# Fundamental Structure of General Stochastic Dynamical Systems: High-Dimension Case

Authors: Haoyu Wang; Xiaoliang Gan; Wenqing Hu; Ping Ao
Journal: Journal of Mathematics
 (2022)

---

## Abstract

No one has proved that mathematically general stochastic dynamical systems have a special structure. Thus, we introduce a structure of a general stochastic dynamical system. According to scientific understanding, we assert that its deterministic part can be decomposed into three significant parts: the gradient of the potential function, friction matrix and Lorenz matrix. Our previous work proved this structure for the low-dimension case. In this paper, we prove this structure for the high-dimension case. Hence, this structure of general stochastic dynamical systems is fundamental.


---

## Body

## 1. Introduction

Stochastic differential equations are widely used to describe random phenomena in complex systems in physics, biology, and chemistry. For such a stochastic dynamical system, researchers usually build an appropriate mathematical model based on basic scientific laws and analyze or simulate it to gain insights about its complex phenomena. However, these models are proposed to solve specific scientific problems [1–3]. Until now, the general theory for stochastic differential equations has been limited.To gain a deeper understanding of the dynamic behaviors of general stochastic dynamical systems requires the exploration of their intrinsic mechanisms. In 2005, Ao [4] proposed Ao decomposition, which demonstrates that the deterministic part of general stochastic dynamical systems can be decomposed into three significant parts: the friction force, gradient of the potential function, and Lorenz force. This inspired much work [5, 6]. We discuss the scientific significance of these three terms.


### 1.1. Potential Function

From the biological point of view, the potential function can be explained by evolution theory. As we know, the fundamental nature of biology is determined by evolution. To explain adaptation and speciation, Darwin [7] formulated the theory of evolution based on natural selection. Accordingly, Fisher [8] proposed the fundamental theorem of natural selection, indicating that the increase rate of mean fitness by natural selection is equal to its genetic variance in fitness. In 1932, Wright [9] proposed the fitness landscape concept, by which evolutionary adaptation may be seen as a hill-climbing process on the mean fitness landscape until a local mean fitness peak is reached. In 1940, Waddington [10] proposed the developmental landscape, which is equivalent to the fitness landscape. Wright’s fitness landscape and Fisher’s fundamental theorem of natural selection have been widely used to interpret adaptation as mean fitness maximization (see Figure 1).Figure 1
Evolution is described as populations moving on landscapes. This is represented as either (a) a maxima on a landscape of fitness; or (b) a minima on a landscape of fitness potential. They are different ways to visualize the same process.
(a)(b)This phenomenon is often illustrated on mathematical landscapes as balls rolling downhill. A ball experiencing gravity tends to a minimum of the gravitational potential energypx as a function of its spatial position x. The force on the ball is given by the local slope, f=−dpx/dx.The potential landscape is also called a potential function or energy function, which has been applied in fields such as physics, biology, and chemistry [11, 12]. It can compare the relative stability of different attractors [13], account for the transition rates between neighboring steady states induced by noise [14], and provide an intuitive picture that reveals the essential mechanism underlying the complex system [15]. In physics, the potential function is closely related to the non-equilibrium thermodynamic framework [16]; in chemistry, it provides useful explanations for protein folding [17, 18]; in biology, it has been used to explore basic problems in evolution such as the robustness, adaptability, and efficiency of real biological networks [19]. Until now, the general existence of the potential function remains unsolved. Researchers such as Prigogine et al. [20–22] have insisted that the potential function does not exist in non-equilibrium systems because they have not found it.


### 1.2. The Friction Matrix

The friction matrix (the frictional force) represents dissipation. In this case, the energy of a dynamical system decreases, so the potential function decreases and the corresponding fitness increases. A system that has only friction is a gradient system (see green thick arrow in Figure1).


### 1.3. The Lorenz Matrix

Interestingly, Wright’s fitness landscape theory [9] cannot explain the Red Queen hypothesis proposed by Valen [23], which illustrates that the biotic interactions between species provide a driving force resulting in endless evolution for some species even if the physical environment is unchanged. This is because he neglects the Lorenz matrix (Lorenz force). If considering the Lorenz force, the population flow on a landscape is not directly down the gradient of the potential function. It also swirls (see red thin arrow in Figure 1).The above scientific understanding indicates that these three components exist in general stochastic dynamical systems, but this decomposition lacks rigorous mathematical proof. This paper is the first to prove that the deterministic part of general high-dimension stochastic dynamical systems can be decomposed into three components: the diffusion force, gradient of the potential function, and curl flux. Our previous work proved this structure for the low-dimension case (when the dimensionn=1,2). On this basis, we prove this structure for the high-dimension case (when n≥4, and nn−1/2 is an even number, i.e., n=4,5,8,9,…,4i,4i+1,…, when i=1,2,3,…). Apart from theoretical significance, our result has important guiding significance for applications in both mathematics and subjects such as biology and physics. The potential function provides intuitive and global landscapes. Real dynamical systems are complex and usually have more than one steady state, so the potential function has a wide range of applications in real dynamical systems. For example, Hu and Xu [24] studied the phenomenon of multi-stable chaotic attractors existing in generalized synchronization for a driving and response system named Rössler system. Angeli and Sontag [25] studied the emergence of multi-stability and hysteresis in those monotone input/output systems that arise, under positive feedback, starting from monotone systems with well-defined steady-state responses. Liu and You [26] studied multi-stability, existence of almost periodic solutions of a class of recurrent neural networks with bounded activation functions and all criteria they proposed can be easily extended to fit many concrete forms of neural networks such as Hopfield neural networks, or cellular neural networks, etc.. The potential function has provided a general and unified perspective for researchers to investigate different types of dynamical systems.The rest of this paper is organized as follows. Section2 introduces Ao decomposition for general stochastic differential equations and proposes our problem: proving the equivalence of the Langevin equation and the equation after Ao decomposition. In Section 3, we reduce this problem to proving the existence of solutions for first-order partial differential equations, and we accomplish this proof in Section 4.


