# Boundary value problems for hypergenic function vectors

## Abstract

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.

## 1 Introduction

AÂ Clifford algebra is an associative and noncommutable algebra [1]. In 1982, Brackx, Delanghe and Sommen [2] established the theoretical basis of Clifford analysis. In recent years, Clifford analysis has been widely used in physics and in mathematics [3â€“5]. Eriksson [6â€“8], Huang [9, 10], Qiao [11, 12], Xie [13â€“17] and Yang [18, 19] have done a lot of work in Clifford analysis. In 1996, Huang [10] studied the nonlinear boundary value problem for biregular functions in Clifford analysis. In 2000, Cai, Huang and Qiao [20] studied the nonlinear boundary value problem for biregular functions vector in Clifford analysis. 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 2009â€“2010, Eriksson and Orelma [6, 7] studied hypergenic functions in the real Clifford algebra $$\mathit {Cl}_{n+1,0}(\mathbb{R})$$ 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.

## 2 Preliminaries

See [6]; let $$\mathit {Cl}_{n+1,0}(\mathbb{R})$$ be a real Clifford algebra and have identity element $$e_{\varnothing}=1$$ and basis elements $$e_{0}, e_{1},\ldots, e_{n};e_{0}e_{1},\ldots,e_{n-1}e_{n}; \ldots ;e_{0}e_{1}\cdots e_{n}$$, and satisfy

$$\textstyle\begin{cases} e_{i}e_{j} = -e_{j}e_{i},\quad i\ne j, i,j=0,1,\ldots,n; \\ e^{2}_{j} = +1,\quad j=0,1,\ldots,n. \end{cases}$$

Any element in $$\mathit {Cl}_{n+1,0}(\mathbb{R})$$ has the form $$a=\sum_{A}a_{A}e_{A}$$, $$e_{A}=e_{\alpha_{1}}e_{\alpha_{2}}\cdots e_{\alpha _{h}}$$ or $$e_{\varnothing}=1$$, where $$A=\{\alpha_{1},\alpha_{2},\ldots,\alpha_{h} \}$$, $$0\le \alpha_{1}<\alpha_{2}<\cdots<\alpha_{h}\le n$$, $$a_{A}\in\mathbb {R}$$. The norm of $$a\in \mathit {Cl}_{n+1,0}(\mathbb{R})$$ is defined as $$\vert a \vert ={(\sum_{A} \vert a_{A} \vert ^{2})}^{\frac{1}{2}}$$. In this paper $$J_{i}$$ ($$i=1,2,\ldots, 32$$) is a positive constant. For any $$a,b\in{ \mathit {Cl}_{n+1,0}(\mathbb{R})}$$, we have

$$\vert {a+b} \vert \leq \vert {a} \vert + \vert {b} \vert ,\quad\quad \vert {ab} \vert \leq{J_{1}} \vert {a} \vert \vert {b} \vert .$$
(1)

If $$a=a_{0}e_{0}+a_{1}e_{1}+\cdots+a_{n}e_{n}$$, it may be observed that $$a^{2}= \vert a \vert ^{2}$$ and when $$a\neq0$$ the inverse of a is $$a^{-1}= {\frac{a}{ \vert a \vert ^{2}}}$$. See [6]; any element $$a\in \mathit {Cl}_{n+1,0}(\mathbb {R})$$ can be uniquely decomposed as $$a=b+e_{0}c$$, where $$b, c\in \mathit {Cl}_{n,0}(\mathbb{R})$$. As regards decomposition we can define the mappings $$P_{0}:\mathit {Cl}_{n+1,0}\rightarrow \mathit {Cl}_{n,0}$$ and $$Q_{0}: \mathit {Cl}_{n+1,0}\rightarrow \mathit {Cl}_{n,0}$$ by $$P_{0}a=b$$, $$Q_{0}a=c$$, where b, c are called the $$P_{0}$$ part and the $$Q_{0}$$ part of a, respectively.

Let $$\Omega_{0}$$ be a nonempty open connected set in $${R^{n+1}}$$. The function $$f:\Omega_{0}\rightarrow \mathit {Cl}_{n+1,0}(\mathbb{R})$$ is denoted by $$f(x)=\sum_{A}f_{A}(x)e_{A}$$, where $$f_{A}\in\mathbb{R}$$. The function $$f:\Omega_{0}\rightarrow \mathit {Cl}_{n+1,0}(\mathbb{R})$$ is said to be continuous on $${\Omega_{0}}$$ if and only if each component $$f_{A}(x)$$ of $$f(x)$$ is continuous on $${\Omega_{0}}$$. Suppose $${C^{r}(\Omega_{0},\mathit {Cl}_{n+1,0}(\mathbb{R}))}=\{f\mid f:\Omega _{0}\rightarrow \mathit {Cl}_{n+1,0}(\mathbb{R}),f(x)=\sum_{A}f_{A}(x)e_{A}$$, where $$f_{A}$$ is r-times continuously differentiable on $$\Omega_{0}$$ and $$r\in\mathbb{N}^{*}\}$$.

For $$f\in{C^{1}(\Omega_{0},\mathit {Cl}_{n+1,0}(\mathbb{R}))}$$, we introduce Dirac operators as follows [6]:

$$Df=\sum_{l=0}^{n}e_{l} \frac{\partial{f}}{\partial{x_{l}}}.$$

### Definition 2.1

([15])

AÂ Lyapunov surface S is a surface satisfying the following three conditions:

1. (1)

Through each point in S, there is a tangent plane.

2. (2)

There is a real constant number d such that, for any $$N_{0}\in {S}$$, 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 $$S^{\prime}$$, 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. (3)

If the angle $$\theta(N_{1},N_{2})$$ 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 Î², Î± ($$0\leq\alpha\leq1$$, $$\beta>0$$) independent of $$N_{1}$$, $$N_{2}$$ such that $$\theta(N_{1},N_{2})\leq\beta{r}^{\alpha}$$.

### Definition 2.2

([15])

The function f: $$\partial\Omega_{0}\longrightarrow \mathit {Cl}_{n+1,0}(\mathbb{R})$$ is said to be HÃ¶lder continuous on $$\Omega _{0}$$ if there exists a positive constant $$M_{0}$$ such that $$\vert f(x_{1})-f(x_{2}) \vert \leq{M_{0} \vert x_{1}-x_{2} \vert ^{\beta}}$$ ($$0<\beta<1$$) holds for any $$x_{1},x_{2}\in\partial\Omega_{0}$$.

The set of all HÃ¶lder continuous functions which are defined on $$\Omega_{0}$$ and valued in $$\mathit {Cl}_{n+1,0}(\mathbb{R})$$ is denoted by $$H(\beta,\partial\Omega_{0},\mathit {Cl}_{n+1,0}(\mathbb{R}))$$.

For any $$f\in{H(\beta,\partial\Omega_{0},\mathit {Cl}_{n+1,0}(\mathbb{R}))}$$, we define the norm of f as $${ \Vert f \Vert _{\beta }}=C(f,\partial\Omega _{0})+H(f,\partial\Omega_{0},\beta)$$, where

$${C(f,\partial\Omega_{0})}=\max_{t\in\partial\Omega_{0}} \bigl\vert f(t) \bigr\vert , \quad\quad{H(f,\partial\Omega_{0}, \beta)}=\sup_{\substack{t_{1}\neq{t_{2}} \\ t_{1},t_{2}\in{\partial\Omega_{0}}}} {\frac { \vert f(t_{1})-f(t_{2}) \vert }{ \vert t_{1}-t_{2} \vert ^{\beta}}} .$$

It is easy to prove that $$H(\beta,\partial\Omega _{0},\mathit {Cl}_{n+1,0}(\mathbb{R}))$$ forms a Banach space.

For any $${f,g}\in{H(\beta,\partial\Omega_{0},\mathit {Cl}_{n+1,0}(\mathbb {R}))}$$, we have

$$\Vert {f+g} \Vert _{\beta}\leq{{ \Vert {f} \Vert _{\beta }}+{ \Vert {g} \Vert _{\beta}}}, \quad\quad { \Vert {fg} \Vert _{\beta}}\leq{{J_{2}} {{ \Vert {f} \Vert _{\beta}} { \Vert {g} \Vert _{\beta}}}}.$$
(2)

In this paper, let Î© be a domain in $$\mathbb{R}^{n+1}_{+}={\{ x\mid (x_{0},x_{1},\ldots,{x_{n}})\in\mathbb{R}^{n+1},x_{0}>0\}}$$, and its boundary âˆ‚Î© be a smooth compact oriented Lyapunov surface. For any $$f\in C^{1}(\Omega, \mathit {Cl}_{n+1,0}(\mathbb{R}))$$, we introduce a modified Dirac operator as follows [6]:

$$Hf=Df- \frac{n-1}{x_{0}}{Q_{0}}f.$$

### Definition 2.3

([6])

$$f(x)$$ is said to be a hypergenic function on Î© if $$f\in {C^{1}(\Omega, \mathit {Cl}_{n+1,0}(\mathbb{R}))}$$ satisfies $$Hf=0$$ onÂ Î©.

In this paper, let $$E_{1}(x,y)= {\frac{x-y}{ \vert x-y \vert ^{n+1} \vert x-\widehat {y} \vert ^{n-1}}}$$, $$E_{2}(x,y)= {\frac{\widehat{x}-y}{ \vert x-y \vert ^{n-1} \vert x-\widehat {y} \vert ^{n+1}}}$$, and $$w_{n+1}$$ is the surface area of the unit hypersphere in $$\mathbb{R}^{n+1}$$.

