Qualitative analysis of eigenvalues and eigenfunctions of one boundary value-transmission problem

R E S E A R C H

Open Access

Qualitative analysis of eigenvalues and

eigenfunctions of one boundary

value-transmission problem

Kadriye Aydemir

_{and Oktay S Mukhtarov}

*_{Correspondence:}

[email protected]

1_{Faculty of Education, Giresun}

University, Giresun, 28100, Turkey Full list of author information is available at the end of the article

Abstract

The aim of this study is to investigate various qualitative properties of eigenvalues and corresponding eigenfunctions of one Sturm-Liouville problem with an interior singular point. We introduce a new Hilbert space and integral operator in it such a way that the problem under consideration can be interpreted as a spectral problem of this operator. By using our own approaches we investigate such properties as uniform convergence of the eigenfunction expansions, the Parseval equality, the Rayleigh-Ritz formula, the minimax principle, and the monotonicity of eigenvalues for the considered boundary value-transmission problem (BVTP).

Keywords: Sturm-Liouville problems; boundary-transmission conditions; eigenvalues; Fourier series of eigenfunctions; minimax principle

1 Introduction

Sturm-Liouville eigenvalue problems appear frequently in solving several classes of par-tial differenpar-tial equations, particularly in solving the heat equation or a wave equation by separation of variables. Other examples of Sturm-Liouville boundary value problems are Hermite equations, Airy equations, Legendre equationsetc.Also, many physical pro-cesses, such as the vibration of strings, the interaction of atomic particles, electrodynamics of complex medium, aerodynamics, polymer rheology or the earth’s free oscillations, yield Sturm-Liouville eigenvalue problems (see, for example, [–] and references therein).

In different areas of applied mathematics and physics many problems arise in the form of boundary value problems involving transmission conditions at the interior singular points. Such problems are called boundary value-transmission problems (BVTPs). For example, in electrostatics and magnetostatics the model problem which describes the heat trans-fer through an infinitely conductive layer is a transmission problem (see [] and retrans-ferences therein). Another completely different field is that of ‘hydraulic fracturing’ (see []) used in order to increase the flow of oil from a reservoir into a producing oil well. Some problems with transmission conditions arise in thermal conduction problems for a thin laminated plate (i.e.a plate composed by materials with different characteristics piled in the thick-ness; see []). Some aspects of spectral problems for differential equations having singu-larities with classical boundary conditions at the endpoints were studied among others in [–] and references therein.

In this paper we shall investigate some qualitative properties of the eigenvalues and the corresponding eigenfunctions of one boundary value problem which consists of the Sturm-Liouville equation,

τ(u) := –u+q(x)u=λu(x), x∈(–π, )∪(,π), ()

together with end-point conditions given by

u:=cosαu(–π) +sinαu(–π) = , ()

u:=cosβu(π) +sinβu(π) = , ()

and with transmission conditions at the interior singular pointx=  given by

tu:=γu

––δu

+= , ()

tu:=γu

––δu

+= , ()

whereq(x) is a real-valued function;δi,γi(i= , ) are real numbers;α,β∈[,π);λis a

complex spectral parameter. Throughout we shall assume thatq(x) is continuous in:=

[–π, ) and:= (,π] with finite one-hand limitsq(±);γγ> , andδδ> .

It is the aim of this study to investigate such important spectral properties as the eigenfunction expansion, Parseval’s equality, the Rayleigh-Ritz formula (minimization principle), the minimax principle, and monotonicity of the eigenvalues for the Sturm-Liouville problem ()-(). The ‘Rayleigh quotient’ is the basis of an important approxi-mation method that is used in solid state physics as well as in quantum mechanics. In the latter, it is used in the estimation of energy eigenvalues of nonsolvable quantum systems. Often in physical problems, the sign of the eigenvalueλis quite important. For exam-ple, the equation dh

dt+λh=  occurs in certain heat flow problems. Here, positiveλ

corre-sponds to exponential decay in time, while negativeλcorresponds to exponential growth. In the vibration problemsd_dth+λh=  only positiveλcorresponds to the ‘usual’ expected oscillations.