## 1.1. Potential Function

From the biological point of view, the potential function can be explained by evolution theory. As we know, the fundamental nature of biology is determined by evolution. To explain adaptation and speciation, Darwin [7] formulated the theory of evolution based on natural selection. Accordingly, Fisher [8] proposed the fundamental theorem of natural selection, indicating that the increase rate of mean fitness by natural selection is equal to its genetic variance in fitness. In 1932, Wright [9] proposed the fitness landscape concept, by which evolutionary adaptation may be seen as a hill-climbing process on the mean fitness landscape until a local mean fitness peak is reached. In 1940, Waddington [10] proposed the developmental landscape, which is equivalent to the fitness landscape. Wright’s fitness landscape and Fisher’s fundamental theorem of natural selection have been widely used to interpret adaptation as mean fitness maximization (see Figure 1).Figure 1
Evolution is described as populations moving on landscapes. This is represented as either (a) a maxima on a landscape of fitness; or (b) a minima on a landscape of fitness potential. They are different ways to visualize the same process.
(a)(b)This phenomenon is often illustrated on mathematical landscapes as balls rolling downhill. A ball experiencing gravity tends to a minimum of the gravitational potential energypx as a function of its spatial position x. The force on the ball is given by the local slope, f=−dpx/dx.The potential landscape is also called a potential function or energy function, which has been applied in fields such as physics, biology, and chemistry [11, 12]. It can compare the relative stability of different attractors [13], account for the transition rates between neighboring steady states induced by noise [14], and provide an intuitive picture that reveals the essential mechanism underlying the complex system [15]. In physics, the potential function is closely related to the non-equilibrium thermodynamic framework [16]; in chemistry, it provides useful explanations for protein folding [17, 18]; in biology, it has been used to explore basic problems in evolution such as the robustness, adaptability, and efficiency of real biological networks [19]. Until now, the general existence of the potential function remains unsolved. Researchers such as Prigogine et al. [20–22] have insisted that the potential function does not exist in non-equilibrium systems because they have not found it.


## 1.2. The Friction Matrix

The friction matrix (the frictional force) represents dissipation. In this case, the energy of a dynamical system decreases, so the potential function decreases and the corresponding fitness increases. A system that has only friction is a gradient system (see green thick arrow in Figure1).


## 1.3. The Lorenz Matrix

Interestingly, Wright’s fitness landscape theory [9] cannot explain the Red Queen hypothesis proposed by Valen [23], which illustrates that the biotic interactions between species provide a driving force resulting in endless evolution for some species even if the physical environment is unchanged. This is because he neglects the Lorenz matrix (Lorenz force). If considering the Lorenz force, the population flow on a landscape is not directly down the gradient of the potential function. It also swirls (see red thin arrow in Figure 1).The above scientific understanding indicates that these three components exist in general stochastic dynamical systems, but this decomposition lacks rigorous mathematical proof. This paper is the first to prove that the deterministic part of general high-dimension stochastic dynamical systems can be decomposed into three components: the diffusion force, gradient of the potential function, and curl flux. Our previous work proved this structure for the low-dimension case (when the dimensionn=1,2). On this basis, we prove this structure for the high-dimension case (when n≥4, and nn−1/2 is an even number, i.e., n=4,5,8,9,…,4i,4i+1,…, when i=1,2,3,…). Apart from theoretical significance, our result has important guiding significance for applications in both mathematics and subjects such as biology and physics. The potential function provides intuitive and global landscapes. Real dynamical systems are complex and usually have more than one steady state, so the potential function has a wide range of applications in real dynamical systems. For example, Hu and Xu [24] studied the phenomenon of multi-stable chaotic attractors existing in generalized synchronization for a driving and response system named Rössler system. Angeli and Sontag [25] studied the emergence of multi-stability and hysteresis in those monotone input/output systems that arise, under positive feedback, starting from monotone systems with well-defined steady-state responses. Liu and You [26] studied multi-stability, existence of almost periodic solutions of a class of recurrent neural networks with bounded activation functions and all criteria they proposed can be easily extended to fit many concrete forms of neural networks such as Hopfield neural networks, or cellular neural networks, etc.. The potential function has provided a general and unified perspective for researchers to investigate different types of dynamical systems.The rest of this paper is organized as follows. Section2 introduces Ao decomposition for general stochastic differential equations and proposes our problem: proving the equivalence of the Langevin equation and the equation after Ao decomposition. In Section 3, we reduce this problem to proving the existence of solutions for first-order partial differential equations, and we accomplish this proof in Section 4.


## 2. A-Type Decomposition for General Stochastic Differential Equations

The Langevin equation in physics, which has the form of a general stochastic differential equation, is usually a more accurate description of physical processes than the purely deterministic one [27–30]. Here, we use the physicists’ notation for the noise, and we can write this equation in the form(1)q˙=fq,t+ζq,t.We discuss this equation inn-dimensional real Euclidean space. The state variable q=q1t,q2t,…,qnt is a function of time t, and the component functions qii=1,2,…,n of the state variable q=q1,q2,…,qn are independent. We assume that fq,t is an infinitely differentiable smooth function. The noise ζq,t is a function of t and the state variable q, and is almost nowhere differentiable. We consider the case that ζq,t is n-dimensional white Gaussian noise with mean(2)ζq,t=0,and covariance(3)ζq,tζτq,t′=2Dqδt−t′.The superscriptτ denotes the transpose of a matrix (vector), δt−t′ is the Dirac delta function, ⋅ indicates the average over the noise distribution, and the diffusion matrix Dq is symmetric and positive semi-definite.Noticing a new formulation of equation (1) [6, 15, 31], we propose Ao decomposition, under which equation (1) can be formally decomposed into(4)Sqt+Aqtq˙=−∇φqt+ξq,t,where Sqt is a symmetric semi-definite matrix (which we call the “friction matrix”), Aqt is an antisymmetric matrix (the “Lorenz matrix”), φq is a real and single-valued function of q1,q2,…,qn, and ξq,t is n-dimensional white Gaussian noise with mean(5)ξq,t=0,and covariance(6)ξq,tξτq,t′=2Sqδt−t′.It should be noted that equations (3) and (6) are manifestations of the fluctuation-dissipation theorem, where Dq and Sq reflect dissipation, and the covariance structures ζq,tζτq,t′ and ξq,tξτq,t′ reflect fluctuation [32].Our main problem is to prove the equivalence of equations (1) and (4). We can also prove that φqt in equation (4) is a potential function.


