Singular integral equations of convolution type

Li Advances in Difference Equations (2017) 2017:360 DOI 10.1186/s13662-017-1413-x

RESEARCH

Open Access

Singular integral equations of convolution type with Hilbert kernel and a discrete jump problem Pingrun Li* *

Correspondence: [email protected] School of Mathematical Science, Qufu Normal University, Jingxuanxi Road 57, Qufu, Shandong 273165, P.R. China

Abstract One class of singular integral equations of convolution type with Hilbert kernel is studied in the space L2 [–π , π ] in the article. Such equations can be changed into either a system of discrete equations or a discrete jump problem depending on some parameter via the discrete Laurent transform. We can thus solve the equations with an explicit representation of solutions under certain conditions. MSC: 45E10; 45E05; 30E25 Keywords: singular integral equation; convolution type; Hilbert kernel; a discrete jump problem

1 Introduction It is well known that singular integral equations and boundary value problems for analytic functions are the main branches of complex analysis and have a lot of applications, e.g., in elasticity theory, fluid dynamics, shell theory, underwater acoustics, and quantum mechanics. The theory is well developed by many authors [–]. Integral equations of convolution type are closely related to boundary value problem for an analytic function. There have been many papers studying integral equations with convolution type or singular type, see, for example, Litvinchuc [], Li [, ], De-Bonis [], Du [], Jiang [], among which a series of valuable achievements have been obtained. In recent years, the author [] discussed some kinds of singular integral equations of convolution type with reflection and translation shifts. Subsequently, the author [] studied one class of generalized boundary value problems for analytic functions and obtained the general solutions and the conditions of solvability. The purpose of this article is to extend further the theory to a periodic singular integral equation of convolution type with Hilbert kernel. We remark that integral equations with periodicity have important applications in the elastic theory. Such equations can be changed into either a system of discrete equations or a discrete jump problem (that is, discrete boundary value problems) depending on some parameter via the discrete Laurent transform. We can thus solve the equations with an explicit representation of solutions in L [–π, π] under certain conditions. This paper improves some results for the references [–]. © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Li Advances in Difference Equations (2017) 2017:360

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We shall consider the following singular integral equation of convolution type with Hilbert kernel and periodicity: a f + b Hf + K ∗ f + M ∗ Hf + ξ (a f + b Hf + K ∗ f + M ∗ Hf ) = (c + c ξ )G,

(.)

where Kj , Mj , G ∈ L [–π, π] are given functions, aj , bj , cj ∈ R for any j = , , and f ∈ L [–π, π] is an unknown function. We denote by ∗ the convolution operator and by Hf the Hilbert type singular integral of f , that is,  Hf = π

π

f (τ ) cot –π

τ –θ dτ . 

Assumption A We shall make the following assumption: (i) a + a = , b + b = . (ii) There exist constants a, b, c, d ∈ R and some m ∈ N such that ξ (θ ) =

a + b tan m θ c + d tan m θ

whenever ad – bc = ; otherwise we have ξ (θ ) = C (constant). We shall also represent ξ (θ ) as

ξ (θ ) =

α + + α – eimθ , β + + β – eimθ

where α ± = a ± ib,

β ± = c ± id.

Let L, L– be the discrete Laurent transform and the inverse transform, respectively. We denote S± = (a a + a c) ± i(a b + a d), T ± = (b c + b a) ± i(b d + b b), (j)

Bk = L– Kj ,

(j)

Ck = L– Mj

and wk =

–k–m , +k

where ± ± ± () ± () ± () ± () ± k = S + β Bk + α Bk + i sgn k T + β Ck + α Ck .


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Assumption B To assure the solvability of Eq. (.), we need to make the assumption S + log – wk eikθj = , S

(.)

k∈Z

where θj =

j π m

for any j = , ±, . . . , ±[ m ]. Via using the method of complex analysis, we can take a con+ + tinuous branch of log( SS– wk ) such that {log( SS– wk )}k∈Z ∈ l . There exist constants ε ∈ (, ) and k ∈ N such that when |k| > k we have + > ε , k

– > ε . k–m

We can thus introduce the equalities Ak =

 – + () – + () + – () β S Bk–m – β + S– B() k + α S Bk–m – α S Bk S– +k

() () sgn(k – m) – β + S– Ck() + α + S– Ck() sgn k . (.) + i β – S+ Ck–m + α – S+ Ck–m

We may make ηk such that ηk = ( + Ak )ηk–m .

