# Dirac System with Discontinuities at Two Points

Authors: Fatma Hıra; Nihat Altınışık
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We deal with a regular Dirac system which has discontinuities at two points and contains eigenparameter in a boundary condition and two transmission conditions. We investigate asymptotic behaviour of eigenvalues and corresponding eigenfunctions of this Dirac system and construct Green’s function.


---

## Body

## 1. Introduction

We consider the Dirac system(1)u2′(x)-p1(x)u1(x)=λu1(x),u1′(x)+p2(x)u2(x)=-λu2(x),x∈I,(2)B1(u):=β1u1(a)+β2u2(a)=0,(3)B2(u):=(λα1′-α1)u1(b)-(λα2′-α2)u2(b)=0,(4)T1(u):=u1(c1-)-δu1(c1+)=0,(5)T2(u):=u2(c1-)-δu2(c1+)+λu1(c1-)=0,(6)T3(u):=δu1(c2-)-γu1(c2+)=0,(7)T4(u):=δu2(c2-)-γu2(c2+)+λu1(c2-)=0,
where I=[a,c1)∪(c1,c2)∪(c2,b], a<c1<c2<b; λ∈ℂ; the real valued functions p1(·) and p2(·) are continuous in [a,c1), (c1,c2) and (c2,b] and have finite limits p1(c1±):=lim⁡x→c1±p1(x), p1(c2±):=lim⁡x→c2±p1(x), p2(c1±):=lim⁡x→c1±p2(x), p2(c2±):=lim⁡x→c2±p2(x); αi,αi′,βi∈ℝ(i=1,2), β12+β22≠0; δ,γ∈ℝ, δ>0, γ≠0; ci-:=ci-0, ci+:=ci+0(i=1,2) and ρ:=(α1′α2-α1α2′)>0.Boundary-value problems with transmission (or discontinuity) conditions inside the interval often appear in mathematical physics, mechanics, electronics, geophysics, and other branches of natural sciences (see [1, 2]). References [3–5] are examples of works with boundary conditions depending linearly on an eigenparameter and transmission conditions at the point of discontinuity for Sturm Liouville problem.The basic and comprehensive results about Dirac operators were given in [6]. The oscillation property and asymptotic formulas for the eigenvalues of Dirac systems were given in [7] and the derivative sampling theorem to compute eigenvalues of Dirac systems was used in [8, 9]. Continuous Dirac systems have been investigated in these works. Works in direction of Dirac systems with an internal point of discontinuity are few (see [10–13]). Sampling theories associated with discontinuous Dirac systems were investigated in [11, 12] (also the problem in [12] contains eigenparameter in a boundary condition). Direct and inverse problems for Dirac operators with discontinuity inside an interval were studied in [13]. Dirac operators with eigenparameter dependent on both boundary conditions and one of the discontinuity conditions were investigated in [10]. In these discontinuous works, Dirac systems had only one point of discontinuity.We examine Dirac system which has two points of discontinuity and contains at the same time an eigenparameter in a boundary condition and two transmission conditions in this paper. Dirac system which has two points of discontinuity does not exist as far as we know. Our investigation will be a good example for discontinuous Dirac systems. Firstly, we give some spectral properties of eigenvalues and eigenfunctions and then obtain asymptotic formulas for eigenvalues and corresponding eigenfunctions. Finally, we construct Green's function of the problem (1)–(7).


