1 Introduction and Preliminaries

Dirichlet problem is one of the boundary value problems. Dirichlet problem is treated for first-order, model, higher-order and higher-order linear equations in various types of domains by many researchers [3,4,5,6,7,8, 10, 11, 13, 14, 19].

In [2], Schwarz problem for nonlinear higher-order equations in upper half plane is investigated. In the present paper, we will study Dirichlet problem for nonlinear higher-order equations in upper half plane. This paper is organized as follows: In Sect. 2, we need to introduce Pompeiu-type operators defined in the upper half plane. The properties of relevant operators are obtained using T and \(\Pi \) operators [9]. In Sect. 3, we transform the Dirichlet problem for nonlinear higher-order equations in upper half plane into the system of integro-differential equations and we obtain the existence of unique solution using Banach fixed point theorem.

Firstly we give some information about notations and then we state the Dirichlet problem for the inhomogeneous polyanalytic equation in upper half plane [11].

Definition 1

If the set of functions w \(\in \) \(W^{k,1}({\mathbb {H}}, {\mathbb {C}})\) satisfy

$$\begin{aligned} \lim _{R \rightarrow \infty } R^v M\left( \partial _{{\bar{z}}}^v w, R\right) =0,\; \; 0 \le v \le k-1, \; k \in {\mathbb {N}} \end{aligned}$$

where

$$\begin{aligned} M\left( \partial _{{\bar{z}}}^v w, R\right) :=\max _{\begin{array}{c} |z|=R \\ 0 \le \text{ Imz } \end{array}}\left| \partial _{{\bar{z}}}^\nu {w}(z)\right| \end{aligned}$$

and

$$\begin{aligned} {\bar{z}}^{k-2} \partial _{{\bar{z}}}^k w \in L_1({\mathbb {H}}, {\mathbb {C}}), \; z \in {\mathbb {H}}, \; k \in {\mathbb {N}} \end{aligned}$$

or for some \(\delta >0\)

$$\begin{aligned} |w(z)|=O\left( |z|^{-k-\delta }\right) \; \text{ as } z \rightarrow \infty , \; z \in {\mathbb {H}}, \; k \in {\mathbb {N}} \end{aligned}$$

This means that w holds regularity condition.

Note that \(L_{p, 2}({\mathbb {H}})\), \(p>2\) is the space of complex valued functions defined in \({\mathbb {H}}\) satisfying the following properties [18]:

$$\begin{aligned} f \in L_{p, 2}({\mathbb {H}}) \Leftrightarrow f(z) \text{ and } |z|^{-2} f\left( \frac{1}{z}\right) \text{ are } \text{ in } L_p\left( \overline{{\mathbb {H}}}_1\right) \end{aligned}$$

where

$$\begin{aligned} \overline{{\mathbb {H}}}_1=\{z \in {\mathbb {C}}:|z| \le 1, {\text {Im}}z \ge 0\}. \end{aligned}$$

The norm for \(f \in L_{p, 2}({\mathbb {H}})\) is defined by

$$\begin{aligned} \left\| f\right\| _{L_{p, 2}({\mathbb {H}})}:=\left\| f\right\| _{L_p\left( \overline{{\mathbb {H}}}_1\right) }+\left\| |z|^{-2} f\left( \frac{1}{z}\right) \right\| _{L_p\left( \overline{{\mathbb {H}}}_1\right) }. \end{aligned}$$

Theorem 1

[11] Let \({\mathcal {F}}_n\) be the space of functions w in \(W^{n, 1}({\mathbb {H}}, {\mathbb {C}})\). For \(w \in {\mathcal {F}}_n\), the Dirichlet problem for the inhomogeneous polyanalytic equation in the upper half plane \({\mathbb {H}}\),

$$\begin{aligned} \partial _{{\bar{z}}}^n w=f \quad \text { in } {\mathbb {H}}, \quad \partial _{{\bar{z}}}^v w=\gamma _v \quad \text { on } {\mathbb {R}}, \quad 0 \le v \le n-1 \end{aligned}$$

