TR/12/94 July 1994 APPROXIMATE SOLUTION OF SECOND KIND INTEGRAL EQUATIONS ON INFINITE CYLINDRICAL SURFACES A.T. Peplow and S.N. Chandler-Wilde To appear in SIAM JOURNAL ON NUMERICAL ANALYSIS

TR/12/94 July 1994

APPROXIMATE SOLUTION OF SECOND KIND INTEGRAL EQUATIONS ON INFINITE

CYLINDRICAL SURFACES A.T. Peplow and S.N. Chandler-Wilde

To appear in SIAM JOURNAL ON NUMERICAL ANALYSIS

w9253524

APPROXIMATE SOLUTION OF SECOND KIND INTEGRAL EQUATIONS ON INFINITE CYLINDRICAL SURFACES

ANDREW T. PEPLOW* AND SIMON N. CHANDLER-WILDE^↑

Abstract. The paper considers second kind integral equations of the formφ(x)=g(x)+ )

( ) ( )

(x,y y ds y

∫ _S k φ (abbreviatedφ =g+Kφ), in which S is an infinite cylindrical surface of arbitrary smooth cross-section. The “truncated equation” (abbreviated φ_a =E_ag +K_aφ_a), obtained by replacing S by Sa, a closed bounded surface of class C², the boundary of a section of the interior of S of length 2a, is also discussed. Conditions on k are obtained (in particular, implying that K commutes with the operation of translation in the direction of the cylinder axis) which ensure that I-K is invertible, that I - Ka is invertible and (I — Ka)^-1 uniformly bounded for all sufficiently large a, and that φ_a converges to φ in an appropriate sense as a→∞. Uniform stability and convergence results for a piecewise constant boundary element collocation method for the truncated equations are also obtained.

A boundary integral equation, which models three-dimensional acoustic scattering from an infi- nite rigid cylinder, illustrates the application of the above results to prove existence of solution (of the integral equation and the corresponding boundary value problem) and convergence of a particular collocation method.

Key words. second kind integral equations, Wiener-Hopf equations, boundary element method, Helmholtz equation, collocation method

AMS subject classifications. 65R20, 45E10, 65N38, 35J05

1. Introduction. We are concerned in this paper with second kind integral equations of the form

(1) φ(x)= g(x)+ ∫ _S k(x,y)φ(y) ds(y), x∈ S,

and their numerical solution, in the case when S is an infinite cylindrical surface with arbitrary cross-section in three-dimensional space. In equation (1)g∈ BC(S) (the space of bounded continuous functions on S) is assumed known and φ ∈BC(S) is to be determined. We abbreviate (1) in operator form as

(2) φ = g + K φ

and make sufficient assumptions on the smoothness of the surface S and on the be- haviour of the kernel k (k is continuous or weakly singular) so that K : BC(S) → BC(S) is bounded but not compact. In particular, we suppose that

}

∈ R

∈ , ) , ( : ) , , {(

= x₁ x₂ x₃ x₁ x₂ x₃

S Γ

where Γ⊂ R² is a Jordan curve of class C².

Integral equations of the form (1) frequently arise when reformulating linear el- liptic boundary value problems in the interior or exterior of S as boundary integral equations. We consider an example of this type in Section 4 of the paper, in which acoustic scattering by an infinite rigid cylinder is investigated, with

(3) ( ),

)

∂ ( 2 ∂

= )

( G x,y

y y n

x, k

* Structural Dynamics Group, Institute of Sound and Vibration Research, University of Southampton, Southampton S09 5NH, UK ([email protected]).

†Department of Mathematics and Statistics, Brunel University, Uxbridge UB8 3PH, UK ([email protected])

1

2 A. T. PEPLOW AND S. N. CHANDLER-WILDE where n(y) is the normal to S at y, directed into the exterior of S, and

(4) x- y

y e x, G

y - x i

π

κ

: 4 )

( =

is a fundamental solution of the Helmholtz equation, Δu+κ²u=0.

When solving (1) numerically it is convenient, as a preliminary stage, to truncate the infinite surface S. Let +~

S and S~_

denote the half cylinders denned by }

> 0 : <

∈ ) , , {(

=

± :

~ x1 x2 x3 S x3

S and, for a ≥ 2, let

.

