On Green’s function for boundary value problem with eigenvalue dependent quadratic boundary condition

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R E S E A R C H

Open Access

On Green’s function for boundary value

problem with eigenvalue dependent

quadratic boundary condition

Haskız Co¸skun

*

_{, Ay¸se Kabata¸s and Elif Ba¸skaya}

*_{Correspondence: [email protected]}

Faculty of Science, Department of Mathematics, Karadeniz Technical University, Trabzon, 61080, Turkey

Abstract

In this paper we obtain Green’s function for a regular Sturm-Liouville problem having the eigenparameter in all boundary conditions in which the left one is in quadratic form. We assume no smoothness condition on the potential.

Keywords: Sturm-Liouville problem; eigenfunction asymptotics; Green’s function

1 Introduction

In this paper, we consider the boundary value problem

–y(t) +q(t) –λy(t) = , (.) y(a) =cλ+dλ+ey(a), (.) by(b) +by(b) =λ

b_y(b) +b_y(b), (.) whereλis real parameter;q(t) is a real-valued function andq(t)∈L[a,b];a,b,b,b,b,b,

c,d,e∈Randc= . This problem is different from the usual regular Sturm-Liouville prob-lem [] in the sense that the eigenvalue parameterλappears in both boundary conditions and in the first boundary condition is in quadratic form. We first derive asymptotic ap-proximations for the eigenfunctions of the problem, and then using these apap-proximations we obtain Green’s function.

One application of the Green’s function is to derive sampling theorems associated with eigenvalue problems containing an eigenvalue parameter in the boundary condition [].

We suppose without loss of generality thatqhas a mean value zero,i.e., b

a

q(t)dt= . (.)

As an illustration of our results we obtain fora≤κ≤ξ≤b,b_= ,

G(κ,ξ,λ) = –λ–/sinλ

/₍_κ_–_a₎_cosλ/₍_b_–_ξ₎

cosλ/₍_b_–_a₎ +

λ– cosλ/₍_b_–_a₎

×

 

κ

a

q(t)dt cosλ/(κ–a)cosλ/(b–ξ)

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– b b_ +

  ξ

q(t)dt sinλ/(κ–a)sinλ/(b–ξ)

+b



b_tanλ

/₍_b_–_a₎_sin_λ/₍_κ_–_a₎_cosλ/₍_b_–_ξ₎

+Oλ–η(λ),

whereη(λ)→ asλ→ ∞. Similar results hold fora≤ξ≤κ≤bexchanging the role of

κandξ.

These types of results with diﬀerent boundary conditions were obtained before in the case whereq(t) is assumed to be continuous on a ﬁnite interval [, ]. In [], the Green’s functions were derived asymptotically with the error term of exponential type, that is, forλ=s_,_G₍_x_,_y_,_{λ) =}_{· · ·}₊_O₍ 

|s|exp[t(y–x)]),a≤y≤x(p.). In our results, we only assume thatq(t)∈L_[_a_,_b_{] and ﬁnd that}_G₍_x_,_y_,_{λ) =}_{· · ·}₊_O_(λ–_{η(λ)) where}_η(λ)_→_{ as}

λ→ ∞. We use a similar approach as in [] in which the spectral functions associated with Sturm-Liouville problems were considered on [,∞) with usual boundary condition not including the eigenparameter forq(t)∈L_[,_∞_).

2 Method

We associate with (.) the Riccati equation

v= –λ+q–v. (.)

It was shown in [] that ifv(t,λ) is a complex-valued solution of (.) and

S(t,λ) :=Rev(t,λ), (.)

T(t,λ) :=Imv(t,λ), (.)

then any nontrivial real-valued solution of (.) can be expressed as

z(t,λ) =cexp

t

a

S(x,λ)dx cos

c+

t

a

T(x,λ)dx

, (.)

with

z(t,λ) =cS(t,λ)exp

t

a

S(x,λ)dx cos

c+

t

a

T(x,λ)dx

–cexp

t

a

S(x,λ)dx sin

c+

t

a

T(x,λ)dx

T(t,λ). (.)

Here,candcare constants to be determined.

