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The semi-tensor product method for special least squares solutions of the complex generalized Sylvester matrix equation

  • In this paper, we are interested in the minimal norm of least squares Hermitian solution and the minimal norm of least squares anti-Hermitian solution for the complex generalized Sylvester matrix equation CXD+EXF=G. By utilizing of the real vector representations of complex matrices and the semi-tensor product of matrices, we first transform solving special least squares solutions of the above matrix equation into solving the general least squares solutions of the corresponding real matrix equations, and then obtain the expressions of the minimal norm of least squares Hermitian solution and the minimal norm of least squares anti-Hermitian solution. Further, we give two numerical algorithms and two numerical examples, and numerical examples illustrate that our proposed algorithms are more efficient and accurate.

    Citation: Fengxia Zhang, Ying Li, Jianli Zhao. The semi-tensor product method for special least squares solutions of the complex generalized Sylvester matrix equation[J]. AIMS Mathematics, 2023, 8(3): 5200-5215. doi: 10.3934/math.2023261

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  • In this paper, we are interested in the minimal norm of least squares Hermitian solution and the minimal norm of least squares anti-Hermitian solution for the complex generalized Sylvester matrix equation CXD+EXF=G. By utilizing of the real vector representations of complex matrices and the semi-tensor product of matrices, we first transform solving special least squares solutions of the above matrix equation into solving the general least squares solutions of the corresponding real matrix equations, and then obtain the expressions of the minimal norm of least squares Hermitian solution and the minimal norm of least squares anti-Hermitian solution. Further, we give two numerical algorithms and two numerical examples, and numerical examples illustrate that our proposed algorithms are more efficient and accurate.



    Variational inequality is a powerful and well-known mathematical tool which has the direct, natural, unified and easily formulation to apply in mathematics such as linear and nonlinear analysis and optimization problem, etc. In addition, it has been used as a tool for studying in many fields such as engineering, industry economics, transportation, social and pure and applied science, see [17,21,30] and the reference therein. As previously mentioned, this implies many researchers developed the variational inequality in the various aspects. For example, the mixed variational inequality has been generalized from the variational inequality where the variational inequality is a special case of the mixed variational inequality and was presented by Lescarret [19] and Browder [2]. Later, Konnov and Volotskaya [16] applied the mixed variational inequality into the general economic equilibrium problems and oligopolistic equilibrium problem. So, many researchers have applied the mixed variational inequality in many fields such as optimization, game theory, control theory, etc., see [1,4,29]. On the other hand, in the study and development of problems, the inverse problem is the interesting one. Because the formatting of a problem from one problem to another is possible, and moreover, the solutions of the two problems are also related. For this reason, some problems, which cannot solve in direct, we can take the concept of the inverse problems for solving that results. Therefore, the inverse problems are applied in many fields such as engineering, finance, economics, transportation and science etc. For example, H. Kunze et al. [13,14,15] studied the inverse problems on many problems, such as the optimization problem, the differential equation and the variational equation etc. In these works, Kunze used the Collage theorem technique for solving these inverse problems and also presented some applications in economics and applied sciences. With the interest mentioned above, we would like to study the variational inequality in the aspect of the inverse problem which many researches of the inverse variational inequality are applied in many branches such as traffic network, economic, telecommunication networks. For example, in 2008, Yang [35] considered and analyzed the dynamic power price problem on both the discrete and evolutionary cases and, moreover, described the characterization of the optimal price by the solution of the inverse variational inequality. In 2010, He et al. [7] proposed some problems in the formulation of the inverse variational inequality on a normative control problem for solving the network equilibrium state in a linearly constrained set, etc. Furthermore, the inverse variational inequality is further developed and studied, where the inverse mixed variational inequality is one that has been improved from the inverse variational inequality and has the inverse variational inequality as a special case, see [3,12]. Later, in 2014, Li et al. [20] studied the inverse mixed variational inequality problem and applied this problem to the traffic network equilibrium problem and the traffic equilibrium control problem. They used the generalized f-projection operators to obtain their results and proposed the properties of the generalized f-projection operator to obtain the convergence of the generalized f-projection algorithm for inverse mixed variational inequality. In 2016, Li and Zou [23] extended the inverse mixed variational inequality into a new class of inverse mixed quasi variational inequality. All of the above, we are interested in studying and developing the problem which is generalized from the inverse mixed variational inequality that, in this paper, will be called the generalized inverse mixed variational inequality and used a generalized f-projection operator for solving our results.