### Definition 2.4

([15])

$$\Psi_{f}(y)={ \frac{{2^{n-1}}{y_{0}^{n-1}}}{w_{n+1}}} { \biggl[ \int_{\partial\Omega}{E_{1}(x,y)\,d\sigma(x)f(x)}}-{ \int_{\partial \Omega}{E_{2}(x,y)\widehat{d\sigma(x)} \widehat{f(x)}} \biggr]}$$
(3)

is called a hypergenic quasi-Cauchy type integral if $$f\in{H(\beta ,\partial\Omega, \mathit {Cl}_{n+1,0}(\mathbb{R}))}$$.

### Lemma 2.1

([14])

If $$y\notin{\partial\Omega}$$, $$f\in{H(\beta,\partial\Omega ,\mathit {Cl}_{n+1,0}(\mathbb{R}))}$$, the hypergenic quasi-Cauchy type integral

$${\Psi_{f}{(y)}={\frac{{2^{n-1}}{y_{0}^{n-1}}}{w_{n+1}}} { \biggl[ \int_{\partial\Omega}{E_{1}(x,y)\,d\sigma(x)f(x)}}-{ \int_{\partial\Omega }{E_{2}(x,y)\widehat{d\sigma(x)} \widehat{f(x)}} \biggr]}}$$

is a hypergenic function on $${{\mathbb{R}}_{+}^{n+1}}\backslash {{\partial\Omega}}$$.

### Remark 2.1

If $$y\notin{\partial\Omega}$$, $$f\in{H(\beta,\partial\Omega ,\mathit {Cl}_{n+1,0}(\mathbb{R}))}$$, the hypergenic quasi-Cauchy type integral

$${\Psi_{f}{(y)}={\frac{{2^{n-1}}{y_{0}^{n-1}}}{w_{n+1}}} { \biggl[ \int_{\partial\Omega}{E_{1}(x,y)\,d\sigma(x)f(x)}}-{ \int_{\partial\Omega }{E_{2}(x,y)\widehat{d\sigma(x)} \widehat{f(x)}} \biggr]}}$$

satisfies $$\Psi_{f}{(\infty)}=0$$.

### Lemma 2.2

([15])

If $$f\in{H(\beta,\partial\Omega, \mathit {Cl}_{n+1,0}(\mathbb{R}))}$$, then $$\Psi_{f}(y)$$ is HÃ¶lder continuous on $$\Omega^{+}\cup\partial \Omega$$ and $$\Omega^{-}\cup\partial\Omega$$.

Let $$\mathbf{B}(\mathbf{y},\delta)$$ be a ball with radius $$\delta>{0}$$, centered at y when $$y\in{\partial\Omega}$$. âˆ‚Î© is divided into two parts by $$\mathbf{B}(\mathbf{y},\delta)$$. The part of âˆ‚Î© lying in the interior of $$\mathbf{B}(\mathbf{y},\delta)$$ is denoted by $$\lambda_{\delta}$$.

### Definition 2.5

([15])

I is called the Cauchy principal of the singular integral value if $$\lim_{\delta\rightarrow{0}}\Psi_{f}(y)=\mathbf{I}$$ exists, and we write directly $$\mathbf{I}=\Phi_{f}(y)$$.

### Lemma 2.3

([15])

If $$y\in\partial\Omega$$, $$f\in{H(\beta,\partial\Omega ,\mathit {Cl}_{n+1,0}(\mathbb{R}))}$$, then the Cauchy principal values of the singular integral (3) exist, and

\begin{aligned}[b] \Phi_{f}(y)& ={ \frac {{2^{n-1}}{y_{0}^{n-1}}}{w_{n+1}}} { \biggl[ \int_{\partial\Omega}{E_{1}(x,y)\,d\sigma(x) \bigl(f(x)-f(y) \bigr)}} \\ &\quad{} -{ \int_{\partial\Omega}{E_{2}(x,y)\widehat{d\sigma(x)} \bigl( \widehat{f(x)}-f(y) \bigr)} \biggr]}+ \frac{1}{2}f(y), \end{aligned}
(4)

when $$f=1$$, we have $$\Phi_{1}(y)= \frac{1}{2}$$.

### Lemma 2.4

([15])

If $$y\in{\partial\Omega}$$, $$f\in{H(\beta,\partial\Omega ,\mathit {Cl}_{n+1,0}(\mathbb{R})})$$, then

$$\textstyle\begin{cases} \Psi^{+}_{f}(y)=\Phi_{f}(y)+ \frac{1}{2}f(y), \\ \Psi^{-}_{f}(y)=\Phi_{f}(y)- \frac{1}{2}f(y). \end{cases}$$
(5)

### Lemma 2.5

([5])

If Î© is a bounded domain in $$\mathbb{R}^{n+1}_{+}$$, $$0<\alpha <n+1$$, for any $$y\in\Omega$$ we have

$$\int_{\Omega} \vert x-y \vert ^{-\alpha}\,dx \leq{M_{1}}(\alpha,\Omega),$$

where $${M_{1}}(\alpha,\Omega)$$ is a positive constant only related to Î±,Â Î©.

### Definition 2.6

$$F(x)=(f_{1}(x),\ldots,f_{q}(x))$$ is called a function vector if $$f_{i}(x):\Omega\rightarrow \mathit {Cl}_{n+1,0}(\mathbb{R})$$ ($$i=1,\ldots,q$$).

For $$F(x)=(f_{1}(x),\ldots,f_{q}(x))$$, $$K(x)=(k_{1}(x),\ldots ,k_{q}(x))$$, define the addition operation and multiplication operation for function vectors as follows:

$$F\oplus K =(f_{1}+k_{1},\ldots,f_{q}+k_{q});F \otimes K =(f_{1}k_{1},\ldots,f_{q}k_{q}).$$

Let $$L(x)$$ be a function valued in Clifford algebra $$\mathit {Cl}_{n+1,0}(\mathbb {R})$$ and $$F(x)$$ be a function vector, then

$$LF=(Lf_{1},\ldots,Lf_{q}),\quad\quad FL=(f_{1}L,\ldots,f_{q}L).$$

Define the model of a function vector as follows: $$\vert F(x) \vert =(\sum_{i=0}^{q} \vert f_{i}(x) \vert ^{2})^{\frac {1}{2}}$$, we have

$$\vert F\oplus K \vert \leq \vert F \vert + \vert K \vert , \quad\quad \vert F\otimes K \vert \leq J_{1} \vert F \vert \vert K \vert . .$$
(6)

### Definition 2.7

$$F(x)=(f_{1}(x),\ldots,f_{q}(x))$$ is called a hypergenic function vector when each component $$f_{i}(x)$$ ($$i=1,\ldots,q$$) is a hypergenic function onÂ Î©.

### Definition 2.8

AÂ hypergenic function vector F is said to be HÃ¶lder continuous on âˆ‚Î© if there is a positive constant $$M_{2}$$ such that

$$\bigl\vert F(x_{1})-F(x_{2}) \bigr\vert = \Biggl(\sum _{i=0}^{q} \bigl\vert f_{i}(x_{1})-f_{i}(x_{2}) \bigr\vert ^{2} \Biggr)^{\frac{1}{2}}\leq M_{2} \vert x_{1}-x_{2} \vert ^{\beta}$$

holds for any $$x_{1},x_{2}\in\Omega$$, where $$0<\beta<1$$ and $$M_{2}$$ is independent of $$x_{i}$$ ($$i=1,2$$).

### Remark 2.2

The hypergenic function vector $$F(x)=(f_{1}(x),\ldots,f_{q}(x))$$ is HÃ¶lder continuous on âˆ‚Î© if and only if each component $$f_{i}(x)$$ ($$i=1,\ldots,q$$) of $$F(x)$$ is HÃ¶lder continuous on âˆ‚Î©.

The set of all HÃ¶lder continuous function vectors which are defined on âˆ‚Î© and valued in $$\mathit {Cl}_{n+1,0}(\mathbb{R})$$ is denoted by $$H_{q}(\beta,\partial\Omega, \mathit {Cl}_{n+1,0}(\mathbb{R}))$$. For any $$F\in{H_{q}(\beta,\partial\Omega, \mathit {Cl}_{n+1,0}(\mathbb{R}))}$$, the norm of F is defined as follows: $${ \Vert F \Vert _{\beta }}=C_{q}(F,\partial\Omega )+H_{q}(F,\partial\Omega,\beta)$$, where

$${C_{q}(F,\partial\Omega)}=\max_{t\in\partial\Omega} \bigl\vert F(t) \bigr\vert ,\quad\quad{H_{q}(F,\partial\Omega, \beta)}=\sup_{\substack{t_{1}\neq{t_{2}} \\ t_{1},t_{2}\in{\partial\Omega}}} {\frac { \vert f(t_{1})-f(t_{2}) \vert }{ \vert t_{1}-t_{2} \vert ^{\beta}}} .$$

It is easy to prove that $$H_{q}(\beta,\partial\Omega ,\mathit {Cl}_{n+1,0}(\mathbb{R}))$$ forms a Banach space.

For any $${F,K}\in{H_{q}(\beta,\partial\Omega, \mathit {Cl}_{n+1,0}(\mathbb {R}))}$$, we have

$$\Vert {F+K} \Vert _{\beta}\leq{{ \Vert {F} \Vert _{\beta }}+{ \Vert {K} \Vert _{\beta}}}, \quad\quad{ \Vert {F \otimes K} \Vert _{\beta}}\leq{{J_{3}} {{ \Vert {F} \Vert _{\beta }} { \Vert {K} \Vert _{\beta}}}} .$$
(7)

## 3 Some properties of hypergenic quasi-Cauchy type integrals

### Theorem 3.1

If $$y\notin{\partial\Omega}$$, $$F(x)=(f_{1}(x),\ldots,f_{q}(x))\in {H_{q}(\beta,\partial\Omega, \mathit {Cl}_{n+1,0}(\mathbb{R}))}$$,

$$\Psi_{F}(y)={ \frac {{2^{n-1}}{y_{0}^{n-1}}}{w_{n+1}}} { \biggl[ \int_{\partial\Omega }{E_{1}(x,y)\,d\sigma(x)F(x)}} -{ \int_{\partial\Omega}{E_{2}(x,y)\widehat{d\sigma(x)}\widehat {F(x)}} \biggr]}$$
(8)

is a hypergenic function vector on $${{\mathbb{R}}_{+}^{n+1}}\backslash {{\partial\Omega}}$$, $$\Psi_{F}{(\infty)}=0$$, and $$\Psi_{F}(y)$$ is HÃ¶lder continuous on $$\Omega^{\pm}\cup\partial\Omega$$.

