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Solvability of some classes of singular integral equations of convolution type via Riemann–Hilbert problem
- Pingrun Li^{1}Email author
https://doi.org/10.1186/s13660-019-1975-0
© The Author(s) 2019
- Received: 29 June 2018
- Accepted: 16 January 2019
- Published: 25 January 2019
Abstract
In this paper, we study methods of solution for some kinds of convolution type singular integral equations with Cauchy kernel. By means of the classical boundary value problems for analytic functions and of the theory of complex analysis, we deal with the necessary and sufficient conditions of solvability and obtain the general solutions and the conditions of solvability for such equations. All cases as regards the index of the coefficients in the equations are considered in detail. Especially, we discuss some properties of the solutions at the nodes. This paper will be of great significance for the study of improving and developing complex analysis, integral equation and boundary value problems for analytic functions (that is, Riemann–Hilbert problems). Therefore, the classical theory of integral equations is extended.
Keywords
- Singular integral equations
- Riemann–Hilbert problems
- Integral operators
- Convolution type
MSC
- 45E05
- 45E10
- 30E25
1 Introduction
There were rather complete investigations on the method of solution for integral equations of Cauchy type and integral equations of convolution type [1–5]. The solvability of a singular integral equation (SIE) of Wiener–Hopf type with continuous coefficients was considered in [6, 7]. For operators with Cauchy principal value integral and convolution, the conditions of their Noethericity were discussed in [8, 9]. Recently, Li [10–16] studied some classes of SIEs with convolution kernels and gave the Noether theory of solvability and the general solutions in the cases of normal type. It is well known that integral equations of convolution type, mathematically, belong to an interesting subject in the theory of integral equations.
In this paper, we study the solvability and the explicit solutions for several classes of SIEs with Cauchy kernel and convolution kernel, in which include equations with one or two convolution kernels, equation of Wiener–Hopf type, and dual equations. Here, we give the new methods of solution for these equations, and our approach of solving the equations is novel and effective, different from the ones in classical cases. Thus, the results in this paper generalize ones in Refs. [1, 2, 10–12], and improve the theory of SIEs and boundary value theory.
- (1)SIEs of dual type$$ \textstyle\begin{cases} a_{1}\omega (t)+\frac{b_{1}}{\pi i}\int _{\mathbb{R}}\frac{\omega ( \tau )}{\tau -t}\,d\tau + \frac{1}{\sqrt{2 \pi }}\int _{\mathbb{R}}k _{1}(t-\tau )\omega (\tau )\,d\tau =g(t),\quad t\in \mathbb{R}^{+}; \\ a_{2}\omega (t)+\frac{b_{2}}{\pi i}\int _{\mathbb{R}}\frac{\omega ( \tau )}{\tau -t}\,d\tau + \frac{1}{\sqrt{2 \pi }}\int _{\mathbb{R}}k _{2}(t-\tau )\omega (\tau )\,d\tau =g(t),\quad t\in \mathbb{R}^{-}. \end{cases} $$
- (2)SIE of Wiener–Hopf type$$ a\omega (t)+\frac{b}{\pi i} \int _{\mathbb{R}}\frac{\omega (\tau )}{ \tau -t}\,d\tau + \frac{1}{\sqrt{2 \pi }} \int _{\mathbb{R}}k(t-\tau ) \omega (\tau )\,d\tau =g(t),\quad t\in \mathbb{R}^{+}. $$
- (3)SIEs with one convolution kernel$$ a\omega (t)+\frac{b}{\pi i} \int _{\mathbb{R}}\frac{\omega (\tau )}{ \tau -t}\,d\tau + \frac{1}{\sqrt{2 \pi }} \int _{\mathbb{R}}k(t-\tau ) \omega (\tau )\,d\tau =g(t),\quad t\in \mathbb{R}. $$
- (4)SIEs with two convolution kernels$$\begin{aligned} &a\omega (t)+ \frac{b}{\pi i} \int _{\mathbb{R}}\frac{\omega (\tau )}{ \tau -t}\,d\tau + \frac{1}{\sqrt{2 \pi }} \int _{\mathbb{R}^{+}}k_{1}(t- \tau )\omega (\tau )\,d\tau \\ &\quad{}+\frac{1}{\sqrt{2 \pi }} \int _{\mathbb{R} ^{-}}k_{2}(t-\tau )\omega (\tau )\,d\tau =g(t), \quad t\in \mathbb{R}, \end{aligned}$$
2 Definitions and lemmas
The concepts of classes \(\{\{0\}\}\) (\(((0))\), \(\langle \!\langle 0 \rangle \!\rangle \)) and \(\{0\}\) (\((0)\), \(\langle0\rangle \)) are introduced as follows.