## 2. Spectral Properties

To formulate a theoretic approach to problem (1)–(7), let L:=L2(a,c1)⊕L2(c1,c2)⊕L2(c2,b) and ℂ3:=ℂ⊕ℂ⊕ℂ and we define the Hilbert space H=L⊕ℂ3 with an inner product
(8)〈U(·),V(·)〉H:=∫ac1u⊤(x)v¯(x)dx+δ2∫c1c2‍u⊤(x)v¯(x)dx+γ2∫c2b‍u⊤(x)v¯(x)dx+γ2ρh1k1¯+h2k2¯+δh3k3¯,
where ⊤ denotes the matrix transpose,
(9)U(x)=(u(x)h1h2h3),V(x)=(v(x)k1k2k3)∈H,u(x)=(u1(x)u2(x)),v(x)=(v1(x)v2(x))∈L,ui(·),vi(·)∈L2(a,b)(i=1,2),hj,kj∈ℂ(j=1,2,3).For functionsu(x), which are defined on I and have finite limit u(c1±):=lim⁡x→c1±u(x) and u(c2±):=lim⁡x→c2±u(x), by u(1)(x), u(2)(x), and u(3)(x), we denote the functions
(10)u(1)(x)={u(x),x∈[a,c1),u(c1-),x=c1,u(2)(x)={u(c1+),x=c1,u(x),x∈(c1,c2),u(c2-),x=c2,u(3)(x)={u(x),x∈(c2,b],u(c2+),x=c2,
which are defined on I1:=[a,c1], I2:=[c1,c2], and I3:=[c2,b], respectively.Lemma 1.
The eigenvalues of the problem (1)–(7) are real.Proof.
Assume the contrary thatλ0 is a nonreal eigenvalue of problem (1)–(7). Let (u1(x)u2(x)) be a corresponding (nontrivial) eigenfunction. By (1), we have
(11)ddx{u1(x)u¯2(x)-u¯1(x)u2(x)}=(λ¯0-λ0){|u1(x)|2+|u2(x)|2},x∈I.
Integrating the above equation through[a,c1], [c1,c2], and [c2,b], we obtain
(12)(λ¯0-λ0)(∫ac1‍(|u1(x)|2+|u2(x)|2)dx)=u1(c1-)u¯2(c1-)-u¯1(c1-)u2(c1-)-(u1(a)u¯2(a)-u¯1(a)u2(a)),(λ¯0-λ0)(∫c1c2‍(|u1(x)|2+|u2(x)|2)dx)=u1(c2-)u¯2(c2-)-u¯1(c2-)u2(c2-)-(u1(c1+)u¯2(c1+)-u¯1(c1+)u2(c1+)),(λ¯0-λ0)(∫c2b‍(|u1(x)|2+|u2(x)|2)dx)=u1(b)u¯2(b)-u¯1(b)u2(b)-(u1(c2+)u¯2(c2+)-u¯1(c2+)u2(c2+)).
Then from boundary conditions (2)-(3) and transmission conditions (4)–(7), we have, respectively,
(13)u1(a)u¯2(a)-u¯1(a)u2(a)=0,(14)u1(b)u¯2(b)-u¯1(b)u2(b)=-ρ(λ¯0-λ0)|u2(b)|2|λ0α1′-α1|2,(15)u1(c1-)u¯2(c1-)-u¯1(c1-)u2(c1-)=δ2(u1(c1+)u¯2(c1+)-u¯1(c1+)u2(c1+)),(16)δ2(u1(c2-)u¯2(c2-)-u¯1(c2-)u2(c2-))=γ2(u1(c2+)u¯2(c2+)-u¯1(c2+)u2(c2+)).
Sinceλ¯0≠λ0, it follows from (13)–(16) and (12) that
(17)∫ac1‍(|u1(x)|2+|u2(x)|2)dx+δ2∫c1c2‍(|u1(x)|2+|u2(x)|2)dx+γ2∫c2b‍(|u1(x)|2+|u2(x)|2)dx=-ργ2|u2(b)|2|λ0α1′-α1|2.
This contradicts the conditions∫ac1‍(|u1(x)|2+|u2(x)|2)dx+δ2∫c1c2‍(|u1(x)|2+|u2(x)|2)dx+γ2∫c2b‍(|u1(x)|2+|u2(x)|2)dx>0 and ρ>0. Consequently, λ0 must be real.Lemma 2.
