Loading [MathJax]/jax/output/SVG/jax.js
Commentary Special Issues

Beyond Panglossian Optimism: Larger N2 Amplitudes Probably Signal a Bilingual Disadvantage in Conflict Monitoring

  • In this special issue on the brain mechanisms that lead to cognitive benefits of bilingualism we discussed six reasons why it will be very difficult to discover those mechanisms. Many of these problems apply to the article by Fernandez, Acosta, Douglass, Doshi, and Tartar that also appears in the special issue. These concerns include the following: 1) an overly optimistic assessment of the replicability of bilingual advantages in behavioral studies, 2) reliance on risky small samples sizes, 3) failures to match the samples on demographic characteristics such as immigrant status, and 4) language group differences that occur in neural measures (i.e., N2 amplitude), but not in the behavioral data. Furthermore the N2 amplitude measure in general suffers from valence ambiguity: larger N2 amplitudes reported for bilinguals are more likely to reflect poorer conflict resolution rather than enhanced inhibitory control.

    Citation: Kenneth R. Paap, Oliver M. Sawi, Chirag Dalibar, Jack Darrow, Hunter A. Johnson. Beyond Panglossian Optimism: Larger N2 Amplitudes Probably Signal a Bilingual Disadvantage in Conflict Monitoring[J]. AIMS Neuroscience, 2015, 2(1): 1-6. doi: 10.3934/Neuroscience.2015.1.1

    Related Papers:

    [1] Zubair Ahmad, Zahra Almaspoor, Faridoon Khan, Sharifah E. Alhazmi, M. El-Morshedy, O. Y. Ababneh, Amer Ibrahim Al-Omari . On fitting and forecasting the log-returns of cryptocurrency exchange rates using a new logistic model and machine learning algorithms. AIMS Mathematics, 2022, 7(10): 18031-18049. doi: 10.3934/math.2022993
    [2] Naif Alotaibi, A. S. Al-Moisheer, Ibrahim Elbatal, Salem A. Alyami, Ahmed M. Gemeay, Ehab M. Almetwally . Bivariate step-stress accelerated life test for a new three-parameter model under progressive censored schemes with application in medical. AIMS Mathematics, 2024, 9(2): 3521-3558. doi: 10.3934/math.2024173
    [3] Jumanah Ahmed Darwish, Saman Hanif Shahbaz, Lutfiah Ismail Al-Turk, Muhammad Qaiser Shahbaz . Some bivariate and multivariate families of distributions: Theory, inference and application. AIMS Mathematics, 2022, 7(8): 15584-15611. doi: 10.3934/math.2022854
    [4] Alanazi Talal Abdulrahman, Khudhayr A. Rashedi, Tariq S. Alshammari, Eslam Hussam, Amirah Saeed Alharthi, Ramlah H Albayyat . A new extension of the Rayleigh distribution: Methodology, classical, and Bayes estimation, with application to industrial data. AIMS Mathematics, 2025, 10(2): 3710-3733. doi: 10.3934/math.2025172
    [5] Baishuai Zuo, Chuancun Yin . Stein’s lemma for truncated generalized skew-elliptical random vectors. AIMS Mathematics, 2020, 5(4): 3423-3433. doi: 10.3934/math.2020221
    [6] Aisha Fayomi, Ehab M. Almetwally, Maha E. Qura . A novel bivariate Lomax-G family of distributions: Properties, inference, and applications to environmental, medical, and computer science data. AIMS Mathematics, 2023, 8(8): 17539-17584. doi: 10.3934/math.2023896
    [7] Walid Emam . Benefiting from statistical modeling in the analysis of current health expenditure to gross domestic product. AIMS Mathematics, 2023, 8(5): 12398-12421. doi: 10.3934/math.2023623
    [8] Qian Li, Qianqian Yuan, Jianhua Chen . An efficient relaxed shift-splitting preconditioner for a class of complex symmetric indefinite linear systems. AIMS Mathematics, 2022, 7(9): 17123-17132. doi: 10.3934/math.2022942
    [9] Yanting Xiao, Wanying Dong . Robust estimation for varying-coefficient partially linear measurement error model with auxiliary instrumental variables. AIMS Mathematics, 2023, 8(8): 18373-18391. doi: 10.3934/math.2023934
    [10] Guangshuai Zhou, Chuancun Yin . Family of extended mean mixtures of multivariate normal distributions: Properties, inference and applications. AIMS Mathematics, 2022, 7(7): 12390-12414. doi: 10.3934/math.2022688
  • In this special issue on the brain mechanisms that lead to cognitive benefits of bilingualism we discussed six reasons why it will be very difficult to discover those mechanisms. Many of these problems apply to the article by Fernandez, Acosta, Douglass, Doshi, and Tartar that also appears in the special issue. These concerns include the following: 1) an overly optimistic assessment of the replicability of bilingual advantages in behavioral studies, 2) reliance on risky small samples sizes, 3) failures to match the samples on demographic characteristics such as immigrant status, and 4) language group differences that occur in neural measures (i.e., N2 amplitude), but not in the behavioral data. Furthermore the N2 amplitude measure in general suffers from valence ambiguity: larger N2 amplitudes reported for bilinguals are more likely to reflect poorer conflict resolution rather than enhanced inhibitory control.