Definition 2.1
- (1)
\(F(s)\in \hat{H}\), that is, it satisfies the Hölder condition on \(\mathbb{R}\cup \{\infty \}\) (for the notation Ĥ, cf. [2]).
- (2)
\(F(s)\in L^{2}(\mathbb{R})\).
Definition 2.2
Definition 2.3
Let \(F(s)\) be a continuous function on \(\mathbb{R}\). If
(1) \(F(s)\in \hat{H}\). (2) \(F(s)=O(|s|^{-\rho })\), \(\rho >\frac{1}{2}\), where \(|s|\) is sufficiently large.
Then we write \(F(s)\in ((0))\) or \(((0))^{\rho }\).
If \(F(s)\in ((0))\) or \(((0))^{\rho }\), we write \(f(t)\in (0)\) or \((0)^{\rho }\).
Definition 2.4
If (1) \(F(s)\in \hat{H}\); (2) \(F(s)\in H^{ \rho }(N_{\infty })\), \(\rho >\frac{1}{2}\), i.e., it belongs to H in the neighborhood \(N_{\infty }\) of ∞, and \(F(\infty )=0\).
Then we write \(F(s)\in \langle \!\langle 0 \rangle \!\rangle \) or \(\langle \!\langle 0 \rangle \!\rangle ^{\rho }\), and \(f(t)\in \langle0\rangle \) or \(\langle0\rangle ^{\rho }\).
We denote by \(H_{0}\) the class of Hölder continuous functions on any closed interval exterior to \(s=0\).
Lemma 2.1 is obvious fact and we omit its proof here.
Lemma 2.1
- (1)
If \(k,f\in \{0\}\) (\(\langle0\rangle \)), then \(k\ast f \in \{0\}\) (\(\langle0\rangle \)).
- (2)
If \(f\in \{0\}\) and \(k\in (0)\) (\(\langle0\rangle \)), then \(k\ast f\in (0)\) (\(\langle0\rangle \)).
Lemma 2.2 plays an important role in our paper.
Lemma 2.2
Proof
Lemma 2.3
If \(f\in \{0\}\) and \(F(0)=0\), then \(Tf\in \{0\}\).
Proof
Through the above discussion, we obtain \(F(s)\operatorname{sgn}s\in \{\{0\}\}\) and then \(Tf\in \{0\}\). □
In Lemma 2.3, note that \(F(0)=0\) is a necessary condition, otherwise the lemma is invalid. Similarly, we can prove that, if \(f\in (0)\) (\(\langle0\rangle \)), and \(F(0)=0\), then \(Tf\in (0)\) (\(\langle0\rangle \)).
In order to transform the above-mentioned SIEs into Riemann–Hilbert problems, we give Lemma 2.4.
Lemma 2.4
- (1)
\(\operatorname{Im}z>0\), \(\tilde{\varOmega }(z)=\frac{1}{\sqrt{2\pi }} \int _{\mathbb{R^{+}}}\omega (t) e^{i t z}\,dt\).
- (2)
\(\operatorname{Im}z<0\), \(\tilde{\varOmega }(z)=-\frac{1}{\sqrt{2 \pi }}\int _{\mathbb{R^{-}}}\omega (t) e^{i t z}\,dt\).
- (3)
\(\operatorname{Im}z=0\), \(\tilde{\varOmega }(z)=\frac{1}{\sqrt{2\pi }} \int _{\mathbb{R}}\omega (t) e^{i t z} \operatorname{sgn}t \,dt\).
Proof
Remark 2.1
If \(\omega (t)\in L^{1}(\mathbb{R})\), then \(\varOmega (0)=0\) if and only if \(\int _{\mathbb{R}}\omega (t)\,dt=0\).
Remark 2.2
Note that, for the class \((0)\) or \(\langle0\rangle \), the index ρ is invariant, provided \(\frac{1}{2}<\rho <1\).
In Sects. 3–6, we shall study the Noether theory of solvability and methods of solution for some classes of SIEs of convolution type with Cauchy kernel.
3 Dual equations
We remark that the Riemann boundary value problem (3.7) can be directly solved by the methods in [1] under certain conditions through Fredholm integral equations (see also Muskhelishvili [17]). But in this paper we shall apply Fourier theory to solve (3.7), which may enable us to deal with other equations.
Now we discuss the behaviors of the solution near \(s=0\).
Conversely, if (3.35) is fulfilled, then (3.32) and (3.33) are valid, and \(F^{\pm }(s), F(s) \in H\) in the neighborhood of \(s=0\). Therefore, we have \(F^{\pm }(s), F(s) \in \{\{0\}\}\). In conclusion, it is necessary that \(G(0)=0\).