We suppose that for eacht∈[a,b] there exist

_tbeiλ/xq(x)dx≤A(t)η(λ), (.)

where

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Forq(t)∈L_[_a_,_b_{]the existence of the}_A_and_η_{functions may be established, for}_λ_positive

as follows.

We note that, avoiding the trivial case_tb|q(x)|dx= ,

_tbeiλ/xq(x)dx≤ b

t

q(x)dx<∞.

So, if we deﬁne

F(t,λ) := ⎧ ⎨ ⎩

|b t ei

λ/x_q₍_x₎_dx_|_/b

t |q(x)|dx, if

b

t |q(x)|dx= ,

, if _tb|q(x)|dx= , (.)

then ≤F(t,λ)≤, and we setη(λ) :=sup_a_≤_t_≤_bF(t,λ). Hereη(λ) is well deﬁned by (.) and goes to zero asλ→ ∞[].

Our method of approximating a solution of (.) is similar to that of []. We consider (.) on [a,b] and set

v(t,λ) :=iλ/+

∞

n=

vn(t,λ). (.)

Substitution of (.) into (.) and rearrangement gives

v_+ iλ/v+v+ iλ/v+

∞

n=

v_n+ iλ/vn

=q–v_–

∞

n=

v_n_–+ vn– n–

m=

vm

.

We choose thevnso that

v_+ iλ/v=q,

v_+ iλ/v= –v,

and forn= , , . . . ,

v_n+ iλ/vn= –

v_n_–+ vn– n–

m=

vm

.

Solving forvn,n= , , , . . . ,

v(t,λ) = –e–iλ

/_t b

t

eiλ/xq(x)dx, (.)

v(t,λ) =e–iλ

/_t b

t

eiλ/x_v

(x,λ)dx, (.)

vn(t,λ) =e–iλ

/_t b

t

eiλ/x

v_n_–+ vn– n–

m=

vm

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It is shown in [] that∞_n_=vn(t,λ) is uniformly absolutely convergent for allλ≥λand

for allt∈[a,b]. It also follows from the choice ofvn,n= , , , . . . , that

∞

n=vn(t,λ) is

uniformly absolutely convergent. The seriesiλ/+∞_n_=vn(t,λ) is therefore a solution of

(.) and

S(t,λ) =Re

∞

n=

vn(t,λ), (.)

T(t,λ) =λ/+Im

∞

n=

vn(t,λ). (.)

The asymptotic forms ofS(t,λ) andT(t,λ) are obtained in [] as follows:

S(t,λ) = –sinλ/t+ζt

+Oη(λ) (.)

and

T(t,λ) =λ/–cosλ/t+ζt

+Oη(λ), (.)

where

sinζt=

b

t

q(x)cosλ/x dx, cosζt=

b

t

q(x)sinλ/x dx. (.)

Also,

t

a

S(x,λ)dx=  λ/

cosλ/t+ζt

–cosλ/a+ζa

+Oλ–/η(λ), (.)

t

a

T(x,λ)dx=λ/(t–a) –  λ/

sinλ/t+ζt

–sinλ/a+ζa

+

t

a

q(x)dx

+Oλ–/η(λ). (.)

The last two equalities are obtained in [].

3 Approximations for the eigenfunctions

In this section we obtain approximations for the solution of (.)-(.). We deﬁne two so-lutions,(t,λ) and(t,λ) of (.)-(.) with the initial conditions

(a,λ) = , (a,λ) =cλ+dλ+e, (.)

and

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Theorem  Let(t,λ)and(t,λ)be the solutions of(.)satisfying(.)and(.), respec-tively.Then we ﬁnd

(t,λ) = 

cos{cot–_[ T(a,λ) –cλ_–_d_λ_–_e₊_S₍_a_,_λ₎]}

exp

t

a

S(x,λ)dx

×cos

cot–

T(a,λ) –cλ_–_d_λ_–_e₊_S₍_a_,_λ)

+

t

a

T(x,λ)dx

and

(i) ifb_= ,

(t,λ) = (b–b

λ)exp(–

b

t S(x,λ)dx)

cos{tan–_F (b,λ)}

cos

tan–F(b,λ) –

b

t

T(x,λ)dx

,

where

F(b,λ) :=

 T(b,λ)

S(b,λ) + b

λ–b

b_λ–b

,

(ii) ifb_= ,

(t,λ) = bexp(– b

t S(x,λ)dx)

cos{cot–_F (b,λ)}

cos

cot–F(b,λ) –

b

t

T(x,λ)dx

,

where

F(b,λ) :=

bT(b,λ)

–b_λ+b+bS(b,λ)

.