### Proof

\begin{aligned} \Psi_{F}(y)&= \biggl(\frac{{2^{n-1}}{y_{0}^{n-1}}}{w_{n+1}} \biggl[ \int_{\partial\Omega}{E_{1}(x,y)\,d\sigma(x)f_{1}(x)}- \int_{\partial\Omega}{E_{2}(x,y)\widehat{d\sigma(x)}\widehat {f_{1}(x)}} \biggr], \\ &\quad\ \cdots, {\frac{{2^{n-1}}{y_{0}^{n-1}}}{w_{n+1}}} \biggl[ \int_{\partial \Omega}{E_{1}(x,y)\,d\sigma(x)f_{q}(x)}- \int_{\partial\Omega }{E_{2}(x,y)\widehat{d\sigma(x)} \widehat{f_{q}(x)}} \biggr] \biggr) \\ &= \bigl(\Psi_{f_{1}}(y),\ldots,\Psi_{f_{q}}(y) \bigr). \end{aligned}

It follows from $$F(x)=(f_{1}(x),\ldots,f_{q}(x))\in H_{q}(\beta ,\partial\Omega, \mathit {Cl}_{n+1,0}(\mathbb{R}))$$ that $$f_{i}\in H_{q}(\beta ,\partial\Omega, \mathit {Cl}_{n+1,0}(\mathbb{R}))$$ ($$i=1,\ldots,q$$). From LemmaÂ 2.1, $$\Psi_{f_{i}}(y)$$ ($$i=1,\ldots,q$$) is a hypergenic function on $$R^{n+1}_{+}\setminus\partial\Omega$$. Hence $$\Psi_{F}(y)$$ is a hypergenic function vector on $$R^{n+1}_{+}\setminus\partial\Omega$$. By LemmaÂ 2.2 and RemarkÂ 2.2 $$\Psi_{F}(y)$$ is HÃ¶lder continuous on $$\Omega^{\pm}\cup \partial\Omega$$. By RemarkÂ 2.1 we conclude $$\Psi_{f_{i}}{(\infty)}=0$$ ($$i=1,\ldots,q$$), thus $$\Psi_{F}{(\infty)}=0$$.â€ƒâ–¡

### Theorem 3.2

If $$y\in\partial\Omega_{,} F\in{H_{q}(\beta,\partial\Omega ,\mathit {Cl}_{n+1,0}(\mathbb{R})})$$, then

\begin{aligned}[b] \Phi_{F}(y)&={ \frac {{2^{n-1}}{y_{0}^{n-1}}}{w_{n+1}}} { \biggl[ \int_{\partial\Omega}{E_{1}(x,y)\,d\sigma(x) \bigl(F(x)-F(y) \bigr)}} \\ &\quad{} -{ \int_{\partial\Omega}{E_{2}(x,y)\widehat{d\sigma(x)} \bigl( \widehat{F(x)}-F(y) \bigr)} \biggr]}+ \frac{1}{2}F(y). \end{aligned}
(9)

From LemmaÂ 2.4 we conclude to the following theorem.

### Theorem 3.3

$$\textstyle\begin{cases} \Psi^{+}_{F}(y)=\Phi_{F}(y)+ \frac{1}{2}F(y), \\ \Psi^{-}_{F}(y)=\Phi_{F}(y)- \frac{1}{2}F(y). \end{cases}$$
(10)

### Lemma 3.1

([15])

If $$x,y\in\mathbb{R}^{n+1}$$ ($$n\geq2$$), m (â‰¥0) is an integer, then

$$\biggl\vert \frac{x}{ \vert x \vert ^{m+2}}-\frac{y}{ \vert y \vert ^{m+2}} \biggr\vert \leq \frac {P_{m}(x,y)}{ \vert x \vert ^{m+1} \vert y \vert ^{m+1}} \vert x-y \vert ,$$

where

$$P_{m}(x,y)= \textstyle\begin{cases} \sum_{l=0}^{m} \vert x \vert ^{m-l} \vert y \vert ^{l},& m\neq0;\\ 1,& m=0. \end{cases}$$

### Theorem 3.4

If $$y\in{\partial\Omega}$$, $$F\in{H_{q}(\beta,\partial\Omega, \mathit {Cl}_{n+1,0}(\mathbb{R})})$$, $$Q(y)= \frac{1}{2}F(y)-\Phi_{F}(y)$$, then

$$\bigl\Vert {Q}(y) \bigr\Vert _{\beta}\leq{J_{31} \Vert {F} \Vert _{\beta}}.$$

### Proof

Similar to Ref. [15], we have

$$\bigl\vert d\sigma(x) \bigr\vert \leq{M_{3}}\rho^{n-1}\,d \rho,$$

where $$M_{3}$$ is a positive constant.

From TheoremÂ 3.2, LemmaÂ 2.3 and LemmaÂ 2.4, we get

\begin{aligned} & \bigl\vert {Q(y)} \bigr\vert \\ &\quad= \biggl\vert {{ {\frac{{2^{n-1}}{y_{0}^{n-1}}}{w_{n+1}}}} { \biggl[ \int_{\partial\Omega}{E_{1}(x,y)\,d\sigma(x)F(y)}- \int_{\partial \Omega}{E_{2}(x,y)\widehat{d\sigma(x)}F(y)} \biggr]}} \\ & \quad\quad {} - { {\frac{{2^{n-1}}{y_{0}^{n-1}}}{w_{n+1}}}} { \biggl[ \int_{\partial\Omega}{E_{1}(x,y)\,d\sigma(x)F(x)}- \int_{\partial\Omega }{E_{2}(x,y)\widehat{d\sigma(x)} \widehat{F(x)}} \biggr]} \biggr\vert \\ &\quad\leq\biggl\vert { {\frac{{2^{n-1}}{y_{0}^{n-1}}}{w_{n+1}}}} \biggr\vert \biggl[ \biggl\vert { \int_{\partial\Omega}{E_{1}(x,y)\,d\sigma(x) \bigl(F(y)-F(x) \bigr)}} \biggr\vert + \biggl\vert { \int_{\partial\Omega }{E_{2}(x,y)\widehat{d\sigma(x)} \bigl( \widehat{F(x)}-F(y) \bigr)}} \biggr\vert \biggr] \\ &\quad\leq J_{4}H_{q}(F,\partial\Omega,\beta) \int_{\partial \Omega} \bigl\vert E_{1}(x,y) \bigr\vert \bigl\vert \,d\sigma(x) \bigr\vert \vert {y-x} \vert ^{\beta}+ 2\max _{x\in\partial\Omega} \bigl\vert {F(x)} \bigr\vert \int_{\partial\Omega } \bigl\vert E_{2}(x,y) \bigr\vert \bigl\vert \widehat{d\sigma(x)} \bigr\vert \\ &\quad\leq J_{5}H_{q}(F,\partial\Omega,\beta) \int_{\partial \Omega} {\frac{1}{ \vert {x-y} \vert ^{n}}} \bigl\vert {d\sigma{(x)}} \bigr\vert \vert {y-x} \vert ^{\beta }+2\max_{x\in\partial\Omega} \bigl\vert {F(x)} \bigr\vert \int_{\partial\Omega} {\frac {1}{ \vert {x-y} \vert ^{n-1}}} \bigl\vert {\widehat{d \sigma(x)}} \bigr\vert \\ &\quad\leq J_{6}H_{q}(F,\partial\Omega,\beta )+J_{7}C_{q}(F,\partial\Omega) \\ &\quad\leq{J_{8} \Vert {F} \Vert _{\beta}}. \end{aligned}

So

$$C_{q}(Q,\partial\Omega)\leq{J_{8} \Vert {F} \Vert _{\beta}}.$$
(11)

Next we consider $$H_{q}(Q,\partial\Omega,\beta)$$.

There is a ball with radius 3Î´, centered at $$y^{1}$$ when $$y^{1},y^{2}\in{\partial\Omega}$$ and $$6\delta< d$$, $$\delta= \vert y^{1}-y^{2} \vert$$. Remark that $$\partial\Omega_{1}$$ is located inside the ball and the rest of âˆ‚Î© is $$\partial\Omega_{2}$$.