    On the other hand, a neural network (also known as a dynamical system in the mathematical literature) is the problem related to time and is a powerful tool which is used to apply in the signal processing, pattern recognition, associative memory and other engineering or scientific field, see [5,18,24,27,39]. By the characteristic of nature of parallelization and distributed information process, the neural networks have served as the promising computational models for real time applications. So, the neural networks have been designed to solve the mathematic programming and the related optimization problems, see [22,33,38] and the reference therein. From the foregoing, it will be interesting to study and develop the neural network further and can be also seen from the continuous development of research in artificial neural networks such as the following research: In 1996, A. Nagurney [28] studied the projected dynamical system and variational inequalities and also presented some applications of these problems in economics and transportation. In 2002, Xia et al. [34] presented a neural network, which has a single-layer structure and has amenable to parallel implementation and proposed the equivalence of the neural network and the variational inequality for solving the nonlinear formulation and the stability of such network. In 2015, Zou et al. [37] presented a neural network which possesses a simple one-layer structures for solving the inverse variational inequality problem and proved the stability of such network and, moreover, proposed some numerical examples. In the same year, M. A. Noor et al. [26] proposed the dynamical systems for the extended general quasi variational inequalities and proved the convergence of globally exponentially of the dynamical system. In 2021, Vuong et al. [31] considered the projected neural network for solving inverse variational inequalities and proposed the stability of the neural network. Moreover, they presented the applications of such neural network in transportation science. Later, in 2022, D. Hu et al. [6] used the neural network for solving about the optimization problems by proposing a modified projection neural network and used this network to solve the non-smooth, nonlinear and constrained convex optimization problems. Then, the existence of the solution and the stability in the Lyapunov sense of the modified projection neural network was proved. In addition, the application of the neural network in the optimization problem is not the only one mentioned here, there are many other studies that have applied the neural network into the optimization problem, see [11,36] and the reference therein. All of these neural networks, we see that it is important and interesting tool to study and develop further applications. Therefore, in this paper, we are interested to propose a neural network which associated with a generalization of the inverse mixed variational inequality problem and consider the stability of such neural network.

    Based on the above, we would like to present the main objectives of this article as follows:

    ● The generalized inverse mixed variational inequality problem is presented and the existence and uniqueness of the problem are proved.

    ● The neural network associated with the generalized inverse mixed variational inequality is proposed. The existence and stability of such neural network are proved.

    ● Finally, we introduce the iterative methods which arises from the previous neural network and also display a numerical example by using such iterative methods.

    The paper is organized as follows: In Section 2, we recall some basic definitions and theorems of the generalized f-projector operator and the neural network. In Section 3, we consider and study a generalized inverse mixed variational inequality by using the generalized f-projection operator for solving the existence and uniqueness of the generalized inverse mixed variational inequality. In Section 4, the neural network of the generalized inverse mixed variational inequality is proposed and the Wiener-Hopf equation, which the solution of the equation is equivalent to the solution of the generalized inverse mixed variational inequality, is also considered. Then, the existence and stability of such neural network are proved. Finally, author will present some iterative schemes which are constructed from the neural network and display a numerical example by using such algorithms to understand all of our theorems in this paper in Sections 5 and 6.

    Throughout for this paper, we let H be a real Hilbert space whose inner product and norm are denoted by , and , respectively. Let 2H be denoted for the class of all nonempty subset of H and K be a nonempty closed and convex subset of H. For each KH we denote by d(,K) for the usual distance function on H to K, that is d(u,K)=infvKuv, for all uH. In this paper, we will study the generalized inverse mixed variational inequality which is a generalization of the variational inequality, so we will use the generalization of the projection operator for considering this results. Then, we will introduce the concept of the generalized f-projector operator which was introduced by Wu and Huang [32].

    Definition 2.1. [32] Let H be a real Hilbert space and K be a nonempty closed and convex subset of H. We say that Pf,ρK:H2K is a generalized f-projection operator if

    Pf,ρK(x)={uKG(x,u)=infξKG(x,ξ)},for allxH

    where G:H×KR{+} is a functional defined as follows:

    G(x,ξ)=x22x,ξ+ξ2+2ρf(ξ),

    with xH,ξK,ρ is a positive number and f:KR{+} is a proper, convex and lower semicontinuous function for the set of real numbers denoted by R.

    Remark 2.1. By the definition of a generalized f-projection operator, if we let f=0 then the Pf,ρK is the usual projection operator. That is, if f=0 then G(x,ξ)=x22x,ξ+ξ2=xξ2. This implies Pf,ρK(x)={uKG(x,u)=infξKG(x,ξ)}={uKG(x,u)=infξKxξ}=PK(x).

    Later, in 2014, Li et al. [20] presented the properties of the operator Pf,ρK in Hilbert spaces as follows.

    Lemma 2.1. [20] Let H be a real Hilbert space and K be a nonempty closed and convex subset of H. Then, the following statements hold:

    (i) Pf,ρK(x) is nonempty and Pf,ρK is a single valued mapping;

    (ii) for all xH,x=Pf,ρK(x) if and only if

    xx,yx+ρf(y)ρf(x)0,yK;

    (iii) Pf,ρK is continuous.

    Theorem 2.1. [20] Let H be a real Hilbert space and K be a nonempty closed and convex subset of H. Let f:KR{+} be a proper, convex and lower semicontinuous function. Then, the following statements hold:

    (vPf,ρK(v))(uPf,ρK(u))2vu2Pf,ρK(v)Pf,ρK(u)2,

    and

    (vPf,ρK(v))(uPf,ρK(u))vu,

    for all u,vH.