## 3. Reduction of Problem into Partial Differential Equations (PDEs)

To show that equation (1) is equivalent to equation (4), we first show that equation (4) implies equation (1). To this end, we assume that the function matrix Sq+Aq is invertible and the components qii=1,2,…,n of the state variable q=q1,q2,…,qnτ are independent. If they are not independent, the dimension can be reduced to n′,n′<n, until they are independent. Therefore, the equations of this system are linearly independent. Equation (4) can be straightforwardly transformed to(7)q˙=−Sqt+Aqt−1∇φqt+ζq,t,where ζq,t is noise that takes the form ζq,t=Sq+Aq−1ξq,t. To match equation (1), we can then set fq=−Sq+Aq−1∇φq. Notice that with the explicit representation of ζq,t in terms of Sq,Aq, and ξq,t, as well as equation (6), we can calculate(8)ζq,t=Sq+Aq−1ξq,t=Sq+Aq−1ξq,t=0,ζq,tζτq,t′=Sq+Aq−1ξq,tξτq,t′Sq+Aq−τ=Sq+Aq−1ξq,tξτq,t′Sq+Aq−τ=2Sq+Aq−1SqSq+Aq−τδt−t′.Comparing the above two calculations with equations (2) and (6), we see that we have(9)Dq=Sq+Aq−1SqSq+Aq−τ,which gives an explicit representation of Dq.Next, we consider the problem of whether equation (1) implies equation (4). In fact, transforming from equations (1) to (4) requires much more effort. In this case, we need to obtain Sq, Aq, and φq from the general dynamic equation (1). We propose heuristic inference. While not a rigorous mathematical proof, it can lead to a reformulation of the problem into PDEs. The main idea of this heuristic inference is that equations (1) and (4) can describe the same dynamical behaviors in ℝn. Hence we may replace q˙ in equation (4) by the right side of equation (1) to obtain(10)Sqt+Aqtfqt+ζqt,t=−∇φqt+ξqt,t.Regardingt as a parameter in qt, the above equation can be written as(11)Sq+Aqfq+ζq,t=−∇φq+ξq,t,which has a deterministic part that is differentiable up to an arbitrary order, and a random part that is nondifferentiable everywhere. From the point of view of physics, the two kinds of noises ζq,t and ξq,t have the same source. Inspired by this, we may assume that we can establish a classification,(12)Sq+Aqfq=−∇φq,(13)Sq+Aqζq,t=ξq,t.This subjective decomposition is the key to understanding Ao decomposition, which results in the consistency of stable points between a stochastic dynamical system and the corresponding dynamical system. Therefore, the generalized Lyapunov function of the stochastic dynamical system is equivalent to the Lyapunov function of the corresponding dynamical system. It must be noted that the A-type integral derived from Ao decomposition is a new integral that is different from the Itô and Stratonovich integrals. In one dimension, the A-type integral is simplified to theα-type [33], where α=1 (Itô corresponds to α=0, and Stratonovich to α=0.5). In the high-dimensional case, the A-type integral is not usually the α-type [34].Combining equations (3) and (6) with (13), we obtain(14)ξq,tξτq,t′=Sq+Aqζq,tζτq,t′Sq+Aqτ=2Sq+AqDqSq+Aqτδt−t′=2Sq+AqDqSq−Aqδt−t′=2Sqδt−t′,which implies(15)Sq+Aqfq=−∇φq,(16)Sq+AqDqSq−Aq=Sq.From the physical point of view, equation (16) is a generalized Einstein relation in greater than one dimension. From equation (16), we have(17)Dq=Sq+Aq−1⋅12Sq+Aq+Sq−Aq⋅Sq−Aq−1=12Sq−Aq−1+Sq+Aq−1=12Sq+Aq−τ+Sq+Aq−1,where the symmetric part of Sq+Aq−1 is 1/2Sq+Aq−1+Sq+Aq−τ. This is the diffusion matrix Dq defined in equation (3). Hence we can rewrite the identity(18)Sq+AqSq+Aq−1=I,as(19)Sq+AqDq+Qq=I,where Qq is an anti-symmetric unknown matrix function and I is the identity matrix. Substituting equation (19) in equation (15), we obtain(20)Dq+Qq−1fq=−∇φq.From equation (15), it is easy to obtain that if q˙=fq∗=0, φq∗=0. Moreover, by equation (15) we have(21)ddtφq=q˙τ∇φq=−q˙τSq+Aqq˙=−q˙τSqq˙≤0.Thusφq satisfies φ˙q≤0 for all q∈ℝn, and it is proven that φq is a potential function.Assuming equation (1) holds true, equation (3) is given, and thus Dq is known. We see that to obtain equation (4), we just have to show that there exists an anti-symmetric matrix Qq and potential function φq that satisfy equation (20). Assuming basic integrability conditions on fq,Dq, and Qq, by the classical Helmholtz decomposition we obtain that it suffices to show that the curl part of the vector field Dq+Qq−1fq vanishes, i.e.,(22)∇×Dq+Qq−1fq=0,where equation (22) is a family of nn−1/2 first–order quasilinear partial differential equations for the coefficients of Qq in equation (20).We notice that according to the above heuristic inference, equation (22) is a sufficient condition for equations (1) ⇒ (4). In fact, if equation (22) holds true, then by Helmholtz decomposition, there exists a function φ=φq such that equation (20) holds true, with the anti-symmetric matrix Qq from (22). Moreover, with Dq from equation (3) and Qq from equation (22), we can construct the matrix Sq+Aq=Dq+Qq−1, where Sq is symmetric and Aq is anti-symmetric, and Sq and Aq satisfy equations (15) and (16). Thus we can construct the noise ξq,t from equation (13), which, together with equation (12), implies that we can construct equation (4) from equation (1).Our problem has now been reduced to proving the existence of solutionQq to equation (22), which is a first-order PDE. The rest of the paper is dedicated to the investigation of this first-order PDE system in dimension n≥4.