(.)

Assumption C To assure the solvability, we need also Hk k∈Z

ηk

ikθj

e

= ,

(.)

where ηk is determined by (.), Hk =

 + – + E gk + E gk–m , k

{gk }k∈Z = L– G

and E± = (c c + c a) ± i(c b + c d). In order to illustrate that Eq. (.) has a solution, at the end of Section , we shall present an example and satisfy the above conditions (Assumptions A-C), then we can conclude that a solution set of (.) is not empty. Now we can state our main result, and we will prove it in Section . Theorem . Under Assumptions A-C, Eq. (.) has a solution in L [–π, π].


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(i) When +k = , –k–m =  for any k ∈ Z, the solution of Eq. (.) is given by

f (θ ) =

+∞

fk eikθ ,

k=–∞

and the coefficients fk are determined uniquely by the formula +k fk + –k–m fk–m = E+ gk + E– gk–m .

(.)

(ii) When +nj = , –nj –m =  (|nj | ≤ k ; j = , , . . . , p ), then E+ gnj + E– gnj –m =  must be satisfied, and fnj can be taken to be an arbitrary constant cnj ( ≤ j ≤ p ), then the solution of Eq. (.) is

f (θ ) =

p

inj θ

+∞

+

cnj e

j=

fk eikθ .

k=–∞,k=nj

(iii) When +n = , –n –m =  (|nj | ≤ k ; j = , , . . . , p ), fnj are given by the following j

j

formula:  + E gnj +m + E– gnj , –n

fnj =

(.)

j

then the solution of Eq. (.) is

f (θ ) =

p

inj θ

f e nj

+∞

+

fk eikθ .

k=–∞,k=nj

j=

(iv) When +n = , –n –m =  (|nj | ≤ k ; j = , , . . . , p ), fnj are given by the following j

j

formula: fnj =

 + – + E gnj + E gnj –m , n

(.)

j

then the solution of Eq. (.) is

f (θ ) =

p j=

inj θ

fnj e

+

+∞

fk eikθ .

(.)

k=–∞,k=nj

In (ii)-(iv), when k = n , n , . . . , np , n , n , . . . , np , n , n , . . . , np , the coefficients fk are determined uniquely by (.).


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2 Hilbert transform and its discrete Laurent transform The crucial tool to study the Hilbert transform is to calculate its discrete Laurent transform. For any f ∈ L [–π, π], its Hilbert transform is defined as Hf (θ ) =

 π

π

f (τ ) cot –π

τ –θ dτ . 

˜ defined by Associated to the operator H is the operator H ˜ (θ ) =  Hf π

π

f (τ ) cot –π

τ +θ dτ , 

that is, ˜ (θ ) = Hf (–θ ). Hf ˜ is a self-map of the space L [–π, π] It is well known that the Hilbert transform H as well H in virtue of the Riesz theorem (see, e.g., []). Let l (Z) be a linear space consisting of sequences {fk }k∈Z for which +∞

|fk | < +∞.

k=–∞

Definition . The discrete Laurent transform L : l (Z) −→ L [–π, π] is defined by +∞

L{fk }k∈Z =

fk eikθ =: f (θ )

(.)

k=–∞

for any f = {fk }k∈Z ∈ l (Z). Its inverse transform is clearly given by L– [f ]k = {fk }k∈Z

(.)

with fk =

 π

π

f (θ )e–ikθ dθ ,

∀k ∈ Z.

(.)

–π

Now we come to calculate the inverse discrete Laurent transform of a function which is a Hilbert transform of a given function. The result shall be crucial to studying the singular integral equations. Lemma . Let f (θ ) ∈ L [–π, π] with its discrete Laurent transform {fk }k∈Z = L– f (θ ).