    Next, the following definition is mappings which are used for solving our results.

    Definition 2.2. [23] Let H be a real Hilbert space and g,A:HH be two single valued mappings.

    (i) A is said to be a λ-strongly monotone on H if there exists a constant λ>0 such that

    AxAy,xyλxy2,x,yH.

    (ii) A is said to be a γ-Lipschitz continuous on H if there exists a constant γ>0 such that

    AxAyγxy,x,yH.

    (iii) (A,g) is said to be a μ-strongly monotone couple on H if there exists a positive constant μ>0 such that

    AxAy,g(x)g(y)μxy2,x,yH.

    On the other hand, we will recall the following well known concepts of the neural network (also known as dynamical system in the literature).

    A dynamical system

    ˙x=f(x),for allxH, (2.1)

    where f is a continuous function form H into H. A solution of (2.1) is a differentiable function x:IH where I is some intervals of R such that for all tI,

    ˙x(t)=f(x(t)).

    The following definitions, we will propose the equilibrium and the stability of the solution of the neural network as follows.

    Definition 2.3. [9]

    a) A point x is an equilibrium point for (2.1) if f(x)=0;

    b) An equilibrium point x of (2.1) is stable if, for any ε>0, there exists δ>0 such that, for every x0B(x,δ), the solution x(t) of the dynamical system with x(0)=x0 exists and is contained in B(x,ε) for all t>0, where B(x,r) denotes the open ball with center x and radius r;

    c) A stable equilibrium point x of (2.1) is asymptotically stable if there exists δ>0 such that, for every solution x(t) with x(0)B(x,δ), one has

    limtx(t)=x.

    Definition 2.4. [28] Let x(t) in (2.1). For any xK, where K is a closed convex set, let L be a real continuous function defined on a neighborhood N(x) of x, and differentiable everywhere on N(x) except possibly at x. L is called a Lyapunov function at x, if it satisfies:

    i) L(x)=0 and L(x)>0, for all xx,

    ii) ˙L(x)0 for all xx where

    ˙L(x)=ddtL(x(t))t=0. (2.2)

    Notice that, the equilibrium point x, which satisfies Definition 2.4 ii), is stable in the sense of Lyapunov.

    Definition 2.5. [26] A neural network is said to be globally convergent to the solution set X of (2.1) if, irrespective of initial point, the trajectory of neural network satisfies

    limtd(x(t),X)=0. (2.3)

    If the set X has a unique point x, then (2.3) satisfies limtx(t)=x. If the neural network is still stable at x in the Lyapunov sense, then the neural network is globally asymptotically stable at x.

    Definition 2.6. [26] The neural network is said to be globally exponentially stable with degree ω at x if, irrespective of the initial point, the trajectory of the neural network x(t) satisfies

    x(t)xc0x(t0)xexpω(tt0)

    for all tt0, where c0 and ω are positive constants independent of initial point. Notice that, if it is a globally exponentially stability then it is a globally asymptotically stable and the neural network converges arbitrarily fast.

    Lemma 2.2. [25](Gronwall) Let ˆu and ˆv be real valued nonnegative continuous functions with domain {ttt0} and let α(t)=α0(|tt0|), where α0 is a monotone increasing function. If for all tt0,

    ˆu(t)α(t)+tt0ˆu(s)ˆv(s)ds,

    then,

    ˆu(t)α(t)exptt0ˆv(s)ds.

    In this section, we will propose the generalized inverse mixed variational inequality. Let g,A:HH be two continuous mappings and f:KR{+} be a proper, convex and lower semicontinuous function. The generalized inverse mixed variational inequality is: to find an xH such that A(x)K and

    g(x),yA(x)+ρf(y)ρf(A(x))0,for allyK. (3.1)

    Remark 3.1. The generalized inverse mixed variational inequality (3.1) can be reduced to the following problems:

    (i) If g is the identity mapping. Then, (3.1) collapses to the inverse mixed variational inequality which was studied by Li et al. [20] as follows: find an xH such that A(x)K and

    x,yA(x)+ρf(y)ρf(A(x))0,yK.

    (ii) If H=Rn, where Rn denotes the real n-dimensional Euclidean space, g is the identity mapping and f(x)=0 for all xRn, then (3.1) collapses to the following inverse variational inequality: find an xRn such that A(x)K and

    x,yA(x)0,yK,

    which was proposed by He and Liu [8].

    (iii) If H=Rn,A is the identity mapping and f(x)=0 for all xRn, then (3.1) becomes the classic variational inequality, that is, to find an xRn such that

    g(x),yx0,yK.

    By using Lemma 2.1 (ii), we obtain the following t heorem.

    Theorem 3.1. Let H be a real Hilbert space and K be a nonempty closed and convex subset of H. Let f:KR{+} be a proper, convex and lower semicontinuous function. Then x is a solution of the generalized inverse mixed variational inequality (3.1) if and only if x satisfies

    A(x)=Pf,ρK[A(x)g(x)]. (3.2)

    Proof. () Let x be a solution of (3.1), that is, A(x)K and

    g(x),yA(x)+ρf(y)ρf(Ax)0

    for all yK. We have

    AxAx+g(x),yAx+ρf(y)ρf(Ax)0

    for all yK. By Lemma 2.1 (ii), we obtain

    Ax=Pf,ρK(Axg(x)).