≤ }

∈ : ) , , ( {

=

~ :

3 3

2

1 x x S x a

x Sa

Let E_ be the surface

(5) E_:= {(x₁,x₂,x₃):(x₁,x₂)∈ Ω,x₃ = f(x₁,x₂)}

where Ω is the interior of , and f is any given continuous function on Γ Ω satisfying

∈ , ) , ( , 0

=

∈ , ) , ( , 0 ) >

, (

2 1

2 2 1

1 Γ

Ω x

x x x x

x f and such that S₋ = S~₋ ∪ E₋

: is a smooth surface of class C² (see Figure 1). Let

~ ∪ .

= : and )}

, (

=

∈ , ) , ( : ) , , {(

=

: _{1 2 3 1 2 3 1 2 + + +}

+ x x x x x x f x x S S E

E Ω

FIG. 1. Cross-section through the surface S_.

For V ⊂ R³ andx ∈ R³, let V + x denote the translation of the set V by the vector x, and let e₃ ∈R³ be the unit vector in the x3 direction. Then, for a ≥ 2, define

Sa := ~ ~

= _ )

∪ ( ) +

∪ (E_ ae₃ E₊ ae₃ S ∪ {(x₁,x₂x₃) :(x₁,x₂)∈ , x₃ =

S_{a a} Ω

(see Figure 2). Note that Sa is a smooth closed surface of class C². ))}

, ( (^a+∫ x₁ x₂

±

Let ∑ : = {S, S+, S_-} }∪ {S_a+be₃:a ≥ 2,b∈R and, for T ⊂ S^*∈ ∑, define the integral operator K_T on BC(T) by

(6) K_Tψ (x) = ∫ _T k(x,y)ψ (y)ds(y) x∈ T.

INTEGRAL EQUATIONS ON CYLINDRICAL SURFACES 3

FIG. 2. Cross-section through the surface Sa.

L e t K±:K _± and K := K , for a ≥ 2.

Sa

a S

A crucial requirement for the theory developed is a translation invariant assump- tion on the kernel function k, that, for all (x, y) ∈I := {(x,y) ∈ T: T ∈ ∑},

.

∈ R ), (

= ) + , +

(x te₃ y te₃ k x,y t k

We analyse, in Section 2 of this paper, the approximation to (1) obtained by replacing the infinite surface S by the finite closed surface Sa and the convergence to φ, of the solution φ_a ∈ BC(S_a) of the approximate equation, as a→∞. To make precise the definition of φ_a and the sense in which φ_a ∈ BC(S_a)approximates

)

∈ BC(S

φ , introduce mappings between these two spaces. For a ≥1, let F_a ∈ C^∞ (R) be an even function satisfying 0 ≤ F_a(t) ≤1, t∈R,, and

. 1

≥ + , 0

≤ ,

≤ 0 ,

= 1 )

( t a

a t t

F_a

For a ≥ 2 define E_a : BC(S) → BC(S_a) by

(7)

⎪⎩

⎪⎨

⎧ =

=

~ ,

∈ \ ,

0

~ ,

∈ ) , , ( ),

( ) : (

)

( ^{1 3 1 2 3}

a a

a -

a

a x S S

S x x x x x

x x F

E ψ

ψ

and by R_a : BC(S_a) → BC(S)

(8)

⎪⎪

⎩

⎪⎪⎨

⎧

≤

=

=

=

.

~ ,

∈ \ ) , , ( ), ( )) , , ((

≥ ,

~ ,

∈ \ ) , , ( ),

( )) , , ((

~ ,

∈ ),

( :

) (

3 3

2 1 3

2 1

3 3

2 1 3

2 1

_a x S S x x x x x F _a x x

a x S S x x x x x F a x x

S x x

x R

a a

a a

a a

ψ ψ ψ ψ

Then )φ_a ∈BC(S_a is defined by

(9) φ_a = E_ag + K_aφ_a, which we will refer to as the “truncated” version of (1).

In Section 2 we construct a partial theory of the solvability of equation (1) and of the truncated equation (9). Theorem 2.8 suggests that the existence of a solution

4 A. T. PEPLOW AND S. N. CHANDLER-WILDE

to the truncated equation for all sufficiently large a depends not just on the unique solvability of the original integral equation (1) but also on that of the “half cylinder”

equations obtained by replacing S by S±. Specifically, it shows that I - Ka is invertible and uniformly bounded for all sufficiently large a provided I – K, I - K+ and I – K_

a r e i n j e ct i v e . T h e s e c o n d i t i o n s a l s o e n su r e t h e i n v er t i b i li t y o f t h e o r i g i n a l o p er a t o r I - K ( Co ro l l a ry 2 .9 ) , s o t h a t t h e s p e c t r u m o f K i s co nta i n ed in the u ni on o f { 0 } and the sets of eigenvalues of K, K+, and K-. This result may be powerful for establishing existence of a solution to equation (1) and, in the case when (1) is a boundary integral equation, for establishing existence of solution for the corresponding boundary value problem formulation. These points are illustrated by the example in Section 4.

