EXISTENCE OF SOLUTIONS FOR DUAL SINGULAR INTEGRAL EQUATIONS WITH CONVOLUTION KERNELS IN CASE OF NON-NORMAL TYPE

This paper is devoted to the study of dual singular integral equations with convolution kernels in the case of non-normal type. Via using the Fourier transforms, we transform such equations into Riemann boundary value problems. To solve the equation, we establish the regularity theory of solvability. The general solutions and the solvable conditions of the equation are obtained. Especially, we investigate the asymptotic property of solutions at nodes. This paper will have a significant meaning for the study of improving and developing complex analysis, integral equations and Riemann boundary value problems.


Introduction
It is well known that there are rather complete investigations on the method of solution for equations of Cauchy type as well as integral equations of convolution type. Singular integral equations and Riemann boundary value problems have a lot of applications, e.g. in elasticity theory, fluid dynamics, quantum mechanics. In recent years, many mathematicians have studied singular integral equations and formed a relatively systematic theoretical system (see [1,4,6,7,10,29,30] and references therein). [5] first began to study singular integral equation of Wiener-Hopf type with continuous coefficients. [11] discussed the Noether theory of singular integral equations of convolution type. [14,16,18,19,[25][26][27] dealt with the invertibility of singular integral operators with discontinuous coefficients, and then considered the solvability theory and the general solutions for some classes of singular integral equations with convolution kernels on the whole real axis (or, on the unit circle) in the case of normal-type. For operators containing both Cauchy principal value integral and convolution, the conditions of their Noethericity were discussed in [8,23,28,33] in more general cases. For applications, the problems to find their solutions is very important. Therefore, singular integral equations of convolution type, mathematically, belong to an interesting subject in the theory of integral equations. Motivated by the above works, we investigate the existence of solutions for one class of dual singular integral equations with convolution kernels in the cases of non-normal type. In the process of studying equations, we find that the methods used in [5,10,29] are no longer suitable for the case of non-normal type, that is, it is difficult to use only the Fourier transform technique to study the case of non-normal type. Hence, we shall introduce a new method to complete our research. In this paper we apply Fourier analysis theory and boundary value method in the theory of analytic functions to deal with the solvability of the equations. Our approach is novel and effective, different from the ones in classical cases. Therefore, this paper generalizes and improves the theories of integral equations and the classical Riemann boundary value problems.
Definition 2.1. The Fourier transform F and the inverse transform F −1 are defined as follows  where |s| is sufficiently large, then we call F (s) ∈ ((0)) σ or ((0)).
Definition 2.6. For two functions k(t) and f (t), their convolution is defined by we denote it as k * f . It is well known that [5,29] F(k * f (t)) = Fk(t) · Ff (t) = K(s)F (s).
Definition 2.7. We also introduce the operator of Cauchy principal value integral (2.4) It follows from [8,16,33] that maps {0} and < 0 > into themselves respectively and 2 = I (identity). Definition 2.8. We define operators N and S as follows (2.5) Lemmas 2.1 and 2.2 are obvious facts and we omit their proof here.
Lemma 2.2 (see [8,33]). The operators F, F −1 , , N, S are as the before, then we have The following lemma 2.3 plays an important role and it is used to get our some results in this paper.
Proof. By Lemma 2.3 and assumptions, and note that thus Lemma 2.4 can be proved. In Lemma 2.4, note that F (0) = 0 is a necessary condition, otherwise the lemma is invalid.
In the following section, we shall focus on the theory of Noether solvability and the methods of solution for dual singular integral equations with convolution kernels in the non-normal type case.

Singular integral equations of dual type
Consider the equation where a j , b j (j = 1, 2) are constants and b 1 , b 2 are not equal to zero simultaneously. k 1 , k 2 , g ∈< 0 > β ( or (0) β , 0 < β < 1) and the unknown function ω(t) is requred to be in {0}. After simplification, (3.1) may be written as Extending t in the first equation of (3.2) to −∞ < t < 0, and in the second one of (3.2) to 0 < t < +∞, i.e., we add −φ − (t) and +φ + (t) to (3.2), then (3.2) can be rewritten as We firstly use the Fourier transform to convert Eq.(3.3) into a Riemann boundary value problem. By Lemmas 2.2 and 2.3, we get where Note that, from equation (3.3) to equation (3.4), by taking the Fourier transform Thus, we should only solve the following Riemann boundary value problem (3.6) in place of (3.1). Ψ + (s) = E(s)Ψ − (s) + W (s), −∞ < s < +∞, (3.6) in which Now we assume that E 1 (s) has some zero-points e 1 , e 2 , · · · , e n with the orders ξ 1 , ξ 2 , · · · , ξ n respectively; E 2 (s) has some zero-points c 1 , c 2 , · · · , c q with the orders η 1 , η 2 , · · · , η q respectively, where ξ j , η j are the non-negative integers. In this case, we say that (3.6) is a Riemann boundary value problem of non-normal type. Put then we can rewrite (3.6) in the form where E(s) = V2(s) V1(s) D(s) and D(s) = 0. In view of the values of a j ± b j , we have the following several cases.
(1) If a 1 ± b 1 = 0, and a 2 ± b 2 are not equal to zero simultaneously, then (3.6) is a Riemann boundary value problem with nodes s = 0, ∞.

(3.9)
We denote Next we discuss the solvability of (3.8). We first define a sectionally holomorphic function X(z): (3.12) in which we have put and here we have taken the definite branch of provided we have chosen ln t+i t−i | t=±0 = ±iπ. It is easy to verify that X(z) is a canonical function and its boundary values satisfy (3.14) Thus, (3.8) could also be rewrite as We again put where ln D(s) is taken to be continuous branch for s > 0 and s < 0 respectively such that it is continuous at s = ∞, and 0 ≤ α ∞ < 1. Without loss of generality, we assume a 1 b 2 = a 2 b 1 , then γ ∞ = 0. If a 1 b 2 = a 2 b 1 , the only difference lies in that γ ∞ and γ may be zero, then in which cases the analysis will be even simpler, here we do not discuss it. We first consider the homogeneous problem of (3.15) given by Via using the principle of analytic continuation [16,27], we obtain an analytic solution of (3.17): in (3.18), when ϑ ≥ 0, P ϑ (z) is a polynomial of degree ϑ with arbitrary complex coefficients; when ϑ < 0, P ϑ (z) ≡ 0, where ϑ = κ − n 1 − n 2 . Now we solve the non-homogeneous problem (3.15) in class {0}. To do this, we consider the following function We will apply SokhotskiPlemelj formula and generalized Liouville theorem [16,33] to the boundary value problem (3.15), which has a singularity at e j and c k . Therefore, we need to construct a Hermite interpolation polynomial H ρ (z) with the degree ρ, and we can assume that A j z ρ−j , ρ = n 1 + n 2 − 1 which has some zero-points of the orders ξ j , η k (1 ≤ j ≤ n, 1 ≤ k ≤ q) at e j , c k , respectively, where A l (0 ≤ l ≤ ρ) are constants. Making use of (3.19) and H ρ (z), we can define the following function: , z ∈ C + ; (3.20) By means of the classical Riemann boundary value problem, we can verify that (3.20) is the particular solution of (3.15). In view of the solvability of linear equations, we obtain a general solution of (3.15): (3.21) From (3.18) and (3.20), Ψ(z) can also be written as the explicit solution: (3.22) In the following, we discuss the conditions of solvability and the properties of solution for Eq. (3.15).
Finally, we give the following two remarks.