The Rayleigh quotient cannot be used to explicitly determine the eigenvalue since the eigenfunction is unknown. However, interesting and significant results can be obtained from the Rayleigh quotient without solving the differential equation. Particularly, it can be quite useful in estimating the eigenvalues.

2 Preliminary results about eigenvalues and eigenfunctions

In the direct sum of the Lebesgue spacesH:=L()⊕L() we shall define a new inner

product in terms of the coefficients of the considered transmission conditions as follows:

f,gH:=γγ

–

–π

f(x)g(x)dx+δδ

π

+ f(x)g(x)dx. ()

Lemma . Let u andυbe eigenfunctions of BVTP()-()corresponding to distinct eigen-valuesλandμ,respectively.Ifλ=μthen u andυ are orthogonal in the Hilbert space H, i.e.

γγ

–

–π

uυdx+δδ

π

+ u

υdx= .

Proof Sinceτ(u) =λuandτ(υ) =μυ,

(λ–μ)u,υH=λu,υH–u,μυH

=τ(u),υ H–

u,τ(υ)_H. ()

By using the Lagrange identity we have

τ(u),υ H–

u,τ(υ)_H=γγW(u,υ;x)|

–

–π+δδW(u,υ;x)|π+, ()

whereW(u,υ;x) denotes the Wronskians ofuandυ. The boundary conditions () and () impliesW(u,υ; –π) =W(u,υ;π) = . Further the transmission conditions () and () imply

γγW

u,υ; –=δδ.W

u,υ; +. ()

By using these equations we get (λ–μ)u,υH. Thus,λ=μimpliesu,υH= , which

completes the proof.

Theorem . All eigenvalues of the BVTP()-()are real.

Proof Let (λ,u(x)) be any eigen-pair of the problem ()-(). Taking the

complex-conjugate of the BVTP ()-() we see that the pair (λ,u(x)) is also an eigen-pair of this

problem. From the boundary-transmission conditions ()-() it follows easily that

W(u,u; –π) =W(u,u;π) =  ()

and

γγW

u,u; –

–δδW

u,u; +

= . ()

Putting these equalities in the equality () we have (λ–λ) u = . This implies that

λ–λ= ,i.e.λis real.

Remark . Letλbe an eigenvalue of ()-() with corresponding eigenfunctionu(x) =

υ(x) +iω(x), whereυ(x) andω(x) are real-valued. Then bothυ(x) andω(x) are also

eigenfunctions corresponding to the same eigenvalueλ. Indeed, puttingu=u=υ+iω

andλ=λin ()-() and in view ofλbeing real, we have

τ(υ) +iτ(ω) = (λυ) +i(λω),

from which it follows that bothυ(x) andω(x) are eigenfunctions corresponding to the

same eigenvalueλ.

Theorem . There exists only one independent eigenfunction corresponding to each eigenvalue of the BVTP()-(),i.e. each of eigenvalues of this problem is geometrically sim-ple.

Proof By way of contradiction suppose that there exist two linearly independent eigen-functionsu(x) andυ(x) corresponding to the same eigenvalueλ. The boundary

con-ditions ()-() imply thatW(u,υ;π) =  and consequentlyW(u,υ;x) =  for allx∈

[–π, ). Sinceu(x) andυ(x) satisfy equation (),u(x) andυ(x) are linearly dependent

onby the well-known theorem of ordinary differential equation theory,i.e.there

ex-ists a constantc=  such thatu(x) =cυ(x) for allx∈. Similarly, from the second

boundary condition it follows that there exists a constantc=  such thatu(x) =cυ(x)

for allx∈. Hence

u(x) =

cυ(x) forx∈,

cυ(x) forx∈.

()

Substituting the transmission conditions ()-() we have

(c–c)γυ

–= (c–c)δυ

+= 

and

(c–c)γυ

–= (c–c)δυ

+= .

From these equalities we getc–c= . Consequentlyu(x) andυ(x) are linearly

depen-dent on the whole=∪. Hence we have obtained a contradiction, which completes

the proof.