In Section 3 we consider the numerical solution of the integral equation on Sa, defining φ⁽ⁿ⁾ ∈ BC(S_an) by

(10) φ⁽ⁿ⁾ = E_ang + K⁽ⁿ⁾φ⁽ⁿ⁾,

w h e r e K ⁿ K_a P_n a n d i s a n i n t e r p o l a t o r y p r o j e c t i o n o p e r a t o r o n t o a s p a c e

= n

) (

Pn

o f p i e c e w i s e c o n s t a n t f u n c t i o n s o n a f i n i t e e l e m e n t m e s h o n N o t e t h a t an

S P_nφ⁽ⁿ⁾

is a piecewise constant collocation method approximation to .

an

φ and φ⁽ⁿ⁾ is the i t e r a t e d c o l l o c a t i o n m e t h o d a p p r o x i m a t i o n o f S l o a n [ 1 2 ] . T h e r e s u l t s o f S e c t i o n 2 are extended to show that the operators (I-K⁽ⁿ⁾)⁻¹ are uniformly bounded for all s u f f i c i e n t l y l a r g e n a n d t h a t ⁽ⁿ⁾ c o n v e r g e s t o

an

R φ φ u n i f o r m l y o n c o m p a c t s u b s e t s o f S p r o v i d e d t h a t a_n → + ∞ a n d h_n → 0 a s n → ∞ , w h e r e hn i s t h e d i a m e t e r o f the largest element of the mesh on . Further, if

an

S φ(x) → 0 as x → ∞, R_anφ⁽ⁿ⁾ converges to φ uniformly on S,

The integral equation (9) may seem a perverse choice as approximation for (1): the approximation obtained by replacing S by S~_a

in (1) may seem more obvious: indeed, this alternative approximation can be analysed in a similar (in fact simpler) manner.

However, the resulting theory appears to be inapplicable in practical situations in which (1) is a boundary integral equation. In such applications it is generally the case that (1) with S replaced by S_± is still a boundary integral equation so that the

injectivity of I−k_± can be established in a similar manner to that of I - K (cf.

Section 4). Equation (1) with S replaced by S~_± is generally not a boundary integral equation, and it is not clear, in practical cases, how the injectivity of I k_S^~_± (which the alternative theory requires) might be established.

Integral equations on smooth closed bounded surfaces in R³ and their numerical t r e a t me n t h a v e a w i d e l i t e r a t u r e: s e e C o l t o n an d K r e ss [ 9 , 1 0 ] f o r t h a t p ar t r e l e v an t to the acoustic scattering example of Section 4. The piecewise constant collocation method discussed in Section 3 is the most commonly used boundary element method (Brebbia et al. [6]). For integral equations of the class discussed in this paper on smooth bounded surfaces, the stability and convergence of this boundary element method can be analysed using standard results from the collectively compact operator theory of Anselone [1],

To the best of our knowledge, this paper is the first attempt to develop a theory for integral equations of the form (1) and their numerical solution. Our arguments generalise ones in collectively compact operator theory [1]. Our assumptions, results, and methods of proof are closest to those of Atkinson [5], Anselone and Sloan [2-4], and Chandler-Wilde [8], who consider the approximate solution of integral equations on the real line.

2. The Original and Truncated Equations. Let k_x(y)=k(x,y),(x,y) ∈I. W e s u p p o s e t h r o u g h o u t t h a t , f o r a l l a ≥ 2 a n d x∈S_a,k_x ∈L₁(S_a), a n d t h a t k

INTEGRAL EQUATIONS ON CYLINDRICAL SURFACES 5 satisfies the following assumptions, A1-A4:

A1. For all (x, y) .∈ I ={(x,y)∈ T:T ∈ ∑}, k(x+te₃,y+te₃)=k(x,y), t∈ R

A2. : sup ( ) ( ) ∞.