(2) Let \(s=0\) be a special node. In this case, \(\delta =0\) and \(\gamma =i\eta _{0}\).
If \(\eta _{0}=0\), that is, \(\gamma =0\), then \(\varPhi (s)\) must be continuous at \(s=0\). It follows from \(b\neq 0\) that \(K_{1}(0)=K_{2}(0)=0\), so \(E(0)=1\), \(\varUpsilon (0)=0\), and \(F(0)=0\). Thus, \(F(s)\) is continuous at \(s=0\), and \(s=0\) is not a nodal point at all. There is no boundary value problem in this case.
Moreover, we have the following results.
Theorem 3.1
- (1)
Let \(s=0\) be an ordinary node, when \(\kappa \geq 0\), (3.1) always has a solution; when \(\kappa <0\), provided that (3.22) are fulfilled, (3.1) has the unique solution (3.21).
- (2)
Let \(s=0\) be a special node, when \(K_{1}(0)=K_{2}(0)\), the results obtained in (1) remain true; when \(K_{1}(0)\neq K_{2}(0)\), if \(\kappa > 0\) and (3.38) is fulfilled, (3.1) has a solution; if \(\kappa \leq 0\), when (3.22) and (3.39) are fulfilled, (3.1) is solvable.
Under the above suppositions, Eq. (3.1) is solvable in class \(\{0\}\) and has the solution \(\omega (t)=\mathbb{F}^{-1}\varOmega (s)\), where \(\varOmega (s)\) is given by (3.6). Obviously, \(F(s) \in \{\{0\}\}\) and so \(\omega (t)\in \{0\}\).
Remark 3.1
In Eq. (3.1), if \(k_{j}\in \{0\}\) (\(j=1,2\)), \(g\in (0)\), then \(\omega \in (0)\); if \(k_{j},g\in \langle0\rangle\) (\(j=1,2\)), then \(\omega \in \langle0\rangle \). Similarly, we can also obtain that, if \(k_{j}, g \in (0)^{\rho }\) (\(j=1,2\)), then \(\omega \in (0)^{\rho }\), and if \(k_{j},g\in \langle0\rangle ^{\rho }\) (\(j=1,2\)), then \(\omega \in \langle0\rangle ^{\rho }\), provided \(0<\rho <1\).
4 Equation of Wiener–Hopf type
Then we choose an integer κ, the index of (4.4), such that \(0\leq \delta =\delta _{0}-\kappa <1\). Denote \(\gamma =\gamma _{0}- \kappa =\delta +i\eta _{0}\). Note that \(\varOmega ^{+}(\infty )=0\), then we also have \(\varOmega ^{-}(\infty )=\varOmega (\infty )=0\). Therefore, it is necessary that \(\varOmega (\infty )=0\). Since \(b\neq 0\), we get \(\gamma \neq 0\). It is seen from the above discussion that both \(s=0\) and \(s=\infty \) are nodes of (4.4).
- (1)Let \(s=\infty \) be an ordinary node, we have the following two cases.
- (a)
When \(0\leq \delta _{\infty }\leq \frac{1}{2}\), we easily find that \(W(s)[Y(s)]^{-1} \) \(\in H_{2}\) since \(W(s)\in H_{2} \).
In order to guarantee that \(\varOmega (s)\in \{\{0\}\}\), when \(\kappa \geq 0\), the constant term \(e_{0}\) of \(P_{\kappa -1}(z)\) should take the valueand when \(\kappa < 0\), we have the following conditions of solvability:$$ e_{0}=-\frac{1}{2 \pi i} \int _{\mathbb{R}}\frac{W(t)}{Y^{+}(t)(t- \bar{z}_{0})}\,dt; $$(4.14)$$ \int _{\mathbb{R}}\frac{W(t)}{Y^{+}(t)t(t-\bar{z}_{0})^{j}}\,dt=0, \quad j=1,2,\ldots, {-}\kappa . $$(4.15) - (b)When \(\frac{1}{2}<\delta _{\infty }<1\), if \(\rho \leq \delta _{ \infty }\), by [20] we havewhere \(\varepsilon >0\) is arbitrarily small such that \(\delta _{\infty }-\varepsilon >\frac{1}{2}\); if \(\rho >\delta _{\infty }\), then \(M(s)\) is bounded and so$$ Y(s)M(s)=O\bigl( \vert s \vert ^{-\delta _{\infty }+\varepsilon }\bigr), $$near \(s=\infty \). Thus we also obtain \(\varOmega ^{\pm }(s) \in H_{2}\) and so \(\varOmega (s)\in H_{2}\). In conclusion, in any case, we have \(F(s)=O(|s|^{-\mu })\), where \(\mu =\min \{\rho , \delta _{\infty }- \varepsilon \}\), obviously, \(\mu >\frac{1}{2}\).$$ Y(s)M(s)=O\bigl( \vert s \vert ^{-\delta _{\infty }}\bigr) $$
- (a)
- (2)
Let \(s=\infty \) be a special node, then \(\delta _{\infty }=0\) and \(\gamma _{\infty }=i\eta _{\infty }\neq 0\). In this case, the discussion is the same as that in (1), and we can obtain \(\varOmega (s)\in H\) and \(\varOmega (s)\in H_{2}\).