Letλ and μ be two different eigenvalues of problem (1)–(7). Then the corresponding eigenfunctions u(x,λ) and v(x,μ) of this problem satisfy the following equality:
(18)∫ac1‍u⊤(x,λ)v(x,μ)dx+δ2∫c1c2‍u⊤(x,λ)v(x,μ)dx+γ2∫c2b‍u⊤(x,λ)v(x,μ)dx+γ2ρh1k1+h2k2+δh3k3=0.One must note that, since the eigenfunctions are real valued, the left hand side in the last expression coincides with the inner product inH.Now one will construct a special fundamental system of solutions of (1). By virtue of Theorem 1.1 in [14], one will define two solutions of (1),
(19)ϕ(·,λ)=(ϕ1(·,λ)ϕ2(·,λ)),χ(·,λ)=(χ1(·,λ)χ2(·,λ)),
where
(20)ϕ1(x,λ)={ϕ11(x,λ),x∈[a,c1),ϕ12(x,λ),x∈(c1,c2),ϕ13(x,λ),x∈(c2,b],ϕ2(x,λ)={ϕ21(x,λ),x∈[a,c1),ϕ22(x,λ),x∈(c1,c2),ϕ23(x,λ),x∈(c2,b],(21)χ1(x,λ)={χ11(x,λ),x∈[a,c1),χ12(x,λ),x∈(c1,c2),χ13(x,λ),x∈(c2,b],χ2(x,λ)={χ21(x,λ),x∈[a,c1),χ22(x,λ),x∈(c1,c2),χ23(x,λ),x∈(c2,b],
as follows. Let one consider the next initial value problem:
(22)u2′(x)-p1(x)u1(x)=λu1(x),u1′(x)+p2(x)u2(x)=-λu2(x),x∈(a,c1)(23)u1(a)=β2,u2(a)=-β1.By virtue of Theorem 1.1 in [14], this problem has a unique solution u=(ϕ11(x,λ)ϕ21(x,λ)), which is an entire function of λ∈ℂ for each fixed x∈[a,c1]. Similarly, employing the same method as in the proof of Theorem 1.1 in [14], one sees that the problem
(24)u2′(x)-p1(x)u1(x)=λu1(x),u1′(x)+p2(x)u2(x)=-λu2(x),x∈(c2,b)u1(b)=λα2′-α2,u2(b)=λα1′-α1,
has a unique solution u=(χ13(x,λ)χ23(x,λ)), which is an entire function of parameter λ∈ℂ for each fixed x∈[c2,b].Now the functionsϕ1i(x,λ) and ϕ2i(x,λ) are defined in terms of ϕ1j(x,λ) and ϕ2j(x,λ), i=2,3,  j=1,2, respectively, as follows. The initial value problem
(25)u2′(x)-p1(x)u1(x)=λu1(x),u1′(x)+p2(x)u2(x)=-λu2(x),x∈(c1,c2),(26)u1(c1)=1δϕ11(c1-,λ),u2(c1)=1δ{ϕ21(c1-,λ)+λϕ11(c1-,λ)}
has a unique solution u=(ϕ12(x,λ)ϕ22(x,λ)) for each λ∈ℂ, and the initial value problem
(27)u2′(x)-p1(x)u1(x)=λu1(x),u1′(x)+p2(x)u2(x)=-λu2(x),x∈(c2,b),(28)u1(c2)=δγϕ12(c2-,λ)u2(c2)=1γ{δϕ22(c2-,λ)+λϕ12(c2-,λ)},
has a unique solution u=(ϕ13(x,λ)ϕ23(x,λ)) for each λ∈ℂ.Similarly, the functionsχ1j(x,λ) and χ2j(x,λ) are defined in terms of χ1i(x,λ) and χ2i(x,λ), i=2,3, j=1,2, respectively, as follows. The initial value problem
(29)u2′(x)-p1(x)u1(x)=λu1(x),u1′(x)+p2(x)u2(x)=-λu2(x),x∈(c1,c2),(30)u1(c2)=γδχ13(c2+,λ),u2(c2)=γδ{χ23(c2+,λ)-λδχ13(c2+,λ)}
has a unique solution u=(χ12(x,λ)χ22(x,λ)) for each λ∈ℂ, and the initial value problem
(31)u2′(x)-p1(x)u1(x)=λu1(x),u1′(x)+p2(x)u2(x)=-λu2(x),x∈(a,c1),(32)u1(c1)=δχ12(c1+,λ),u2(c1)=δ{χ22(c1+,λ)-λχ12(c1+,λ)}
has a unique solution u=(χ11(x,λ)χ21(x,λ)) for each λ∈ℂ.Hence,ϕ(·,λ) satisfies (1) on I, boundary condition (2), and transmission conditions (4)–(7) and χ(·,λ) satisfies (1) on I, boundary condition (3), and transmission conditions (4)–(7).LetW(ϕ,χ)(·,λ) denote the Wronskians of ϕ(·,λ) and χ(·,λ) defined by
(33)W(ϕ,χ)(·,λ):=|ϕ1(·,λ)ϕ2(·,λ)χ1(·,λ)χ2(·,λ)|.It is obvious that the Wronskians(34)ωi(λ):=W(ϕ,χ)(·,λ)=ϕ1i(x,λ)χ2i(x,λ)-ϕ2i(x,λ)χ1i(x,λ),x∈Ii
are independent of x∈Ii(i=1,2,3) and are entire functions. Taking into account (26), (28), (30), and (32), a short calculation gives
(35)ω1(λ)=δ2ω2(λ)=γ2ω3(λ)
for each λ∈ℂ.Corollary 3.
