumed that the functions p, q and w satisfy Weyl’s limit-circle case conditions at singular pointcn+1. Weyl’s theory is well known and there are several sufficient conditions in which Weyl’s limit-circle case holds for adifferential expression [2,5,8,12,13,16,17].

Now let us consider the solutions ϕ(x, λ) = {ϕ1(x, λ), ϕ2(x, λ), . . . , ϕn+1(x, λ)} and ψ(x, λ) = {ψ1(x, λ), ψ2(x, λ),. . . , ψn+1(x, λ)}, where ϕk(x, λ) andψk(x, λ) are the parts of the functions ϕ(x, λ) andψ(x, λ), respectively, defined on theintervalΛk (k = 1, 2, . . . , n + 1), of the equation

− (p(x)y′)′ + q(x)y = λw(x)y, x ∈ Λ, (1.1)

where λ is some complex parameter, satisfying the conditions [1,4,10,11,14],ϕ1(c0, λ) = cosα, ϕ

[1]1 (c0, λ) = sinα,

ψ1(c0, λ) = − sinα, ψ[1]1 (c0, λ) = cosα,

and ϕm+1(cm+, λ) =

1κmϕm(cm−, λ), ϕ

[1]m+1(cm+, λ) =

1κ ′mϕ[1]m (cm−, λ),

ψm+1(cm+, λ) =1κmψm(cm−, λ), ψ

[1]m+1(cm+, λ) =

1κ ′mψ [1]

m (cm−, λ),

where α, κm and κ ′m are some real numbers with κmκ ′

m > 0 and m = 1, 2, . . . , n. Since Weyl’s limit-circle case holds atsingular point cn+1 for η, the solutions ϕ(x, λ) and ψ(x, λ) (x ∈ Λ) belong to L2w(Λ).

Let z(x) = {z1(x), z2(x), . . . , zn+1(x)} and u(x) = {u1(x), u2(x), . . . , un+1(x)} be the solutions of η(y) = 0 (x ∈ Λ)satisfying the conditions

z1(c0) = cosα, z[1]1 (c0) = sinα,

u1(c0) = − sinα, u[1]1 (c0) = cosα,

and zm+1(cm+) =

1κm

zm(cm−), z[1]m+1(cm+) =

1κ ′mz[1]m (cm−),

um+1(cm+) =1κm

um(cm−), u[1]m+1(cm+) =

1κ ′mu[1]m (cm−),

where α, κm and κ ′m are some real numbers with κmκ ′

m > 0 and m = 1, 2, . . . , n. It is clear that z(x) = ϕ(x, 0) (x ∈ Λ) andu(x) = ψ(x, 0) (x ∈ Λ). Hence z(x) and u(x) belong to L2w(Λ). Further they belong to D. This implies that for each y ∈ D, atsingular point cn+1, the values [y, z]cn+1 and [y, u]cn+1 exist and are finite.

It should be noted that [y, χ ]x (x ∈ Λ) denotes the Wronskian of the solutions y = y(x, λ) and χ = χ(x, λ) of (1.1).For y ∈ D, let us consider the following boundary and transmission conditions

y(c0) cosα + y[1](c0) sinα = 0, (1.2)[y, z]cn+1 − h[y, u]cn+1 = 0, (1.3)

y(cm−) = κmy(cm+), (1.4)

y[1](cm−) = κ ′

my[1](cm+), (1.5)

where α, κm and κ ′m are real numbers with κmκ ′

m > 0, m = 1, 2, . . . , n, and h is some complex number such thath = ℜh + iℑhwith ℑh > 0.

The spectral analysis and some properties of the regular symmetric (selfadjoint) boundary value transmission problems(BVTPs) have been studied in [1,10,11,14]. On the other hand singular dissipative boundary value transmission and boundaryvalue problems have been investigated in [3,4].

It is known that all eigenvalues of the dissipative operators lie in the closed upper half-plane but this analysis is so weak.To complete the analysis of a dissipative operator there are some methods. One of them is about Livšic’s theorem. In thispaper, using Livšic’s theorem the spectral analysis of the problem (1.1)–(1.5) is investigated. Further it should be noted thatthe results in this paper are the generalization of the results of [4].