Proof of Theorem From (.), (.) and (.), we have

(a,λ) =ccosc= , (.)

(a,λ) =cS(a,λ)cosc–csincT(a,λ) =cλ+dλ+e. (.)

From (.), we obtain

c=

 cosc

. (.)

Using this equation in (.),

c=cot–

T(a,λ) –cλ_–_d_λ_–_e₊_S₍_a_,_λ)

. (.)

Hence

c=

 cos{cot–_[ T(a,λ)

–cλ_–_d_λ_–_e₊_S₍_a_,_λ₎]}

. (.)

Substituting the values ofcandcinto (.), we evaluate(t,λ) as required. In order to

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(i) ifb_= , we obtain from (.), (.) and (.)

(b,λ) =cexp

b

a

S(x,λ)dx cos

c+

b

a

T(x,λ)dx

=b–bλ, (.)

(b,λ) =cS(b,λ)exp

b

a

S(x,λ)dx cos

c+

b

a

T(x,λ)dx

–cexp

b

a

S(x,λ)dt sin

c+

b

a

T(x,λ)dx

T(b,λ)

=b_λ–b. (.)

From the last two equalities we have

c=

b–bλ

exp(_abS(x,λ)dx)cos{tan–_F (b,λ)}

, (.)

c=tan–F(b,λ) –

b

a

T(x,λ)dx. (.)

Substituting the values ofcandcinto (.) proves the result and,

(ii) ifb_= , we obtain similarly from (.), (.) and (.)

(b,λ) =cexp

b

a

S(x,λ)dx cos

c+

b

a

T(x,λ)dx

=b, (.)

(b,λ) =cS(b,λ)exp

b

a

S(x,λ)dx cos

c+

b

a

T(x,λ)dx

–cexp

b

a

S(x,λ)dt sin

c+

b

a

T(x,λ)dx

T(b,λ)

=b_λ–b. (.)

Hence from the last two equalities we evaluate

c=

b

exp(_abS(x,λ)dx)cos{cot–_F (b,λ)}

, (.)

c=cot–F(b,λ) –

b

a

T(x,λ)dx. (.)

Substituting the values ofcandcinto (.) proves the theorem.

Theorem  Let(t,λ)and(t,λ)be the solutions of(.)satisfying(.)and(.), re-spectively.Then we ﬁnd asλ→ ∞,

(t,λ) =cλ/sinλ/(t–a) – c λ

t

a

q(x)dx cosλ/(t–a) +Oλη(λ),

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(i) ifb_= ,

(t,λ) = –b_λcosλ/(b–t) –λ/

b_+b



 b

t

q(x)dx

×sinλ/(b–t) +Oλ/η(λ),

(ii) ifb= ,

(t,λ) = –b_λ/sinλ/(b–t) +

b+

b_ 

b

t

q(x)dx

×cosλ/(b–t) +Oη(λ),

whereη(λ)is as in(.).

Proof of Theorem We evaluate the terms in Theorem  asλ→ ∞. Firstly, using (.) and (.) together with the series expansion, we ﬁnd

T(a,λ) –cλ_–_d_λ_–_e₊_S₍_a_,_λ)

= λ

/_–_cos(λ/_a₊_ζ

a) +O(η(λ))

–cλ_–_d_λ_–_e_–_sin(λ/_a₊_ζ

a) +O(η(λ))

= λ

/_–_cos(λ/_a₊_ζ

a) +O(η(λ))

–cλ_{[ +}d cλ–+

e cλ–+



cλ–sin(λ/a+ζa) +O(λ–η(λ))]

=

– cλ

–/₊

cλ

–_cos_λ/_a₊_ζ a

+Oλ–η(λ)

×

 –d cλ

–_–e

cλ

–_–

cλ

–_sin_λ/_a₊_ζ a

+Oλ–η(λ)

= – cλ

–/₊_O_λ–_η(λ)_.