From equality (9) and (1), we have

\begin{aligned}& \bigl\vert {Q \bigl(y^{1} \bigr)-Q \bigl(y^{2} \bigr)} \bigr\vert \\& \quad= \biggl\vert { \frac{1}{2}F \bigl(y^{1} \bigr)- \Phi_{F} \bigl(y^{1} \bigr)- \biggl(\frac {1}{2}F \bigl(y^{2} \bigr)-\Phi_{F} \bigl(y^{2} \bigr) \biggr)} \biggr\vert \\& \quad= \biggl\vert {{ \frac {{2^{n-1}}{(y^{1}_{0})^{n-1}}}{w_{n+1}}}} \biggl[ \int_{\partial\Omega }{E_{1} \bigl(x,y^{1} \bigr)\,d \sigma(x) \bigl(F \bigl(y^{1} \bigr)-F(x) \bigr)} \\& \quad\quad{} + \int_{\partial\Omega }{E_{2} \bigl(x,y^{1} \bigr) \widehat{d\sigma(x)} \bigl(\widehat{F(x)}-F \bigl(y^{1} \bigr) \bigr)} \biggr] \\& \quad\quad{} - {{ \frac {{2^{n-1}}{(y^{2}_{0})^{n-1}}}{w_{n+1}}}} \biggl[ \int_{\partial\Omega }{E_{1} \bigl(x,y^{2} \bigr)\,d \sigma(x) \bigl(F \bigl(y^{2} \bigr)-F(x) \bigr)} \\& \quad\quad{}+ \int_{\partial\Omega }{E_{2} \bigl(x,y^{2} \bigr) \widehat{d\sigma(x)} \bigl(\widehat{F(x)}-F \bigl(y^{2} \bigr) \bigr)} \biggr] \biggr\vert \\& \quad= \biggl\vert {{ \frac {{2^{n-1}}{(y^{1}_{0})^{n-1}}}{w_{n+1}}}} \biggl[ \int_{\partial\Omega _{1}}{E_{1} \bigl(x,y^{1} \bigr)\,d \sigma(x) \bigl(F \bigl(y^{1} \bigr)-F(x) \bigr)} \\& \quad\quad{}+ \int_{\partial \Omega_{1}}{E_{2} \bigl(x,y^{1} \bigr) \widehat{d\sigma(x)} \bigl(\widehat{F(x)}-F \bigl(y^{1} \bigr) \bigr)} \biggr] \\& \quad\quad{} + {{ \frac {{2^{n-1}}{(y^{1}_{0})^{n-1}}}{w_{n+1}}}} \biggl[ \int_{\partial\Omega _{2}}{E_{1} \bigl(x,y^{1} \bigr)\,d \sigma(x) \bigl(F \bigl(y^{1} \bigr)-F(x) \bigr)} \\& \quad\quad{}+ \int_{\partial \Omega_{2}}{E_{2} \bigl(x,y^{1} \bigr) \widehat{d\sigma(x)} \bigl(\widehat{F(x)}-F \bigl(y^{1} \bigr) \bigr)} \biggr] \\& \quad\quad{} - {{ \frac {{2^{n-1}}{(y^{2}_{0})^{n-1}}}{w_{n+1}}}} \biggl[ \int_{\partial\Omega _{1}}{E_{1} \bigl(x,y^{2} \bigr)\,d \sigma(x) \bigl(F \bigl(y^{2} \bigr)-F(x) \bigr)} \\& \quad\quad{}+ \int_{\partial \Omega_{1}}{E_{2} \bigl(x,y^{2} \bigr) \widehat{d\sigma(x)} \bigl(\widehat{F(x)}-F \bigl(y^{2} \bigr) \bigr)} \biggr] \\& \quad\quad{} - {{ \frac {{2^{n-1}}{(y^{2}_{0})^{n-1}}}{w_{n+1}}}} \biggl[ \int_{\partial\Omega _{2}}{E_{1} \bigl(x,y^{2} \bigr)\,d \sigma(x) \bigl(F \bigl(y^{2} \bigr)-F(x) \bigr)} \\& \quad\quad{}+ \int_{\partial \Omega_{2}}{E_{2} \bigl(x,y^{2} \bigr) \widehat{d\sigma(x)} \bigl(\widehat{F(x)}-F \bigl(y^{2} \bigr) \bigr)} \biggr] \biggr\vert \\& \quad\leq\biggl\vert \frac{{2^{n-1}}{(y^{1}_{0})^{n-1}}}{w_{n+1}} \biggr\vert \biggl\vert { \int_{\partial\Omega_{1}}{E_{1} \bigl(x,y^{1} \bigr)\,d\sigma (x) \bigl(F \bigl(y^{1} \bigr)-F(x) \bigr)}} \\& \quad\quad{}+{ \int_{\partial\Omega _{1}}{E_{2} \bigl(x,y^{1} \bigr) \widehat{d\sigma(x)} \bigl(\widehat{F(x)}-F \bigl(y^{1} \bigr) \bigr)}} \biggr\vert \\& \quad\quad{} + \biggl\vert \frac {{2^{n-1}}{(y^{2}_{0})^{n-1}}}{w_{n+1}} \biggr\vert \biggl\vert { \int_{\partial\Omega_{1}}{E_{1} \bigl(x,y^{2} \bigr)\,d\sigma (x) \bigl(F \bigl(y^{2} \bigr)-F(x) \bigr)}} \\& \quad\quad{}+{ \int_{\partial\Omega _{1}}{E_{2} \bigl(x,y^{2} \bigr) \widehat{d\sigma(x)} \bigl(\widehat{F(x)}-F \bigl(y^{2} \bigr) \bigr)}} \biggr\vert \\& \quad\quad{} + \biggl\vert {{ \frac {{2^{n-1}}{(y^{1}_{0})^{n-1}}}{w_{n+1}}}} \biggl[ \int_{\partial\Omega _{2}}{E_{1} \bigl(x,y^{1} \bigr)\,d \sigma(x) \bigl(F \bigl(y^{1} \bigr)-F(x) \bigr)} \\& \quad\quad{}+ \int_{\partial \Omega_{2}}{E_{2} \bigl(x,y^{1} \bigr) \widehat{d\sigma(x)} \bigl(\widehat{F(x)}-F \bigl(y^{1} \bigr) \bigr)} \biggr] \\& \quad\quad{}- {{ \frac{{2^{n-1}}{(y^{2}_{0})^{n-1}}}{w_{n+1}}}} \biggl[ \int_{\partial\Omega_{2}}{E_{1} \bigl(x,y^{2} \bigr)\,d\sigma (x) \bigl(F \bigl(y^{2} \bigr)-F(x) \bigr)} \\& \quad\quad{}+ \int_{\partial\Omega _{2}}{E_{2} \bigl(x,y^{2} \bigr) \widehat{d\sigma(x)} \bigl(\widehat{F(x)}-F \bigl(y^{2} \bigr) \bigr)} \biggr] \biggr\vert \\& \quad=I_{1}+I_{2}+I_{3}; \\& I_{1} \leq{J_{9} \biggl[ \int_{\partial\Omega _{1}} \bigl\vert {E_{1} \bigl(x,y^{1} \bigr)} \bigr\vert \bigl\vert {d\sigma(x)} \bigr\vert \bigl\vert {F \bigl(y^{1} \bigr)-F(x)} \bigr\vert + \int_{\partial \Omega_{1}} \bigl\vert {E_{2} \bigl(x,y^{1} \bigr)} \bigr\vert \bigl\vert {\widehat{d\sigma(x)}} \bigr\vert \bigl\vert {\widehat{F(x)}-F \bigl(y^{1} \bigr)} \bigr\vert \biggr]} \\& \hphantom{I_{1}}\leq{J_{10}} \int^{3\delta}_{0} {\frac {1}{ \vert {x-y^{1}} \vert ^{n} \vert {x-\widehat{y^{1}}} \vert ^{n-1}}}\rho ^{n-1}H_{q}(F,\partial\Omega,\beta) \bigl\vert {y^{1}-x} \bigr\vert ^{\beta}{d\rho} \\& \hphantom{I_{1}}\quad{} +2{J_{10}} \int^{3\delta}_{0} {\frac {1}{ \vert {x-y^{1}} \vert ^{n-1} \vert {x-\widehat{y^{1}}} \vert ^{n}}}\rho ^{n-1}C_{q}(F,\partial\Omega){d\rho} \\& \hphantom{I_{1}} \leq{J_{11}H_{q}(F,\partial\Omega,\beta) \int^{3\delta}_{0} {\frac{1}{ \vert {x-y^{1}} \vert ^{n-\beta}}} \rho^{n-1}\,d\rho}+{J_{12}C_{q}(F,\partial\Omega) \int^{3\delta}_{0} {\frac {1}{ \vert {x-y^{1}} \vert ^{n-1}}} \rho^{n-1}\,d\rho} \\& \hphantom{I_{1}}\leq{J_{11}H_{q}(F,\partial\Omega,\beta) \int^{3\delta}_{0}\rho^{\beta-1}\,d \rho}+{J_{12}C_{q}(F,\partial\Omega) \int^{3\delta}_{0}\,d\rho} \\& \hphantom{I_{1}} \leq{J_{13} \bigl(H_{q}(F,\partial\Omega, \beta)+C_{q}(F,\partial\Omega) \bigr) \bigl\vert {y^{1}-y^{2}} \bigr\vert ^{\beta}} \\& \hphantom{I_{1}} \leq{J_{13} \Vert {F} \Vert _{\beta } \bigl\vert {y^{1}-y^{2}} \bigr\vert ^{\beta}}, \end{aligned}

that is,

$$I_{1}\leq{J_{13} \Vert {F} \Vert _{\beta} \bigl\vert {y^{1}-y^{2}} \bigr\vert ^{\beta}}.$$
(12)