E. Uğurlu / J. Math. Anal. Appl. 409 (2014) 567–575 569

2. Preliminaries

Let g(λ) be an entire function. If for each ϵ > 0 there exists a finite constant Dϵ such that

|g(λ)| ≤ Dϵ exp(ϵ |λ|), λ ∈ C, (2.1)

then g(λ) is called the entire function of order ≤ 1 of growth and minimal type.From (2.1) one gets that

lim|λ|→∞

sup1|λ|

ln |g(λ)| ≤ 0. (2.2)

It is known that [9] each entire function satisfying (2.1) and g(0) = 1, has the representation

g(λ) = limr→∞

|λj|≤r

1 −

λ

λj

(2.3)

and the limit limr→∞

|λj|≤r

1λj

exists and is finite.Let L denote a linear nonselfadjoint operator in the Hilbert space H with the domain D(L). The element y ∈ D(L), y = 0,

is called a root function of the operator L corresponding to the eigenvalue λ0, if all powers of L are defined on this elementand (L − λ0I)n y = 0 for some n > 0. The set of all root functions of L corresponding to the some eigenvalue λ0 with y = 0forms a linear set Nλ0 and is called the root lineal. The dimension of the lineal Nλ0 is called the algebraic multiplicity of theeigenvalue λ0.

The functions y1, y2, . . . , yk are called the associated functions of the eigenfunction y0 if they belong to D(L) and theequalities Lyj = λ0yj + yj−1, j = 1, 2, . . . , k, hold.

The completeness of the system of all eigen and associated functions of L is equivalent to the completeness of the systemof all root functions of this operator.

If for the operator Lwith the dense domainD(L), the inequalityℑ (Ly, y) ≥ 0 (y ∈ D(L)) holds, then L is called dissipative.

Theorem 2.1 ([15]). Let L be an invertible operator. Then,−L is dissipative if and only if the inverse operator L−1 of L is dissipative.

Letσ1 andσ2 denote the class of all nuclear andHilbert–Schmidt operators inH , respectively. Letµj(L)

ν(L)j=1 be a sequence

of all nonzero eigenvalues of L ∈ σk, k = 1, 2, arrangedby considering algebraic sumof algebraicmultiplicities of all nonzeroeigenvalues of L.

The following results can be found in [7].If L ∈ σ1, then

ν(L)j=1 µj(L) is called the trace of L and is denoted by spL.

For L ∈ σ1, the determinant

det (I − µL) =

ν(L)j=1

1 − µµj(L)

,

is called the characteristic determinant and is denoted by DL(µ). Since for any L ∈ σ1,

ν(L)j=1

µj(L) < ∞,

DL(µ) is an entire function of µ.For L ∈ σ2, the regularized characteristic determinant is defined by

DL(µ) =

ν(L)j=1

1 − µµj(L)

eµµj(L). (2.4)

If the operator I − µL has a bounded inverse defined on the whole space H , then the complex number µ is called theF-regular point for L.

Let L and K be linear bounded operators in H and K − L ∈ σ1. If the point µ is a F-regular point of K , then the equality

(I − µL) (I − µK)−1= I − µ(L − K) (I − µK)−1

holds, where µ(L − K) (I − µK)−1∈ σ1. Consequently, the determinant

DL/K (µ) = det(I − µL) (I − µK)−1

is meaningful and is called the determinant of perturbation of K by the operator L − K .

570 E. Uğurlu / J. Math. Anal. Appl. 409 (2014) 567–575

Theorem 2.2 ([7, p. 172]). If K , L ∈ σ2, K − L ∈ σ1 and µ is a F-regular point of K , then

DL/K (µ) =

DL(µ)DK (µ)eµsp(K−L).

Theorem 2.3 ([7, p. 177]). Let L and K be bounded dissipative operators (in particular, one of them or both may be selfadjoint)and L − K ∈ σ1. Then for any β0 (0 < β0 <

π2 ) the relation

limρ→∞

1ρ

lnDL/K (ρeiβ)

= 0

holds uniformly with respect to β in the sectorλ : λ = ρeiβ , 0 < ρ < ∞,

π2

− β

< β0

.