The zeros of the functionsω1(λ), ω2(λ), and ω3(λ) coincide.Then, one may take into consideration the characteristic functionω(λ) as
(36)ω(λ):=ω1(λ)=δ2ω2(λ)=γ2ω3(λ).Lemma 4.
All eigenvalues of problem (1)–(7) are just zeros of the function ω(λ). Moreover, every zero of ω(λ) has multiplicity one.Proof.
Since the functionsϕ1(x,λ) and ϕ2(x,λ) satisfy the boundary condition (2) and transmission conditions (4)–(7), to find the eigenvalues of problem (1)–(7), we have to insert the functions ϕ1(x,λ) and ϕ2(x,λ) in the boundary condition (3) and find the roots of this equation.
By (1), we obtain, for λ,μ∈ℂ, λ≠μ,
(37)ddx{ϕ1(x,λ)ϕ2(x,μ)-ϕ1(x,μ)ϕ2(x,λ)}=(μ-λ){ϕ1(x,λ)ϕ1(x,μ)+ϕ2(x,λ)ϕ2(x,μ)}.
Integrating the above equation through[a,c1], [c1,c2], and [c2,b] and taking into account the initial conditions (23), (26), (28), (30), and (32), we obtain
(38)(λ-μ){δ2ϕ12(c1+,μ)ϕ12(c1+,λ)+γ2ϕ13(c2+,λ)ϕ13(c2+,μ)}+γ2(ϕ13(b,λ)ϕ23(b,μ)-ϕ13(b,μ)ϕ23(b,λ))=(μ-λ){∫ac1‍(ϕ11(x,λ)ϕ11(x,μ)kkkkkkkkkkkkkkk+ϕ21(x,λ)ϕ21(x,μ))dxkkkkkkkkkkk+δ2∫c1c2‍(ϕ12(x,λ)ϕ12(x,μ)kkkkkkkkkkkkkkkkkk+ϕ22(x,λ)ϕ22(x,μ))dxkkkkkkkkkkk+γ2∫c2b‍(ϕ13(x,λ)ϕ13(x,μ)kkkkkkkkkkkkkkkkkk+ϕ23(x,λ)ϕ23(x,μ))dx∫c2b‍}.
Dividing both sides of (38) by (λ-μ) and letting μ→λ, we reach the relation
(39)δ2(ϕ12(c1+,λ))2+γ2(ϕ13(c2+,λ))2+γ2(ϕ23(b,λ)∂ϕ13(b,λ)∂λ-ϕ13(b,λ)∂ϕ23(b,λ)∂λ)=-{∫ac1‍(|ϕ11(x,λ)|2+|ϕ21(x,λ)|2)dxkkkkkkl+δ2∫c1c2‍(|ϕ12(x,λ)|2+|ϕ22(x,λ)|2)dxkkkkkkl+γ2∫c2b‍(|ϕ13(x,λ)|2+|ϕ23(x,λ)|2)dx}.