In a similar way, we have

\begin{aligned}& I_{2}\leq{J_{14} \Vert {F} \Vert _{\beta} \bigl\vert {y^{1}-y^{2}} \bigr\vert ^{\beta}} , \\& I_{3} = \biggl\vert {{ \frac{{2^{n-1}}{(y^{1}_{0})^{n-1}}}{w_{n+1}}}} \int_{\partial\Omega_{2}} \bigl[E_{1} \bigl(x,y^{1} \bigr)-E_{1} \bigl(x,y^{2} \bigr) \bigr]\,d\sigma(x) \bigl(F \bigl(y^{1} \bigr)-F(x) \bigr) \\& \hphantom{I_{3}} \quad{} + {{ \frac {{2^{n-1}}{(y^{1}_{0})^{n-1}}}{w_{n+1}}}} \int_{\partial\Omega _{2}}E_{1} \bigl(x,y^{2} \bigr)\,d \sigma(x) \bigl(F \bigl(y^{1} \bigr)-F(x) \bigr) \\& \hphantom{I_{3}} \quad{}+ {{ \frac {{2^{n-1}}{(y^{1}_{0})^{n-1}}}{w_{n+1}}}} \int_{\partial\Omega _{2}} \bigl[E_{2} \bigl(x,y^{1} \bigr)-E_{2} \bigl(x,y^{2} \bigr) \bigr]\widehat{d\sigma(x)} \bigl(\widehat{F(x)}-F \bigl(y^{1} \bigr) \bigr) \\& \hphantom{I_{3}} \quad{}+ {{ \frac {{2^{n-1}}{(y^{1}_{0})^{n-1}}}{w_{n+1}}}} \int_{\partial\Omega _{2}}E_{2} \bigl(x,y^{2} \bigr) \widehat{d \sigma(x)} \bigl(\widehat{F(x)}-F \bigl(y^{1} \bigr) \bigr) \\& \hphantom{I_{3}} \quad{}- {{ \frac {{2^{n-1}}{(y^{2}_{0})^{n-1}}}{w_{n+1}}}} \int_{\partial\Omega _{2}}E_{1} \bigl(x,y^{2} \bigr)\,d \sigma(x) \bigl(F \bigl(y^{2} \bigr)-F(x) \bigr) \\& \hphantom{I_{3}} \quad{}- {{ \frac {{2^{n-1}}{(y^{2}_{0})^{n-1}}}{w_{n+1}}}} \int_{\partial\Omega _{2}}E_{2} \bigl(x,y^{2} \bigr) \widehat{d \sigma(x)} \bigl(\widehat{F(x)}-F \bigl(y^{2} \bigr) \bigr) \biggr\vert \\& \hphantom{I_{3}} \leq{J_{15}} \biggl\vert \int_{\partial\Omega _{2}} \bigl[E_{1} \bigl(x,y^{1} \bigr)-E_{1} \bigl(x,y^{2} \bigr) \bigr]\,d\sigma(x) \bigl(F \bigl(y^{1} \bigr)-F(x) \bigr) \biggr\vert \\& \hphantom{I_{3}} \quad{}+{J_{15}} \biggl\vert \int_{\partial\Omega _{2}} \bigl[E_{2} \bigl(x,y^{1} \bigr)-E_{2} \bigl(x,y^{2} \bigr) \bigr]\widehat{d\sigma(x)} \bigl(\widehat{F(x)}-F \bigl(y^{1} \bigr) \bigr) \biggr\vert \\& \hphantom{I_{3}} \quad{}+ \biggl\vert {{ \frac {{2^{n-1}}{(y^{1}_{0})^{n-1}}}{w_{n+1}}}} \int_{\partial\Omega _{2}}E_{1} \bigl(x,y^{2} \bigr)\,d \sigma(x) \bigl(F \bigl(y^{1} \bigr)-F(x) \bigr) \\& \hphantom{I_{3}} \quad{}-{ {\frac {{2^{n-1}}{(y^{2}_{0})^{n-1}}}{w_{n+1}}}} \int_{\partial\Omega _{2}}E_{1} \bigl(x,y^{2} \bigr)\,d \sigma(x) \bigl(F \bigl(y^{2} \bigr)-F(x) \bigr) \biggr\vert \\& \hphantom{I_{3}} \quad{}+ \biggl\vert {{ \frac {{2^{n-1}}{(y^{1}_{0})^{n-1}}}{w_{n+1}}}} \int_{\partial\Omega _{2}}E_{2} \bigl(x,y^{2} \bigr) \widehat{d \sigma(x)} \bigl(\widehat{F(x)}-F \bigl(y^{1} \bigr) \bigr) \\& \hphantom{I_{3}} \quad{}-{ {\frac {{2^{n-1}}{(y^{2}_{0})^{n-1}}}{w_{n+1}}}} \int_{\partial\Omega _{2}}E_{2} \bigl(x,y^{2} \bigr) \widehat{d \sigma(x)} \bigl(\widehat{F(x)}-F \bigl(y^{2} \bigr) \bigr) \biggr\vert \\& \hphantom{I_{3}} ={J_{15}}(I_{4}+I_{5})+I_{6}+I_{7}, \end{aligned}
(13)

that is,

$$I_{3}\leq{J_{15}}(I_{4}+I_{5})+I_{6}+I_{7} .$$
(14)

Because $$x\in{\partial\Omega_{2}\setminus\lambda_{3\delta}}$$, and $$y^{1},y^{2}\in\partial\Omega_{1}$$, $$\vert \frac {x-y^{2}}{x-y^{1}} \vert ^{l+1}$$ and $$\vert \frac {x-y^{1}}{x-y^{2}} \vert ^{l+1}$$ ($$l=0,1,\ldots,n$$) are continuous on $$\partial\Omega_{2}$$, there is a positive constant $$J_{16}$$, such that

$${ \biggl\vert \frac{x-y^{2}}{x-y^{1}} \biggr\vert ^{l+1}}\leq {J_{16}}, \quad\quad \biggl\vert \frac{x-y^{1}}{x-y^{2}} \biggr\vert ^{l+1}\leq{J_{16}} \quad(l=0,1,\ldots,n).$$
(15)

From inequality (1), the Hile lemma and inequality (15), we get

\begin{aligned} I_{4}&= \biggl\vert \int_{\partial\Omega _{2}} \bigl[E_{1} \bigl(x,y^{1} \bigr)-E_{1} \bigl(x,y^{2} \bigr) \bigr]\,d\sigma(x) \bigl(F \bigl(y^{1} \bigr)-F(x) \bigr) \biggr\vert \\ &= \biggl\vert \int_{\partial\Omega_{2}} \biggl( {\frac {x-y^{1}}{ \vert {x-y^{1}} \vert ^{n+1} \vert {x-\widehat {y^{1}}} \vert ^{n-1}}-\frac {x-y^{2}}{ \vert {x-y^{2}} \vert ^{n+1} \vert {x-\widehat {y^{2}}} \vert ^{n-1}}} \biggr)\,d\sigma(x) \bigl(F \bigl(y^{1} \bigr)-F(x) \bigr) \biggr\vert \\ &\leq{J_{17}} \int_{\partial\Omega_{2}} \biggl\vert \frac {x-y^{1}}{ \vert {x-y^{1}} \vert ^{n+1} \vert {x-\widehat {y^{1}}} \vert ^{n-1}}-\frac {x-y^{2}}{ \vert {x-y^{2}} \vert ^{n+1} \vert {x-\widehat {y^{2}}} \vert ^{n-1}} \biggr\vert \bigl\vert {d\sigma(x)} \bigr\vert \bigl\vert {F \bigl(y^{1} \bigr)-F(x)} \bigr\vert \\ &\leq{J_{17}} \int_{\partial\Omega_{2}} {\frac{1}{ \vert {x-\widehat {y^{1}}} \vert ^{n-1}} \biggl\vert \frac{x-y^{1}}{ \vert {x-y^{1}} \vert ^{n+1}}- \frac {x-y^{2}}{ \vert {x-y^{2}} \vert ^{n+1}} \biggr\vert } \bigl\vert {d\sigma(x)} \bigr\vert \bigl\vert {F \bigl(y^{1} \bigr)-F(x)} \bigr\vert \\ &\quad{} +{J_{17}} \int_{\partial\Omega_{2}} {\frac {1}{ \vert {x-y^{2}} \vert ^{n+1}} \biggl\vert \frac{x-y^{2}}{ \vert {x-\widehat {y^{1}}} \vert ^{n-1}}- \frac{x-y^{2}}{ \vert {x-\widehat{y^{2}}} \vert ^{n-1}} \biggr\vert } \bigl\vert {d\sigma(x)} \bigr\vert \bigl\vert {F \bigl(y^{1} \bigr)-F(x)} \bigr\vert \\ &\leq \int_{\partial\Omega_{2}} \Biggl(J_{18}\sum _{l=0}^{n-1} \biggl\vert \frac{x-y^{2}}{x-y^{1}} \biggr\vert ^{l+1}\frac{ \vert {y^{1}-y^{2}} \vert }{ \vert {x-y^{2}} \vert ^{n+1}} +J_{19}\frac{ \vert {y^{1}-y^{2}} \vert }{ \vert {x-y^{2}} \vert ^{n}} \Biggr) \bigl\vert {d\sigma(x)} \bigr\vert \bigl\vert {F \bigl(y^{1} \bigr)-F(x)} \bigr\vert \\ &\leq{J_{20}}H_{q}(F,\partial\Omega,\beta) \int_{\partial\Omega _{2}} \biggl( {\frac{1}{ \vert {x-y^{2}} \vert ^{n+1}}+\frac {1}{ \vert {x-y^{2}} \vert ^{n}}} \biggr) \bigl\vert {d\sigma(x)} \bigr\vert \bigl\vert {y^{1}-x} \bigr\vert ^{\beta } \bigl\vert {y^{1}-y^{2}} \bigr\vert \\ &\leq H_{q}(Q,\partial\Omega,\beta) \biggl(J_{21} \int^{L}_{3\delta}\rho^{\beta-2}\,d \rho+J_{22} \int^{L}_{3\delta}\rho^{\beta-1}\,d\rho\biggr) \bigl\vert {y^{1}-y^{2}} \bigr\vert \\ &\leq{J_{23}}H_{q}(F,\partial\Omega,\beta) \bigl\vert {y^{1}-y^{2}} \bigr\vert ^{\beta} \\ &\leq{J_{24}} \Vert {F} \Vert _{\beta} \bigl\vert {y^{1}-y^{2}} \bigr\vert ^{\beta}, \end{aligned}

that is,

$$I_{4}\leq{J_{24}} \Vert {F} \Vert _{\beta } \bigl\vert {y^{1}-y^{2}} \bigr\vert ^{\beta}.$$
(16)