Livšic’s Theorem ([7, p. 226]). Let L be a compact dissipative operator as L = ℜL + iℑL with ℑL ∈ σ1. In order that the systemof all root functions of L be complete, it is necessary and sufficient that

ν(L)j=1

ℑµj(L) = spℑL.

3. The spectral analysis of the BVTP (1.1)–(1.5)

To analyze the BVTP (1.1)–(1.5) we should define a suitable inner product in the Hilbert space L2w(Λ).Let H =

n+1k=1 Hk, Hk = L2wk

(Λk), be the Hilbert space with the inner product

⟨y, χ⟩H =(y1, χ1)H1 (y2, χ2)H2 · · · (yn+1, χn+1)Hn+1

×1 κ1κ

′

1 · · · κ1κ′

1 · · · κnκ′

n

T,

where (yk, χk)Hk=Λkwk(x)yk(x)χk(x)dx, k = 1, 2, . . . , n + 1, y(x) = {y1(x), y2(x), . . . , yn+1(x)} ∈ H, χ(x) =

{χ1(x), χ2(x), . . . , χn+1(x)} ∈ H, w(x) = {w1(x), w2(x), . . . , wn+1(x)} and [.]T denotes the transpose of the matrix [.].It should be noted that the space H is, in fact, the direct sum space L2w(Λ).Let D(ℵ) be the set of all functions y ∈ H such that y, y[1] are locally absolutely continuous functions on all Λk, k =

1, 2, . . . , n + 1, satisfying η(y) ∈ H, R−(y) = 0, R+(y) = 0, Rm(y) = 0 and R′m(y) = 0, where R−(y) = y(c0) cosα +

y[1](c0) sinα, R+(y) = [y, z]cn+1 − h[y, u]cn+1 , Rm(y) = y(cm−)− κmy(cm+) and R′m(y) = y[1](cm−)− κ ′

my[1](cm+), m =

1, 2, . . . , n.Let ℵ be the operator defined on D(ℵ) as ℵy = η(y)(x ∈ Λ). Then the BVTP (1.1)–(1.5) can be handled in H as

ℵy = λy, y ∈ D(ℵ), x ∈ Λ.

It is clear that the eigenvalues and the root lineals of ℵ and the BVTP (1.1)–(1.5) coincide.Let us consider the solution v(x) = {v1(x), v2(x), . . . , vn+1(x)} of η(y) = 0 (x ∈ Λ), where v(x) = z(x)− hu(x) (x ∈ Λ)

and vk(x) = zk(x)−huk(x) (x ∈ Λk) (k = 1, 2, . . . , n+1). It is clear that v(x) satisfies the boundary-transmission condition(1.3)–(1.5). On the other hand u(x) satisfies the boundary and transmission conditions (1.2), (1.4) and (1.5).

If we set∆k = [uk, vk]x (x ∈ Λk), k = 1, 2, . . . , n + 1, then the equalities

∆1 = −1, ∆2 = −1κ1κ

′

1, . . . ,∆n+1 = −

1κ1κ

′

1 · · · κnκ ′n

(3.1)

hold [3,4].From (3.1), for arbitrary y, χ ∈ D(ℵ) the following equalities are obtained

[y1, χ1]x = [y1, z1]x[χ1, u1]x − [y1, u1]x[χ1, z1]x, x ∈ Λ1,

[y2, χ2]x = κ1κ′

1

[y2, z2]x[χ2, u2]x − [y2, u2]x[χ2, z2]x

, x ∈ Λ2,

...

[yn+1, χn+1]x = κ1κ′

1 · · · κnκ′

n{[yn+1, zn+1]x[χn+1, un+1]x − [yn+1, un+1]x[χn+1, zn+1]x}, x ∈ Λn+1.

(3.2)

Theorem 3.1. The operator ℵ is dissipative in H.