Remark . By virtue of Theorem . the eigenfunctions of a BVTP ()-() can be chosen to be real-valued. Indeed, letλbe an eigenvalue with the eigenfunctionu(x) =υ(x) +

iω(x). By Remark . bothυ(x) andω(x) are also eigenfunctions corresponding to the

same eigenvalue. By Theorem . there is a complex numberC=  such thatω(x) =

Cυ(x). Henceu(x) =υ(x) +iω(x) = ( +iC)υ(x),i.e. here is only one real-valued

eigenfunction, except for a constant factor, corresponding to each eigenvalue. In view of this fact, from now on we can assume that all eigenfunctions of the BVTP ()-() are real-valued.

Now from Lemma ., Theorem ., and Remark . we have the next corollary.

Corollary . Let uand ube eigenfunctions of BVTP()-()corresponding to distinct

eigenvaluesλandλ.Then uand uare orthogonal in the sense of the following equality:

γγ

–

–π

u(x)u(x)dx+δδ

π

+ u

3 Reduction of (1)-(5) to the integral equation with the Green kernel

Letu(x,λ) be the solution of equation () on the left interval(the so-called left-hand

solution) satisfyingu(–π) =sinα,u(–π) = –cosα. Next we proceed fromu(x,λ) to

de-fine the right-hand solutionu(x,λ) of equation () on the right-hand intervalby the

initial conditions

u() = δ

γ

u

–,λ, u() =δ

γ

u_–,λ.

Now, letv(x,λ) be the solution of equation () on the right-hand intervalsatisfying the

initial conditionsv(–π) =sinβ,v(–π,λ) = –cosβ. Similarly we proceed fromv(x,λ) to

define the left-hand solutionv(x,λ) of equation () on the left-hand intervalby the

initial conditions

v() = δ

γ

v

+,λ, v_() = δ

γ

v

+,λ.

The existence of the solutionsuiandvi(i= , ) is obvious. Moreover, by using totally

sim-ilar arguments as in [] we can prove that each of these solutions is an entire function of the parameterλ∈Cfor each fixedx. Since the WronskianW[ui(x,λ),vi(x,λ)] is

inde-pendent of the variablex∈i(i= , ), we can denoteωi(λ) :=W[υi(·,λ),νi(·,λ)] (i= , ).

Using the transmission conditions ()-() it is easy to see thatγγω(λ) =δδω(λ). Both

sides of this equality we shall denote byω(λ). Now consider the following nonhomoge-neous BVTP:

τ(u) –λu=f, i(u) =ti(u) = , i= , . ()

Let us define a Banach space⊕Ck_{() as}

⊕Ck() :=

f=

f()(x) forx∈,

f()(x) forx∈

: f()(x)∈Ck[–π, ],f()(x)∈Ck[,π]

(k= , , , . . .) with the norm f _⊕Ck₍₎:=max{ f_() Ck_[–π,], f_() Ck_[,π]}. Below instead of

⊕C_{() we shall write⊕C().}

Theorem . Let f ∈ ⊕C().Then forλnot an eigenvalue,the nonhomogeneous BVTP ()has a unique solution uf for which the following formula holds:

uf(x,λ) =



ω(λ) ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

γγ{v(x,λ)

x

–πu(ξ,λ)f(ξ)dξ+u(x,λ) –

x v(ξ,λ)f(ξ)dξ}

+δδu(x,λ)

π

+v(ξ,λ)f(ξ)dξ for x∈[–π, ),

δδ{v(x,λ)

x

+u(ξ,λ)f(ξ)dξ+u(x,λ)

π

x v(ξ,λ)f(ξ)dξ}

+γγv(x,λ)

–

–π u(ξ,λ)f(ξ)dξ for x∈(,π].

()

Proof By differentiating equation () twice we can easily see thatτ(u) =λu+f,i(uf) =

ti(uf) =  (i= , ) so the functionuf given by () is the solution of the problem. We

shall prove the uniqueness by way of contradiction. Suppose that there are two different solutionsu andυ of the system () corresponding to the sameλ, which is not an

λis an eigenvalue with the corresponding eigenfunctionω. So we get a contradiction,

which completes the proof.