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Then we have L– Hf (θ ) = {ifk sgn k}k∈Z , ˜ (θ ) = {–if–k sgn k}k∈Z . L– Hf Proof Since

cot

e–iτ τ –θ eiτ – , = iτ  e – eiθ e–iτ – e–iθ

by Definition . we have i Hf (θ ) = π

π

eiτ e–iτ f (τ ) iτ – e – eiθ e–iτ – e–iθ –π

dτ .

For any f (θ ) ∈ L [–π, π], we already know that Hf (θ ) ∈ L [–π, π] so that L– (Hf (θ )) ∈ l (Z). We denote L– Hf (θ ) = {gk }k∈Z . Then gk =

 π

π

Hf (θ )e–ikθ dθ –π

i = π 

π –π

π

eiτ e–ikθ i f (τ ) iτ dτ dθ – iθ  e – e π –π

π

π

f (τ ) –π

–π

e–iτ e–ikθ dτ dθ e–iτ – e–iθ

=: I – I . For the first term, we have I =

i π 

π π –π

–π

e–ikθ dθ f (τ )eiτ dτ . eiτ – eiθ

By a change of variable t = exp(iθ ), the inner integral above becomes

π

M(τ ) := –π

e–ikθ dθ = eiτ – eiθ

|t|=

t –k dt. (eiτ – t)it

Applying the extended residue theory, we thus obtain ⎧ ⎨ M(τ ) =

π , ei(k+)τ ⎩– π ei(k+)τ

k ≥ , , k < .

Therefore, when k ≥ , we have I =

i π 

π –π

i f (τ )eiτ dτ = ei(k+)τ π π

π

f (τ )e–ikτ dτ = ifk –π


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and similarly I = –ifk ,

k < .

In other words, I = ifk sgn k,

∀k ∈ Z.

Similarly, we have ∀k ∈ Z.

I = –ifk sgn k, As a result,

gk = I – I = ifk sgn k,

∀k ∈ Z.

The other equality can be proven similarly. Lemma . Let f (θ ) =

∞

ikθ k=–∞ fk e

and f ∈ l , then f (θ ) ∈ L [–π, π] if and only if f ∈ l .

Proof Since f (θ ) ∈ L [–π, π] and f ∈ l , then formly. It is easy to see that

π

f (θ ) dθ =

–π

π

π

f (θ )f (θ ) dθ =

–π π

=

∞

k=–∞ fk

fk eikθ

is convergent absolutely and uni-

f¯j e–ijθ dθ

–π ∞

fk f¯j ei(k–j)θ dθ = π

–π k,j=–∞

∞

|fk | .

(.)

k=–∞

The proof of Lemma . is complete. Finally, we remark that

L– f ∗ g(θ ) = {fk gk }k∈Z

for any f , g ∈ L [–π, π] with the discrete Laurent transforms {fk }k∈Z and {gk }k∈Z , respectively. Here the convolution in L [–π, π] is defined by (f ∗ g)(θ ) =

 π

π

f (θ – τ )g(τ ) dτ .

(.)

–π

3 Problem presentation and solution In this section, we study the method of solution for Eq. (.). By Euler’s formula eimθ = cos mθ + i sin mθ , Eq. (.) can be written as S+ f + T + Hf + β + K ∗ f + α + K ∗ f + β + M ∗ Hf + α + M ∗ Hf – E+ G + eimθ S– f + T – Hf + β – K ∗ f + α – K ∗ f + β – M ∗ Hf + α – M ∗ Hf – E– G = .

(.)