Proof. For y ∈ D(ℵ), we have

⟨ℵy, y⟩H − ⟨y,ℵy⟩H = [y, y]c1−c0+ + κ1κ′

1[y, y]c2−c1+ + · · · + κ1κ

′

1 · · · κnκ′

n[y, y]cn+1−cn+ . (3.3)

E. Uğurlu / J. Math. Anal. Appl. 409 (2014) 567–575 571

On the other hand, the conditions R−(y) = 0, Rm(y) = 0 and R′m(y) = 0 (m = 1, 2, . . . , n) give

[y, y]c0+ = 0, [y, y]c1− = κ1κ′

1[y, y]c1+, . . . , [y, y]cn− = κnκ′

n[y, y]cn+1 . (3.4)

From the formula (3.2) and R+(y) = 0, we get that

[y, y]cn+1 = κ1κ′

1 · · · κnκ′

n2iℑh[y, u]cn+1

2 . (3.5)

Substituting (3.4) and (3.5) in (3.3) we have

ℑ ⟨ℵy, y⟩H =κ1κ

′

1 · · · κnκ′

n

2ℑh[y, u]cn+1

2 (3.6)

and this completes the proof. �

From Theorem 3.1 we get that all eigenvalues of ℵ lie in the closed upper half-plane.

Theorem 3.2. The operator ℵ has no real eigenvalue.

Proof. Let us assume the contrary, i.e., let λ0 be a real eigenvalue of ℵ and ψ = ψ(x, λ0) (x ∈ Λ) be the correspondingeigenfunction of λ0. A direct calculation gives

ℑ ⟨ℵψ,ψ⟩H = ℑλ0 ∥ψ∥

2H

. (3.7)

Since λ0 is a real number, from (3.7) and (3.6) we get that [ψ, u]cn+1 = 0. Hence from (1.3) we have [ψ, z]cn+1 = 0.Setting ϕ = ϕ(x, λ0) (x ∈ Λ) and using (3.2) one gets that

1 = κ1κ′

1 · · · κnκ′

n[ϕ,ψ]cn+1

=κ1κ

′

1 · · · κnκ′

n

2{[ϕn+1, zn+1]cn+1 [ψn+1, un+1]cn+1 − [ϕn+1, un+1]cn+1 [ψn+1, zn+1]cn+1} = 0.

This contradiction completes the proof. �

From Theorem 3.2 it is obtained that all eigenvalues of ℵ lie in the open upper half-plane.In particular zero is not an eigenvalue of ℵ.For y ∈ D(ℵ), the equation ℵy = f (x) (x ∈ Λ) is equivalent to the nonhomogeneous differential equation

η(y) = f (x), x ∈ Λ,

subject to the conditions

y(c0) cosα + y[1](c0) sinα = 0,[y, z]cn+1 − h[y, u]cn+1 = 0, ℑh > 0,y(cm−) = κmy(cm+),

y[1](cm−) = κ ′

my[1](cm+),

where κmκ ′m > 0, m = 1, 2, . . . , n and f (x) = {f1(x), f2(x), . . . , fn+1(x)} ∈ L2w(Λ).

The general solution of the homogeneous differential equation can be represented as y(x) = {s1u1(x) + p1v1(x),. . . , sn+1un+1(x)+ pn+1vn+1(x)}, where all sk and pk (k = 1, 2, . . . , n + 1) are arbitrary constants [1,4].

By applying the standard method of variation of parameters the general solution is obtained as

y(x) =

u1(x)

κ1κ

′

1

c2

c1f2v2w2dξ + · · · + κ1κ

′

1 · · · κnκ′

n

cn+1

cnfn+1vn+1wn+1dξ +

c1

xf1v1w1dξ

+ v1(x)

x

c0f1u1w1dξ, u2(x)

κ1κ

′

1κ2κ′

2

c3

c2f3v3w3dξ + · · ·

+ κ1κ′

1 · · · κnκ′

n

cn+1

cnfn+1vn+1wn+1dξ + κ1κ

′

1

c2

xf2v2w2dξ

+ v2(x)

c1

c0f1u1w1dξ + κ1κ

′

1

x

c1f2u2w2dξ

, . . . , κ1κ

′

1 · · · κnκ′

nun+1(x) cn+1

xfn+1vn+1wn+1dξ

+ vn+1(x)

c1

c0f1u1w1dξ + κ1κ

′

1

c2

c1f2u2w2dξ + · · · + κ1κ

′

1 · · · κn−1κ′

n−1

cn

cn−1

fnunwndξ

+ κ1κ′

1 · · · κnκ′

n

x

cnfn+1un+1wn+1dξ

.