Let us introduce to the consideration the functionG(x,ξ,λ) given by

G(x,ξ;λ) = 

ω(λ) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

u(x,λ)v(ξ,λ) for –π≤ξ≤x< ,

u(ξ,λ)v(x,λ) for –π≤x≤ξ< ,

u(ξ,λ)v(x,λ) for –π≤x<  <ξ≤π,

u(x,λ)v(ξ,λ) for –π≤ξ<  <x≤π,

u(x,λ)v(ξ,λ) for  <ξ≤x≤π,

u(ξ,λ)v(x,λ) for  <x≤ξ≤π.

()

Then equation () can be written in the following form:

uf(x,λ) =γγ

–

–π

G(x,ξ;λ)f(ξ)dξ+δδ

π

+ G(x,

ξ;λ)f(ξ)dξ, ()

i.e. uf(x,λ) =G(x,·,λ),f(·)H. Consequently the functionG(x,ξ,λ) given by () is the

Green’s function for the considered BVTP. Now suppose thatλ=  is not an eigenvalue and letf∈ ⊕C() be an arbitrary function. DenotingG(x,ξ) =G(x,ξ; ) we have

τ(u) =f, i(u) =ti(u) = , i= , , ()

has an unique solutionu=u(x) given by

u(x) =γγ

–

–π

G(x,ξ)f(ξ)dξ+δδ

π

+ G(x,ξ)f(ξ)dξ. ()

Puttingf =λuin equation () we have the following integral equation with Green’s kernel:

u(x) =λ

γγ

–

–π

G(x,ξ)u(ξ)dξ+δδ

π

+ G(x,

ξ)u(ξ)dξ

. ()

4 Uniform and mean-square convergence of the eigenfunction expansions Let us define the integral operatorFby

(Fu)(x) =γγ

–

–π

G(x,ξ)u(ξ)dξ+δδ

π

+ G(x,

ξ)u(ξ)dξ. ()

Then the BVTP ()-() converts to the spectral problem for the integral operatorFgiven by

(I–λF)u= ,

whereIis the identity operator. Since the kernelG(x,ξ) of the integral operatorFis sym-metric and continuous we can apply the well-known extremal principle (see, for example, []). Let{λn}be a sequence of eigenvalues of the integral operatorFdetermined by the extremal principles and{φn(x)}be the corresponding sequence of orthonormal

Lemma . Let g∈ ⊕C().Then

lim

m→∞

γγ

–

–π

Fg–

m

i=

ci(Fg)φi



dx

+δδ

π

+

Fg–

m

i=

ci(Fg)φi



dx

= , ()

where ci(Fg) =Fg,φiHdenote the Fourier coefficients ofFg with respect to the orthonormal

set(φi).

Proof Denotegm(x) =g(x) –

m

i=g,φiHφi. Since{φn} is the orthonormal system inH, gm,φiH=  fori= , . . . ,m. From the fact that the eigenvaluesλnare determined by the

extremal principle with the corresponding sequence of orthonormal eigenfunctions{φn}

we have Fgm H≤ |λm+| gm H. Sinceλm+→, Fgm H→. Then we have

Fg=Fgm+ m

i=

g,φiHFφi=Fgm+ m

i=

λig,φiHφi

=Fgm+ m

i=

g,FφiHφi=Fgm+ m

i=

Fg,φiHφi ()

for arbitrarym= , , . . . . Lettingm→ ∞we get

Fg= ∞

i=

Fg,φiHφi, ()

where the convergence is in the Hilbert spaceH,i.e.the equality () holds.

Corollary . If g∈ ⊕C()then the Parseval equality

Fg _H= ∞

i=

c_i(Fg)

holds.

Corollary . The set of orthonormal eigenfunction of the integral operatorFis complete in the range of the integral operatorFgiven by

R(F) =h∈ ⊕C()|there exists g∈ ⊕C()such that h=Fg.