Page 8 of 13

In view of Lemma ., by applying L– to both sides of (.), we see that (.) is readily reduced to the equation +k fk – E+ gk + –k–m fk–m – E– gk–m = ,

(.)

where {fk }k∈Z = L– f , {gk }k∈Z = L– G, (j) (j) Bk k∈Z = L– Kj , Ck k∈Z = L– Mj with any j = , ; k ∈ Z. Since by assumption, for each j = , , (j) Bk k∈Z ∈ l (Z),

(j) Ck k∈Z ∈ l (Z),

it follows that (j)

lim Bk = ,

k→∞

(j)

lim Ck = .

k→∞

Therefore, lim +k = S+ ± iT + = ,

k→∞

lim –k = S– ± iT – = ,

k→∞

which means that for any ε sufficiently small, there exists k >  such that when |k| > k we have + > ε , k

– > ε . k–m

Case : |k| > k . Since |+k | > ε , |–k–m | > ε for |k| > k , it follows from (.) that fk = –

–k–m  + – + fk–m + + E gk + E gk–m , k k

∀|k| > k ,

(.)

so that Eq. (.) can be rewritten as fk = –wk fk–m + Hk ,

∀|k| > k .

(.)

Denote ρ=

S– , S+

wk = ρ( + Ak ).

When |k| is large enough, we see that sgn(k – m) and sgn k equal  or – simultaneously so that S+ T – sgn(k – m) – S– T + sgn k = .


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Consequently, Ak =

 S– +k

– + () – + () + – () β S Bk–m – β + S– B() k + α S Bk–m – α S Bk

() () + i β – S+ Ck–m + α – S+ Ck–m sgn(k – m) – β + S– Ck() + α + S– Ck() sgn k . (.)

Therefore, (.) becomes a discrete jump problem fk = –ρ( + Ak )fk–m + Hk ,

∀|k| > k .

(.)

In order to solve (.), we may make ηk such that ηk = ( + Ak )ηk–m

(.)

with ε < |ηk | < ε– . First, we need to construct ηk . By taking logarithms on both sides of (.) and denoting Mk = log( + Ak ),

ok = log ηk ,

we get ok = ok–m + Mk ,

(.)

where we have taken a continuous branch of log( + Ak ) so that {log( + Ak )}k∈Z ∈ l . Taking the Laurent transform L on both sides of (.) yields ˜ ), O(θ ) = eimθ O(θ ) + M(θ

(.)

that is,

˜ ),  – eimθ O(θ ) = M(θ

where O(θ ) = Lo,

˜ ) = LM, M(θ

o = {ok }k∈Z ,

M = {Mk }k∈Z .

˜ j ) =  or, equiva˜ j ) = , Eq. (.) is not solvable. This means M(θ Notice that when M(θ lently, Assumption B becomes the solvability conditions of Eq. (.), where θj are  + [ m ] roots on [–π, π] for equation  – eimθ = . Finally, from (.) we get

ηk = exp ok ,

ok = L–

˜ ) M(θ .  – eimθ

(.)


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Now we come to solve Eq. (.). Denote pk =

 fk , ηk

qk =

 Hk ηk

and rewrite (.) as pk = –ρpk–m + qk .

(.)

Due to {Hk }k∈Z ∈ l (Z), {ηk }k∈Z ∈ l (Z), we can know that {pk }k∈Z ∈ l (Z), {qk }k∈Z ∈ l (Z). Taking the Laurent transform L on both sides of (.), we thus obtain 

P(θ ) = –ρeimθ P(θ ) + Q(θ ),

(.)

where P(θ ) = Lp and Q(θ ) = Lq with p = {pk }k∈Z and q = {qk }k∈Z . Owing to |ρ| = , we know that  + ρeimθ has a finite number of zero points, say θ , θ , . . . , θn in [–π, π]. The same approach as in the discussion of Eq. (.) shows that (.) is not solvable if Q(θj ) = . Therefore, Q(θj ) =  (j = , , . . . , n), or equivalently, Assumption C becomes the solvability conditions of Eq. (.). Under Assumption C, (.) becomes P(θ ) =

Q(θ ) .  + ρeimθ

(.)

This determines pk so does fk = pk ηk . Case : |k| ≤ k . We split the situation into four cases. (a) +k =  and –k–m =  for some |k| ≤ k . By (.), we have E+ gk = –E– gk–m

(.)

and fk can be taken to be any constant. (b) +k =  and –k–m =  for some |k| ≤ k . By (.), we get fk =

 + E gk+m + E– gk . –k

(.)

(c) +k =  and –k–m =  for some |k| ≤ k . By (.), we get fk =

 + – + E gk + E gk–m . k

(.)

(d) +k =  and –k–m =  for some |k| ≤ k . In this situation, fk can be determined as in the proof of Case (i). In the following, we give the proof of Theorem .. Proof of Theorem . From the above discussion, we only need to prove that the function f (θ ) obtained by (.) belongs to L [–π, π]. Obviously, Eqs. (.), (.) and (.) are equiv±  alent to each other. Since {Ak }, {Bk }, {Ck }, {gk } ∈ l , then {± k } ∈ l and k = . It follows


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ikθ from (.) that {fk } ∈ l is a bounded sequence and ∞ is convergent. Thus, by k=–∞ fk e Lemma ., Eq. (.) has a unique solution f (θ ) = Lf , and f (θ ) ∈ L [–π, π]. Finally, in order to illustrate that Eq. (.) has a solution, we shall present an example. For example, suppose that a = b = c = a = b = c = , K (θ ) = K (θ ) = sin θ ,

a = d = ,

M (θ ) = M (θ ) = ,

b = c, G(θ ) = sin θ – cos θ ,

then ξ (θ ) = tan m θ , and Eq. (.) can be transformed into f (θ ) +

 π

π

 t–θ dt + √  π

f (t) cot –π

= sin θ – cos θ ,

π

sin(t – θ )f (t) dt –π

θ ∈ [–π, π].

(.)

Equation (.) is often used in engineering mechanics. It is easy to verify that Eq. (.) satisfies the above conditions (Assumptions A-C). Via using the methods of Section , we can obtain the exact solution of Eq. (.): √   π –θ  sin θ , f (θ ) =  π +θ

θ ∈ [–π, π].

(.)

As for the solving method of (.), we will not elaborate. We can verify that (.) is indeed the solution of (.). Therefore, we conclude that a solution set of (.) is non-empty.

4 Homogenous equation and some specific equation In this section we consider the homogenous equation and some specific equation. First we consider the homogenous equation (that is, G(θ ) ≡ ) a f + b Hf + K ∗ f + M ∗ Hf + ξ (a f + b Hf + K ∗ f + M ∗ Hf ) = .

(.)

Via the Laurent transform, it can be reduced to the equation +k fk + –k–m fk–m = .

(.)

That is, fk = –wk fk–m ,

wk =

–k–m . +k

(.)

Again we apply the same approach as the discussion for Eq. (.) to deduce that fk ≡  for all k so that f (θ ) ≡ . As a result, the homogeneous equation (.) has only a trivial solution. Next we consider the specific case that ξ (θ ) is a constant. Since ad – bc = , Eq. (.) can be expressed in the form a f + b Hf + K ∗ f + M ∗ Hf = G.

(.)


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Via transform L– , (.) can be written as

a + ib sgn k + Bk + i sgn kCk fk = gk ,

(.)

where a , b are constants and {fk }k∈Z = L– f , {Bk }k∈Z = L– K , {Ck }k∈Z = L– M, {gk }k∈Z = L– G. ikθ One can solve out fk from (.) and get the solution f (θ ) = +∞ k=–∞ fk e .

5 Conclusions In this paper, we first proposed one class of singular integral equations of convolution type with Hilbert kernel and periodicity. Applying the discrete Laurent transform and its properties, such an equation can be changed into a discrete boundary value problem depending on some parameter, here we call it ‘a discrete jump problem’. In this article, our method is different from the ones of the classical boundary value problem, and it is novel and simple. The exact solution, denoted by series, of Eq. (.) and the conditions of solvability are obtained. We remark that our approach is also effective to some other classes of equations such as the equations of dual type with periodicity and Hilbert kernel, the Wiener-Hopf type equations, and the equations with periodicity and cosecant kernel. Thus, this paper generalizes the classical theory of boundary value problems and singular integral equations. One can also consider a similar problem in the setting of Clifford analysis (see, e.g., [– ]). Acknowledgements The author is grateful to the referees for many suggestions to improve the exposition of the paper. Competing interests The author declares no conflicts of interest. Authors’ contributions Author read and approved the final manuscript.

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