The solvability of a kind of generalized Riemann–Hilbert problems on function spaces H∗$H_{\ast }$

*Correspondence: lipingrun@163.com 1School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, China Abstract In this paper, we study a kind of generalized Riemann–Hilbert problems (R-HPs) with several unknown functions in strip domains. We mainly discuss methods of solving R-HPs with two unknown functions and obtain general solutions and conditions of solvability on function spaces H∗. At the end of this paper, we consider in detail the behavior of the solution at ∞ and in different domains. Thus the results in this paper generalize and improve the theory of the classical Riemann–Hilbert problems.


Introduction
It is well known that Riemann-Hilbert problems are a powerful mathematical tool widely applied in physics, fracture mechanics, engineering mechanics, engineering and technology, and many other fields [1][2][3]. Especially, the problem of finding solutions for some kinds of singular integral equations is often transformed to solving Riemann-Hilbert problems [4][5][6][7][8][9][10][11][12][13][14][15]. In [1,2] the Riemann-Hilbert problems on an infinite straight line has systematically been studied, and the Riemann-Hilbert problems with unknown function on two parallel lines was further described. So far, the results of the boundary value problems for analytic functions have been mostly confined to the case of only one unknown function.
Motivated by the above researches, the main purpose of this paper is extending the theory to the R-HPs with n ≥ 2 unknown functions on n parallel straight lines, and we mainly discuss the case n = 2. Using the classical boundary value theory and principle of analytic continuation, we investigate the analytic solutions and the conditions of solvability on function spaces H * (the notation H * can be found in Sect. 2). At nodal points the asymptotic behavior of a solution of such a problem is discussed in detail. Our method of solving problems is innovate, different from those in classical cases. Meanwhile, this paper also improves some results of [2][3][4]7].

Definitions and lemmas
In this section, we introduce some definitions and lemmas. Definition 2.1 Let F(x) be a continuous function in the real number field R. A function F(x) belongs toĤ if the following two conditions are fulfilled: (1) there exists B ∈ R + such that on the neighborhood N ∞ of ∞, that is, there exists a sufficiently large M > 0 such that (2.1) is satisfied for all x 1 , where H is the class of Hölder continuous functions (for the notation H, see [1]).

Definition 2.2 A function F belongs to H * if it satisfies:
(1) F ∈Ĥ, (2) F ∈ L 2 (R) (see [1,6] for the definition of L 2 (R)). With respect to the function spaces H * , one of its important properties is closedness under pointwise multiplication.
If a function F satisfies the Hölder condition on a neighborhood N ∞ of ∞, then we write F ∈ H(N ∞ ).

Definition 2.3
Let f (t) ∈ H * . We define its Fourier transform F and the inverse Fourier transform F -1 as follows: Lemma 2.1 Let functions 1 and 2 be analytic in the upper half-plane C + and the lower half-plane Cexcept their poles z 0 = ∞ and z k (k = 1, 2, . . . , n). Suppose that the boundary values of 1 and 2 on Im z = 0 are equal. The main parts of the Laurent expansion of 1 and 2 at z 0 = ∞ are where c 0 m = 0 and c 0 k (1 ≤ k ≤ m) are constants. The main parts of the Laurent expansion for 1 and 2 at every pole z k (k = 1, 2, . . . , n) are where c k l (1 ≤ l ≤ p) are constants, and c k p = 0, p ≥ 1. Then 1 and 2 can be represented by the same function in the complex plane C, namely,

5)
where c 0 is a complex constant.
The following lemmas are obvious facts, and we omit their proofs.
Then for z ∈ C + , we have

7)
and for z ∈ C -, we have It is easy to prove that + (z) and -(z) are analytical in z ∈ C + and z ∈ C -, respectively.

Problem presentation and solution
We now propose boundary value problems for analytic functions with n ≥ 2 unknown functions on n parallel lines, and then we discuss the methods of solution of such problems. Suppose that n lines γ j (1 ≤ j ≤ n) are parallel to the X-axis and denote them by L = n j=1 γ j , where γ j (1 ≤ j ≤ n) take the direction from left to right as the positive direction and can be expressed by ξ j = x + iR j (R n < · · · < R 2 < R 1 ), where x, R j ∈ R. Our goal is to obtain functions F j (z)(1 ≤ j ≤ n) such that F j (z) are analytic in R j < Im z < R j-1 (2 ≤ j ≤ n) and F 1 (z) is analytic in Im z > R 1 and Im z < R n , and the following boundary value conditions are fulfilled: when j = n, denote F -n+1 (ξ ) as F -1 (ξ ), where the given functions A j (ξ ), B j (ξ )(1 ≤ j ≤ n) belong to H * on γ j . Obviously, R-HP (3.1) can also be written in the following form: · · · · · · · · · · · · F + n-1 (ξ ) -A n-1 (ξ )Fn (ξ ) = B n-1 (ξ ), ξ ∈ γ n-1 , It follows from (3.1) that the orders of F j (z)(1 ≤ j ≤ n) are equal to each other at ∞. Thus, when the orders of F j (z) are m at ∞, (3.1) can be denoted as R m . In fact, the problem R 0 and problem R -1 are frequently discussed. On the problem R 0 , F j (z) are supposed to be finite and nonzero at ∞. On problem R -1 , F j (z) are assumed to be zero at ∞. When A j (ξ )(1 ≤ j ≤ n) are not zero on L, problem (3.1) is said to be of normal type; otherwise, it is called of nonnormal type or of exception type.
Note that since the positive direction of γ j (1 ≤ j ≤ n) is the direction from left to right, when the observer moves from left to right on γ j , the boundary values of left domain of γ j are positive, that is, the positive boundary values of F j (z)(1 ≤ j ≤ n) are the boundary values above γ j , and the negative boundary values of F j (z) are those below γ j .
Without loss of generality, in this paper, we only discuss the case n = 2. As for R-HP with n > 2 unknown functions on n parallel lines, there is no essential difference for the methods of solution with the case n = 2.
When n = 2, R-HP (3.1) can be stated as follows.
Problem Assume that γ 1 : ξ = x + iβ and γ 2 : ξ = x + iα are two oriented lines, where α and β are real numbers with α < β. Similarly to the above statement, we take the direction of γ 1 and γ 2 from left to right as the positive direction. We want to obtain functions F 1 (z) and α < Im z < β} and satisfies the following boundary value conditions on γ 1 and γ 2 : In fact, (3.3) is the R-HP on two parallel straight lines Im z = β and Im z = α with z = ∞ as a pole, and it is a generalization of the classical R-HP.

The conditions of solvability of R-HP (3.3)
Now we are concerned about solution (3.36) and the conditions of solvability of R-HP (3.3).