From the last equality we have

cot–

T(a,λ) –cλ_–_d_λ_–_e₊_S₍_a_,_λ)

=π

 +  cλ

–/₊_O_λ–_η(λ)_, _(.)

and from that

cos

cot–

T(a,λ) –cλ_–_d_λ_–_e₊_S₍_a_,_λ)

= –sin

 cλ

–/₊_O_λ–_η(λ)

= – cλ

–/₊_O_λ–_η(λ)_, _(.)

sin

cot–

T(a,λ) –cλ_–_d_λ_–_e₊_S₍_a_,_λ)

=cos

 cλ

–/₊_O_λ–_η(λ)

=  –  cλ

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and

 cos{cot–_[ T(a,λ)

–cλ_–_d_λ_–_e₊_S₍_a_,_λ₎]}

= 

–_cλ–/₊_O_(λ–_η(λ))

= –cλ/+Oλη(λ). (.)

Using_atS(x,λ)dxgiven by (.) we get

exp

t

a

S(x,λ)dx =  +  λ/

cosλ/_t₊_ζ t

–cosλ/_a₊_ζ a

+Oλ–/η(λ) (.)

and also, using_atT(x,λ)dxgiven by (.),

sin

t

a

T(x,λ)dx =sin

λ/(t–a) –  λ/

t

a

q(x)dx

+Oλ–/η(λ), (.)

cos

t

a

T(x,λ)dx =cos

λ/(t–a) –  λ/

t

a

q(x)dx

+Oλ–/η(λ). (.)

Hence

cos

cot–

T(a,λ)

–cλ_–_d_λ_–_e₊_S₍_a_,_λ)+

t

a

T(x,λ)dx

= –sin

λ/(t–a) –  λ/

t

a

q(x)dx

– cλ

–/

×cos

λ/(t–a) –  λ/

t

a

q(x)dx

+Oλ–/η(λ). (.)

Substituting the values of (.), (.) and (.) into(t,λ) as in Theorem  and using trigonometric expansions, we obtain(t,λ) as required. In order to ﬁnd(t,λ) asλ→ ∞:

(i) ifb_= , using (.) and (.), we obtain S(b,λ)

T(b,λ)=

O(η(λ))

λ/_{[ +}_O_(λ–/_η_(λ))]=O

λ–/η(λ) (.)

and

b_λ–b

(b_λ–b)T(b,λ)

= b

λ–b

b_λ/_–_b

λ/+O(λη(λ))

= b

λ–b

b_λ/_{[ –}b

bλ

–₊_O_(λ–/_η_(λ))]

=

b_ b_λ

–/_–b

b_λ

–/

×

 +b

b_λ

–₊_O_λ–/_η_(λ)

=b



b_λ

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Substituting the values of (.) and (.) intoF(b,λ) as in Theorem (i)), we get

F(b,λ) =

b_ b_λ

–/₊_O_λ–/_η_(λ)_.

From that

tan–F(b,λ) =

b_ b_λ

–/₊_O_λ–/_η_(λ)_, _(.)

costan–F(b,λ)

=  –



b_ b_



λ–+Oλ–η(λ) (.)

and

sintan–F(b,λ)

=b



b_λ

–/₊_O_λ–/_η_(λ)_. _(.)

From (.), we ﬁnd

b–bλ

cos{tan–_F (b,λ)}

= b–b

λ

 –_(b

b_)λ–+O(λ–η(λ))

=b–bλ

×

 +



b_ b_



λ–+Oλ–η(λ)

= –b_λ+

b–

(b_)

b_

+Oη(λ). (.)

Using_atS(x,λ)dxgiven by (.) we get

exp

– b

t

S(x,λ)dx =  +  λ/cos

λ/_t₊_ζ t

+Oλ–/η(λ) (.)

and also, using_atT(x,λ)dxgiven by (.)

b

t

T(x,λ)dx= b

a

T(x,λ)dx– t

a

T(x,λ)dx

=λ/(b–t) +  λ/

sinλ/t+ζt

–

b

t

q(x)dx

+Oλ–/η(λ). (.)