    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 $ \langle \cdot, \cdot \rangle $ and $ \lVert \cdot \rVert $, respectively. Let $ 2^H $ be denoted for the class of all nonempty subset of $ H $ and $ K $ be a nonempty closed and convex subset of $ H. $ For each $ K \subseteq H $ we denote by $ d(\cdot, K) $ for the usual distance function on $ H $ to $ K, $ that is $ d(u, K) = \inf_{v \in K}\lVert u - v \rVert, $ for all $ u \in H. $ 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 $ P^{f, \rho}_{K}: H \rightarrow 2^{K} $ is a generalized $ f $-projection operator if

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

    where $ G: H \times K \rightarrow \mathbb{R} \cup \{+\infty\} $ is a functional defined as follows:

    $ G(x,ξ)=x22x,ξ+ξ2+2ρf(ξ), $

    with $ x \in H, \xi \in K, \rho $ is a positive number and $ f: K \rightarrow \mathbb{R} \cup \{+\infty\} $ is a proper, convex and lower semicontinuous function for the set of real numbers denoted by $ \mathbb{R}. $

    Remark 2.1. By the definition of a generalized $ f $-projection operator, if we let $ f = 0 $ then the $ P^{f, \rho}_{K} $ is the usual projection operator. That is, if $ f = 0 $ then $ G(x, \xi) = \lVert x \rVert^{2} - 2\langle x, \xi \rangle + \lVert \xi \rVert^{2} = \lVert x - \xi \rVert^{2}. $ This implies $ P^{f, \rho}_{K}(x) = \left\{ u \in K \arrowvert G(x, u) = \inf_{\xi \in K}G(x, \xi) \right\} = \left\{ u \in K \arrowvert G(x, u) = \inf_{\xi \in K}\lVert x - \xi\rVert \right\} = P_{K}(x). $

    Later, in 2014, Li et al. [20] presented the properties of the operator $ P^{f, \rho}_{K} $ 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) $ P^{f, \rho}_{K}(x) $ is nonempty and $ P^{f, \rho}_{K} $ is a single valued mapping;

    (ii) for all $ x \in H, x^{*} = P^{f, \rho}_{K}(x) $ if and only if

    $ xx,yx+ρf(y)ρf(x)0,yK; $

    (iii) $ P^{f, \rho}_{K} $ 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: K \rightarrow \mathbb{R}\cup\{+\infty\} $ 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, v \in H. $

    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: H \rightarrow H $ be two single valued mappings.

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

    $ AxAy,xyλxy2,x,yH. $

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

    $ AxAyγxy,x,yH. $

    (iii) $ (A, g) $ is said to be a $ \mu $-strongly monotone couple on $ H $ if there exists a positive constant $ \mu > 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: \mathbb{I} \rightarrow H $ where $ \mathbb{I} $ is some intervals of $ \mathbb{R} $ such that for all $ t \in \mathbb{I}, $

    $ ˙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 $ \varepsilon > 0, $ there exists $ \delta > 0 $ such that, for every $ x_{0} \in B(x^{*}, \delta), $ the solution $ x(t) $ of the dynamical system with $ x(0) = x_{0} $ exists and is contained in $ B(x^{*}, \varepsilon) $ 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 $ \delta > 0 $ such that, for every solution $ x(t) $ with $ x(0) \in B(x^{*}, \delta), $ one has