In a similar way, we have

\begin{aligned}& I_{5}\leq{J_{25}} \Vert {F} \Vert _{\beta } \bigl\vert {y^{1}-y^{2}} \bigr\vert ^{\beta}, \\& I_{6}\leq\biggl\vert {\frac {2^{n-1}[(y^{2}_{0})^{n-1}-(y^{1}_{0})^{n-1}]}{w_{n+1}}} \int_{\partial \Omega_{2}} {\frac{x-y^{2}}{ \vert {x-y^{2}} \vert ^{n+1} \vert {x-\widehat {y^{2}}} \vert ^{n-1}}}\,d\sigma(x)F(x) \biggr\vert \\& \hphantom{I_{6}}\quad{} + \biggl\vert {\frac {{2^{n-1}}{(y^{1}_{0})^{n-1}}}{w_{n+1}}} \int_{\partial \Omega_{2}} {\frac{x-y^{2}}{ \vert {x-y^{2}} \vert ^{n+1} \vert {x-\widehat {y^{2}}} \vert ^{n-1}}}\,d\sigma(x) \bigl(F \bigl(y^{1} \bigr)-F \bigl(y^{2} \bigr) \bigr) \biggr\vert \\& \hphantom{I_{6}}\quad{}+ \biggl\vert {\frac {2^{n-1}[(y^{1}_{0})^{n-1}-(y^{2}_{0})^{n-1}]}{w_{n+1}}} \int_{\partial \Omega_{2}} {\frac{x-y^{2}}{ \vert {x-y^{2}} \vert ^{n+1} \vert {x-\widehat {y^{2}}} \vert ^{n-1}}}\,d\sigma(x)F \bigl(y^{2} \bigr) \biggr\vert . \end{aligned}
(17)

Because $$\lim_{\delta\rightarrow0} {\frac {x-y^{2}}{ \vert {x-y^{2}} \vert ^{n+1} \vert {x-\widehat {y^{2}}} \vert ^{n-1}}}$$ exists, there is a constant $$\delta_{2}>0$$, when $$0<\delta<\delta_{2}$$, such that

$$\biggl\vert \int_{\partial\Omega_{2}}\frac {(x-y^{2})}{ \vert {x-y^{2}} \vert ^{n+1} \vert {x-\widehat {y^{2}}} \vert ^{n-1}}\,d\sigma(x) \biggr\vert \leq{J_{26}}.$$

Hence

\begin{aligned} I_{6}&\leq{J_{27}} \bigl[2C_{q}(F,\partial\Omega ) \bigl\vert \bigl(y^{1}_{0} \bigr)^{n-1}- \bigl(y^{2}_{0} \bigr)^{n-1} \bigr\vert + \bigl\vert {F \bigl(y^{1} \bigr)-F \bigl(y^{2} \bigr)} \bigr\vert \bigr] \\ &\leq{J_{28}} \bigl[C_{q}(F,\partial\Omega)+H_{q}(F, \partial\Omega,\beta) \bigr] \bigl\vert {y^{1}-y^{2}} \bigr\vert ^{\beta} \\ &={J_{28}} \Vert {F} \Vert _{\beta} \bigl\vert {y^{1}-y^{2}} \bigr\vert ^{\beta}, \end{aligned}

that is,

$$I_{6}\leq{J_{28}} \Vert {F} \Vert _{\beta } \bigl\vert {y^{1}-y^{2}} \bigr\vert ^{\beta}.$$
(18)

In a similar way, we have

$$I_{7}\leq{J_{29}} \Vert {F} \Vert _{\beta } \bigl\vert {y^{1}-y^{2}} \bigr\vert ^{\beta}.$$
(19)

From inequalities (14), (16), (17), (18) and (19), we have

$$\vert I_{3} \vert \leq\bigl[J_{15}(J_{24}+J_{25})+J_{28}+J_{29} \bigr] \Vert {F} \Vert _{\beta} \bigl\vert y^{1}-y^{2} \bigr\vert ^{\beta } .$$
(20)

From inequalities (12), (13) and (20), we have

$${\frac{ \vert {Q(y^{1})-Q(y^{2})} \vert }{ \vert {y^{1}-y^{2}} \vert ^{\beta}}}\leq\bigl[J_{13}+J_{14}+J_{15}(J_{24}+J_{25})+J_{28}+J_{29} \bigr] \Vert {F} \Vert _{\beta} ={J_{30}} \Vert {F} \Vert _{\beta}.$$

So

$$H_{q} \bigl(Q(y),\partial\Omega,\beta\bigr) \leq{J_{30}} \Vert {F} \Vert _{\beta}.$$
(21)

From inequalities (11) and (21), we have $$\Vert {Q(y)} \Vert _{\beta}\leq(J_{8}+J_{30}) \Vert {F} \Vert _{\beta}=J_{31} \Vert {F} \Vert _{\beta}$$.â€ƒâ–¡

### Remark 3.1

If $$y\in\partial\Omega$$, $$F\in{H_{q}(\beta,\partial\Omega ,\mathit {Cl}_{n+1,0}(\mathbb{R})})$$, then

$$\bigl\Vert \Phi_{F}(y) \bigr\Vert _{\beta} \leq{J_{32}} \Vert {F} \Vert _{\beta}.$$

### Remark 3.2

If $$y\in\partial\Omega$$, $$F\in{H_{q}(\beta,\partial\Omega ,\mathit {Cl}_{n+1,0}(\mathbb{R})})$$, then

$$\textstyle\begin{cases} \Vert {\Psi^{+}_{F}(y)} \Vert _{\beta}\leq{J_{33}} \Vert {F} \Vert _{\beta}, \\ \Vert {\Psi^{-}_{F}(y)} \Vert _{\beta}\leq{J_{33}} \Vert {F} \Vert _{\beta}. \end{cases}$$
(22)

## 4 The existence of the solution to the nonlinear boundary value problem for the hypergenic function vector

Let $$A(y),B(y),G(y)\in{H_{q}(\beta,\partial\Omega, \mathit {Cl}_{n+1,0}(\mathbb {R})})$$ be HÃ¶lder continuous function vectors on âˆ‚Î©, we find a function vector $$\Psi_{F}^{*}(y)$$, which is hypergenic on $$\Omega^{+} \cup\Omega^{-}$$, and continuous on $$\Omega^{+}\cup \partial\Omega$$ and $$\Omega^{-}\cup\partial\Omega$$, satisfying $$\Psi_{F}^{*}(\infty)=0$$, and the nonlinear boundary value condition:

$$A(y)\otimes{\Psi^{*+}_{F}}(y)+B(y)\otimes{\Psi ^{*-}_{F}}(y)=G(y)\otimes P \bigl({\Psi^{*+}_{F}}(y),{ \Psi^{*-}_{F}}(y) \bigr),$$
(23)

where $$P(\Psi^{*+}_{F}(y),\Psi^{*-}_{F}(y))$$ is a HÃ¶lder continuous function vector on âˆ‚Î© which is related to $$\Psi^{*+}_{F}(y)$$, $$\Psi^{*-}_{F}(y)$$.

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.

By TheoremÂ 3.1, $$\Psi_{F}(y)$$ is hypergenic on $$\Omega^{+} \cup \Omega^{-}$$, and $$\Psi_{F}(y)$$ is continuous on $$\Omega^{+}\cup \partial\Omega$$ and $$\Omega^{-}\cup\partial\Omega$$, and $$\Psi_{ F}(\infty)=0$$. If $$P(\Psi^{+}_{F}(y),\Psi^{-}_{F}(y))$$ satisfies equality (23) under certain conditions, then $$\Psi_{F}(y)$$ is a solution to the nonlinear boundary value problem SR.

Putting (10) into (23), we have

$$A(y)\otimes\biggl(\Phi_{F}(y)+\frac{1}{2}F(y) \biggr)+B(y) \otimes\biggl(\Phi_{F}(y)-\frac{1}{2}F(y) \biggr)=G(y)\otimes P \bigl(\Psi^{+}_{F}(y),\Psi^{-}_{F}(y) \bigr).$$
(24)

Let

\begin{aligned}[b] NF(y)& = \bigl(A(y)+B(y) \bigr)\otimes\biggl(- \frac{1}{2}F(y)+\Phi_{F}(y) \biggr)+ \bigl(1+A(y) \bigr)\otimes F(y) \\ &\quad{} -G(y)\otimes P \bigl(\Psi^{+}_{F}(y), \Psi^{-}_{F}(y) \bigr), \end{aligned}
(25)

and equality (23) is transformed into the following singular integral equation:

$$NF=F .$$
(26)

### Theorem 4.1

If $$A(y),B(y),G(y)\in{H_{q}(\beta,\partial\Omega, \mathit {Cl}_{n+1,0}(\mathbb {R})})$$, for any $$y^{1},y^{2}\in\partial\Omega$$, $$P(\Psi^{+}_{F}(y), \Psi^{-}_{F}(y))$$ satisfies

$$\begin{gathered}[b] \bigl\vert {P} \bigl(\Psi^{+}_{F} \bigl(y^{1} \bigr),\Psi^{-}_{F} \bigl(y^{1} \bigr) \bigr)-P \bigl(\Psi^{+}_{F} \bigl(y^{2} \bigr),\Psi^{-}_{F} \bigl(y^{2} \bigr) \bigr) \bigr\vert \\ \quad\leq J_{34} \bigl\vert \Psi^{+}_{F} \bigl(y^{1} \bigr)-\Psi^{+}_{F} \bigl(y^{2} \bigr) \bigr\vert +J_{35} \bigl\vert \Psi^{-}_{F} \bigl(y^{1} \bigr)- \Psi^{-}_{F} \bigl(y^{2} \bigr) \bigr\vert , \end{gathered}$$
(27)

where $$J_{34}$$ and $$J_{35}$$ are positive constants independent of $$y^{i}$$ ($$i=1,2$$) andÂ F. If $$P(0,0)=0$$, $$0<\gamma=J_{36}( \Vert {A+B} \Vert _{\beta}+ \Vert 1+A \Vert _{\beta})<1$$, $$\Vert {G(y)} \Vert _{\beta}<\delta$$, when $$0<\delta\leq{\frac{1-\gamma }{J_{3}J_{41}}}$$, Problem SR has at least one solution and the integral expression of the solution is (8).