From (.), (.) and (.) we see that

cos

tan–F(b,λ) –

b

t

T(x,λ)dx

=cos

λ/(b–t) –  λ/

b

t

q(x)dx

+b



b_λ

–/_sin

λ/(b–t) –  λ/

b

t

q(x)dx

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Finally, substituting the values of (.), (.) and (.) into(t,λ) as in Theorem (i)) and using trigonometric expansions, we complete the proof.

(ii) Ifb_= , the prove is similar to (i).

4 Approximations for Green’s function

In this section, we obtain asymptotic approximations for Green’s function of (.)-(.). LetWt(,) be the Wronskian of(t,λ) and(t,λ). We deﬁnew(λ) as follows:

w(λ) :=Wt(,) =(t,λ)(t,λ) –(t,λ)(t,λ). (.)

It is well known ([]) that the Green’s function of problem (.)-(.) is

G(κ,ξ,λ) = ⎧ ⎨ ⎩

(κ,λ)(ξ,λ)

w(λ) , a≤κ≤ξ≤b,

(κ,λ)(ξ,λ)

w(λ) , a≤ξ≤κ≤b.

(.)

Theorem  For a≤κ≤ξ≤b,we ﬁnd Green’s function of problem(.)-(.)asλ→ ∞:

(i) if◦_= ,

G(κ,ξ,λ) = –λ–/sinλ

/₍_κ_–_a₎_cosλ/₍_b_–_ξ₎

cosλ/₍_b_–_a₎ +

λ–

cosλ/₍_b_–_a₎

×

 

κ

a

q(t)dt cosλ/(κ–a)cosλ/(b–ξ)

–

b_ b_ +

 

b

ξ q(t)dt

sinλ/(κ–a)sinλ/(b–ξ)

+b



b

tanλ/(b–a)sinλ/(κ–a)cosλ/(b–ξ)

+Oλ–η(λ),

(ii) ifb_= ,

G(κ,ξ,λ) = –λ–/sinλ

/₍_κ_–_a₎_sin_λ/₍_b_–_ξ₎

sinλ/₍_b_–_a₎ +

λ– sinλ/₍_b_–_a₎

×

 

κ

a

q(t)dt cosλ/(κ–a)sinλ/(b–ξ)

+

b

b_ +  

b

ξ q(t)dt

sinλ/(κ–a)cosλ/(b–ξ)

–b b_cotλ

/₍_b_–_a_)sin_λ/₍_κ_–_a₎_sin_λ/₍_b_–_ξ₎

+Oλ–η(λ),

whereη(λ)is as in(.).Similar results hold fora≤ξ≤κ≤bexchanging the role of κandξ.

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and (.) into (.) and evaluate the terms asλ→ ∞. Hence

(t,λ) =cλcosλ/(t–a) + c 

t

a

q(x)dx λ/sinλ/(t–a)

+Oλ/η(λ). (.)

Also substituting (.) and (.) into (.) and evaluating the terms asλ→ ∞, we have

(t,λ) = –b_λ/sinλ/(b–t) +

b_+b



 b

t

q(x)dx

λcosλ/(b–t)

+Oλη(λ). (.)

Using(t,λ),(t,λ) as in Theorem (i)) and (.), (.) in (.), we get

w(λ) =cb_λcosλ/(b–a) +cb_λ/sinλ/(b–a)

+Oλ/η(λ). (.)

From this

 w(λ)=

 {cb_λ_cosλ/₍_b_–_a_{)[ +}b

b_λ–/tanλ/(b–a) +O(λ–/η(λ))]}

= 

cb_cosλ/₍_b_–_a₎λ

–_– b

c(b_)λ

–/ sinλ/(b–a)

cos_λ/₍_b_–_a₎

+Oλ–/η(λ). (.)

Theorem (i) is proved by substituting(t,λ),(t,λ) as in Theorem (i)) and (.) into (.). The other part (ii) can be proved similarly.

Acknowledgements

The authors would like to thank the referees for invaluable comments and insightful suggestions.

Funding

The current work is partially supported by Karadeniz Technical University.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read carefully and approved the ﬁnal version of the manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 22 February 2017 Accepted: 2 May 2017

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