∈

≥ , <

= ∫ k x,y ds y c

a a

S S x 2 a

A3. ( ):= sup ( )_ ( ) ( )→ 0 as → 0.

≤

∈ , ' , ,

≥2 ∫ k x,y k x',y ds y h

h a

a

S h x' x S x x a

Δ

A4. For x = (x₁,x₂,x₃) ∈ S_±, ∫ k(x,y)ds(y) → 0 as x₃ → ± ∞.

E±

Let BT := {ψ ∈ L_∞(T): ψ _∞ ≤ 1}, for all T ∈ ∑. Note that Assumptions A1-A3 imply that

(11) _{T T} bounded and equicontinuous,

T

B

U

K

∑

∈

in that

K_T c

BT T

≤

sup _∞

∈

∑,

∈ ψ

ψ

and

).

≥ ( ) ( sup, ∈ , ≤

∈

∑,

∈ K_T x' h

h x' x T x' T x, B T

Δ

ψ ψ

Thus, if Assumptions A1-A3 are satisfied, then K_Tis a bounded operator from L∞(T) to BC(T) for each T ∈ ∑ (indeed, KT is compact if T is bounded), and

(12) sup ≤ .

∑

∈ K_T c

T

From A1-A3 we have also the following technical lemma:

LEMMA 2.1. Define Φ₊(t), Φ₋(t), for t ≥ 0, by ), ( ) ( sup

: )

( _~ ^~

∈ ∫ k x,y ds y

t S±

te3 x Γm

Φ_± =

Where Γ~ : ={(x₁,x₂,0):(x₁,x₂)∈ Γ}, and Φ(t) byΦ(t):= max (Φ ₊ (t),Φ_-(t)). Then .

0 )

(t → as t → +∞ Φ

Proof. Consider Φ₊(t). From Assumption A2, sup_x∈S∫ _S k(x,y)ds(y)<∞, so that Φ₊ is well denned and

∫^~ k(x,y)ds(y)→ 0

3 ++te S

as t→+∞ with x fixed. Applying Al, it follows that (13) ∫^~ k(x,y) ds(y)→ 0

S+

as t→+∞ with x = (x₁,x₂,−t) and (x₁,x₂) ∈ Γ fixed. But, by Assumption A3 uniformly continuous on S. It follows that the convergence (13) )

( )

~ k(x,y ds y S+

∫

6 A. T. PEPLOW AND S. N. CHANDLER-WILDE

is uniform in (x₁,x₂) ∈Γ, so that Φ+(t)→0 as t→+∞. Similarly, Φ−(t)→0 as

∞. +

→

t □

Assumption A4 gives rise to a similar result concerning the integral over E±. LEMMA 2.2. Define Ψ₊(t), Ψ (t), for t ≥0, by

), ( )

~ (

∈sup :

)

( k x,y ds y

E±

∫ te3 Γ x

t = ±

Ψ±

and Ψ(t)byΨ(t):=max(Ψ+(t),Ψ−(t)). Then Ψ (t) →0as t→+∞.

Proof. From Assumption A2, Ψ is well defined and, from Assumptions A2 and A3, )

( ) (x,y ds y

± k

∫E is uniformly continuous on S±. It follows that Ψ(t) → 0 from Assumption A4.

□

We shall employ principally the following notions of convergence. For a sequence {T_n}

∑

⊂ and T ∈ ∑ we shall write Tn →T if, for all A > 0, {x ∈Tn: x ≤ A} = {x∈T: x ≤ A}for all sufficiently large n. For a sequence

{ }

ψ_n , with ψn ∈L∞ (Tn), we shall write

, 0 all

for , and , sup

, if

)

( → ∞ < ∞ >

∈ ∞

→ L T Tn T n n A

n ψ ψ

ψ

ess supx∈T,x≤Aψn(x)_ψ(x) → 0 as n → ∞.

REMARK 2.1. If Tn→T then, for all sufficiently large n,Tn \T ⊂ (E+ −bn) ∪ ),

(E−+cn ^where bn,cn ≥ 0 ^and bn,cn → +∞as n→ ∞. Thus, if T is bounded, Tn= T for all sufficiently large n, so that, if ψ∈ L_∞(T), then ψ_n →ψ ⇔ ψ_n −ψ _∞ → 0.

REMARK 2.2. If ψ_n∈ BC(T_n ) for each n then ψ ∈ BC(T). REMARK 2.3. If ψ_n → ψ then ψ _∞ ≤ sup _n ψ_{n ∞} .