    $ limtx(t)=x. $

    Definition 2.4. [28] Let $ x(t) $ in (2.1). For any $ x^{*} \in K, $ 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 $ x \neq x^{*}, $

    ii) $ \dot{L}(x) \leq 0 $ for all $ x \neq x^{*} $ 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 $ \lim\limits_{t \rightarrow \infty} x(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 $ \omega $ 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 $ t \geq t_{0}, $ where $ c_{0} $ and $ \omega $ 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 $ \hat{u} $ and $ \hat{v} $ be real valued nonnegative continuous functions with domain $ \{t \arrowvert t \geq t_{0}\} $ and let $ \alpha(t) = \alpha_{0}(\lvert t - t_{0} \rvert), $ where $ \alpha_{0} $ is a monotone increasing function. If for all $ t \geq t_{0}, $

    $ 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: H \rightarrow H $ be two continuous mappings and $ f: K \rightarrow \mathbb{R} \cup \{+\infty\} $ be a proper, convex and lower semicontinuous function. The generalized inverse mixed variational inequality is: to find an $ x^* \in H $ such that $ A(x^*) \in 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 $ x^* \in H $ such that $ A(x^*) \in K $ and

    $ x,yA(x)+ρf(y)ρf(A(x))0,yK. $

    (ii) If $ H = \mathbb{R}^{n}, $ where $ \mathbb{R}^{n} $ denotes the real $ n $-dimensional Euclidean space, $ g $ is the identity mapping and $ f(x) = 0 $ for all $ x \in \mathbb{R}^{n}, $ then (3.1) collapses to the following inverse variational inequality: find an $ x^* \in \mathbb{R}^{n} $ such that $ A(x^*) \in K $ and

    $ x,yA(x)0,yK, $

    which was proposed by He and Liu [8].

    (iii) If $ H = \mathbb{R}^{n}, A $ is the identity mapping and $ f(x) = 0 $ for all $ x \in \mathbb{R}^{n}, $ then (3.1) becomes the classic variational inequality, that is, to find an $ x^* \in \mathbb{R}^{n} $ 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: K \rightarrow \mathbb{R}\cup\{+\infty\} $ 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. $ (\Rightarrow) $ Let $ x^* $ be a solution of (3.1), that is, $ A(x^*) \in K $ and

    $ g(x),yA(x)+ρf(y)ρf(Ax)0 $

    for all $ y \in K. $ We have

    $ AxAx+g(x),yAx+ρf(y)ρf(Ax)0 $

    for all $ y \in K. $ By Lemma 2.1 (ii), we obtain

    $ Ax=Pf,ρK(Axg(x)). $

    $ (\Leftarrow) $ Let $ Ax^* = P^{f, \rho}_{K}(Ax^* - g(x^*)). $ By Lemma 2.1 (ii), we have

    $ Ax(Ax+g(x)),yAx+ρf(y)ρf(Ax)0 $

    for all $ y \in K. $ This implies that

    $ g(x),yA(x)+ρf(y)ρf(Ax)0 $

    for all $ y \in K. $ 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: H \rightarrow H $ be Lipschitz continuous on $ H $ with constants $ \alpha $ and $ \beta, $ respectively. Let $ f: K \rightarrow \mathbb{R}\cup\{+\infty\} $ be a proper, convex and lower semicontinuous function. Assume that

    (i) $ g $ is a $ \lambda $-strongly monotone and $ (A, g) $ is a $ \mu $-strongly monotone couple on $ H; $

    (ii) the following condition holds

    $ \sqrt{\beta^{2} - 2\mu + \alpha^{2}} + \sqrt{1 -2\lambda + \alpha^{2}} < 1, $

    where $ \mu < \frac{\beta^{2} + \alpha^{2}}{2} $ and $ \lambda < \frac{1 + \alpha^{2}}{2}. $

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

    Proof. Let $ F: H \rightarrow H $ be defined as follows: for any $ u \in H, $

    $ F(u)=uAu+Pf,ρK(Aug(u)). $

    For any $ x, y \in H, $ denote $ \bar{x} = Ax - g(x) $ and $ \bar{y} = Ay - g(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 $ \lambda $-strongly monotone and $ \alpha $-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 $ \beta $-Lipschitz continuous, $ g $ is a $ \alpha $-Lipschitz continuous and $ (A, g) $ is a $ \mu $-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 $ \theta = \sqrt{1 - 2\lambda + \alpha^{2}} + \sqrt{\beta^2 - 2\mu + \alpha^{2}}. $ By the assumption (ii), we have $ 0 < \theta < 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^* = P^{f, \rho}_{K}(Ax^* - g(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: H \rightarrow H $ be two continuous mappings and $ K $ be a nonempty closed and convex subset of $ H $. Let $ f: K \rightarrow \mathbb{R} \cup \{+\infty\} $ 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 $ x^{*} \in H $ such that