572 E. Uğurlu / J. Math. Anal. Appl. 409 (2014) 567–575

Let us set

G(x, ξ) =

u(x)v(ξ), c0 ≤ x ≤ ξ ≤ cn+1; x = ck, ξ = ck; k = 1, 2, . . . , nu(ξ)v(x), c0 ≤ ξ ≤ x ≤ cn+1; x = ck, ξ = ck; k = 1, 2, . . . , n. (3.8)

Then the general solution can be represented as

y(x) =(G(x, ξ), f (ξ))H1 (G(x, ξ), f (ξ))H2 · · · (G(x, ξ), f (ξ))Hn+1

×1 κ1κ

′

1 · · · κ1κ′

1 · · · κnκ′

n

T,

that is,

y(x) =G(x, ξ), f (ξ)

H ,

where f ∈ L2w(Λ).Let L denote the integral operator defined by the formula

Lf =G(x, ξ), f (ξ)

H , (3.9)

where f ∈ L2w(Λ).It is clear that L = ℵ

−1. Consequently the root lineals of the operators L and ℵ coincide. Hence the completeness of thesystem of all eigen and associated functions of L is equivalent to the completeness of those for ℵ in H .

Since u, v ∈ L2w(Λ),L is a Hilbert–Schmidt operator. Since the algebraicmultiplicity of nonzero eigenvalues of a compactoperator is finite, each eigenfunction of L may have only a finite number of linear independent associated functions.

Let us consider the functions

ζ1(x, λ) = [ψn+1(x, λ), zn+1(x)]x, ζ2(x, λ) = [ψn+1(x, λ), un+1(x)]x.

If we set

ζ (λ) := ζ1(cn+1, λ)− hζ2(cn+1, λ),

then the zeros of ζ (λ) coincide with the eigenvalues of the operator ℵ.

Theorem 3.3. The function ζ (λ) is an entire function of order ≤ 1 of growth and minimal type.

Proof. For the solution y(x, λ) = {y1(x, λ), y2(x, λ), . . . , yn+1(x, λ)} of the Eq. (1.1), it is possible to get for x ∈ Λn+1 that

yn+1(x, λ) = κ1κ′

1 · · · κnκ′

n([yn+1(x, λ), un+1(x)]xzn+1(x)− [yn+1(x, λ), zn+1(x)]xun+1(x)). (3.10)

Let say

Ψ1(x, λ) = [yn+1(x, λ), zn+1(x)]x, Ψ2(x, λ) = [yn+1(x, λ), un+1(x)]x, x ∈ Λn+1. (3.11)

Following [6], for x ∈ Λn+1 we have

∂

∂xΨ1(x, λ) = λyn+1(x, λ)zn+1(x)wn+1(x),

∂

∂xΨ2(x, λ) = λyn+1(x, λ)un+1(x)wn+1(x).

(3.12)

Substituting (3.10) in (3.12) and using (3.11) we get that

∂

∂xΨ (x, λ) = λA(x)Ψ (x, λ), x ∈ Λn+1, (3.13)

where

Ψ (x, λ) =

Ψ1(x, λ)Ψ2(x, λ)

,

A(x) =

−κ1κ

′

1 · · · κnκ′

nzn+1(x)un+1(x)wn+1(x) κ1κ′

1 · · · κnκ′

nz2n+1(x)wn+1(x)

−κ1κ′

1 · · · κnκ′

nu2n+1(x)wn+1(x) κ1κ

′

1 · · · κnκ′

nzn+1(x)un+1(x)wn+1(x)

.

Since zn+1, un+1 ∈ L2wn+1(Λn+1), the elements of A(x) are in L1(Λn+1).

It is known [13,17] that for fixed d ∈ Λ1, the functions ψ1(d, λ) and ψ[1]1 (d, λ) are entire functions of λ of order 1

2 . Fromtransmission conditions (1.4), (1.5), all ψk(e, λ) and ψ

[1]k (e, λ), e ∈ Λk, k = 2, 3, . . . , n + 1, are entire functions of λ of

order 12 for fixed e ∈ Λk. Hence ζj(b, λ) (j = 1, 2) are entire functions of λ of order 1

2 for fixed b, cn ≤ b < cn+1.