We show that equation(40)ω(λ)=γ2((λα1′-α1)ϕ13(b,λ)-(λα2′-α2)ϕ23(b,λ))=0
has only simple roots. Assume the converse; that is, (40) has a double root λ0. Then the following two equations hold:
(41)(λ0α1′-α1)ϕ13(b,λ0)-(λ0α2′-α2)ϕ23(b,λ0)=0,α1′ϕ13(b,λ0)+(λ0α1′-α1)∂ϕ13(b,λ0)∂λ-α2′ϕ23(b,λ0)-(λ0α2′-α2)∂ϕ23(b,λ0)∂λ=0.
Sinceρ≠0 and λ0 is real, then (λ0α1′-α1)2+(λ0α2′-α2)2≠0, from (41),
(42)ϕ13(b,λ0)=(λ0α2′-α2)(λ0α1′-α1)ϕ23(b,λ0),∂ϕ13(b,λ0)∂λ=ρϕ23(b,λ0)(λ0α1′-α1)2+(λ0α2′-α2)(λ0α1′-α1)∂ϕ23(b,λ0)∂λ.
Combining (42) and (39), with λ=λ0, we obtain
(43)δ2(ϕ12(c1+,λ))2+γ2(ϕ13(c2+,λ))2+ργ2(ϕ23(b,λ0))2(λ0α1′-α1)2=-{∫ac1‍(|ϕ11(x,λ)|2+|ϕ21(x,λ)|2)dxkkkkkk+δ2∫c1c2‍(|ϕ12(x,λ)|2+|ϕ22(x,λ)|2)dxkkkkkk+γ2∫c2b‍(ϕ13(x,λ)ϕ13(x,μ)kkkkkkkkkkkk+ϕ23(x,λ)ϕ23(x,μ))dx∫c2b‍},
contradicting the assumption ρ>0.


## 3. Asymptotic Approximate Formulas

Now using variation of constants (see [14]), we will transform (1), (20),(23), (26), and (28) into the integral equations
(44)ϕ11(x,λ)=β2cos⁡(λ(x-a))+β1sin(λ(x-a))-∫ax‍sin(λ(x-y))p1(y)ϕ11(y,λ)dy-∫ax‍cos⁡(λ(x-y))p2(y)ϕ21(y,λ)dy,ϕ21(x,λ)=β2sin(λ(x-a))-β1cos⁡(λ(x-a))+∫ax‍cos⁡(λ(x-y))p1(y)ϕ11(y,λ)dy-∫ax‍sin(λ(x-y))p2(y)ϕ21(y,λ)dy,ϕ12(x,λ)=1δϕ11(c1-,λ)cos⁡(λ(x-c1))-1δ(ϕ21(c1-,λ)+λϕ11(c1-,λ))sin(λ(x-c1))-∫c1x‍sin(λ(x-y))p1(y)ϕ12(y,λ)dy-∫c1x‍cos⁡(λ(x-y))p2(y)ϕ22(y,λ)dy,ϕ22(x,λ)=1δϕ11(c1-,λ)sin(λ(x-c1))+1δ(ϕ21(c1-,λ)+λϕ11(c1-,λ))cos⁡(λ(x-c1))+∫c1x‍cos⁡(λ(x-y))p1(y)ϕ12(y,λ)dy-∫c1x‍sin(λ(x-y))p2(y)ϕ22(y,λ)dy,ϕ13(x,λ)=δγϕ12(c2-,λ)cos⁡(λ(x-c2))-1γ(δϕ22(c2-,λ)+λϕ12(c2-,λ))sin(λ(x-c2))-∫c2x‍sin(λ(x-y))p1(y)ϕ13(y,λ)dy-∫c2x‍cos⁡(λ(x-y))p2(y)ϕ23(y,λ)dy,ϕ23(x,λ)=δγϕ12(c2-,λ)sin(λ(x-c2))+1γ(δϕ22(c2-,λ)+λϕ12(c2-,λ))cos⁡(λ(x-c2))+∫c2x‍cos⁡(λ(x-y))p1(y)ϕ13(y,λ)dy-∫c2x‍sin(λ(x-y))p2(y)ϕ23(y,λ)dy.For|λ|→∞, the solutions ϕ1i(x,λ) and ϕ2i(x,λ)(i=1,2,3) have the following asymptotic representation uniformly with respect to x, x∈I, cf. [14]:
(45)ϕ11(x,λ)=[β2cos⁡(λ(x-a))+β1sin(λ(x-a))]+O(1λeτ(x-a)),ϕ21(x,λ)=[β2sin(λ(x-a))-β1cos⁡(λ(x-a))]+O(1λeτ(x-a)),ϕ12(x,λ)=-λδ[β2cos⁡(λ(c1-a))+β1sin(λ(c1-a))]×sin(λ(x-c1))+O(eτ(x-a)),ϕ22(x,λ)=λδ[β2cos⁡(λ(c1-a))+β1sin(λ(c1-a))]×cos⁡(λ(x-c1))+O(eτ(x-a)),ϕ13(x,λ)=λ2δγ[β2cos⁡(λ(c1-a))+β1sin(λ(c1-a))]×sin(λ(c2-c1))sin(λ(x-c2))+O(λeτ(x-a)),ϕ23(x,λ)=-λ2δγ[β2cos⁡(λ(c1-a))+β1sin(λ(c1-a))]×sin(λ(c2-c1))cos⁡(λ(x-c2))+O(λeτ(x-a)),
where τ=|Im⁡λ|.By substituting the obtained asymptotic formulas forϕ1i(x,λ) and ϕ2i(x,λ)(i=1,2,3) in the definitions of ω(λ), we can establish the following theorem.Theorem 5.
Letτ=|Im⁡λ|.Then the characteristic function ω(λ) has the following asymptotic representation:
(46)ω(λ)=λ3γδ{α1′[β2cos⁡(λ(c1-a))+β1sin(λ(c1-a))]kkklkkkk×sin(λ(c2-c1))sin(λ(b-c2))kkkkkkkl+α2′[β2cos⁡(λ(c1-a))+β1sin(λ(c1-a))]kkkkkkk×sin(λ(c2-c1))cos⁡(λ(b-c2))α1′}+O(λ2eτ(b-a)).By puttingsinz=(eiz-e-iz)/2i and cos⁡z=(eiz+e-iz)/2 in (46), one derives that
(47)ω(λ)=ω~(λ)+O(λ2eτ(b-a)),
where
(48)ω~(λ)=λ3γδ{e2λi(b-a)(-α1′β28-α1′β18i+α2′β28i-α2′β18)kkkkkkll+e2λi(b-a+c1-c2)(α1′β28+α1′β18i-α2′β28i+α2′β18)kkkkkkll+e2λi(c2-a)(α1′β28+α1′β18i+α2′β28i-α2′β18)kkklkkkl+e2λi(c1-a)(-α1′β28-α1′β18i-α2′β28i+α2′β18)kkklkkkl+e2λi(b-c1)(-α1′β28+α1′β18i+α2′β28i+α2′β18)kkklkkkl+e2λi(c2-c1)(α1′β28-α1′β18i+α2′β28i+α2′β18)kkklkkkl+e2λi(b-c2)(α1′β28-α1′β18i-α2′β28i-α2′β18)kkklkkkl+(-α1′β28+α1′β18i-α2′β28i-α2′β18)}.Lemma 6.
Let{αi}i=1p be the set of real numbers satisfying the inequalities α0>α1>⋯>αp-1>0 and {βi}i=1p the set of complex numbers. If βp≠0, then the roots of the equation
(49)eα0λ+β1eα1λ+⋯+βp-1eαp-1λ+βp=0
have the form
(50)λn=2nπiα0+Ψ(n)(n=0,±1,…),
where Ψ(n) is a bounded sequence [15].Theorem 7.
The eigenvaluesλn of problem (1)–(7) have the following asymptotic formula:
(51)λn=nπ(b-a)+O(1).Proof.