REMARK 2.4. If ψ_n∈ BC(T) and ψ_n∈ BC(T). for each n, then the convergence

ψ_n → ψ is strict convergence in the sense of Buck [7].

REMARK 2.5. A useful test of convergence is: ψ_n →ψ ⇔ every subsequence of

{ }

ψ_n

has a subsequence that converges to ψ .

Our next two results are, respectively, a collective compactness and a convergence property of the operator sequence

{ }

KT_n ,in the case T_n → T.

LEMMA 2.3. Suppose that {T_n}⊂Σ,T_n T∈ Σ,ψ_n∈L_∞(T_n)for each n, and . T h e n , f o r s o m e s u b s e q u e n c e { },K ψ _m → ψ∈ BC(T)

m nm n

n T

ψ .

∞

<

sup_n ψ_{n ∞}

Proof. Define T_A^*:={(y₁,y₂,y₃)∈T |;y₃|≤A}, for A > 0, and let x_n = K ψ _n . Note

Tn

that, for all A > 0, T_A^* ⊂T_n for all sufficiently large n.

By (11),

{ }

x_n is bounded and equicontinuous. By the Arzela-Ascoli theorem and the above remarks, it follows that {x_n} has a subsequence, {x_n⁽¹⁾} which is a Cauchy sequence on

1*

T . Similarly, {x_n^{(m 1)}} has a subsequence, {x_n⁽²⁾} , which is a Cauchy sequence on T₂^* Continuing the argument, we may construct, for each m Є N, a subsequence {x_n^(m)} of

}

{x⁽_n^m-1) which is a Cauchy sequence on T_m^*. Then {x_n⁽ⁿ⁾} is a Cauchy sequence on T_m^* for each m Є N and thus converges to an element of BC(T). □

LEMMA 2.4. Suppose that {T_n}⊂Σ, T∈ Σ, ψ_n∈ L_∞ (T_n) for each n, and ψ_n →ψ. Then K ψ_n K_Tψ

Tn → .

), T ( L_∞ ψ∈

Proof. If T is bounded then, by Remark 2.1, the above is no more than a statement that K_T is continuous. Suppose that T is unbounded. Since, by (11),

{

K_Tnψ_n

}

is bounded and equicon- tinuous, to show K_T ψ_n K_Tψ we need only show pointwise convergence of to

n → K _T ψ _n

n

ψ K_T

Let C := supn ψ_{n ∞}, choose x = (x1, x2, x3,) Є T, and define T_A^* as in the previous proof. For all A > |x₃| and provided T_A^*⊂T_n (true for all sufficiently large

INTEGRAL EQUATIONS ON CYLINDRICAL SURFACES 7 n),

(y)ds(y) ψ

(y y)(ψ k(x,

∫* x K x

K_Tnψ_n( ) _Tψ_n( ) ≤ _{TA n} )) _n

) (

| ) , ( )

(

| ) ,

( \

\ |k x y ds y C ∫ |k x y ds y

∫

C _*

TA

* T TA

Tn +

+

) (

| ) , (

| )

( )

≤ (

sup _{) ( - )}

3

y ds y x

∫ k C y y

A y

c _n + _{(E+ bn ∪ E +Cn}

≤ ψ ψ

) (

| ) (

|

2C ∫\ _* x,y ds y

TA

+ T

f o r a l l s u f f i c i e n t l y l a r g e n, w h e r e b_n,c_n ≥ 0 a n d b_n,c_n → +∞ ( T o o b t a i n this last inequality we have used Assumption A2, that, T_n\T_A^*⊂(T_n \T)∪(T\T_A^*), and that, by Remark 2,1,Tn\T C (E+ - bn) ∪ (E- + cn) for all sufficiently large n.) Applying Assumption Al and recalling the definitions of Ψ and Φ we obtain

)) (

) (

( ) (

≤ (y)

|

| sup

≤ (x)

(x) y C b +x₃ c x₃

3 A c y K

K_{T n T n n n}

nψ ψ ψ ψ + Ψ +Ψ

+2C(Φ(A+x₃)+Φ(A x₃)).