is uniquely solvable for \(f \in L_{p, 2}({\mathbb {H}}, {\mathbb {C}}), p>2\) satisfying regularity conditions above and

$$\begin{aligned} t^\lambda \gamma _\lambda (t) \in L_p({\mathbb {R}}, {\mathbb {C}}) \cap C({\mathbb {R}}, {\mathbb {C}}), \quad 0 \le \lambda \le n-1 \end{aligned}$$

if and only if for \(z \in {\mathbb {H}}\)

$$\begin{aligned}{} & {} \sum _{\lambda =v}^{n-1} \frac{1}{2 \pi i} \frac{(-1)^{\lambda -v}}{(\lambda -v) !} \int _{-\infty }^{\infty } \gamma _\lambda (t)(t-z)^{\lambda -v} \frac{\textrm{d} t}{t-{\bar{z}}}+\frac{(-1)^{n-v}}{(n-v-1) !} \frac{1}{\pi }\\{} & {} \qquad \int _{{\mathbb {H}}} f(\zeta )({\bar{\zeta }}-z)^{n-1-v} \frac{\textrm{d} \xi \textrm{d} \eta }{(\zeta -{\bar{z}})}=0. \end{aligned}$$

for \(0 \le v \le n-1\). The solution is given by

$$\begin{aligned} w(z)=&\sum _{\lambda =0}^{n-1} \frac{1}{2 \pi i} \frac{(-1)^\lambda }{\lambda !} \int _{-\infty }^{\infty } \gamma _\lambda (t)(t-{\bar{z}})^\lambda \frac{\textrm{d} t}{(t-z)} \nonumber \\&+\frac{(-1)^n}{(n-1) !} \frac{1}{\pi } \int _{{\mathbb {H}}} f(\zeta )(\overline{\zeta -z})^{n-1} \frac{\textrm{d} \xi \textrm{d} \eta }{\zeta -z}. \end{aligned}$$
(1)

2 The Properties of Integral Operators

We consider the solution of inhomogeneous polyanalytic equation in the upper half plane \({\mathbb {H}}\) (1). Under the homogeneous boundary conditions, assuming \(n=1\), we obtain the integral operator \(T_{{\mathbb {H}}}\):

$$\begin{aligned} w(z)=T_{{\mathbb {H}}}f=-\frac{1}{\pi }\int \nolimits _{{\mathbb {H}}}f(\zeta )\frac{d\xi d\eta }{ \zeta -z}. \end{aligned}$$
(2)

This operator is extensively studied by Begehr and Hile [9], many properties of this operator are known.

\(T_{{\mathbb {H}}}f\) has generalized derivatives with respect to \({\overline{z}}\) and z:

  1. (i)

    \(T_{{\mathbb {H}}} f\) has a generalized first order derivative with respect to \({\overline{z}}\),

    $$\begin{aligned} \partial _{{\overline{z}}}T_{{\mathbb {H}}}f=f \end{aligned}$$

    if \(f \in L_{p, 2}({\mathbb {H}})\),  \(p>2\).

  2. (ii)

    The generalized derivative of \(T_{{\mathbb {H}}} f\) with respect to z is

    $$\begin{aligned} \partial _{{z}}T_{{\mathbb {H}}}f:=\Pi _{{\mathbb {H}}}f=-\frac{1}{\pi } \int \nolimits _{{\mathbb {H}}}f(\zeta )\frac{d\xi d\eta }{\left( \zeta -z\right) ^{2}}, \ \ z\in {\mathbb {H}}. \end{aligned}$$

We know that \(T_{{\mathbb {H}}}^{k}:=T_{0,k}\) and \(T_{{\mathbb {H}}}^{m}:=T_{m,0}\) are obtained by iteration [9]; that is

$$\begin{aligned} T_{{\mathbb {H}}}^{s+1}f=T_{{\mathbb {H}}}(T_{{\mathbb {H}}}^{s}f), \; 0\le s<k. \end{aligned}$$

\(T_{{\mathbb {H}}}^{s}\), \(s\le k\) are weakly singular operators. Using Arzela-Ascoli theorem we observe that \(T_{{\mathbb {H}}}^{s}\), \(s\le k\) are compact operators [9].