Theorem . Let the hypotheses and notation of Lemma.hold.Then,for any h∈R(F),

h= ∞

i=

γγ

–

–π

hφidx+δδ

π

+ h

φidx

φi(x),

Proof Leth=Fg. Then for anyn,pwe have

n+p

i=n

λig,φiHφi=F

_n+p

i=n

g,φiHφi

. ()

In view of the fact that the integral operatorFis a bounded linear operator in the Banach space⊕C() we get from ()

n+p

i=n

λig,φiHφi

≤C

_n+p

i=n

_g,φiH

/

()

for some constantCindependent ofn. By Bessel’s inequality, the right-hand side of this inequality tends to zero asn→ ∞uniformly. Thus the series

∞

i=

Fg,φiHφi(x) ()

converges in the Banach space⊕C(). Leth(x) be the sum of the last series. Consequently˜

˜

h∈ ⊕C() and

˜

h(x) = ∞

i=

Fg,φiHφi(x). ()

From () and () it follows that Fg–h˜ H= ,i.e. h(x) =h(x) almost everywhere. Since˜ his also continuous inwe haveh(x) =h(x) for all˜ x∈. Thus

h= ∞

i=

h,φiHφi, ()

where the series converges with respect to the norm of⊕C(),i.e.uniformly on.

Theorem . The set of all nonzero eigenvalues of the integral operatorFcoincide with the set of the eigenvalues(λn)which are obtained from the extremal principle.

Proof By way of contradiction, suppose there is a nonzero eigenvalueλ∗distinct from all eigenvalues (λn). Letu∗be the eigenfunction corresponding to the eigenvalueλ∗. Then

from Theorem . we get

λ∗u∗=Fu∗= ∞

i=

Fu∗,φi

Hφi=λ

∗∞

i=

u∗,φi

Hφi=  ()

sinceu∗,φiH=  for alli= , , . . . by Theorem .. Thus we get a contradiction.

Theorem . Let f ∈ ⊕C_{()and satisfy the boundary-transmission conditions()-().}

Then the Fourier series of f with respect to{φi}converges uniformly on∪,i.e.

lim

n→∞

sup

x∈ f(x) –

n

i=

γγ

–

–π

fφidx+δδ

π

+ f

φidx



φi(x)

Proof Letf ∈ ⊕C_{() satisfy the boundary-transmission conditions ()-() and denote}

g=τ(f). Theng∈ ⊕C(). By virtue of () and () we havef =Fg. From Lemma .,

f =Fg= ∞

i=

Fg,φiHφi=

∞

i=

f,φiHφi, ()

where the series is convergent in the Banach space⊕C().

Theorem . The set of eigenfunctions{φi(x)}is a complete orthonormal set in the Hilbert

space H.

Proof Denote by⊕Ck_() the set of all functionsf ∈Ck() which vanishes at some neigh-borhoods of the pointsx= –π,x= , andx=π. Letf ∈Hand>  be given. Then there exists a functiong∈ ⊕C

() such that f –g H<

 since the set⊕C() is dense in the

Hilbert spaceH,i.e.⊕C

() =H(see, for example, []). It is clear that

f – m i=

f,φiHφi

H

≤ f–g H+ g– m i=

g,φiHφi

H + m i=

(g–f),φi

Hφi

H

()

for arbitrarym. By Bessel’s inequality we have

m i=

(g–f),φi

Hφi

 H = m i=

(g–f),φi

H 

≤ f –g _H<

 

and, by Theorem ., there exists an integern=n() such that, form>n,

g– m i=

g,φiHφi

H < . ()

Finally, from () and () we get f – m i=

f,φiHφi

H

<

form>n. The proof is complete.

Now we are ready to prove the next important result.

Theorem . The set of eigenfunctions(φi(x))of the problem()-()form an orthonormal

basis in the Hilbert space H and for any f ∈H the Parseval equality

f _H= ∞

i=

γγ

–

–π

fφidx+δδ

π

+ fφidx



Proof Without loss of generality we shall assume thatλ=  is not an eigenvalue. Other-wise, we can select a realλ=  such that the problemτu=λu,i(u) =ti(u) = ,i= , 

has no nontrivial solutions. Then denotingλ˜=λ–λandq(x) =˜ q(x) –λwe see that the

problem

–u+q(x)u˜ =λ˜u, i(u) =ti(u) = , i= , , ()