E. Uğurlu / J. Math. Anal. Appl. 409 (2014) 567–575 573

Let y(x, λ) = ψ(x, λ) (x ∈ Λ). Then from (3.13) it is obtained that

ζ (x, λ) =ζ (b, λ)+ λ

x

bA(t)ζ (t, λ)dt, x ∈ Λn+1, (3.14)

where

ζ (x, λ) =

ζ1(x, λ)ζ2(x, λ)

.

Using Gronwall inequality, from (3.14) we arrive atζ (x, λ) ≤ζ (b, λ) exp|λ|

x

b∥A(t)∥ dt

, x ∈ Λn+1. (3.15)

From (3.14) and (3.15), we get for x ∈ Λn+1 thatζ (cn+1, λ)−ζ (b, λ) ≤ |λ|ζ (b, λ) cn+1

b∥A(t)∥ dt

exp

|λ|

cn+1

cn∥A(t)∥ dt

, (3.16)

and ζ (cn+1, λ) ≤

ζ (b, λ) exp|λ|

cn+1

b∥A(t)∥ dt

, x ∈ Λn+1. (3.17)

From (3.16) as b → cn+1, ζ (b, λ) →ζ (cn+1, λ), uniformly in λ in each compact set. Hence ζ1(b, λ) and ζ2(b, λ) are entirefunctions of λ.

For b = cn in (3.17) one gets thatζ (cn+1, λ) ≤ (|ζ1(cn+, λ)| + |ζ2(cn+, λ)|) exp

|λ|

cn+1

cn∥A(t)∥ dt

, x ∈ Λn+1. (3.18)

(3.18) shows thatζ (cn+1, λ) is of not higher than first order and from (3.17)ζ (cn+1, λ) is of minimal type. �

Theorem 3.3 shows that all zeros of ζ (λ) (all eigenvalues of ℵ) are discrete and possible limit points of these zeros(eigenvalues of ℵ) can only occur at infinity.

Since v(x) = z(x)− (ℜh + iℑh)u(x)we get from (3.9) in view of (3.8) that L = ℜL + iℑL, where

ℜL =ℜG(x, ξ), f (ξ)

H , ℑL =

ℑG(x, ξ), f (ξ)

H ,

and

ℜG(x, ξ) =

u(x) [z(ξ)− ℜhu(ξ)] , c0 ≤ x ≤ ξ ≤ cn+1; x = ck, ξ = ck; k = 1, 2, . . . , nu(ξ) [z(x)− ℜhu(x)] , c0 ≤ ξ ≤ x ≤ cn+1; x = ck, ξ = ck; k = 1, 2, . . . , n

ℑG(x, ξ) = −ℑhu(x)u(ξ), ℑh > 0.

The operator ℜL is the selfadjoint Hilbert–Schmidt operator in H and ℑL is the one-dimensional selfadjoint operator in H .Let ℜℵ denote the operator in H generated by the differential expression η and the boundary-transmission conditions

y(c0) cosα + y[1](c0) sinα = 0,[y, z]cn+1 − ℜh[y, u]cn+1 = 0,y(cm−) = κmy(cm+),

y[1](cm−) = κ ′

my[1](cm+),

where κmκ ′m > 0, m = 1, 2, . . . , n. It is easy to see that ℜL is the inverse of ℜℵ : ℜL = (ℜℵ)−1. Let us consider the

function

ℜζ (λ) := ζ1(cn+1, λ)− ℜhζ2(cn+1, λ).

The zeros of ℜζ (λ) coincide with the eigenvalues of ℜℵ.Now consider the operator −L, − L = −ℜL − iℑL. Since ℵ is a dissipative operator, hence from Theorem 2.1 −L

is also a dissipative operator. If λj and γk denote the eigenvalues of the operators ℵ and ℜℵ, respectively, then −1λj

and

−1γk

will denote the eigenvalues of −L and −ℜL, respectively. It should be noted that, since ℜℵ is a selfadjoint operator,ℑγk = 0, for all k.