On the other hand,

$$\begin{aligned} \Pi _{{\mathbb {H}}}f:=\partial _{{z}}T_{{\mathbb {H}}}f=-\frac{1}{\pi } \int \nolimits _{{\mathbb {H}}}f(\zeta )\frac{d\xi d\eta }{\left( \zeta -z\right) ^{2}}, \ \ z\in {\mathbb {H}} \end{aligned}$$

are Calderon-Zygmund type strongly singular bounded operators. Similar to the operations in Çelebi-Gökgöz [12], we can obtain these properties.

3 Dirichlet Problem for Nonlinear Higher-Order Equations in Upper Half Plane

The boundary value problems for nonlinear elliptic equations have been studied by several researchers, for example, Akal and Begehr [1], Aksoy-Begehr-Çelebi [2], Mshimba [15,16,17], in the book by Wen [20] and others.

In this section we consider the Dirichlet boundary value problem for nonlinear higher-order equations in upper half plane \({\mathbb {H}}\).

Thus we start with elliptic equation

$$\begin{aligned} \partial _{{\bar{z}}}^n w=F\left( z, w, D^{\alpha _1} w, \ldots , D^{\alpha _n} w\right) \end{aligned}$$
(3)

of order n in upper half plane \({\mathbb {H}}\). As in [2] we employ the notations

$$\begin{aligned} D=\left( \partial _z, \partial _{{\bar{z}}}\right) , \alpha _j=(k, l),\left| \alpha _j\right| =k+l=j, \; j=0,1,2, \ldots n \end{aligned}$$

with restriction \((k, l) \ne (0, n)\). To simplify the notation, we take

$$\begin{aligned} \left\{ D^{\alpha _1} w, D^{\alpha _2} w, \ldots , D^{\alpha _n} w\right\} =\left\{ w_{k, l}\right\} . \end{aligned}$$

Hence Eq. (3) may be written as

$$\begin{aligned} \partial _{{\bar{z}}}^n w=F\left( z,\left\{ w_{k, l}\right\} \right) \end{aligned}$$

To discuss the Dirichlet boundary value problem, we need some assumptions on \(F\left( z,\left\{ w_{k, l}\right\} \right) \) defined in \({\mathbb {H}}\).

Assumptions

  1. (i)

    The function \(F\left( z,\left\{ w_{k, l}\right\} \right) \) is continuous with respect to all of its variables,

  2. (ii)

    For multiples \(\left( z,\left\{ w_{k, l}\right\} \right) \) if \(\left\{ w_{k, l}\right\} \in L_p({\mathbb {H}}), p>2\), then \(F\left( z,\left\{ w_{k, l}\right\} \right) \in L_p({\mathbb {H}})\),

  3. (iii)

    The function \(F\left( z,\left\{ w_{k, l}\right\} \right) \) satisfies the Lipschitz condition

    $$\begin{aligned} \left| F\left( z,\left\{ w_{k, l}\right\} \right) -F\left( z,\left\{ w_{k, l}^*\right\} \right) \right| \le \sum _{k+l=0}^n L_{k l}\left| w_{k, l}-w_{k, l}^*\right| \le L \sum _{k+l=0}^n\left| w_{k, l}-w_{k, l}^*\right| \end{aligned}$$

    where \(L=\max _{k+l \le n} L_{k l},(k, l) \ne (0, n)\) in which \(L_{k l}\) are constants.

Now we state the relevant Dirichlet problem.