    () Let Ax=Pf,ρK(Axg(x)). By Lemma 2.1 (ii), we have

    Ax(Ax+g(x)),yAx+ρf(y)ρf(Ax)0

    for all yK. This implies that

    g(x),yA(x)+ρf(y)ρf(Ax)0

    for all yK. We conclude that x is a solution of (3.1).

    The next theorem, we consider the existence and uniqueness of the generalized inverse mixed variational inequality (3.1) as follows.

    Theorem 3.2. Let H be a real Hilbert space and K be a nonempty closed convex subset of H,g,A:HH be Lipschitz continuous on H with constants α and β, respectively. Let f:KR{+} be a proper, convex and lower semicontinuous function. Assume that

    (i) g is a λ-strongly monotone and (A,g) is a μ-strongly monotone couple on H;

    (ii) the following condition holds

    β22μ+α2+12λ+α2<1,

    where μ<β2+α22 and λ<1+α22.

    Then, the generalized inverse mixed variational inequality (3.1) has a unique solution in H.

    Proof. Let F:HH be defined as follows: for any uH,

    F(u)=uAu+Pf,ρK(Aug(u)).

    For any x,yH, denote ˉx=Axg(x) and ˉy=Ayg(y), we have

    F(x)F(y)=xAx+Pf,ρK(Axg(x))y+AyPf,ρK(Ayg(y))=xyg(x)+g(y)(ˉxPf,ρK(ˉx)[ˉyPf,ρK(ˉy)])xyg(x)+g(y)+ˉxPf,ρK(ˉx)[ˉyPf,ρK(ˉy)].

    Since g is a λ-strongly monotone and α-Lipschitz continuous, we see that

    xyg(x)+g(y)2=xy22g(x)g(y),xy+g(x)g(y)2(12λ+α2)xy2, (3.3)

    and, by Theorem 2.1, we obtain

    (ˉxPf,ρK(ˉx))(ˉyPf,ρK(ˉy))ˉxˉy=A(x)g(x)A(y)+g(y).

    Since A is a β-Lipschitz continuous, g is a α-Lipschitz continuous and (A,g) is a μ-strongly monotone couple on H. Then,

    A(x)g(x)A(y)+g(y)2=A(x)A(y)22A(x)A(y),g(x)g(y)+g(x)g(y)2β2xy22μxy2+α2xy2=(β22μ+α2)xy2. (3.4)

    By (3.3) and (3.4), then

    F(x)F(y)12λ+α2xy+β22μ+α2xy=(12λ+α2+β22μ+α2)xy=θxy

    where θ=12λ+α2+β22μ+α2. By the assumption (ii), we have 0<θ<1. This implies that F is a contraction mapping in H. So, F has a unique fixed point in H. Therefore if x is a fixed point, then

    x=xAx+Pf,ρK(Axg(x)).

    Hence, Ax=Pf,ρK(Axg(x)). By Theorem 3.1, we conclude that x is a solution of the generalized inverse mixed variational inequality (3.1).

    In this part, firstly, we will propose the Wiener Hopf equation which the solution of the equation is equivalent to the solution of the generalized inverse mixed variational inequality (3.1). Then, we will present the neural network associated with the generalized inverse mixed variational inequality. Finally, the existence and stability of the solution of such neural network are proved as follows.

    Let g,A:HH be two continuous mappings and K be a nonempty closed and convex subset of H. Let f:KR{+} be a proper, convex and lower semicontinuous function. The Wiener Hopf Equation which is equivalent to the generalized inverse mixed variational inequality (3.1) as follows: find xH such that

    QK(A(x)g(x))+g(x)=0, (4.1)

    where QK=IPf,ρK with I is an identity operator.

    The following lemma, we will present the equivalent solution of the Wiener Hopf Equation (4.1) and the generalized inverse mixed variational inequality (3.1) problem.

    Lemma 4.1. x is a solution of the generalized inverse mixed variational inequality (3.1) if and only if x is a solution of the Wiener Hopf Equation (4.1).

    Proof. () Assume that xH is a solution of (3.1). By Theorem 3.2, we obtain that

    A(x)=Pf,ρK(A(x)g(x)).

    Since QK=IPf,ρK, we have

    QK(A(x)g(x))=(IPf,ρK)(A(x)g(x))=A(x)g(x)Pf,ρK(A(x)g(x))=A(x)g(x)A(x)=g(x).

    Then, QK(A(x)g(x))+g(x)=0.

    () Since xH is a solution of the Wiener Hopf equation (4.1), that is,

    QK(A(x)g(x))+g(x)=0.

    Since QK=IPf,ρK, we get

    (IPf,ρK)(A(x)g(x))+g(x)=0.

    Then,

    A(x)=Pf,ρK(A(x)g(x)).

    Therefore, x is a solution of (3.1).