574 E. Uğurlu / J. Math. Anal. Appl. 409 (2014) 567–575

Theorem 3.4. The equalityj

ℑ

−

1λj

= sp (−ℑL)

holds.

Proof. Taking L = −ℜL, K = −L in Theorem 2.2, we get that

D−ℜL/−L(µ) =

D−ℜL(µ)D−L(µ)eiµsp(−ℑL). (3.19)

Using (2.4) we have

D−L(µ) =

j

1 +

µ

λj

e−µλj , D−ℜL(µ) =

k

1 +

µ

γk

e−

µγk .

On the other hand the equalities

ζ1(cn+1, λ) = ζ1(cn+, λ)+ λ

Λn+1

ψn+1(x, λ)zn+1(x)wn+1(x)dx,

and

ζ2(cn+1, λ) = ζ2(cn+, λ)+ λ

Λn+1

ψn+1(x, λ)un+1(x)wn+1(x)dx

hold. Hence we have

ζ (0) = ζ1(cn+, 0)− hζ2(cn+, 0)

and

ℜζ (0) = ζ1(cn+, 0)− ℜhζ2(cn+, 0).

Now letζ (λ) =1ζ (0)

ζ (λ), ℜζ (λ) =1

ℜζ (0)ℜζ (λ).

The eigenvalues of ℵ and ℜℵ coincide with the zeros of the functionsζ (λ) and ℜζ (λ), respectively. From Theorem 3.3 onegets thatζ (λ) and ℜζ (λ) are entire functions of λ of order ≤ 1 and minimal type. Furtherζ (0) = ℜζ (0) = 1. Therefore

ζ (µ) =

j

1 −

µ

λj

, ℜζ (µ) =

k

1 −

µ

γk

by (2.3). Hence we have

D−L(µ) =ζ (−µ)e−µ

j

1λj

, D−ℜL(µ) = ℜζ (−µ)e−µ

k

1γk

.

From (3.19) we get that (γk ∈ R)

D−ℜL/−L(µ) =

ℜζ (−µ)ζ (−µ) exp

µ

j

1λj

− µk

1γk

+ iµsp(−ℑL)

.

Taking µ = it (0 < t < ∞) one gets that

1tlnD−ℜL/−L(it)

=1tlnℜζ (−it)

− 1tlnζ (−it)

−j

ℑ

1λj

− sp(−ℑL). (3.20)

From (2.2) and Theorem 2.3 we have

limt→∞

1tlnD−ℜL/−L(it)

= 0 (3.21)

and

lim supt→∞

1tlnζ (−it)

≤ 0, lim supt→∞

1tlnℜζ (−it)

≤ 0. (3.22)

E. Uğurlu / J. Math. Anal. Appl. 409 (2014) 567–575 575

From the equalities1 +itλj

2 ≥ 1,1 +

itγk

2 ≥ 1,

we haveζ (−it)

≥ 1 andℜζ (−it)

≥ 1 for all t > 0. Consequently

1tlnζ (−it)

≥ 0,1tlnℜζ (−it)

≥ 0

and it follows from (3.22) that

limt→∞

1tlnζ (−it)

= limt→∞

1tlnℜζ (−it)

= 0. (3.23)

Hence from (3.20), (3.21) and (3.23) we findj

ℑ

−

1λj

= sp (−ℑL) . �

So all conditions of Livšic’s theorem are satisfied for the operator −L. Hence we have the following theorem.

Theorem 3.5. The system of all root functions of −L (also L) is complete in H.

Since the completeness of the system of all root functions (eigen and associated functions) of L is equivalent to thecompleteness of those for ℵ, and from all conclusions throughout the paper we obtain the following theorem.

Theorem 3.6. All eigenvalues of the BVTP (1.1)–(1.5) lie in the open upper half-plane and they are purely discrete. The limitpoints of these eigenvalues can only occur at infinity. The system of all eigen and associated functions of the BVTP (1.1)–(1.5) iscomplete in L2w(Λ).

Acknowledgment

I would like to thank my supervisor Professor Elgiz Bairamov for his helpful suggestions during the preparation of thiswork.

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