Boundary value problems for hypergenic function vectors

This article mainly studies the boundary value problems for hypergenic function vectors in Clifford analysis. Firstly, some properties of hypergenic quasi-Cauchy type integrals are discussed. Then, by the Schauder fixed point theorem the existence of the solution to the nonlinear boundary value problem is proved. Finally, using the compression mapping principle the existence and uniqueness of the solution to the linear boundary value problem are proved.

In 2003, Xie, Qiao and Jiao [20] studied a nonlinear boundary value problem for a generalized biregular function vector. In 2005, Qiao [11] discussed a linear boundary value problem for hypermonogenic functions in Clifford analysis. In 2009-2010, Eriksson and Orelma [6,7] studied hypergenic functions in the real Clifford algebra Cl n+1,0 (R) and its Cauchy integral formula was given. In 2014, Xie [14,15] studied the Cauchy integral for dual k-hypergenic functions and the boundary properties of the hypergenic quasi-Cauchy integral in real Clifford analysis were given. In 2016, Xie, Zhang and Tang [17] discussed some properties of k-hypergenic functions.
On the basis of the above, the boundary value problems for hypergenic function vectors are proved.
Let 0 be a nonempty open connected set in R n+1 . The function f : where f A is r-times continuously differentiable on 0 and r ∈ N * }.
For f ∈ C 1 ( 0 , Cl n+1,0 (R)), we introduce Dirac operators as follows [6]: Definition 2.1 ([15]) A Lyapunov surface S is a surface satisfying the following three conditions: (1) Through each point in S, there is a tangent plane.
(2) There is a real constant number d such that, for any N 0 ∈ S, E is a ball with radius d, centered at N 0 , and E is divided into two parts by S, the part of S lying in the interior of E is denoted by S , the other is in the exterior of S: and each straight line parallel to the normal direction of S at N 0 intersects it at one point. (3) If the angle θ (N 1 , N 2 ) between outward normal vectors through N 1 , N 2 is an acute angle and r is the distance between N 1 and N 2 , then there are two numbers β, α The set of all Hölder continuous functions which are defined on 0 and valued in Cl n+1,0 (R) is denoted by H(β, ∂ 0 , Cl n+1,0 (R)).
, the hypergenic quasi-Cauchy type integral satisfies f (∞) = 0. Let B(y, δ) be a ball with radius δ > 0, centered at y when y ∈ ∂ . ∂ is divided into two parts by B(y, δ). The part of ∂ lying in the interior of B(y, δ) is denoted by λ δ .

Definition 2.5 ([15]) I is called the Cauchy principal of the singular integral value if
lim δ→0 f (y) = I exists, and we write directly I = f (y).
, then the Cauchy principal values of the singular integral (3) exist, and

Lemma 2.5 ([5]) If is a bounded domain in R n+1
where M 1 (α, ) is a positive constant only related to α, .
, define the addition operation and multiplication operation for function vectors as follows: Let L(x) be a function valued in Clifford algebra Cl n+1,0 (R) and F(x) be a function vector, then Define the model of a function vector as follows: . . , f q (x)) is called a hypergenic function vector when each component f i (x) (i = 1, . . . , q) is a hypergenic function on .
There is a ball with radius 3δ, centered at y 1 when y 1 , y 2 ∈ ∂ and 6δ < d, δ = |y 1y 2 |. Remark that ∂ 1 is located inside the ball and the rest of ∂ is ∂ 2 .

The existence of the solution to the nonlinear boundary value problem for the hypergenic function vector
Let A(y), B(y), G(y) ∈ H q (β, ∂ , Cl n+1,0 (R)) be Hölder continuous function vectors on ∂ , we find a function vector * F (y), which is hypergenic on + ∪ -, and continuous on + ∪ ∂ and -∪ ∂ , satisfying * F (∞) = 0, and the nonlinear boundary value condition: where P( * + F (y), * -F (y)) is a Hölder continuous function vector on ∂ which is related to * + F (y), * -F (y). The above problem is called the nonlinear boundary value problem SR. If P = 1, then the above problem is called the linear boundary value problem SR.
Putting (10) into (23), we have and equality (23) is transformed into the following singular integral equation:

Theorem 4.1 If A(y), B(y), G(y)
∈ H q (β, ∂ , Cl n+1,0 (R)), for any y 1 , y 2 ∈ ∂ , P( + F (y), -F (y)) satisfies where J 34 and J 35 are positive constants independent of y i (i = 1, 2) and F. If P(0, 0) = 0, Problem SR has at least one solution and the integral expression of the solution is (8). 4 and F is uniformly Hölder continuous on ∂ , that is, to say, there is a positive constant M 2 , for any x 1 , Obviously T is a convex subset of the continuous function vector space C q (∂ ).
(1) We prove that N maps the set T to itself. From inequality (7), Theorem 3.1 and Remark 3.2, it follows that By inequality (27) and Remark 3.2, we have By inequality (27) and Remark 3.2, we have then As If F is uniformly Hölder continuous on ∂ , then F (y), + F , -F are uniformly Hölder continuous on ∂ . So NF is uniformly Hölder continuous on ∂ .
Hence N maps the set T to itself.
(2) We prove that N is a continuous mapping. Any F n ∈ T, {F n } uniformly converges to F on ∂ . As for ε > 0, when n is fully large and |F n -F| is sufficiently small. There is a ball with radius 3δ, centered at y when 6δ d, δ 0, and remark that ∂ 1 is located inside the ball and the rest of ∂ is ∂ 2 By inequality (27), Theorem 3.3, we have = I 10 + I 11 ; that is, that is, From inequality (33) and (34), we have In a similar way, we get From inequality (32), (35) and (36), we get Select a sufficiently small positive number δ such that J 51 δ β < ε 2 ; and let n be large enough such that J 52 F n -F β < ε 2 . So for any y ∈ ∂ , we have |P( + F n (y), -F n (y)) -P( + F (y), -F (y))| < ε, thus |NF n (y) -NF(y)| < ε, then N is a continuous mapping which maps T to itself.
From the Arzela-Ascoli theorem we conclude that T is a compact set in C q (∂ ).
As the continuous mapping N maps the closed convex set T to itself, N(T) is compact in C q (∂ ). From the Schauder fixed point principle it follows that there is at least F ∈ H q (β, ∂ , Cl n+1,0 (R)) that satisfies (26). Hence the nonlinear boundary value problem SR has at least one solution F (y), and the expression of the solution is (8).
Proof Let T be as in Theorem 4.1. N is a continuous mapping which maps T to itself from Theorem 4.1.
From inequalities (7), (25) and Remark 3.1, we get There is only one solution to the equation NF = F by the compression mapping principle. So there is a unique solution to the linear boundary value problem SR, and the integral expression of the solution is (8).

Conclusions
In this paper, we prove the existence of the solution to the nonlinear boundary value problem for the hypergenic function vector by virtue of the Arzela-Ascoli theorem and prove the existence and uniqueness of the solution to the linear boundary value problem for the hypergenic function vector by the compression mapping principle.