has the same properties for the eigenfunctions and eigenvalues as the considered problem ()-(). Namely, the pair (λ˜,u(x)) is the eigen-pair of the problem () if and only if the pair (λ,u(x)) is an eigen-pair of ()-(). Clearly,λ˜ =  is not an eigenvalue of the problem (). Hence, without loss of generality we can assume thatλ=  is not an eigenvalue of the considered BVTP ()-(). Moreover, ifλ= , then the pair (λ,u(x)) is the eigen-pair of the BVTP ()-() if and only if the pair (_λ,u(x)) is the eigen-pair of the integral oper-atorF. Consequently the setφiform an orthonormal set of eigenfunctions either forF

and ()-(). Moreover, this set is complete by Theorem .. It is well known that any com-plete orthonormal set in a Hilbert space forms an orthonormal basis. Consequently, every functionf ∈Hmay be expanded in a Fourier series with respect to the orthonormal set of eigenfunctions (φi),i.e.the equality

f = ∞

i=

f,φiHφi

holds, where the series converges with respect to the norm of the Hilbert spaceH. Further, the Parseval equality follows immediately from the last equality.

5 The Rayleigh-Ritz principle for the BVTP (1)-(5)

In the last sections of this study we will investigate some extremal properties of the eigen-values and corresponding eigenfunctions of the considered BVTP ()-() by using some variational methods.

Lemma . Let q(x)≥for all x∈.Then all eigenvalues of the problem()-()are non-negative.

Proof Let (λ,u(x)) be any eigen-pair of the problem ()-(). Multiplying () byu(x) and integrating by parts fromx= –πtox= , and fromx=  tox=π, we have

γγ

–

–π

uu–qu+λudx+δδ

π

+ u

u–qu+λudx

=γγ

–

–π

–u–qu+λudx+δδ

π

+

–u–qu+λudx

+γγuu|

–

–π+δδuu|π+= . ()

By using the equalities ()-(), we getγγuu|

–

–π+δδuu|π+= . Hence

γγ

–

–π

–u–qu+λudx+δδσ

π

+

Consequently

λ=γγ –

–π[u+qu]dx+δδ π

+[u+qu]dx

γγ

–

–πudx+δδ π

+udx

≥ ()

sinceq≥ on∪by assumption.

Theorem . Let q(x)≥for all x∈.Then,all the eigenvalues of the problem()-() are positive if any one of the following conditions holds:

() q≡,i.e. there exists at least onex∈such thatq(x) > ; () cos_α+cos_{β= .}

Proof Letλbe the first eigenvalue with the corresponding eigenfunctionu(x). Show that

λ> .

() Sinceq(x) is continuous inthere areδ>  andq>  such thatq(x)≥qfor all

x∈[x–δ,x+δ]⊂. Then from () it follows immediately thatλ> .

() Suppose that it possible thatλ= . Then from (),u(x) =  for allx∈,i.e. u(x)

is a constant function in each ofand. Putting in () and () we havecosαu(–π) =

cosβu(π) = . Consequently at least one ofu(–π) andu(π) is equal to zero and therefore

uis identically zeroor. Then by applying the transmission conditions () and ()

we see thatuis identically zero on the whole=∪. We have a contradiction, which

completes the proof.

Theorem . Suppose that any one of the following conditions holds:

() q≡andq(x)≥;

() q(x)≥andcos_α+cos_{β= .}

Letλ<λ<· · · be the sequence of eigenvalues with the corresponding normalized

eigen-functionsφ(x),φ(x), . . .and let

Sn=

u(x)|u∈ ⊕C();u≡;i(u) =ti(u) = for i= , ;

u,φkH= for k= , , . . . ,n– , n= , , . . .

(naturally by Swe mean S={u(x)|u∈ ⊕C();u≡;i(u) =ti(u) = for i= , }).Then

for all n= , , . . .we have

λn=min

I(u)|u∈Sn

,

where the functional I(u)is given by

I(u) =  u _H

γγ

–

–π

u+qudx+δδ

π

+

u+qudx

. ()

Moreover,the minimizing function isφn,i.e.λn=I(φn).