By using Lemma6, ω~(λ)=0 has an infinite number of roots λ~n with asymptotic expression
(52)λ~n=nπ(b-a)+Ψ(n),sup⁡n|Ψ(n)|<∞,
where Ψ(n)=O(1/n). By applying well-known Rouche's theorem, which assets that if f(s) and g(s) are analytic inside and on a closed contour Γ, and |g(s)|<|f(s)| on Γ, then f(s) and f(s)+g(s) have the same number of zeros inside Γ, and then we obtain
(53)λn=λ~n+O(1),λn=nπ(b-a)+O(1).By putting (51) in the (45), we obtain the following asymptotic formulae of the eigen-vector-functions ϕ(·,λn):(54)ϕ(x,λn)={((β2cos⁡(λn(x-a))+β1sin(λn(x-a)))+O(1n)(β2sin(λn(x-a))-β1cos⁡(λn(x-a)))+O(1n)),x∈[a,c1),(-λnδ(β2cos⁡(λn(c1-a))+β1sin(λn(c1-a)))sin(λn(x-c1))+O(1)λnδ(β2cos⁡(λn(c1-a))+β1sin(λn(c1-a)))cos⁡(λn(x-c1))+O(1)),x∈(c1,c2),(λn2δγ(β2cos⁡(λn(c1-a))+β1sin(λn(c1-a)))sin(λn(c2-c1))sin(λn(x-c2))+O(n)-λn2δγ(β2cos⁡(λn(c1-a))+β1sin(λn(c1-a)))sin(λn(c2-c1))cos⁡(λn(x-c2))+O(n)),x∈(c2,b],where
(55)ϕ(x,λn)={(ϕ11(x,λn)ϕ21(x,λn)),x∈[a,c1),(ϕ12(x,λn)ϕ22(x,λn)),x∈(c1,c2),(ϕ13(x,λn)ϕ23(x,λn)),x∈(c2,b].


## 4. Green’s Function

Letf(·)=(f1(·)f2(·)) be a continuous vector-valued function. Now, we derive Green's function of problem (1)–(7). Consider the inhomogeneous eigenvalue problem consisting of the differential system,
(56)u2′(x)-(p1(x)+λ)u1(x)=f1(x),u1′(x)+(p2(x)+λ)u2(x)=-f2(x),x∈I,
and boundary conditions (2), (3) and transmission conditions (4)–(7) with λ are not an eigenvalue of problem (1)–(7).Now we can represent the general solution of (56) in the following form:
(57)u(x,λ)={A1(ϕ11(x,λ)ϕ21(x,λ))+B1(χ11(x,λ)χ21(x,λ)),x∈[a,c1),A2(ϕ12(x,λ)ϕ22(x,λ))+B2(χ12(x,λ)χ22(x,λ)),x∈(c1,c2),A3(ϕ13(x,λ)ϕ23(x,λ))+B3(χ13(x,λ)χ23(x,λ)),x∈(c2,b].We applied the standard method of variation of the constants to (57); thus, the functions Ai(x,λ) and Bi(x,λ)(i=1,2,3) satisfy the linear system of equations