    Next, we will propose the neural network (known as dynamical system in the literature) associated with the generalized inverse mixed variational inequality. Let H be a real Hilbert space and K be a nonempty closed and convex subset of H. Let A:HK be a Lipschitz continuous with constants β and g:HH be a Lipschitz continuous with constants α and f:KR{+} be a proper, convex and lower semicontinuous function. By Theorem 3.2, we know that the solution of the generalized inverse mixed variational inequality (3.1) exists and Lemma 4.1, we have the equivalence of the solution of (3.1) with the solution of Wiener Hopf equation (4.1). So, we obtain the following result.

    Since QK(A(x)g(x))+g(x)=0 and QK=IPf,ρK, we have

    (IPf,ρK)(A(x)g(x))+g(x)=0.

    This implies that

    A(x)Pf,ρK(A(x)g(x))=0.

    Now, we define the residue vector R(x) by the relation

    R(x)=A(x)Pf,ρK(A(x)g(x)). (4.2)

    Then, by the previous article, we see that xK is a solution of the generalized inverse mixed variational inequality if and only if xK is a zero of the equation

    R(x)=0. (4.3)

    By the equivalent formulation (3.2), we will propose the neural network associated with the generalized inverse mixed variational inequality as follows:

    dxdt=η{Pf,ρK(A(x)g(x))A(x)}, (4.4)

    which x(t0)=x0 and η is a positive constant with a positive real number t0.

    Notice that the right-hand side is related to the projection operator and is discontinuous of the boundary of K. It is clear from the definition that the solution to the neural network associated with the generalized inverse mixed variational inequality always stay in K. This implies that the qualitative results such as the existence of the solution on the given data to such neural network can be studied.

    Now, we will present the existence and uniqueness of the solution of the neural network associated with the generalized inverse mixed variational inequality (4.4).

    Theorem 4.1. Let g:HH be a Lipschitz continuous with constants α and A:HK be a Lipschitz continuous with constants β. Let f:KR{+} be a proper, convex and lower semicontinuous function. Assume that all of assumption of Theorem 3.2 hold. Then, for each x0H, there exists the unique continuous solution x(t) of the neural network associated with the generalized inverse mixed variational inequality (4.4) with x(t0)=x0 over the interval [t0,).

    Proof. Let η be a positive constant and define the mapping F:HK by

    F(x)=η{Pf,ρK[A(x)g(x)]A(x)},

    for all xH. By using Theorem 2.1 and (3.4), we obtain

    F(x)F(y)=η{Pf,ρK[A(x)g(x)]A(x)}η{Pf,ρK[A(y)g(y)]A(y)}=η{Pf,ρK[A(x)g(x)]A(x)}{Pf,ρK[A(y)g(y)]A(y)}=η[(A(y)g(y))Pf,ρK[A(y)g(y)]][(A(x)g(x))Pf,ρK[A(x)g(x)]]+g(y)g(x)η{[(A(y)g(y))Pf,ρK[A(y)g(y)]][(A(x)g(x))Pf,ρK[A(x)g(x)]]+g(y)g(x)}η{(A(y)g(y))(A(x)g(x))+g(y)g(x)}=η(β22μ+α2+α)xy.

    By the assumption (ii) of Theorem 3.2, we know that η(β22μ+α2+α)>0. Thus, F is a Lipschitz continuous. This implies that, for each x0H, there exists a unique continuous solution x(t) of (4.4), defined in initial t0t<Γ with the initial condition x(t0)=x0.

    Let [t0,Γ) be its maximal interval of existence, we will show that Γ=. Under the assumption, we obtain that (3.1) has a unique solution (say x) such that A(x)K and

    A(x)=Pf,ρK[A(x)g(x)].

    Let xH. We have

    F(x)=η{Pf,ρK[A(x)g(x)]A(x)}=ηPf,ρK[A(x)g(x)]Pf,ρK[A(x)g(x)]+A(x)A(x)=ηPf,ρK[A(x)g(x)]Pf,ρK[A(x)g(x)]+A(x)g(x)A(x)+g(x)+g(x)g(x)η{[(A(x)g(x))Pf,ρK[A(x)g(x)][(A(x)g(x))Pf,ρK[A(x)g(x)]]+g(x)g(x)}η{A(x)g(x)A(x)+g(x)+g(x)g(x)}η{(β22μ+α2)xx+αxx}=η{(β22μ+α2+α)xx}η(β22μ+α2+α)x+η(β22μ+α2+α)x.

    Hence,

    x(t)x(t0)+tt0F(s)dsx(t0)+tt0η(β22μ+α2+α)xds+tt0η(β22μ+α2+α)x(s)ds=x(t0)+η(β22μ+α2+α)x(tt0)+η(β22μ+α2+α)tt0x(s)ds=x(t0)+k1(tt0)+k2tt0x(s)ds,

    where k1=η(β22μ+α2+α)x and k2=η(β22μ+α2+α). By Gronwall's Lemma, we obtain that

    x(t){x(t0)+k1(tt0)}expk2(tt0),

    where t[t0,Γ). Therefore, the solution x(t) is bounded on [t0,Γ), if Γ is finite. We conclude that Γ=.