### Proof

Let $$T=\{F\mid { \Vert F \Vert }_{\beta}\leq M_{4}$$ and F is uniformly HÃ¶lder continuous on âˆ‚Î©, that is, to say, there is a positive constant $${M_{2}}$$, for any $${x_{1},x_{2}\in\partial\Omega}$$, $${F}\in{H_{q}(\beta,\partial\Omega, \mathit {Cl}_{n+1,0}(\mathbb{R})})$$, we have $$\vert F(x_{1})-F(x_{2}) \vert \leq{M_{2} \vert x_{1}-x_{2} \vert ^{\beta}}\}$$. Obviously T is a convex subset of the continuous function vector space $$C_{q}(\partial\Omega)$$.

1. (1)

We prove that N maps the set T to itself.

From inequality (7), TheoremÂ 3.1 and RemarkÂ 3.2, it follows that

\begin{aligned}& \Vert NF \Vert \\& \quad\leq J_{3} \bigl\Vert A(y)+B(y) \bigr\Vert _{\beta} \biggl\Vert {-\frac {1}{2}}F(y)+\Phi_{F}(y) \biggr\Vert _{\beta}+J_{3} \bigl\Vert 1+A(y) \bigr\Vert _{\beta} \Vert F \Vert _{\beta}+J_{3} \Vert G \Vert _{\beta} \Vert P \Vert _{\beta} \\& \quad\leq J_{3} \bigl\Vert A(y)+B(y) \bigr\Vert _{\beta}J_{31} \Vert F \Vert _{\beta}+J_{3} \bigl\Vert 1+A(y) \bigr\Vert _{\beta} \Vert F \Vert _{\beta}+J_{3} \Vert G \Vert _{\beta} \Vert P \Vert _{\beta} \\& \quad\leq J_{36} \bigl( \Vert {A+B} \Vert _{\beta}+ \Vert 1+A \Vert _{\beta} \bigr) \Vert F \Vert _{\beta}+J_{3} \Vert G \Vert _{\beta} \Vert P \Vert _{\beta} \\& \quad\leq\gamma \Vert F \Vert _{\beta}+J_{3}\delta \Vert P \Vert _{\beta}. \end{aligned}

By inequality (27) and RemarkÂ 3.2, we have

\begin{aligned}& \bigl\vert P \bigl(\Psi^{+}_{F}(y),\Psi^{-}_{F}(y) \bigr) \bigr\vert \\& \quad= \bigl\vert P \bigl(\Psi^{+}_{F}(y), \Psi^{-}_{F}(y) \bigr)-P(0,0) \bigr\vert \\& \quad\leq J_{34} \bigl\vert \Psi^{+}_{F}(y) \bigr\vert +J_{35} \bigl\vert \Psi^{-}_{F}(y) \bigr\vert \\& \quad\leq J_{34}J_{33} \Vert F \Vert _{\beta}+J_{35}J_{33} \Vert F \Vert _{\beta} \\& \quad=J_{37} \Vert F \Vert _{\beta}. \end{aligned}

So

$$C_{q}(P,\partial\Omega,\beta)=\max_{y\in{\partial\Omega}} \vert P \vert \leq J_{37} \Vert F \Vert _{\beta}.$$
(28)

By inequality (27) and RemarkÂ 3.2, we have

\begin{aligned}& \bigl\vert {P} \bigl(\Psi^{+}_{F} \bigl(y^{1} \bigr),\Psi^{-}_{F} \bigl(y^{1} \bigr) \bigr)-P \bigl(\Psi^{+}_{F} \bigl(y^{2} \bigr),\Psi^{-}_{F} \bigl(y^{2} \bigr) \bigr) \bigr\vert \\& \quad\leq J_{34} \bigl\vert \Psi^{+}_{F} \bigl(y^{1} \bigr)-\Psi^{+}_{F} \bigl(y^{2} \bigr) \bigr\vert +J_{35} \bigl\vert \Psi^{-}_{F} \bigl(y^{1} \bigr)- \Psi^{-}_{F} \bigl(y^{2} \bigr) \bigr\vert \\& \quad\leq J_{34}H_{q} \bigl(\Psi^{+}_{F}(y), \partial\Omega,\beta\bigr) \bigl\vert y^{1}-y^{2} \bigr\vert ^{\beta}+J_{35}H_{q} \bigl( \Psi^{+}_{F}(y),\partial\Omega,\beta\bigr) \bigl\vert y^{1}-y^{2} \bigr\vert ^{\beta} \\& \quad\leq\bigl[J_{34} \bigl\Vert \Psi^{+}_{F}(y) \bigr\Vert _{\beta }+J_{35} \bigl\Vert \Psi^{+}_{F}(y) \bigr\Vert _{\beta} \bigr] \bigl\vert y^{1}-y^{2} \bigr\vert ^{\beta} \\& \quad\leq\bigl(J_{38} \Vert F \Vert _{\beta}+J_{39} \Vert F \Vert _{\beta} \bigr) \bigl\vert y^{1}-y^{2} \bigr\vert ^{\beta} \\& \quad\leq J_{40} \Vert F \Vert _{\beta} \bigl\vert y^{1}-y^{2} \bigr\vert ^{\beta}, \end{aligned}
(29)

then

$$H_{q}(P,\partial\Omega,\beta)\leq J_{40} \Vert F \Vert _{\beta}.$$

So

\begin{aligned}[b] \Vert P \Vert _{\beta}&=C_{q}(P, \partial\Omega,\beta)+H_{q}(P,\partial\Omega,\beta) \\ &\leq J_{37} \Vert F \Vert _{\beta}+J_{40} \Vert F \Vert _{\beta} \\ &\leq J_{41} \Vert F \Vert _{\beta}. \end{aligned}
(30)

As $$\gamma=J_{35}( \Vert {A+B} \Vert _{\beta}+ \Vert 1+A \Vert _{\beta})<1$$,

\begin{aligned}[b] \Vert NF \Vert _{\beta}&\leq\gamma \Vert F \Vert _{\beta}+J_{3}\delta \Vert P \Vert _{\beta} \\ &\leq\gamma M_{4}+J_{3} {\frac{1-\gamma}{J_{3}J_{41}}}J_{41}M_{4}=M_{4}. \end{aligned}
(31)

If F is uniformly HÃ¶lder continuous on âˆ‚Î©, then $$\Phi_{F}(y)$$, $$\Psi^{+}_{F}$$, $$\Psi^{-}_{F}$$ are uniformly HÃ¶lder continuous on âˆ‚Î©. So NF is uniformly HÃ¶lder continuous on âˆ‚Î©.

Hence N maps the set T to itself.

1. (2)

We prove that N is a continuous mapping.

Any $$F_{n}\in{T}$$, $$\{F_{n}\}$$ uniformly converges to F on âˆ‚Î©. As for $$\varepsilon>0$$, when n is fully large and $$\vert {F_{n}-F} \vert$$ is sufficiently small. There is a ball with radius 3Î´, centered at y when $$6\delta \langle d,\delta\rangle0$$, and remark that $$\partial\Omega_{1}$$ is located inside the ball and the rest of âˆ‚Î© is $$\partial\Omega_{2}$$

By inequality (27), TheoremÂ 3.3, we have

\begin{aligned}& \bigl\vert P \bigl(\Psi^{+}_{F_{n}}(y), \Psi^{-}_{F_{n}}(y) \bigr)-P \bigl(\Psi^{+}_{F}(y), \Psi^{-}_{F}(y) \bigr) \bigr\vert \\& \quad\leq{J_{34}} \bigl\vert \Psi^{+}_{F_{n}}(y)- \Psi^{+}_{F}(y) \bigr\vert +{J_{35}} \bigl\vert \Psi^{-}_{F_{n}}(y)-\Psi^{-}_{F}(y) \bigr\vert \\& \quad=J_{34} \biggl\vert \Phi_{F_{n}}(y)- \Phi_{F}(y)+ \frac {1}{2} \bigl(F_{n}(y)-F(y) \bigr) \biggr\vert +J_{35} \biggl\vert \Phi_{F_{n}}(y)- \Phi_{F}(y)+ \frac {1}{2} \bigl(F(y)-F_{n}(y) \bigr) \biggr\vert \\& \quad\leq(J_{34}+J_{35}) \bigl\vert \Phi_{F_{n}}(y)-\Phi_{F}(y) \bigr\vert +(J_{34}+J_{35}) \biggl\vert \frac{1}{2} \bigl(F_{n}(y)-F(y) \bigr) \biggr\vert \\& \quad\leq(J_{34}+J_{35}) \biggl\vert { \frac {{2^{n-1}}{y_{0}^{n-1}}}{w_{n+1}}} \int_{\partial\Omega }E_{1}(x,y)\,d\sigma(x) \bigl[ \bigl(F_{n}(x)-F_{n}(y) \bigr)+ \bigl(F(y)-F(x) \bigr) \bigr] \biggr\vert \\& \quad\quad{} +(J_{34}+J_{35}) \biggl\vert \frac {{2^{n-1}}{y_{0}^{n-1}}}{w_{n+1}} \int_{\partial\Omega }E_{2}(x,y)\widehat{d\sigma(x)} \bigl[ \bigl( \widehat{F(x)}-\widehat{F(y)} \bigr)+ \bigl(\widehat{F_{n}(y)}- \widehat{F_{n}(x)} \bigr) \bigr] \biggr\vert \\& \quad\quad{} +(J_{34}+J_{35}) \vert F_{n}(y)-F(y) \vert \\& \quad\leq(J_{34}+J_{35}) \bigl(I_{8}+I_{9}+ \Vert F_{n}-F \Vert _{\beta} \bigr), \end{aligned}