(58)mA1′(x,λ)ϕ21(x,λ)+B1′(x,λ)χ21(x,λ)=f1(x),A1′(x,λ)ϕ11(x,λ)+B1′(x,λ)χ11(x,λ)=-f2(x),mmmmmmmmmmmmmmmmmmmllx∈[a,c1),mA2′(x,λ)ϕ22(x,λ)+B2′(x,λ)χ22(x,λ)=f1(x),A2′(x,λ)ϕ12(x,λ)+B2′(x,λ)χ12(x,λ)=-f2(x),mmmmmmmmmmmmmmmmmmmlx∈(c1,c2),mA3′(x,λ)ϕ23(x,λ)+B3′(x,λ)χ23(x,λ)=f1(x),A3′(x,λ)ϕ13(x,λ)+B3′(x,λ)χ13(x,λ)=-f2(x),mmmmmmmmmmmmmmmmmmllllllx∈(c2,b].Sinceλ is not an eigenvalue and ωi(λ)≠0(i=1,2,3), each of the linear systems in (58) has a unique solution which leads to
(59)A1(x,λ)=1ω(λ)∫xc1‍χ⊤(ξ,λ)f(ξ)dξ+A1,B1(x,λ)=1ω(λ)∫ax‍ϕ⊤(ξ,λ)f(ξ)dξ+B1,mmmmmmmmmmmmmmmmlx∈[a,c1),A2(x,λ)=δ2ω(λ)∫xc2‍χ⊤(ξ,λ)f(ξ)dξ+A2,B2(x,λ)=δ2ω(λ)∫c1x‍ϕ⊤(ξ,λ)f(ξ)dξ+B2,mmmmmmmmmmmmmmmmlx∈(c1,c2),A3(x,λ)=γ2ω(λ)∫xb‍χ⊤(ξ,λ)f(ξ)dξ+A3,B3(x,λ)=γ2ω(λ)∫c2x‍ϕ⊤(ξ,λ)f(ξ)dξ+B3,mmmmmmmmmmmmmmmlllx∈(c2,b],
where Ai,Bi(i=1,2,3) are arbitrary constants and
(60)ϕ(ξ,λ)={(ϕ11(ξ,λ)ϕ21(ξ,λ)),ξ∈[a,c1),(ϕ12(ξ,λ)ϕ22(ξ,λ)),ξ∈(c1,c2),(ϕ13(ξ,λ)ϕ23(ξ,λ)),ξ∈(c2,b],χ(ξ,λ)={(χ11(ξ,λ)χ21(ξ,λ)),ξ∈[a,c1),(χ12(ξ,λ)χ22(ξ,λ)),ξ∈(c1,c2),(χ13(ξ,λ)χ23(ξ,λ)),ξ∈(c2,b].Substituting (59) into (57), we obtain the solutions of (56),
(61)ϕ(x,λ)={ϕ(x,λ)ω(λ)∫xc1‍χ⊤(ξ,λ)f(ξ)dξ+χ(x,λ)ω(λ)∫ax‍ϕ⊤(ξ,λ)f(ξ)dξ+A1ϕ(x,λ)+B1χ(x,λ),x∈[a,c1),δ2ϕ(x,λ)ω(λ)∫xc2‍χ⊤(ξ,λ)f(ξ)dξ+δ2χ(x,λ)ω(λ)∫c1x‍ϕ⊤(ξ,λ)f(ξ)dξ+A2ϕ(x,λ)+B2χ(x,λ),x∈(c1,c2),γ2ϕ(x,λ)ω(λ)∫xb‍χ⊤(ξ,λ)f(ξ)dξ+γ2χ(x,λ)ω(λ)∫c2x‍ϕ⊤(ξ,λ)f(ξ)dξ+A3ϕ(x,λ)+B3χ(x,λ),x∈(c2,b].Then, from boundary conditions (2), (3) and transmission conditions (4)–(7), we get
(62)A1=γ2ω(λ)∫c2b‍χ⊤(ξ,λ)f(ξ)dξ-δ2ω(λ)∫c1c2‍χ⊤(ξ,λ)f(ξ)dξ,B1=0,A2=γ2ω(λ)∫c2b‍χ⊤(ξ,λ)f(ξ)dξ,B2=-1ω(λ)∫ac1‍ϕ⊤(ξ,λ)f(ξ)dξ,A3=0,B3=δ2ω(λ)∫c1c2‍ϕ⊤(ξ,λ)f(ξ)dξ-1ω(λ)∫ac1‍ϕ⊤(ξ,λ)f(ξ)dξ.Then (61) can be written as
(63)u(x,λ)=∫ac1‍G(x,ξ,λ)f(ξ)dξ+δ2∫c1c2‍G(x,ξ,λ)f(ξ)dξ+γ2∫c2b‍G(x,ξ,λ)f(ξ)dξ,
where
(64)G(x,ξ,λ)={χ(x,λ)ϕ⊤(ξ,λ)ω(λ),a≤ξ≤x≤b,x≠c1,c2,ξ≠c1,c2,χ⊤(ξ,λ)ϕ(x,λ)ω(λ),a≤x≤ξ≤b,x≠c1,c2,ξ≠c1,c2.The functionG(x,ξ,λ) is called Green's function of problem (1)–(7). Obviously, G(x,ξ,λ) is a meromorphic function of λ, for every (x,ξ)∈I2, which has poles only at the eigenvalues.



---

*Source: 927693-2014-03-17.xml*