    Theorem 4.2. Assume that all of the assumptions of Theorem 4.1 hold and satisfy the following condition

    β22μ+α2<λ. (4.5)

    Then, the neural network associated with the generalized inverse mixed variational inequality (4.4) is globally exponentially stable and also globally asymptotically stable to the solution of the generalized inverse mixed variational inequality (3.1).

    Proof. By Theorem 4.1, we known that (3.1) has a unique continuous solution x(t) over [t0,Γ) for any fixed x0H. Let x0(t)=x(t,t0;x0) be the solution of the initial value problem (4.4) and x(t) be a solution of (3.1).

    Define the Lyapunov function L:HR by

    L(x)=12xx2,

    for all xH. We obtain that

    dLdx=dLdxdxdt=xx,dxdt=xx,η{Pf,ρK[A(x)g(x)]A(x)}=ηxx,Pf,ρK[A(x)g(x)]A(x)=ηxx,Pf,ρK[A(x)g(x)]A(x)+A(x)A(x)=ηxx,(A(x)g(x))Pf,ρK[A(x)g(x)](A(x)g(x))+Pf,ρK[A(x)g(x)]+ηxx,g(x)g(x)ηxx(A(x)g(x))Pf,ρK[A(x)g(x)](A(x)g(x))+Pf,ρK[A(x)g(x)]ηxx,g(x)g(x)ηxx(A(x)g(x))(A(x)g(x))ηλxx2ηβ22μ+α2xx2ηλxx2=η(β22μ+α2λ)xx2.

    By the assumption (4.5), we obtain that β22μ+α2λ<0. We have

    x(t)xx0x+tt0L(x(s))dsx0x+tt0η(β22μ+α2λ)x(s)xdsx0xexpθ(tt0),

    where θ=η(β22μ+α2λ). Since θ<0, this implies that (4.4) is a globally exponentially stable with degree θ at x. Moreover, we see that L(x) is a global Lyapunov function for the (4.4) and (4.4) is stable in the sense of Lyapunov. Thus, the neural network is also globally asymptotically stable. We conclude that the solution of the (4.4) converges to the unique solution of (3.1).

    By the previous article, we proposed the neural network associated with the generalized inverse mixed variational inequality. Next, we will suggest and analyze some iterative schemes which will be used for solving the solution of the generalized inverse mixed variational inequality (3.1). By using the concept of the forward difference scheme, we obtain the discretization of the neural network (4.4) with respect to the time variable t, with step size hn>0 and initial point x0H such that

    xn+1(t)xn(t)hn=η{Pf,ρK[A(xn(t))g(xn(t))]A(xn(t))}, (5.1)

    where x(t0)=x0 and η is a positive constant with a positive real number t0.

    If we let hn=1 then, by (5.1), we obtain the following iterative scheme:

    xn+1(t)=xn(t)+η{Pf,ρK[A(xn(t))g(xn(t))]A(xn(t))}, (5.2)

    where x(t0)=x0 and η is a positive constant with a positive real number t0. Hence, we will consider the following algorithm which is introduced by (5.2).

    Algorithm 1: Choose the starting point x0H with A(x0)K and fixed ρ,η are positive constants.

    Then, we compute {xn} through the following iterative scheme: Set n=0.

    Step 1: Compute

    xn+1=xn+η{Pf,ρK[A(xn)g(xn)]A(xn)}.

    If xn+1=xn, then STOP and xn is a solution. Otherwise, update n to n+1 and go to Step 1.

    Furthermore, if we use the inertial type predictor and corrector technique. Then, Algorithm 1 can be written in the following algorithm.

    Algorithm 2: Choose the starting point x0,x1H with

    y1=x1+ω1(x1x0)H

    where 0ω11 and A(y1)K. Then, we compute {xn} through the following iterative scheme: Set n=1.

    Step 1: Compute: fixed ρ and η are positive constants,

    xn+1=yn+η{Pf,ρK[A(yn)g(yn)]A(yn)}.

    If xn+1=xn, then STOP and xn is a solution. Otherwise, go to next step.

    Step 2: Set

    yn+1=xn+1+ωn+1(xn+1xn)

    where 0ωn1 and update n to n+1 and go to Step 1.

    The following theorem, we will present the convergence of the previous algorithms which converges to the solution of (3.1).

    Theorem 5.1. Assume that all of the assumptions of Theorem 3.2 hold and satisfy the following condition:

    Δ+η(Δ+α)22<λ<Δ+η(Δ+α)22+12η (5.3)

    where Δ=β22μ+α2. Then, the sequence {xn} generated by Algorithm 1 converges strongly to the unique solution of the generalized inverse mixed variational inequality (3.1).

    Proof. By the assumption of Theorem 3.2, we have (3.1) has a unique solution and we let x be a unique solution of (3.1). By Algorithm 1, we have

    xn+1=xn+η{Pf,ρK[A(xn)g(xn)]A(xn)}.