Given Є > 0, by Lemma 2.1, the final term on the right hand side is ≤ Є/2 if A is chosen sufficiently large enough. Then, since ψ_n →ψ and by Lemma 2.2, the remaining terms are ≤∈/2 for all sufficiently large n. Thus K_Tnψ_n

( )

x K_Tψ_n

( )

x → 0 as n→∞ for every fixed x. □

We apply the above compactness and continuity properties first of all to give a condition for the continuous dependence of φ on g in equation (1), in the case

(I K)BC( )S

∈

g .

THEOREM 2.5. If I - K is injective then (I — K)^-1 exists and is a bounded operator on (I – K)BC(S).

{ }

_n ^⊂

Proof. Suppose that the theorem is false. Then there exists a sequence ψ BC(S) with ψ_{n ∞}=1 for each n s uc h t ha t ψ_n Kψ_{n ∞} → 0.. Since ψ_{n ∞} =1, we can find a sequence {an} R such that ⊂ sup_x∈Γ^~ χ_n(x) ≥ ₂¹, where {χ_n}⊂BC(S) is defined by χ_n(x)=ψ_n(x a_ne₃),x∈ S. From Assumption Al it follows that

Thus ).

( )

(x a e₃ K x Kψ_{n n} = ^χ_n

(14) χ_n Kχ_{n ∞} = ψ_n Kψ_{n ∞} → 0 as n→ ∞.

Since {χn} is bounded in BC(S), by Lemma 2.3 we can choose a subsequence }

{χnm and χ Є BC(S) such that → χ.

nm

Kχ From (14) it follows that Kχ_nm → χ and then, from Lemma 2.4, that Kχ_nm → Kχ. Thus χ = Kχ. Since χ _∞ ≥

( ) _≥ ¹₂

Γ n

inf_x∈~ χ x and I - K is injective we have a contradiction. □

In the next two theorems we commence an examination of conditions for (I – K)BC(S) = BC(S) and of the convergence of ,φ_a the solution of the integral equation (9) on Sa, to φ as a→ ∞

THEOREM 2.6. Suppose that I - K is injective and that, for some A > 0 and all a≥ A,

(

I K_a

)

¹ exists and is a bounded operator on BC(Sa), with C :=

( )

<∞.

≥

supa A I K_a ¹ Then (I — K)^-1 exists as an operator on BC(S) with (I K) ¹ ≤C Moreover, if g Є BC(S), {an} ⊂ R⁺ and a_n → ∞^, then (I –

( ) .

→

) ¹E g I K ¹g

K _an

an

8 A. T. PEPLOW AND S. N. CHANDLER-WILDE

Proof. Let g ∈ BC(S) and define the sequence {φ_n}, where φ_n∈ BC(S_an) for

each n, by g. Then

n

n a

a E

K -

n I

) 1

=( φ

(15) g.

n

n a

a E

K _n

n = φ +

φ

Since {φ_n} is a bounded sequence, by Lemma 2.3 there exists a subsequence { }

nm

φ and BC(S) such that

φ∈

φ.

φ g →

nm m n

n a

a +E

K m

Thus, from (15), φ → φ

nm Hence, from Lemma 2.4, K φ Kφ

nm nm

a → , and thus

. K E

Kanmφn_m + anmg→ φ+g T h u s φ =Kφ+g a n d s o g Є (I – K)B C(S) and we have shown that I - K is surjective, so that (I - K)^-1 exists as an operator on BC(S).

By the same argument, given g Є BC(S), every subsequence of g has

an

E K - I )^-1 (

a subsequence converging to (I-K_an)^-1g Thus, by Remark 2.5, g

g ^-1

1

- → ( )

)

(I-K E I-K

n

n a

a

for every g Є BC(S). From this and Remark 2.3 it follows that ||(I - K)^{- 1}|| ≤ C.

□

THEOREM 2.7. Suppose that the conditions of the previous theorem are satisfied.

Define g and If

n n:=(I-Kan)^-1Ea

φ φ:=(I-K)^-1g. φ(x)→ 0 as x→ ∞ then as n→ ∞

0

R _n

an →

φ ∞

φ

Proof. By definition,

(16) g

n

n a

a E

K _n

n φ =

φ and φ Kφ =g, so that

(17) E K E .

n n

n a a

a φ φn = g F r o m ( 1 6 ) a n d ( 1 7 ) i t f o l l o w s t h a t

, ) (

-E_an I K_an ^-1xn

n φ = −

φ

Where x K E E K ,

n n

n a a

a

n = φ− φ so that

(18) || _n -E_a ||∞ ≤C||X_n||∞.

nφ φ

For each n, let A_n :=

(

a_n - 1

)

2. By Assumptions Al and A2, and from the definition of Φ ,

sup E K(x) ≤ sup K (x)

_1 an S n\ a x∈S _1

an S n\ Sa

x∈ ^an

φ

≤ sup k( , )ds( ) _∞ c sup ( )

1

y y

y

∫ x

An An

S S\

_ y San n\ Sa

x∈ ^S φ φ

+ ∈

≤ sup (t)|| || c sup (y).