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

    where $ Q_{K} = I - P^{f, \rho}_{K} $ 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. $ (\Rightarrow) $ Assume that $ x^{*} \in H $ is a solution of (3.1). By Theorem 3.2, we obtain that

    $ A(x)=Pf,ρK(A(x)g(x)). $

    Since $ Q_{K} = I - P^{f, \rho}_{K}, $ 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, $ Q_{K}(A(x^{*}) - g(x^{*})) + g(x^{*}) = 0. $

    $ (\Leftarrow) $ Since $ x^{*} \in H $ is a solution of the Wiener Hopf equation (4.1), that is,

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

    Since $ Q_{K} = I - P^{f, \rho}_{K}, $ 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: H \rightarrow K $ be a Lipschitz continuous with constants $ \beta $ and $ g: H \rightarrow H $ be a Lipschitz continuous with constants $ \alpha $ and $ f: K \rightarrow \mathbb{R}\cup\{+\infty\} $ 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 $ Q_{K}(A(x^{*}) - g(x^{*})) + g(x^{*}) = 0 $ and $ Q_{K} = I - P^{f, \rho}_{K}, $ 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 $ x \in K $ is a solution of the generalized inverse mixed variational inequality if and only if $ x \in K $ 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(t_{0}) = x_{0} $ and $ \eta $ is a positive constant with a positive real number $ t_{0}. $

    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: H \rightarrow H $ be a Lipschitz continuous with constants $ \alpha $ and $ A: H \rightarrow K $ be a Lipschitz continuous with constants $ \beta. $ Let $ f: K \rightarrow \mathbb{R}\cup\{+\infty\} $ be a proper, convex and lower semicontinuous function. Assume that all of assumption of Theorem 3.2 hold. Then, for each $ x_{0} \in H, $ there exists the unique continuous solution $ x(t) $ of the neural network associated with the generalized inverse mixed variational inequality (4.4) with $ x(t_{0}) = x_{0} $ over the interval $ [t_{0}, \infty). $

    Proof. Let $ \eta $ be a positive constant and define the mapping $ F: H \rightarrow K $ by

    $ F(x)=η{Pf,ρK[A(x)g(x)]A(x)}, $

    for all $ x \in H. $ 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 $ \eta \left(\sqrt{\beta^{2} - 2\mu + \alpha^{2}} + \alpha\right) > 0. $ Thus, $ F $ is a Lipschitz continuous. This implies that, for each $ x_{0} \in H, $ there exists a unique continuous solution $ x(t) $ of (4.4), defined in initial $ t_{0} \leq t < \Gamma $ with the initial condition $ x(t_{0}) = x_{0}. $

    Let $ [t_{0}, \Gamma) $ be its maximal interval of existence, we will show that $ \Gamma = \infty. $ Under the assumption, we obtain that (3.1) has a unique solution (say $ x^{*} $) such that $ A(x^{*}) \in K $ and

    $ A(x)=Pf,ρK[A(x)g(x)]. $

    Let $ x \in H. $ 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 $ k_{1} = \eta (\sqrt{\beta^{2} - 2\mu + \alpha^{2}} + \alpha)\lVert x^{*} \rVert $ and $ k_{2} = \eta (\sqrt{\beta^{2} - 2\mu + \alpha^{2}} + \alpha). $ By Gronwall's Lemma, we obtain that

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

    where $ t \in [t_{0}, \Gamma). $ Therefore, the solution $ x(t) $ is bounded on $ [t_{0}, \Gamma), $ if $ \Gamma $ is finite. We conclude that $ \Gamma = \infty. $

    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 $ [t_{0}, \Gamma) $ for any fixed $ x_{0} \in H. $ Let $ x_{0}(t) = x(t, t_{0}; x_{0}) $ be the solution of the initial value problem (4.4) and $ x^{*}(t) $ be a solution of (3.1).