## 4. Existence of Solutions to First-Order Quasilinear Partial Differential Equations in Dimensionn≥4

Obviously,q=q1,q2,…,qn,n≥4. Assume that(23)Qq=0Q12q⋯Q1nq−Q12q0⋯Q2nq⋮⋮⋱⋮−Q1nq−Q2nq⋯0.From equation (3), we can assume that(24)Dq=d11qd12q⋯d1nqd12qd22q⋯d2nq⋮⋮⋱⋮d1nqd2nq⋯dnnq.Let vectorQ=Q12,Q13,…,Q1n,Q23,…,Q2n,…,Qn−1,n.By equation (22), we can assume D+Q−1=R,(25)Rq=R11QR12Q⋯R1nQR21QR22Q⋯R2nQ⋮⋮⋱⋮Rn1QRn2Q⋯RnnQ.Therefore, according to the matrix-valued cross-product rule, equation (22) can be transformed to(26)∂∂q1f1R21+f2R22+⋯+fnR2n−∂∂q2f1R11+f2R12+⋯+fnR1n=0,⋮∂∂qn−1f1Rn1+f2Rn2+⋯+fnRnn−∂∂qnf1Rn−1,1+⋯+fnRn−1,n=0.According to the composite function derivation rule, we obtain(27)f11R21+f1R211+⋯−f12R11−f1R112−⋯−fn2R1n−fnR1n2=0,⋮f1n−1Rn1+f1Rn1n−1+⋯−f1nRn−1,1−f1Rn−1,1n−⋯−fnRn−1,nn=0,where superscripts 1,2,…,n denote the partial derivatives corresponding to q1,q2,…,qn, respectively. This is obviously a first-order n-dimensional quasilinear system of partial differential equations consisting of nn−1/2 equations and nn−1/2 unknown functions. Our goal is to prove the existence of solutions for PDEs (27).We first consider the matrix form of PDEs (27),(28)Rxx,y+Ax,yRyx,y+Bx,yRx,y+Cx,y=0.The independent variables arex=q1,…,qi and y=qi,…,qn, where 1<i<n. If A has no real eigenvalues at any point in a region, PDEs (28) are elliptic in this region, and they obey the rule that the equations do not explicitly contain time. Because A is a real matrix, this may occur only when nn−1/2 is an even number. Obviously, when the multiplicities of eigenvalues of matrix A at each point x,y are constant in the whole region, then the order of every sub-block of the Jordan standard form of A is constant in the whole region, such that there exists a nonsingular matrix Tx,y satisfying(29)J=TAT−1=diagJ0,J¯,…,Jε,J¯ε,where(30)J0=diagλ1x,y,…,λpx,y,Ji=λp+jx,y⋯⋯01λp+jx,y⋯0⋮⋱⋱00⋯1λp+jx,y.Then we assume the following:(i)
T and A belong to function space Cα1G;(ii)
The order of every Jordan sub-block of matrixJ=TAT−1 is constant in the whole region G;(iii)
The eigenvalueλjx,y∈CαG¯.Letv=TR. Equation (28) can be transformed to(31)vx+Jvy+zero−order term=0.Then we study the solution of system(32)vx+Jvy=0.This system can be decomposed into a sub-system with the form(33)∂vj1∂x+Jj∂vj1∂y=0,∂vj2∂x+J¯j∂vj2∂y=0,where j=0,1,…,s, and vj1,vj2 are real vector functions whose dimensions equal the order of matrix Ji. Let ωj=vj1+ivj2, where i is the imaginary unit. Then equation (33) can be written in the form(34)∂ωj∂x+Jj∂ωj∂y=0,j=0,1,…,s.Using operator notation,(35)∂∂z¯=12∂∂x+i∂∂y,∂∂z=12∂∂x−i∂∂y.Whenj=0, the sub-system can be decomposed into p equations like(36)1−iλωz¯+1+iλωz=0,where the value of λ is selected from λ1,…,λp. This case has been solved [35].Whenj≠0, the sub-system can be transformed to(37)1−iλp+jωjz¯+1+iλp+jωjz+ieωjz−ωjz¯=0,j=1,2,…,s,where(38)e=00⋯010⋯0⋮⋱⋱⋮0⋯10.Therefore, we consider the system(39)1−iλωz¯+1+iλωz+ieωz−ωz¯=0,where e has the form of equation (38). We assume that the pair of eigenvalues λ,λ¯ is r-fold and the corresponding linear independent eigenvector is only one, so the complex vector ω in PDEs (39) is r-dimensional. Obviously, without loss of generality, we can assume λ in PDEs (39) satisfies Imλ>0, and PDEs (39) are uniformly elliptic,(40)Imλz≥ε0>0,z∈G.We divide both sides of PDEs (39) by 1−iλ to obtain(41)I−i1−iλeωz¯+1+iλ1−iλI+i1−iλeωz=0,where I is a unit matrix of order r. Because the linear transformation 1+iξ/1−iξ maps the half-plane Imξ>ε0 to disk ξ≤ρ<1, we obtain the function(42)q˜0z=1+iλ1−iλ,which satisfies q˜0z≤ρ<1, where ρ is some positive constant.Definition 1.
Thenn−1/2×nn−1/2 matrix A=ajk is called quasi-diagonal if(43)ajk=0,whenj<k,aj+m,k+m,when1≤k+m≤j+m≤n.
A quasi-diagonal matrix is a lower triangular matrix, we set(i)
a0 represents an element of the main diagonal;(ii)
aj,1≤j≤n−1, represent elements of the jth diagonal under the main diagonal.
Because the coefficient matrix ofωz¯ and ωz in PDEs (41) is quasi-diagonal, PDEs (41) can be written as(44)ωz¯+Q˜zωz=0,where Q˜z is quasi-diagonal, and the element q˜0 in main diagonal of Q˜z. The first equation of PDEs (41) is(45)ω1z¯+Q˜zω1z=0,which is a Beltrami equation. Because λ∈CαG¯, q˜0∈CαG¯, we can extend q˜0 such that it belongs to Cαθ and maintain 0 outside a large enough circle. With q˜0≤ρ<1, we obtain the solution ξz∈Cα1θ of Beltrami equation (45) [36].
Under the coordinate transformationξ=ξz, PDEs (44) change to standard form,(46)ωξ¯+ξ¯zI−q˜0Q˜−1ξz−q˜0I+Q˜ωξ=0.
If we letQ represent the coefficient matrix of ωξ, and still use z as the independent variable, we have(47)ωz¯+Qωz=0.
A. Douglis derived the quasi-diagonal form ofQ by introducing the algebra of hypercomplex numbers [36, 37].Definition 2.
(see [36]). a=∑k=0r−1akek is called a hypercomplex number, where e is defined by equation (38), ak is a complex number, and a0 is the complex number part of a, and ∑k=1r−1akek is the nilpotent part of a. Note that a=∑k=0r−1ak, where ak is the kth component of a. A hypercomplex function is a map from the plane into this algebra, and it has the form(48)ωx,y=∑k=0r−1ωkx,yek,where each ωk is complex-valued.
Using Definition2, we can write PDEs (47) as(49)ω0z¯+∑k=1r−1ekωz¯k+∑j=0k−1qk−jωjz=0.
Let(50)ω=∑k=0r−1ωkek,q=∑k=1r−1qkek.
Note the differential operator,(51)D≡∂∂z¯+qm∂∂z,where qm is the mth column vector of q. Using the nilpotency of e, we have(52)Dω=∑k=0r−1ekωkz¯+∑k=1r−1∑j=0r−1ej+kqkωjz=∑k=0r−1ekωkz¯+∑k=1r−1ek∑j=0k−1qk−jωjz.
Therefore, PDEs (49) can be written as(53)Dω=0.
We define a generating solution.Definition 3.
A hypercomplex function space that has bounded continuous derivatives up to orderk defined in set N is represented by BkN, and a hypercomplex function space whose k-order derivative with index α is Holder¨ continuous in BkN is represented by BαkN. A module of hypercomplex function space Bα0N is represented by ⋅,N0,α,(54)ω,E0,α=supz∈Nωz+supz1≠z2,z1,z2∈Nωz1−ωz2z1−z2α.
A hypercomplex functiontz is called a generating solution of operator D if:(1)
tZ has the form tz=z+∑k=1r−1ektkz=z+Tz;(2)
Tz∈B1θ;(3)
Dtz=0,z∈θ.
We prove the existence of a generating solution to PDEs (32). We assume that(55)qk∈Cα0G,k=1,2,…,r−1,0<a<1,and qkz can be extended to θ such that they all belong to Cα0θ and are equal to zero outside a large enough circle. Let(56)t0z=z,tkz=−∑j=0k−1JGqk−j∂∂ztjz,k=1,2,…,r−1,where JG is an integral operator,(57)JGfz=−1π∫∫Gfξξ−zdξdη.
By the property of operatorJG and the assumption on qkz, we obtain(58)tz=∑k=0r−1ektkz∈Bα1θ.
By equation (52), we obtain(59)Dtz=∂∂z¯t+q∂∂zt=∑k=0r−1ek∂∂z¯tk+∑k=1r−1ek∑j=0k−1qk−j∂∂ztj=∑k=1r−1ek−∑j=0k−1qk−j∂∂ztjz+∑k=1r−1ek∑j=0k−1qk−j∂∂ztj=0.
Therefore,tz is the generating solution.
tz has the property of a positive constant M such that(60)1tζ−tz≤Mζ−z,ζ≠z,where 1/tζ−tz is another sign of tζ−tz−1. The inverse exists because the complex part ζ−z of tζ−tz is not zero.
Next, we consider two corresponding boundary value problems of original nonhomogeneous PDEs (28), the nonlinear Riemann boundary value problem and nonlinear Riemann-Hilbert boundary value problem. PDEs (26) can be written in the following form in the sense of Douglis algebra:(61)Dω=Fz,ω,where differential operator(62)D=∂∂z¯+qz∂∂z,where qz is a known nilpotent hypercomplex function, and(63)Fz,ω=∑k=0r−1Fkz,z¯,ω0,…,ωr−1,ω¯0,…,ω¯r−1ek,where Fk is a known complex value function of all its variables.
Assume thatIz,ω=Iz,z¯,ω,ω¯ is a hypercomplex function of independent variable z and hypercomplex variable ω. Define the Gateaux first-order differential of I about ω,ω¯,(64)δωhIz,ω=limα⟶01αIz,z¯,ω+∂h,ω¯−Iz,z¯,ω,ω¯,δω¯h¯Iz,ω=limα⟶01αIz,z¯,ω,ω¯+αh¯−Iz,z¯,ω,ω¯.
We can similarly define the second-order Gateaux differentialδωh2I,δωhδω¯h¯I,δω¯h¯2I, and so on. We utilize Γ to represent a simple smooth closed contour in the complex plane Γ, whose positive direction is counterclockwise. It divides θ into bounded interior region G+ and external unbounded region G−. Assume that(65)Mτ,ω1,ω2=∑k=0r−1Mkτ,ω01,…,ωr−11,ω02,…,ωr−12ek,is a known hypercomplex function on variable τ∈Γ, with hypercomplex elements(66)ω1=∑k=0r−1ωk1ek,ω2=∑k=0r−1ωk2ek.
Assume thatgτ is a hypercomplex function that satisfies the Holder¨ condition on Γ, whose complex part g0τ is not zero forever on Γ. We also introduce the integer notation(67)n=Indgτ=12πi∫ΓdIng0τ.
Now we can introduce the corresponding nonlinear Riemann boundary value problem.Definition 4.
(Nonlinear Riemann boundary value problem). Assume thatG+ is a bounded and simply connected region in plane θ, whose boundary Γ is a smooth closed curve, and the positive direction of Γ causes G+ to be located to the left. Note the complement of G++Γ as G−, where the origin of coordinates is located in G+. In the whole plane θ, we seek the normal block solution ωz to PDEs (61), such that a nonlinear boundary value in Γ that satisfies(68)ω+τ=gτω−τ+Mτ,ω+τ,ω−τ,τ∈Γ,has definite order m−n at infinity, where m is an integer.
Then we consider the linear Riemann boundary value problem,(69)Dω+Aω+Bω¯=Cz,z∈θ\Γ,ω+τ=ω−τ+γτ,τ∈Γ,ω−∞=0.
Assume thatAz,Bz,Cz∈Lp,2θ,p>2, are known hypercomplex functions. Note(70)α=p−2p,and known hypercomplex function γτ∈Bα0Γ on Γ. Then we represent the generating solution of differential operator D by tz. Before stating our main theorem and proof, we introduce four lemmas. The proofs of these lemmas can refer to Appendix.Lemma 1.
The Cauchy-type integral(71)φz=12πi∫Γγζtζ−tzdtζ,is a block hyperanalytic function that is equal to 0 at infinity,(72)φ+z∈Bα0G++Γ,φ−z∈Bα0G−+Γ,and the estimate establishes(73)φ+,G++Γ0,α≤N1γ,Γ0,α,φ−,G−+Γ0,α≤N1γ,Γ0,α,where N1 is a positive constant related to α,qz, and Γ. In addition, the boundary value condition satisfies on Γ,(74)φ+τ−φ−τ=γτ,τ∈Γ.Lemma 2.
Assume hypercomplex functionsAz,Bz,fz∈Lp,2θ,p>2. The integral operator K˜ is defined by(75)K˜fz=12πi∫∫θtξζi+ebζfζtζ−tzdξdη,where f is a hypercomplex function in the whole plane θ. Hypercomplex functional K˜fz satisfies:(i)
K˜fz≤Mf,θp,2,z∈θ;(ii)
K˜fz1−K˜fz2≤Mf,θp,2z1−z2α,z1,z2∈θ;(iii)
For any real numberR>1, when z≥R,(76)K˜fz≤Mf,θp,2z−α;(iv)
Hypercomplex functionalωz=K˜fz satisfies the following system in the Sobolev sense:(77)Dω+Aω+Bω¯=fz.In (i)–(iii),M is a positive constant only relative to q,A,B, p, and R. The positive number in (ii) and (iii) is(78)α=p−2p.According to Lemma2, operator K˜fz is zero at infinity and continuous in the whole plane θ. Then we can establish the expression and estimate of the solution of boundary value problem (69).Lemma 3.
Boundary value problem (69) has a unique solution,(79)ωz=φz+K˜−Aφ−Bφ¯+C,where(80)φz=12πi∫Γγζtζ−tzdtζ,and K˜ is an integral operator defined by Lemma 2.Next, we introduce two estimates of solutionωz.Lemma 4.
For solutionωz of boundary value problem (69), the following estimates hold true:(81)ω+,G++Γ0,α≤Nγ,Γ0,α+C,θp,2,ω−,G−+Γ0,α≤Nγ,Γ0,α+C,θp,2,where N is a positive constant only relative to p,qz,Az,Bz, and Γ.The same as above, if hypercomplex functionωz∈Bα0G++Γ∩Bα0G−+Γ, define(82)ω,θ0,α=ω+,G++Γ0,α+ω−,G−+Γ0,α.For solutionωz of (69), from estimates (81), we can deduce that(83)ω,θ0,α≤2Nγ,Γ0,α+C,θp,2.Now, we go back to the research on seeking solutions for nonlinear Riemann boundary value problems (61) and (68). The corresponding hypercomplex function is represented by Xz, which means the determined hyperanalytic function defined in the whole plane θ, and it satisfies boundary value condition X+τ=gτX−τ on Γ, and has −n order at infinity. Because it has complex number parts that are not zero everywhere, it has inverse 1/Xz.Making the substitution(84)ωz=Xzω˜z,The new hypercomplex functionω˜z satisfies(85)Dω˜=1XzFz,Xzω˜z,z∈θ\Γ,ω˜+τ=ω˜−τ+1X+τMτ,X+τω˜+τ,X−τω˜−τ,τ∈Γ,ω˜z=Ozm,z⟶∞.By the properties ofXz,(86)1XzFz,Xzω˜z,1X+τMτ,X+τω˜+τ,X−τω˜−τ,have the same behavior as Fz,ω, M (τ,ω+τ, and ω−τ). If m≤−1, when z⟶∞, ω˜z⟶0. If m≥0, there must exist a hyperpolynomial whose order is within m,(87)Pz=∑j=0majtzm−j,where all aj are hypercomplex constants, such that under the transformation(88)ωz=Xzω˜z+Pz.A new hypercomplex functionω˜z satisfies a similar system and boundary value condition, and when z⟶∞, ω˜z⟶0. That only needs adding some degenerating condition on hypercomplex function Fz,ω and its first order and second order Gateaux differential at infinity. We omit the specific condition.Finally, we only need to consider the solution of a nonlinear boundary value problem,(89)Dω=Fz,ω,z∈θ\Γ,ω+τ=ω−τ+Mτ,ω+τ,ω−τ,τ∈Γ,ω−∞=0.We assume the following:(i)
Hypercomplex functionFz,0∈Lp,2θ, p>2. For every fixed z∈θ, the first- and second-order Gateaux differential of Fz,ω related to ω,ω¯ exist and are continuous. For all hypercomplex elements ω,δωhF,δω¯h¯F,δωh2F, δωhδω¯h¯F,δω¯h¯2F, their coefficients about h,h¯,h2,hh¯,h¯2 belong to Lp,2θ,p>2.(ii)
Note thatα=p−2/p. Assume that for any hypercomplex ω1τ,ω2τ∈Bα0Γ, hypercomplex function Mτ,ω1τ,ω2τ as a function of τ belongs to Bα0Γ. There exists a positive constant Q, such that for any hypercomplex function(90)ω1τ,ω2τ,ω˜1τ,ω˜2τ∈Bα0Γ,we have(91)Mτ,ω1τ,ω2τ−Mτ,ω˜1τ,ω˜2τ,Γ0,α≤Qω1τ−ω˜1τ,Γ0,α+ω2τ−ω˜2τ,Γ0,α.Theorem 1.
Under the above assumptions, if positive constantQ in inequality (91) satisfies(92)2NQ<1,where N is the positive constant appearing on the right side of estimate equations (81) or (83), the solution of nonlinear boundary value problem (89), which can be written as ωz∈Bα0G++Γ∩Bα0G−+Γ, must exist, and it can be constructed by a successive approximation and continuity method.Proof.
We introduce the parameterλ,0≤λ≤1, and consider the boundary value problem with parameter λ,(93)Dω=λFz,ω,z∈θ\Γ,ω+τ=ω−τ+λMτ,ω+τ,ω−τ,τ∈Γ,ω−∞=0.
Whenλ=1, we discuss boundary value problem (89). When λ=0, boundary value problem (93) has unique solution ωz=0. Assume that there exists a solution ωz,λ0∈Bα0G++Γ∩Bα0G−+Γ of (93) for value λ0,0≤λ0<1. We want to prove that there exists a certain positive constant ε, independent of λ0, such that for all λ,λ0≤λ≤λ0+ε, (93) has solution ωz,λ∈Bα0G++Γ∩Bα0G−+Γ. Therefore, we can deduce that boundary value problem (93) has a solution when λ=1, starting from λ=0 and by finite steps, i.e., we prove the existence of a solution to PDEs (93). Note that hypercomplex function ω0z=ωz;λ0.
We demonstrate that when condition (92) is established, ω0,θ0,α is bounded. Actually, because ω0 satisfies(94)Dω0=λ0Fz,ω0=λ0Fz,ω0−Fz,0+Fz,0=λ0∫01δωhFz,sω0ds+∫01δω¯h¯Fz,sω0ds+λ0Fz,0,where h=ω0,h¯=ωc¯, we use estimate (83) to obtain(95)ω0,θ0,α≤2NMτ,ω0+τ,ω0−τ,Γ0,α+Fz,0,θp,2≤2NQω0,θ0,α+2NMτ,0,0,Γ0,α+Fz,0,θp,2.
By condition (92), we can obtain(96)ω0,θ0,α≤2N1−2NQMτ,0,0,Γ0,α+Fz,0,θp,2.
Now, we regardω0 as a zeroth order approximation, and successively determine the sequence of hypercomplex function ωlz according to the form(97)Dωl=λδωhFz,ωl−1+δω¯h¯Fz,ωl−1+λFz,ωl−1,z∈θ\Γ,ωl+τ=ωl−τ+λMτ,ωl−1+τ,ωl+1−τ,τ∈Γ,ωl−∞=0,where h=ωl−ωl−1, h¯=ωl−ωl−1¯,l≥1. Since (97) is a linear boundary value problem about ωl, if ωl−1z∈Bα0G++Γ∩Bα0G−+Γ, then by Lemmas 3 and 4, (97) has a solution ωlz that belongs to Bα0G++Γ∩Bα0G−+Γ. We utilize estimate (83) to easily prove that if ωl−1,θ0,α is bounded, ωl,θ0,α is also bounded. Therefore, hypercomplex function sequence ωlz is uniformly bounded, relying on module ⋅,θ0,α.
Now we prove the convergence of sequenceωl−1z. We consider difference(98)ηl−1=ωl−ωl−1,l≥1,satisfying, when l=1,(99)Dη0=λ−λ0Fz,ω0+λδωhFz,ω0+δω¯h¯Fz,ω0,z∈θ\Γ,η0+τ=η0−τ+λ−λ0Mτ,ω0+τ,ω0−τ,τ∈Γ,η0−∞=0,where h=η0, h¯=η0¯. When l>1,(100)Dηl−1=λδωhFz,ωl−1+δω¯h¯Fz,ωl−1+λδω,ωh1,h1fz+2δω,ω¯h1,h1¯fz+δω¯,ω¯h1¯,h1¯fz,z∈θ\Γ,ηl−1+τ=ηl−1−τ+λMτ,ωl−1+τ,ωl−1−τ−Mτ,ωl−2+τ,ωl−2−τ,τ∈Γ,ηl−1−∞=0,where h=ηl−1, h¯=ηl−1¯, h1=ηl−2, h1¯=ηl−2¯, but(101)δω,ω¯h1,h1¯fz=∫011−sδωh12Fz,sωl−1z+1−sωl−2zds.
Similarly, we can defineδω,ω¯h1,h1¯fz,δω¯,ω¯h1¯,h1¯fz. Now we can utilize (83) and assumption (i) on Fz,ω to estimate ηl−1,θ0,αl≥1:(102)η0,θ0,α≤2λ−λ0NFz,ω0,θp,2+Mτ,ω0+τ,ω0−τ,Γ0,α≤2λ−λ0NQω0,θ0,α+Mτ,0,0,Γ0,α+N4ω0,θ0,α+Fz,0,θp,2=λ−λ0N5,where N4,N5 are positive constants independent of λ,λ0. Then, from the property of the hypercomplex function ηl−2ηl−2¯≤ηl−22, we have the recurrent inequality(103)ηl−1,θ0,α≤2Nλ|Qηl−2,θ0,α+N6ηl−2,θ0,α2,where N6 is a positive constant independent of λ,λ0. We can write (103) as(104)ηl−1,θ0,α≤λ2NQ+2NN6ηl−2,θ0,αηl−2,θ0,α,l>1.
Obviously, as the hypercomplex function sequenceωlz converges according to module ⋅,θ0,α, we only need(105)λ2NQ+2NN6η0,θ0,α<1.
If inequality (105) holds, then we only need(106)2NQ+2NN6η0,θ0,α<1,or by estimate (102),(107)2NQ+2NN6λ−λ0N5<1,i.e.,(108)λ−λ0<1−2NQ2NN5N6.
When inequality (108) holds, hypercomplex function sequence ωlz has the limit ωz;λ, and obviously, ωz;λ∈Bα0G++Γ∩Bα0G−+Γ. Now we prove the limit function ωz;λ obtained in this way is the solution of boundary value problem (93). Actually, it is apparent that ωz;λ satisfies the boundary value condition on Γ of (93). Because module ωl,θ0,α is uniformly bounded, Dωl uniformly converges to Dω in any compact subset on θ\Γ, ω satisfies equation (93), and we utilize Lemma 3, assumption (ii) on Mτ,ω+τ,ω−τ, and the uniform boundedness of ωl,θ0,α to prove that when z⟶∞, the relation(109)ωlz=Oz−α,is uniformly established. Therefore, limit function ωz;λ also has the property ω−∞,λ=0. Here we find(110)ε=1−2NQ2NN5N6>0,which is independent of λ0, such that for all λ:λ0≤λ≤λ0+ε, the solution of boundary value problem (93) exists.Definition 5 (Nonlinear Riemann-Hilbert boundary value problem).
To seek a continuous solutionωz to PDEs (61) in the region G+¯, such that the nonlinear boundary value condition(111)Reω=ψz,ω,z∈Γ,12π∫ΓImωζdϕζ=χω,is satisfied in boundary Γ, where ϕζ is a conformal mapping from region G+ to the unit disk, ψz,ω is a known real hypercomplex function of variable z∈Γ and hypercomplex element ω, and χω is a known real hyperfunctional of hypercomplex function ω.
We assume:(i)
Nilpotent functionqz∈Bν0G++Γ,0<ν<1;(ii)
Hypercomplex functionFz,ω and its first- and second-order Gateaux differential δωhF,δω¯h¯F,δωh2F,δωhδω¯h¯F, δω¯h¯2F about ω,ω¯ are continuous for all variable elements, and bounded according to module ⋅,G+∪Γ0, this boundary is noted as M˜ and is a positive constant;(iii)
Real-valued hypercomplex functionψz,ωz, as function of z∈Γ, belongs to Bν1Γ for any hypercomplex element ωz∈Bν1Γ, and for any hypercomplex function ω1z,ω2z of Bν1Γ, we have the equation(112)ψz,ω1z−ψz,ω2z,Γ0,ν≤Lω1z−ω2z,Γ0,ν,whereL is a positive constant independent of ω1z,ω2z. Assume that hypercomplex function ψz,0 is bounded according to module ⋅;Γ0,ν;(iv)
Real-valued hyperfunctionalχω∈Bν1Γ, and any hyperfunction ω1z,ω2z∈Bν1Γ has(113)χω1z−χω2z≤Nω1z−ω2z,Γ0,ν,whereN is a positive constant independent of ω1z,ω2z. Assume that nonnegative number χ0 is finite.Theorem 2.
When the above assumptions (i)–(iv) on nilpotent functionqz, hypercomplex function Fz,ω, real-valued hypercomplex function ψz,ω, and real-valued hyperfunctional χω satisfy, when the positive constant L and N are small enough in inequalities (112) and (113), solution of nonlinear Riemann–Hilbert boundary value problem (61), (111)(114)ωz∈Bν1G++Γ,exists.Before the proof, we establish an estimate equation.Lemma 5.
Assume that hypercomplex functionAz,Bz,g˜z∈Bν0G+Γ, and(115)A,G++Γ0,ν≤M0,B,G++Γ0,ν≤M0,where M0 is a positive constant. For the solution ω∈Bν1G++Γ of linear boundary value problem(116)Dω=Aω+Bω¯+g˜,inG+,Reω|Γ=ψ˜z,z∈Γ,12π∫ΓImωζdϕζ=c,there exist positive constants M1, M2, M3 which are only relevant to region G+, positive integer r, positive constant M0, and ν, such that(117)ω,G++Γ0,ν≤M1ψ˜,Γ0,ν+M2c+M3g˜,G++Γ0,ν,holds, where ψ˜z∈Bν0Γ is a real-valued hypercomplex function and c is a real hypercomplex number.Now we are in position to give proof of Theorem2.Proof.
We consider a family of nonlinear Riemann-Hilbert boundary value problems with real parameterλ,0≤λ≤1:(118)Dω=λFz,ω,z∈G+,Reω|Γ=λψz,ω,z∈Γ,12π∫ΓImωζdϕζ=λχω.
We want to prove that for everyλ:0≤λ≤1, the solution ωz;λ of boundary value problem (118) exists, and belongs to Bν1G++Γ; then ωz;1 is the solution of the above Riemann-Hilbert boundary value problems (61) and (111).
Obviously, whenλ=0, boundary value problem (118) has only the zero solution ωz;0=0. If it has solution ωz;λ∈Bν1G++Γ for λ=λ1,0≤λ1<1, we prove there must exist a positive constant ε<1, independent of λ1, such that (118) has solution ωz;λ2∈Bν1G++Γ for λ2:λ1<λ2<λ1+ε≤1; then we can start from the existence of a solution to (118) for λ=0, through finite steps, to obtain ωz;1.
Now we assume thatωz;λ1∈Bν1G++Γ is the solution of boundary value problem (118) when λ=λ1,0≤λ1<1. Note that ω0z=ωz;λ1,z∈G++Γ, and by the following, a sequence of linear boundary value problems j≥1:(119)Dωj=λδωhFz,ωj−1+δω¯h¯Fz,ωj−1+Fz,ωj−1,z∈G+,Reωj|Γ=λψz,ωj−1,z∈Γ,12π∫ΓImωjζdϕζ=λχωj−1,0<λ<1.
We can determineωjz;λ, one by one, where h=ωj−ωj+1,h¯=ωj¯−ωj−1¯. Similar to the proof of the linear Riemann boundary problem, the solution of linear boundary problem (119) exists [38]. If ωj−1z;λ∈Bν1G++Γ, ωjz;λ∈Bν1G++Γ. Assuming ω0z=ωz;λ∈Bν1G++Γ, the hypercomplex function sequences ωjz;λ1 successively determined by (119) all belong to Bν1G++Γ. Then we prove the convergence of hypercomplex function sequence ωjz;λ according to module ⋅,G++Γ0,ν. So, we consider difference ωj−ωj+1, which satisfies(120)Dωj−ωj−1=λδωhFz,ωj−1+δω¯h¯Fz,ωj−1+λδω,ωh1,h1fz+2δω,ω¯h1,h1¯fz+δω¯,ω¯h1¯,h1¯fz,z∈G+,Reωj−ωj−1|Γ=λψz,ωj−1−ψz,ωj−2,z∈Γ,12π∫ΓImωj−ωj−1ζ;λdϕζ=λχωj−1−χωj−2,j>1,where δω,ωh1,h1 has the same meaning as before. For boundary value problem (120), we use Lemma 5 to obtain(121)ωj−ωj−1,G++Γ0,ν≤M1λψz,ωj−1−ψz,ωj−2,Γ0,ν+M2λχωj−1−χωj−2+M34λM˜ωj−1−ωj−2,G++Γ0,ν2.
We use inequality (112), (113) in assumption condition (iii), (iv), and (121), we obtain recursive estimate equation(122)ωj−ωj−1,G++Γ0,ν≤M1Lλωj−1−ωj−2,Γ0,ν+M2Nλωj−1−ωj−2,Γ0,ν+4λM3M˜ωj−1−ωj−2,G++Γ0,ν2≤λM1L+M2N+4M3M˜ωj−1−ωj−2,G++Γ0,ν⋅ωj−1−ωj−2,G++Γ0,ν=λH1+H2ωj−1−ωj−2,G++Γ0,ν×ωj−1−ωj−2,G++Γ0,ν,where(123)H1=M1L+M2N,H2=4M3M˜,are positive constants independent of j. From inequality (122), if(124)λH1+H2ω1−ω0,G++Γ0,ν<1,ωjz;λ must converge. Assume that the limit function of ωjz;λ is ωz;λ when j⟶∞. From the structure of ωjz;λ, we can prove ωz;λ is a solution of boundary value problem (118), and ωz;λ∈Bν1G++Γ. To establish inequality (124), it must hold true that(125)H1+H2ω1−ω0,G++Γ0,ν<1.
Now we estimate module⋅,G++Γ0,ν of the internal term ω1−ω0. By definition,(126)Dω1−ω0=λδωhFz,ω0+λδω¯h¯Fz,ω0+λ−λ1Fz,ω0,z∈G+,Reω1−ω0|Γ=λ−λ1ψz,ω0,z∈Γ,12π∫ΓImω1−ω0dϕζ=λ−λ1χω0,and we estimate equation (117) to obtain(127)ω1−ω0,G++Γ0,ν≤M1λ−λ1ψz,ω0,Γ0,ν+M2λ−λ1χω0+λ−λ1M3Fz,ω0,G++Γ0,ν≤λ−λ1M1Lω0,Γ0,ν+ψz,0,Γ0,ν+λ−λ1M2Nω0,Γ0,ν+χ0+λ−λ1M3Fz,ω0,G++Γ0,ν≤H3λ−λ1,where H3 is a positive constant independent of λ,λ1. To make inequality (125) hold true, we only need(128)H1+H2H3λ−λ1<1.
We assume that positive constantsL,N in inequalities (112), (113) are small enough that(129)H1=LM1+NM2<1.
From (128), we obtain(130)λ−λ1<1−H1H2H3.
In a word, we obtain from inequality (130) that we can choose any positive number ε that is less than 1−H1/H2H3 and independent of λ1, such that for arbitrary λ,(131)λ1<λ≤λ1+ε≤1.
The solutionωz;λ∈Bν1G++Γ of nonlinear boundary value problem (118) exists by the successive approximation method. Then we start from λ=0 and, after a finite step, we can obtain(132)ωz;1.
Therefore, we have proved the existence of a solution to the initial nonlinear Riemann-Hilbert boundary value problem (61), (111) [39, 40].


## 5. Conclusion

We revealed a fundamental structure of general stochastic dynamical systems proposed by Ao et al.. We demonstrated a scientific understanding of three essential components: the potential function, friction matrixSq, and Lorenz matrix Aq. Our goal was to prove the equivalence between general stochastic differential equations and equations after A-type decomposition, and then we could assert that the above elements are fundamental components of general stochastic dynamical systems. This problem can be transformed to proof of the existence of solutions for first-order quasilinear partial differential equations. Then we mathematically proved the existence of solutions for these equations. Specifically, when dimension n satisfies n≥4, and nn−1/2 is an even number (n=4,5,8,9,…,4i,4i+1,…, when i=1,2,3,…), the existence of the generated solutions of the homogeneous equations corresponding to these equations was proved by introducing the hypercomplex algebra proposed by Douglis. Then, by successive approximation and continuous methods, we proved the existence of solutions of the Riemann boundary value problem and Riemann-Hilbert boundary value problem corresponding to first-order quasilinear partial differential equations. Therefore, we proved this fundamental structure of general stochastic dynamical systems for the high-dimensional case.



---

*Source: 2596074-2022-06-13.xml*