Proof Letϕ(·)∈ ⊕C() withiϕ=tiϕ= ,i= , . Then by Theorem . we have

ϕ(x) = ∞

n=

ϕ(·),φn(·)

where the convergence is uniform on. Then, by integration by parts, we get

τ(ϕ),φn

H=

ϕ,τ(φn)

H=λn

φn(·),ϕ(·)

H. ()

Since{φn(x)}is a complete orthonormal set, by Parseval’s equation

ϕ,ϕH= ∞

n=

c_n(ϕ), ()

wherecn(ϕ) =φn(·),ϕ(·)H. By using () we get

γγ



–π

ϕ+qϕdx+δδ

π



ϕ+qϕdx

=ϕ,τ(ϕ)_H= –

_∞

n=

cnφn

,τ(ϕ)

H = – ∞ n= cn

φn,τ(ϕ)

H = ∞ n=

cnλnϕ,φnH=

∞

n=

λncn≥λ ϕ H. ()

Consequently

λ≤



ϕ _H

γγ



–π

ϕ+qϕdx+δδ

π



ϕ+qϕdx

. ()

Puttingϕ=φin equation () we havecn(ϕ) =φ,φn=  forn= , , . . . and

λ=



φ H

γγ

–

–π

φ_+qφ_dx+δδ

π

+

φ_+qφ_dx

. ()

From ()-() it follows immediately that

λ=min

I(ϕ)|ϕ∈S

and the minimizing function is ϕ=φ(x),i.e. λ=I(φ). Next, let ϕ(x)∈ ⊕C() with

iϕ=tiϕ= ,i= , , andϕ,φn=  forn= , . . . ,k. Then

γγ

–

–π

ϕ+qϕdx+δδ

π

+

ϕ+qϕdx

= ∞

n=k+

λncn(ϕ)≥λk+

∞

n=k+

cn(ϕ)=λk+ ϕ H. ()

Hence, by the same arguments as before, we have

λk+=min

I(ϕ)|ϕ∈Sk+

Remark . By applying the minimization principle directly, it is not possible to deter-mine explicitly the eigenvalues and corresponding eigenfunctions, since we do not know how to minimize over all ‘admissible’ functions. Nevertheless, using the Rayleigh func-tional () with appropriate test functions one can obtain useful approximations for the eigenvalues.

6 The minimax property of eigenvalues

According to the minimization principle which is given by the preceding Theorem . we can find thenth eigen-pair (λn,φn) only after the previous eigenfunctionsφ(x),φ(x), . . . ,

φn–(x) are known. But in many applications it is important to have a characterization of

any eigen-pair (λk,φk) that makes no reference to other eigen-pairs. By applying the

fol-lowing theorem we can determine thenth eigen-pair (λn,φn) without using the preceding

eigenfunctionsφ(x),φ(x), . . . ,φn–(x).

Theorem . Let u(x),u(x), . . . ,un–(x)∈ ⊕C()be arbitrary functions.Denote

Dn–(u,u, . . . ,un–) =

u∈ ⊕C()|j(u) =tj(u) = ,j= , ;

u,ui= for i= , , . . . ,n– ;, n= , , . . .

(naturally by D(u,u, . . . ,un–)we mean the linear manifold

u∈ ⊕C()|j(u) =tj(u) = ,j= , 

.

Then the nth eigenvalue of the BVTP()-()is

λn=max

minI(u)|u∈Dn–(u,u, . . . ,un–)

|φi∈ ⊕C()

for i= , , . . . ,n– . ()

Proof Let  <λ<λ<· · · be the sequence of eigenvalues determined by the extremal

principles andφ(x),φ(x), . . . be the corresponding sequence of orthonormal

eigenfunc-tions and letu,u, . . . ,un–∈ ⊕C() be arbitrary functions. Define

Fn(u,u, . . . ,un–) =inf

I(u)|u∈Dn–(u,u, . . . ,un–)

, n= , , . . . .