that is,

\begin{aligned}& \bigl\vert P \bigl(\Psi^{+}_{F_{n}}(y), \Psi^{-}_{F_{n}}(y) \bigr)-P \bigl(\Psi^{+}_{F}(y), \Psi^{-}_{F}(y) \bigr) \bigr\vert \leq(J_{34}+J_{35}) \bigl(I_{8}+I_{9}+ \Vert F_{n}-F \Vert _{\beta} \bigr), \\& I_{8} \leq\biggl\vert \frac{{2^{n-1}}{y_{0}^{n-1}}}{w_{n+1}} \int_{\partial\Omega}E_{1}(x,y)\,d\sigma(x) \bigl[ \bigl(F_{n}(x)-F_{n}(y) \bigr)+ \bigl(F(y)-F(x) \bigr) \bigr] \biggr\vert \\& \hphantom{I_{8}}\leq\biggl\vert \frac{{2^{n-1}}{y_{0}^{n-1}}}{w_{n+1}} \int_{\partial \Omega_{1}}E_{1}(x,y)\,d\sigma(x) \bigl[ \bigl(F_{n}(x)-F_{n}(y) \bigr)+ \bigl(F(y)-F(x) \bigr) \bigr] \biggr\vert \\& \hphantom{I_{8}}\quad{} + \biggl\vert \frac{{2^{n-1}}{y_{0}^{n-1}}}{w_{n+1}} \int_{\partial\Omega _{2}}E_{1}(x,y)\,d\sigma(x) \bigl[ \bigl(F_{n}(x)-F_{n}(y) \bigr)+ \bigl(F(y)-F(x) \bigr) \bigr] \biggr\vert \\& \hphantom{I_{8}}=I_{10}+I_{11}; \\& I_{10} \leq{J_{42}} \int_{\partial\Omega _{1}} \bigl\vert {E_{1}(x,y)} \bigr\vert \bigl\vert {d\sigma(x)} \bigr\vert \vert {x-y} \vert ^{\beta} \\& \hphantom{I_{10}}\leq{J_{43}} \int_{\partial\Omega_{1}} \frac {1}{ \vert {x-y} \vert ^{n}} \vert {x-y} \vert ^{\beta} \bigl\vert {d\sigma(x)} \bigr\vert \\& \hphantom{I_{10}}\leq{J_{44}} \int^{3\delta}_{0}\rho^{\beta-1}\,d\rho \\& \hphantom{I_{10}}=J_{45}\delta^{\beta}, \end{aligned}
(32)

that is,

\begin{aligned}& I_{10}\leq{J_{45}}\delta^{\beta}, \\& I_{11} = \biggl\vert \frac{{2^{n-1}}{y_{0}^{n-1}}}{w_{n+1}} \int_{\partial \Omega_{2}}E_{1}(x,y)\,d\sigma(x) \bigl[ \bigl(F_{n}(x)-F(x) \bigr)- \bigl(F_{n}(y)-F(y) \bigr) \bigr] \biggr\vert \\& \hphantom{I_{11}} \leq{J_{46}} \biggl\vert \int_{\partial\Omega_{2}}E_{1}(x,y)\,d\sigma(x) \biggr\vert \bigl[ \bigl\vert F_{n}(x)-F(x) \bigr\vert + \bigl\vert F_{n}(y)-F(y) \bigr\vert \bigr] \\& \hphantom{I_{11}}\leq{J_{47}} \Vert F_{n}-F \Vert _{\beta} \biggl\vert \int_{\partial\Omega_{2}}E_{1}(x,y)\,d\sigma(x) \biggr\vert \\& \hphantom{I_{11}} \leq J_{48} \Vert F_{n}-F \Vert _{\beta}, \end{aligned}
(33)

that is,

$$I_{11}\leq{J_{48}} \Vert F_{n}-F \Vert _{\beta}.$$
(34)

From inequality (33) and (34), we have

$$I_{8}\leq{J_{45}}\delta^{\beta}+J_{48} \Vert F_{n}-F \Vert _{\beta}.$$
(35)

In a similar way, we get

$$I_{9}\leq{J_{49}}\delta^{\beta}+J_{50} \Vert F_{n}-F \Vert _{\beta}.$$
(36)

From inequality (32), (35) and (36), we get

\begin{aligned} & \bigl\vert P \bigl(\Psi^{+}_{F_{n}}(y), \Psi^{-}_{F_{n}}(y) \bigr)-P \bigl(\Psi^{+}_{F}(y), \Psi^{-}_{F}(y) \bigr) \bigr\vert \\ &\quad\leq(J_{34}+J_{35}) \bigl[(J_{45}+J_{49}) \delta^{\beta }+(J_{48}+J_{50}) \Vert F_{n}-F \Vert _{\beta} \bigr] \\ &\quad=J_{51}\delta^{\beta}+J_{52} \Vert F_{n}-F \Vert _{\beta}. \end{aligned}

Select a sufficiently small positive number Î´ such that $$J_{51}\delta^{\beta}< {\frac{\varepsilon}{2}}$$; and let n be large enough such that $$J_{52} \Vert F_{n}-F \Vert _{\beta}< {\frac{\varepsilon}{2}}$$. So for any $${y}\in\partial\Omega$$, we have $$\vert P(\Psi^{+}_{F_{n}}(y),\Psi^{-}_{F_{n}}(y))-P(\Psi ^{+}_{F}(y),\Psi^{-}_{F}(y)) \vert <\varepsilon$$, thus $$\vert {NF_{n}(y)-NF(y)} \vert <\varepsilon$$, then N is a continuous mapping which maps T to itself.

From the Arzelaâ€“Ascoli theorem we conclude that T is a compact set in $$\mathbf{C_{q}}(\partial\Omega)$$. As the continuous mapping N maps the closed convex set T to itself, $$N(T)$$ is compact in $$\mathbf {C_{q}}(\partial\Omega)$$. From the Schauder fixed point principle it follows that there is at least $$F\in{H_{q}(\beta,\partial\Omega ,\mathit {Cl}_{n+1,0}(\mathbb{R}))}$$ that satisfies (26). Hence the nonlinear boundary value problem SR has at least one solution $$\Psi_{F}(y)$$, and the expression of the solution is (8).â€ƒâ–¡

## 5 The existence and uniqueness of the solution to the linear boundary value problem for the hypergenic function vector

### Theorem 5.1

If $$A(y),B(y),G(y)\in{H_{q}(\beta,\partial\Omega, \mathit {Cl}_{n+1,0}(\mathbb {R})})$$, when $$0<\gamma=J_{3}(J_{32}+ \frac{1}{2}) \Vert {A+B} \Vert _{\beta}+J_{3} \Vert {1+A} \Vert _{\beta}<1$$, the linear boundary value problem SR has a unique solution.

### Proof

Let T be as in TheoremÂ 4.1. N is a continuous mapping which maps T to itself from TheoremÂ 4.1.

From inequalities (7), (25) and RemarkÂ 3.1, we get

\begin{aligned}& \Vert {N{F_{1}}-N{F_{2}}} \Vert _{\beta} \\& \quad\leq{J_{3}} \Vert {A+B} \Vert _{\beta} \biggl\Vert \frac {1}{2}(F_{2}-F_{1})+\Phi_{F_{1}}- \Phi_{F_{2}} \biggr\Vert _{\beta} +J_{3} \Vert {1+A} \Vert _{\beta} \Vert {F_{1}-F_{2}} \Vert _{\beta} \\& \quad\leq{J_{3}} \Vert {A+B} \Vert _{\beta} \biggl[ \biggl\Vert \frac {1}{2}(F_{1}-F_{2}) \biggr\Vert _{\beta}+ \Vert \Phi_{F_{1}}-\Phi_{F_{2}} \Vert _{\beta} \biggr] +J_{3} \Vert {1+A} \Vert _{\beta} \Vert {F_{1}-F_{2}} \Vert _{\beta} \\& \quad\leq{J_{3}} \Vert {A+B} \Vert _{\beta} \biggl( \frac {1}{2}+J_{31} \biggr) \Vert {F_{1}-F_{2}} \Vert _{\beta} +J_{3} \Vert {1+A} \Vert _{\beta} \Vert {F_{1}-F_{2}} \Vert _{\beta} \\& \quad\leq\biggl(J_{3} \biggl(J_{32}+ \frac{1}{2} \biggr) \Vert {A+B} \Vert _{\beta }+J_{3} \Vert {1+A} \Vert _{\beta} \biggr) \Vert {F_{1}-F_{2}} \Vert _{\beta} \\& \quad=\gamma \Vert {F_{1}-F_{2}} \Vert _{\beta}. \end{aligned}

â€ƒâ–¡

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).

## 6 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 Arzelaâ€“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.

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## Funding

This work was supported by the National Science Foundation of China (No. 11571089, No. 11401164, No. 11401159) and the Doctoral Foundation of Hebei Normal University (No. L2015B04, No. L2015B03) and the Key Foundation of Hebei Normal University (No. L2018Z01).

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Correspondence to Yonghong Xie.

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Zhang, G., Li, C. & Xie, Y. Boundary value problems for hypergenic function vectors. J Inequal Appl 2018, 132 (2018). https://doi.org/10.1186/s13660-018-1725-8