    Then,

    xn+1x2=xn+η{Pf,ρK[A(xn)g(xn)]A(xn)}x2=(xnx)+η{Pf,ρK[A(xn)g(xn)]A(xn)}2=xnx2+2ηxnx,Pf,ρK[A(xn)g(xn)]A(xn)+η2Pf,ρK(xn)[A(xn)g(xn)]A(xn)2.

    By Theorem 2.1, (3.4) and g is an \alpha -Lipschitz continuous, we have

    \begin{eqnarray} & & \left\lVert P^{f, \rho}_{K}[A(x_{n}) - g(x_{n})] - A(x_{n}) \right\rVert\\ & = & \left\lVert P^{f, \rho}_{K}[A(x_{n}) - g(x_{n})] - A(x^{*}) + A(x^{*}) - A(x_{n}) \right\rVert\\ &\leq& \left\lVert ( A(x^{*}) - g(x^{*}) ) - P^{f, \rho}_{K}[A(x^{*}) - g(x^{*})] - (A(x_{n}) - g(x_{n})) + P^{f, \rho}_{K}[A(x_{n}) - g(x_{n})]\right\rVert + \lVert g(x_{n}) - g(x^{*}) \rVert\\ &\leq& \sqrt{\beta^{2} - 2\mu + \alpha^{2}} \lVert x_{n} - x^{*} \rVert + \alpha\lVert x_{n} - x^{*} \rVert\\ & = & \left(\sqrt{\beta^{2} - 2\mu + \alpha^{2}} + \alpha\right)\lVert x_{n} - x^{*} \rVert, \end{eqnarray} (5.4)

    and, by (3.4) and g is a \lambda -strongly monotone, we get that

    \begin{eqnarray} &&\langle x_{n} - x^{*}, P^{f, \rho}_{K}[A(x_{n}) - g(x_{n})] - A(x_{n})\rangle \\ & = & \langle x_{n} - x^{*}, P^{f, \rho}_{K}[A(x_{n}) - g(x_{n})] - P^{f, \rho}_{K}[A(x^{*}) - g(x^{*})] + A(x^{*}) - A(x_{n})\rangle \\ & = & \langle x_{n} - x^{*}, (A(x^{*}) - g(x^{*})) - P^{f, \rho}_{K}[A(x^{*}) - g(x^{*})] - \left[ (A(x_{n}) - g(x_{n})) - P^{f, \rho}_{K}[A(x_{n}) - g(x_{n})]\right] \rangle\\ && + \langle x_{n} - x^{*}, g(x^{*}) - g(x_{n}) \rangle\\ &\leq& \lVert x_{n} - x^{*} \rVert \left\lVert (A(x^{*}) - g(x^{*})) - P^{f, \rho}_{K}[A(x^{*}) - g(x^{*})] - (A(x_{n}) - g(x_{n})) + P^{f, \rho}_{K}[A(x_{n}) - g(x_{n})] \right\rVert \\ & & - \langle x_{n} - x^{*}, g(x_{n}) - g(x^{*}) \rangle\\ &\leq& \lVert x_{n} - x^{*} \rVert \lVert A(x^{*}) - g(x^{*}) - A(x_{n}) + g(x_{n})\rVert - \lambda\lVert x_{n} - x^{*}\rVert^{2}\\ &\leq& \sqrt{\beta^{2} - 2\mu + \alpha^{2}}\lVert x_{n} - x^{*}\rVert^{2} - \lambda\lVert x_{n} - x^{*}\rVert^{2}\\ & = & \left( \sqrt{\beta^{2} - 2\mu + \alpha^{2}} - \lambda \right)\lVert x_{n} - x^{*}\rVert^{2}. \end{eqnarray} (5.5)

    Hence, by (5.4) and (5.5), we obtain that

    \begin{eqnarray} & & \lVert x_{n + 1} - x^{*} \rVert^{2}\\ &\leq& \lVert x_{n} - x^{*} \rVert^{2} + 2\eta( \sqrt{\beta^{2} - 2\mu + \alpha^{2}} - \lambda)\lVert x_{n} - x^{*}\rVert^{2} + \eta^{2}\left( \sqrt{\beta^{2} - 2\mu + \alpha^{2}} + \alpha \right)^{2}\lVert x_{n} - x^{*} \rVert^{2}\\ & = & \left[ 1 + 2\eta( \sqrt{\beta^{2} - 2\mu + \alpha^{2}} - \lambda) + \eta^{2}( \sqrt{\beta^{2} - 2\mu + \alpha^{2}} + \alpha)^{2} \right]\lVert x_{n} - x^{*} \rVert^{2}\\ & = & \left[ 1 + 2\eta( \Delta - \lambda) + \eta^{2}( \Delta + \alpha)^{2} \right]\lVert x_{n} - x^{*} \rVert^{2}, \end{eqnarray}

    where \Delta = \sqrt{\beta^{2} - 2\mu + \alpha^{2}}. This implies that

    \begin{eqnarray} \lVert x_{n + 1} - x^{*} \rVert^{2} \leq \Theta \lVert x_{n} - x^{*} \rVert^{2}, \end{eqnarray}