Now let ψ(x),ψ(x), . . . ,ψn–(x)∈ ⊕C() be any given functions, such that i(ψj) =

ti(ψj) =  (i= , ;j= , , . . . ,n– ). Denotingaij=φi,ψiH fori,j= , , . . . ,n– ,

con-sider a system ofn–  homogeneous linear equations innunknownsz,z, . . . ,zngiven

by

n

i=

aijzi= , j= , , . . . ,n– . ()

Obviously this system ofn–  homogeneous linear equations innunknowns has a nontriv-ial solution. Letα,α, . . . ,αn–be any nontrivial solution of system (). Define a function

υn(x) by

It is easy to see that

υn,ψjH=  forj= , , . . . ,n– . ()

Consequentlyυn∈Dn–(ψ,ψ, . . . ,ψn–) and υ H =α+α+· · ·+αn. Then we have

γγ



–π

υ_n+qυ_ndx+δδ

π



υ_n+qυ_ndx

=

n

i,j=

αiαj

γγ



–π

φ_iφ_j+qφiφj

dx+δδ

π



φ_iφ_j+qφiφj

dx

.

Integrating by parts we get

γγ



–π

φ_iφ_j+qφiφj

dx+δδ

π



φ_iφ_j+qφiφj

dx

=φ_iφj|_––_π+φ_iφj|π_++

φj,τ(φi)

H=λiφj,uφiH=λiδij,

whereδijis the Kronecker delta. Then we have

I(υn) =



υn _H n

i=

λiαi≤λn.

Consequently Fn(u,u, . . . ,un–)≤I(υn)≤λn for allu,u, . . . ,un–∈ ⊕C().

Further-more, by virtue of the preceding theorem

Fn(φ,φ, . . . ,φn–) =λn.

Hence

λn=max

F(u,u, . . . ,un–)|φj∈ ⊕C(),j= , , . . . ,n– 

,

which completes the proof.

Remark . In many problems of mathematical physics, the smallest eigenvalue (the so-called principal eigenvalue) plays an important role. For example, the principal eigenvalue of the simple boundary value problem

–u=λρ(x)u, x∈[–π,π], u(–π) =u(π) = ,

⊕C() be any nontrivial function satisfying the boundary-transmission conditions ()-() called the trial function. Then by virtue of the minimax principle we have the inequality

λ≤



υ _H

γγ



–π

υ+qυdx+δδ

π



υ+qυdx

,

which gives us an upper bound for the principal eigenvalue. By taking the trial function

υ(x) as close as possible to the corresponding eigenfunction we can expect to get the ‘good’ estimation for principal eigenvalueλ. In many special cases the useful trial function can

be found by applying some principles of variational analysis.

7 Dependence of eigenvalues on the potential

The minimax principle of the eigenvalues,i.e.equation () for the eigenvalues makes it possible to study the dependence of the eigenvalues on the coefficients of the differen-tial equation. In this section we shall establish the monotonicity of the eigenvalues with respect to the potentialq(x) for fixed boundary-transmission conditions.

Theorem . Letλn(q)be the nth eigenvalue of the BVTP()-().Thenλn(q)is a

mono-tonically increasing function with respect to the variable q=q(x),i.e. if q(x)≤q(x)for all

x∈thenλn(q)≤λn(q).

Proof Define

Ii(u) =

 u 

H

γγ

–

–π

u+qiu

dx+δδ

π

+

u+qiu

dx

, i= , .

Let the notation of the preceding theorem hold and letu(x),u(x), . . . ,un–(x)∈ ⊕C()

be any arbitrary functions.

Since  ≤ q(x) ≤ q(x) for all x ∈ it is obvious that I(u) ≤I(u) for all u ∈

Dn–(u,u, . . . ,un–). Then by virtue of Theorem ., we find the required inequality

λn(q)≤λn(q). The proof is complete.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors read and approved the final manuscript.

Author details

1_{Faculty of Education, Giresun University, Giresun, 28100, Turkey.}2_{Department of Mathematics, Faculty of Arts and}

Science, Gaziosmanpa¸sa University, Tokat, 60250, Turkey.3_{Institute of Mathematics and Mechanics, Azerbaijan National}

Academy of Sciences, Baku, Azerbaijan.

Acknowledgements

The authors would like to thank the referees for their valuable comments.

Received: 1 January 2016 Accepted: 8 April 2016 References

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