An S~

\ S

∞ y

An

t

φ +

φ Φ

≥ ∈

The same upper bound applies to sup K_anE_an (x)

_1 an an ^`S~

∈S

x φ so that, for x ∈

_1 an an\S S

X (x) ≤2 sup (t) 2c sup (y).

An S~

\ y∈ S

∞

≥A t _n

n Φ φ + φ

INTEGRAL EQUATIONS ON CYLINDRICAL SURFACES 9 For, ,x ∈ S~

_1

an

), ( ) ( ) , ( )

( ) )(

)(

(

= )

(x ∫ ₁ k x,y E y ds y ∫ 1 K x y y ds y

X _{_}

an S

n\ Sa

n an S\S

n_

a φ φ

So that

) y ( sup c

≤ 2 ) y ( sup

c

≤ 2 ) x ( X

An _1

an S~

S\

y S

S\

n y φ φ

∈

∈

Thus

( )

^{t 2c} _sup ^(y).,

2 sup

≤ X

SAn

S y

∞ A

∞ t

n

n Φ φ + φ

∈

≥

so that, by Lemma 2.1, X_{n ∞} → 0 |and, by (18), _ → 0 φ ∞

φ_n E_an as n→ ∞ Now,

) x _ ( ) x ( sup

) x ( R

sup )

x (

≤ sup _ R

n n

n n

n

a n S

\

∈S a x

S

\

∈S x S

\

∈S

∞ x

a _n

a a

n φ φ + φ + φ φ

φ

≤ 2 sup ( ) 2 sup ( )_ ( )

∈ x x _n x

a

a_n ^x∈ S _n

S

\ S x

φ φ

φ +

≤ 3 sup ( ) 2 _ _∞

_1

∈ ~ φ φ _{n n}φ

a

S a

\ S x

E x

n

+ + S i n c e ( ) ( ) _≤ ( ) _{∈ S \S _1}

an

x , x x

_E

x _anφ φ

φ T h u s φ_R_anφ_{n ∞} → 0 a s n→ ∞

□

The next theorem is a criterion for the existence of a solution to equation (9) for all sufficiently large a ≥ A and the uniform stability of the approximate inverse operators.

T H E O R E M 2 . 8 . I f I_K,I _K₊, a n d I_K a r e i n j e c t i v e , t h e n (I_K_a)^_1 exists and is a bounded operator on BC(Sa) for all sufficiently large a ≥ A and

∞. )

K - I ( sup

:

C _a ^-1

A

a <

= _≥

_1

_ ) Ka

I

( exists (i.e. I - Ka is injective) then, since Ka is Proof. Note that if

compact, (I - Ka}^-1 is a bounded operator on BC(Sa} by the Fredholm Alternative.

Suppose now that the theorem is false. Then there exist sequences {a_n}⊂R⁺ and {ψ_n} such that a_n →∞,ψ_nBC(S_an)and ψ_{n ∞} =1 for each n, and

(19) ψ _n − K _anψ _{n ∞} → 0.

Since ψ_{n ∞} =1 we can find a sequence {x^{( )n} } such that x^{( )n} = (x₁( ) ( ) ( )ⁿ ,x₂ⁿ ,x₃ⁿ ) ∈

an

S f o r e a c h n a n d ψ_n(x^{( )n}) ≥ ¹₂.

There are two cases to consider: (a)a_n x₃⁽ⁿ⁾ → ∞ as n→∞; (b) {a_n x₃⁽ⁿ⁾} is bounded.

CASE (a). For each n, define T_n:=S_an_ x₃⁽ⁿ⁾ and X_n∈ BC(T_n) by (20) x_n(x) =ψ_n(x+ x₃⁽ⁿ⁾e₃), x∈ T_n.

Then, by Assumption Al, it is easy to see that

(21) K_anψ_n(x+ x₃⁽ⁿ⁾e₃) = K_Tn X_n(x), x∈ T_n.