    Define the Lyapunov function $ L: H \rightarrow \mathbb{R} $ by

    $ L(x)=12xx2, $

    for all $ x \in H. $ 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 $ \sqrt{\beta^{2} - 2\mu + \alpha^{2}} - \lambda < 0. $ We have

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

    where $ \theta = \eta(\sqrt{\beta^{2} - 2\mu + \alpha^{2}} - \lambda). $ Since $ \theta < 0, $ this implies that (4.4) is a globally exponentially stable with degree $ -\theta $ 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 $ h_{n} > 0 $ and initial point $ x_{0} \in H $ such that

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

    where $ x(t_{0}) = x_{0} $ and $ \eta $ is a positive constant with a positive real number $ t_{0}. $

    If we let $ h_{n} = 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(t_{0}) = x_{0} $ and $ \eta $ is a positive constant with a positive real number $ t_{0}. $ Hence, we will consider the following algorithm which is introduced by (5.2).

    Algorithm 1: Choose the starting point $ x_{0} \in H $ with $ A(x_{0}) \in K $ and fixed $ \rho, \eta $ are positive constants.

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

    Step 1: Compute

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

    If $ x_{n+1} = x_{n}, $ then STOP and $ x_{n} $ 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 $ x_{0}, x_{1} \in H $ with

    $ y1=x1+ω1(x1x0)H $

    where $ 0 \leq \omega_{1} \leq 1 $ and $ A(y_{1}) \in K. $ Then, we compute $ \{x_{n}\} $ through the following iterative scheme: Set $ n = 1. $

    Step 1: Compute: fixed $ \rho $ and $ \eta $ are positive constants,

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

    If $ x_{n+1} = x_{n}, $ then STOP and $ x_{n} $ is a solution. Otherwise, go to next step.

    Step 2: Set

    $ yn+1=xn+1+ωn+1(xn+1xn) $

    where $ 0 \leq \omega_{n} \leq 1 $ 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 $ \Delta = \sqrt{\beta^{2} - 2\mu + \alpha^{2}}. $ Then, the sequence $ \{x_{n}\} $ 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

    $ Pf,ρK[A(xn)g(xn)]A(xn)=Pf,ρK[A(xn)g(xn)]A(x)+A(x)A(xn)(A(x)g(x))Pf,ρK[A(x)g(x)](A(xn)g(xn))+Pf,ρK[A(xn)g(xn)]+g(xn)g(x)β22μ+α2xnx+αxnx=(β22μ+α2+α)xnx, $ (5.4)

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

    $ xnx,Pf,ρK[A(xn)g(xn)]A(xn)=xnx,Pf,ρK[A(xn)g(xn)]Pf,ρK[A(x)g(x)]+A(x)A(xn)=xnx,(A(x)g(x))Pf,ρK[A(x)g(x)][(A(xn)g(xn))Pf,ρK[A(xn)g(xn)]]+xnx,g(x)g(xn)xnx(A(x)g(x))Pf,ρK[A(x)g(x)](A(xn)g(xn))+Pf,ρK[A(xn)g(xn)]xnx,g(xn)g(x)xnxA(x)g(x)A(xn)+g(xn)λxnx2β22μ+α2xnx2λxnx2=(β22μ+α2λ)xnx2. $ (5.5)

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

    $ xn+1x2xnx2+2η(β22μ+α2λ)xnx2+η2(β22μ+α2+α)2xnx2=[1+2η(β22μ+α2λ)+η2(β22μ+α2+α)2]xnx2=[1+2η(Δλ)+η2(Δ+α)2]xnx2, $

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

    $ xn+1x2Θxnx2, $

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

    $ xn+1x2Θxnx2Θ2xn1x2Θ3xn2x2Θn+1x0x2. $

    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.