Next, we consider the behavior of the solution near \(s=0\).
It is seen from the above discussions that \(\varOmega (0)=0\), \(\varOmega (s) \in \{\{0\}\}\). In fact, when \(\delta _{\infty }>\frac{1}{2}\), we can get \(\varOmega (s)\in ((0))\) and hence \(\omega (t)\in (0)\).
Now we can formulate the main results about the solutions of Eq. (4.1) in following form.
Theorem 4.1
- (1)
Let \(s=0\) be a node, if \(\frac{1}{2}<\delta _{\infty }<1\), when \(\kappa \geq 0\), (4.1) always has a solution, and the constant term of \(P_{\kappa -1}(s)\) takes value as (4.18); and when \(\kappa <0\), (4.11) and (4.19) should be supplemented, then Eq. (4.1) has a solution. If \(0\leq \delta _{\infty }\leq \frac{1}{2}\), when \(\kappa \geq 0\), the constant term of \(P_{\kappa -1}(s)\) should be taken as (3.34); when \(\kappa <0\), the conditions of solvability (3.22) and (3.35) are fulfilled.
- (2)
Let \(s=\infty \) be a node, if \(\frac{1}{2}<\delta _{\infty }<1 \), when \(\kappa \geq 0\), (4.1) has a solution; when \(\kappa <0\), provided that (4.11) are satisfied, (4.1) is solvable. If \(0\leq \delta _{ \infty }\leq \frac{1}{2}\), when \(\kappa \geq 0\), (4.1) has the solution; when \(\kappa <0\), the conditions of solvability (4.15) must be augmented.
Remark 4.1
Remark 4.2
5 Equations with two convolution kernels
Similarly, we easily verify that \(s=\infty \) is not a nodal point of (5.4), and \(s=0\) is its unique nodal point. The solution of (5.4) be discussed by using the same method as shown in Sect. 3. The remaining discussions will be omitted also.
6 Equations with one convolution kernel
Thus, we obtain the following conclusions.
Theorem 6.1
7 Comments
Equation (7.5) is a Riemann–Hilbert problem with discontinuous coefficients and nodes \(s=0\), ∞, and its method of solution may be made fully analogous to those in Sect. 4. In order that \(\varOmega (s)\) is continuous at \(s=0\), it is necessary that \(F(s)\) is continuous at \(s=0\) and \(F^{\pm }(0)=-G(0)\). Since \(F^{+}(s)\) is continuous at \(s=0\), we should again get \(G(0)=0\). Hence all the results as stated in Theorem 4.1 remain true and \(\omega (t)=\mathbb{F}^{-1}\varOmega (s)\), in which \(\varOmega (s)\) is given by (7.3). The only difference lies in that \(\gamma _{\infty }\) and γ may be zero, for instance, when \(a_{1}=a_{2}\), \(b_{1}=b_{2}\), we have \(\gamma _{\infty }=0\), then this case may be transformed to that in Sect. 3. Here, we will not elaborate on the solving method of (7.5).
8 Conclusions
In this paper, we study some classes of SIEs with convolution kernels and Cauchy kernels in the different classes of functions. By Fourier transform, these equations are transformed into Riemann–Hilbert problems with discontinuous coefficients. The general solutions denoted by integrals and the solvable conditions are obtained for the equations. Here, our method is different from the classical ones, and it is novel and effective. Thus, this paper generalizes the theory of the classical Riemann–Hilbert problems and SIEs. Meanwhile, we remark that the methods of this paper may be used to solving the above equations in the non-normal case. Indeed, it is possible to study the above-mentioned equation in Clifford analysis, which is similar to that in [24–28]. Further discussion is omitted here.
Declarations
Acknowledgements
The author is very grateful to the anonymous referees for their valuable suggestions and comments, which helped to improve the quality of the paper. This research is supported by the Science and Technology Plan Project of Qufu Normal University (xkj 201606).
Availability of data and materials
Not applicable.
Funding
This work was supported by the Science and Technology Plan Project of Qufu Normal University (xkj 201606).
Authors’ contributions
This entire work has been completed by the author. Analytical solutions were determined by him. The author read and approved the final manuscript.
Competing interests
The author declares to have no competing interests.
Open Access 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.
Authors’ Affiliations
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