“Find a solution \(w\in W^{n,p}({\mathbb {H}})\) with \(2<p<\infty \) of Eq. (3) subject to

$$\begin{aligned} \partial _{{\bar{z}}}^v w=\gamma _v \quad \text { on } {\mathbb {R}}, \quad 0 \le v \le n-1 \end{aligned}$$
(4)

   is uniquely solvable for \(f \in L_{p, 2}({\mathbb {H}}, {\mathbb {C}}), p>2\) satisfying regularity conditions and

$$\begin{aligned} t^\lambda \gamma _\lambda (t) \in L_p({\mathbb {R}}, {\mathbb {C}}) \cap C({\mathbb {R}}, {\mathbb {C}}), \quad 0 \le \lambda \le n-1." \end{aligned}$$

To solve this problem we employ the operators

$$\begin{aligned} {T}_{{\mathbb {H}}}^{n}w(z)=&\frac{(-1)^n}{(n-1) !} \frac{1}{\pi } \int _{{\mathbb {H}}} f(\zeta )(\overline{\zeta -z})^{n-1} \frac{\textrm{d} \xi \textrm{d} \eta }{\zeta -z}. \end{aligned}$$

As in [2] we transform the Dirichlet problem for nonlinear equation into the following system of integro-differential equations:

$$\begin{aligned} w(z)&= \phi _{w}(z)+{T}_{{\mathbb {H}}}^{n}F \left( z,\left\{ w_{k,l}\right\} \right) (z) \nonumber \\ w_{k,l}(z)&= \partial _{z}^{k}\partial _{{\overline{z}}}^{l}\phi _{w}(z)+\partial _{z}^{k}{T}_{{\mathbb {H}}}^{n-l}F \left( z,\left\{ w_{k,l}\right\} \right) (z) \end{aligned}$$

for \(0\le k+l\le n\), with the restriction \((k,l)\ne (0,n)\) and

$$\begin{aligned} \phi _{w}(z)&= \sum _{\lambda =0}^{n-1} \frac{1}{2 \pi i} \frac{(-1)^\lambda }{\lambda !} \int _{-\infty }^{\infty } \gamma _\lambda (t)(t-{\bar{z}})^\lambda \frac{\textrm{d} t}{(t-z)}. \end{aligned}$$
(5)

It is trivial that \(\phi _{w}\) is a polyanalytic function. Now we may discuss the solvability of the Dirichlet problem. We use the notation given in [2]. Firstly we introduce the function space \(\mathfrak {L}_{\mathfrak {p}} ({\mathbb {H}})\), \(p>2\):

$$\begin{aligned} \mathfrak {L}_{\mathfrak {p}} ({\mathbb {H}})=\left\{ \left\{ w_{k, l}\right\} : w_{k, l} \in L_p({\mathbb {H}}), p>2\right\} \end{aligned}$$

where \(0\le k+l\le n\), \((k,l)\ne (0,n)\). \(\mathfrak {L}_{\mathfrak {p}} ({\mathbb {H}})\) is a Banach space with norm

We introduce an operator Q by

$$\begin{aligned} Q\left( \left\{ w_{k,l}\right\} \right) =\left( \left\{ W_{k,l}\right\} \right) \end{aligned}$$

if

$$\begin{aligned} W(z)&= \phi _{w}(z)+{T}_{{\mathbb {H}}}^{n}F \left( z,\left\{ w_{k,l}\right\} \right) (z)\nonumber \\ W_{k,l}(z)&= \partial _{z}^{k}\partial _{{\overline{z}}}^{l}\phi _{w}+\partial _{z}^{k}{T}_{{\mathbb {H}}}^{n-l}F \left( z,\left\{ w_{k,l}\right\} \right) (z) \end{aligned}$$

where \(\phi _{w}(z)\) is given by Eq. (5). Thus we obtain a system of integro-differential equations. Then the fixed point of Q is the solution of Eq. (3), satisfying the Dirichlet boundary conditions (4) subject to the solvability requirements.

Now we try to show that Q has a fixed point. We take the images of \(\left\{ w_{k,l}\right\} \) and \(\left\{ {\overline{w}}_{k,l}\right\} \) in \(\mathfrak {L}_{\mathfrak {p}} ({\mathbb {H}})\) which are denoted by \(\left\{ W_{k,l}\right\} \) and \(\left\{ {\overline{W}}_{k,l}\right\} \) respectively. Then

and

(6)

for \(0\le k+l<n\) and

(7)

for \(k+l=n\). Combining (6) and (7) we obtain

where

Thus the operator Q is contractive if \(M<1\). From Banach fixed point theorem we obtain the existence of the unique solution \(w_{k,l}\).