    where \Theta = 1 + 2\eta(\Delta - \lambda) + \eta^{2}(\Delta + \alpha)^{2}. By this processing, we obtain that

    \begin{eqnarray} \lVert x_{n + 1} - x^{*} \rVert^{2} &\leq& \Theta \lVert x_{n} - x^{*} \rVert^{2}\\ &\leq& \Theta^{2} \lVert x_{n - 1} - x^{*} \rVert^{2}\\ &\leq& \Theta^{3} \lVert x_{n - 2} - x^{*} \rVert^{2}\\ &\vdots& \\ &\leq& \Theta^{n + 1} \lVert x_{0} - x^{*} \rVert^{2}. \end{eqnarray}

    By the condition (5.3), we see that 0 < \Theta < 1. Then, \lVert x_{n+1} - x^{*} \rVert \rightarrow 0 as n \rightarrow \infty. We conclude that \{x_{n}\} converges to the solution of (3.1).

    Remark 5.1 1.) By Theorem 5.1, if we let x_{n+1} = y_{n} - \eta A(y_{n}) + \eta P^{f, \rho}_{K}[A(y_{n})-g(y_{n})] (in Algorithm 2), then we also obtain that Algorithm 2 converges to the solution of (3.1).

    2.) Moreover, by the condition (5.3), we see that the choice of \eta affect to \lambda, this means that if we have the value of a suitable \alpha, \lambda, \mu, \beta, then we can find the value of suitable \eta.

    In this section, we will propose the example for understanding the previous theorems and algorithms as follows.

    Example 6.1. Let H = [0, \infty) and K = H. Assume that A(x) = \frac{x}{2}, g(x) = \frac{3x}{2} and f(x) = x^{2} + 2x + 1. Here, we fix \rho = 1 and, by the condition (5.3), we can choose \eta = 0.1.

    It is easy to show that g is a Lipschitz continuous with constant \frac{3}{2} and strongly monotone with constant \frac{3}{2} and A is a Lipschitz continuous with constant \frac{1}{2}. Moreover, we can show that the definition of g and A satisfies a (A, g) \frac{3}{4} -strongly monotone couple on H. So, we obtain \alpha = \frac{3}{2}, \beta = \frac{1}{2}, \lambda = \frac{3}{2} and \mu = \frac{3}{4}. It is easy to show that the solution of (3.1) is 0 and 0 is also the solution of the neural network associated with the generalized inverse mixed variational inequality.

    The following results, we considered the numerical example by using Algorithm 1. Firstly, if we chose x_{0} = 50 and by the definition of A, it is easily to see that A(x) \in K where x \in K. The computation in SCILAB program and the computer system used was a ASUS located at the Pibulsongkham Rajabhat University at Phitsanulok, Thailand. We had the following results: It was convergent to x^* = 4.45 \times 10^{-323} in the 14514 iterations and, moreover, when we assign x_{0} to any other value, we obtain the following results:

    If we chose x_{0} = 500 then, in the 14590 iterations, it was convergent to x^*.

    If we chose x_{0} = 1000 then, in the 14603 iterations, it was convergent to x^*.

    If we chose x_{0} = 2500 then, in the 14621 iterations, it was convergent to x^*.

    If we chose x_{0} = 5000 then, in the 14635 iterations, it was convergent to x^*, see in Figure 1.

    Figure 1.  Algorithm 1.

    On the other hand, we compute this example by using Algorithm 2 and let \omega_{i} = \frac{1}{i}. If we chose x_{0} = 10 and x_{1} = 15. Then, we had the following results: it was convergent to the same solution, x^* = 4.45 \times 10^{-323}, in the 14517 iterations.

    If we chose x_{0} = 75 and x_{1} = 50 then, in the 14528 iterations, it was convergent to x^*.

    If we chose x_{0} = 100 and x_{1} = 500 then, in the 14591 iterations, it was convergent to x^*.

    If we chose x_{0} = 500 and x_{1} = 400 then, in the 14573 iterations, it was convergent to x^*, see in Figure 2.

    Figure 2.  Algorithm 2.

    Remark 6.1. By the above numerical example:

    1). Observe that in Algorithm 2, we must assume that x_{1} \geq \frac{x_{0}}{2} to guarantee that y_{1} \in H, this implies that x_{i} and y_{i} in H.

    2). if we change \rho (such as \rho = 0.01, \rho = 100 ), then we will obtain the same results with \rho = 1.

    In this work, we presented the concept of the generalized inverse mixed variational inequality and the neural network associated with the generalized inverse mixed variational inequality. The existence and uniqueness of both problems were proved. The stability of the neural network was studied by assuming some condition. For considering our results, we proposed some algorithms and used such algorithms to show our numerical example. The results in this work extend and improve the literature paper.

    The authors would like to thank Pibulsongkram Rajabhat University.

    The authors declare that they have no conflicts of interest.



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    沈阳化工大学材料科学与工程学院 沈阳 110142

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