    [1] Paap KR, Sawi OM, Dalibar C, et al. (2014) The Brain Mechanisms Underlying the Cognitive Benefits of Bilingualism may be Extraordinarily Difficult to Discover. AIMS Neuroscience 1:245-256. doi: 10.3934/Neuroscience.2014.3.245
    [2] Fernandez M, Acosta J, Douglass K, et al. (2014) Speaking Two Languages Enhances an Auditory but not a Visual Neural Marker of Cognitive Inhibition. AIMS Neuroscience 1:145-157.
    [3] Paap KR, Greenberg ZI (2013) There is no coherent evidence for a bilingual advantage in executive processing. Cogn Psychol 66: 232-258. doi: 10.1016/j.cogpsych.2012.12.002
    [4] Paap KR, Johnson HA, Sawi O (2014) Are bilingual advantages dependent upon specific tasks or specific bilingual experiences? J Cogn Psychol 26: 615-639. doi: 10.1080/20445911.2014.944914
    [5] Paap KR (2014) The role of componential analysis, categorical hypothesizing, replicability and confirmation bias in testing for bilingual advantages in executive functioning. J CognPsychol 26:242-255.
    [6] Paap KR, Liu Y (2014) Conflict resolution in sentence processing is the same for bilinguals and monolinguals: The role of confirmation bias in testing for bilingual advantages. J Neurolinguistics 27: 50-74. doi: 10.1016/j.jneuroling.2013.09.002
    [7] Paap KR, Sawi O (2014) Bilingual advantages in executive functioning: problems in convergent validity, discriminant validity, and the identification of the theoretical constructs. Frontiers in Psychology 5: 962.
    [8] Rouder JN, Speckman PL, Sun D, et al. (2009) Bayesian t tests for accepting and rejecting the null hypothesis. Psychonomic Bull Rev 16: 225-237. doi: 10.3758/PBR.16.2.225
    [9] Valian V (2014) Bilingualism and cognition. Biling Lang Cogn 18: 3-24.
    [10] Paap KR (2014) Do many hones dull the bilingual whetstone? Biling Lang Cogn 18: 41-42.
    [11] Fernandez M, Tartar JL, Padron D, et al. (2013) Neurophysiological marker of inhibition distinguishes language groups on a non-linguistic executive function test. Brain Cognition 83:330-336. doi: 10.1016/j.bandc.2013.09.010
    [12] Falkenstein M, Hoormann J, Johnsbein J (1999) ERP components in Go/NoGo tasks and their relation to inhibition. Acta Psychol 101.
    [13] Lamm C, Zelazo PD, Lewis MD (2006) Neural correlates of cognitive control in childhood and adolescence: disentangling the contributions of age and executive function. Neuropsychologia 44:2139-2148. doi: 10.1016/j.neuropsychologia.2005.10.013
    [14] Espinet SD, Anderson JE, Zelazo PD (2012) N2 amplitude as a neural marker of executive function in young children: an ERP study of children who switch versus perseverate on the Dimensional Change Card Sort. Dev Cogn Neurosci 2 Suppl 1: S49-58.
    [15] Kousaie S, Phillips NA (2012) Conflict monitoring and resolution: Are two languages better than one? Brain Res 1446: 71-90. doi: 10.1016/j.brainres.2012.01.052
    [16] Kornblum S, Hasbroucq T, Osman A (1990) Dimensional Overlap: Cognitive Basis for Stimulus-Response Compatibility - A Model and Taxonomy. Psychol Rev 97: 253-270. doi: 10.1037/0033-295X.97.2.253
    [17] Nigg JT (2000) On inhibition/disinhibition in developmental psychopathology: Views from cognitive and personality psychology and a working inhibition taxonomy. Psychol Bulletin 126:220-246. doi: 10.1037/0033-2909.126.2.220
    [18] Bunge SA, Dudukovic NM, Thomason ME, et al. (2002) Immature frontal lobe contributions to cogntive control in children: evidence from fMRI. Neuron 33: 301-311. doi: 10.1016/S0896-6273(01)00583-9
    [19] Luk G, Anderson JA, Craik FI, et al. (2010) Distinct neural correlates for two types of inhibition in bilinguals: response inhibition versus interference suppression. Brain Cogn 74: 347-357. doi: 10.1016/j.bandc.2010.09.004
    [20] Friedman NP, Miyake A (2004) The Relations Among Inhibition and Interference Control Functions: A Latent-Variable Analysis. J ExpPsychol Gen133: 101-135.
    [21] Duckworth L, Kern ML (2011) A meta-analysis of the convergent validity of self-control measures. J Res Personal 45: 259-268. doi: 10.1016/j.jrp.2011.02.004
    [22] Votruba KL, Langenecker SA (2013) Factor structure, construct validity, and age- and education-based normative data for the Parametric Go/No-Go Test. J Clin Exp Neuropsychol 35:132-146. doi: 10.1080/13803395.2012.758239
  • This article has been cited by:

    1. Xiaolin Qu, Wei Li, Chenkai Xing, Xueping Luo, Stability analysis for set-valued inverse mixed variational inequalities in reflexive Banach spaces, 2023, 2023, 1029-242X, 10.1186/s13660-023-03060-7
    2. Pham Ky Anh, Trinh Ngoc Hai, Regularized dynamics for monotone inverse variational inequalities in hilbert spaces, 2024, 25, 1389-4420, 2295, 10.1007/s11081-024-09882-8
  • Reader Comments
  • © 2015 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(5686) PDF downloads(1162) Cited by(9)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog