The influence of an appropriate reporting time and publicity intensity on the spread of infectious diseases

Chang Hou; Qiubao Wang; Chang Hou; Qiubao Wang

doi:10.3934/math.20231199

AIMS Mathematics

2023, Volume 8, Issue 10: 23578-23602. doi: 10.3934/math.20231199

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Research article

The influence of an appropriate reporting time and publicity intensity on the spread of infectious diseases

Chang Hou ,
Qiubao Wang ^,

Department of Mathematical and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China

Received: 10 May 2023 Revised: 17 June 2023 Accepted: 30 June 2023 Published: 31 July 2023
MSC : 37H20, 37N25

We present a stochastic time-delay susceptible-exposed-asymptomatic-symptom-vaccinated-recovered (SEAQVR) model with media publicity effect in this study. The model takes into account the impacts of noise, time delay and public sensitivity on infectious illness propagation. The stochastic dynamics of the system are analyzed at the Hopf bifurcation, using time delay and noise intensity as bifurcation parameters, and the theoretical conclusions are validated using numerical simulation. Increasing the time delay and sensitivity coefficient can effectively delay the occurrence of the peak number of infected individuals and mitigate the extent of infection. Additionally, time delay and noise intensity are shown to have specific thresholds, beyond which periodic infections occur. Notably, heightened public sensitivity reduces the threshold for time delay, and media publicity directly affects public sensitivity. The numerical simulation reveals that increasing media publicity intensity does not always yield better results, and that the sensitivity of the public at present is an important reference index for setting an appropriate publicity intensity.

Keywords:

Citation: Chang Hou, Qiubao Wang. The influence of an appropriate reporting time and publicity intensity on the spread of infectious diseases[J]. AIMS Mathematics, 2023, 8(10): 23578-23602. doi: 10.3934/math.20231199

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Abstract

1. Introduction

Let $M_1, M_2\ldots, M_ s\in\mathbb{R}^{n\times n}$ , and let $u_1(z), u_2(z), \ldots, u_ s(z): \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ , be $s$ nonlinear mappings. The definition of the vertical nonlinear complementarity problem (abbreviated as VNCP) is: Find $z\in\mathbb{R}^{n}$ such that

$\begin{equation} r_i = M_iz+u_i(z), 1\leq i\leq s,\; \; \text{with}\; \; \min\{z,r_1,\ldots,r_ s\} = 0. \end{equation}$

(1.1)

Here, the minimum operation is taken component-wise.

The VNCP has many applications in generalized Leontief input-output models, control theory, nonlinear networks, contact problems in lubrication, and so on; e.g., see ^{[11,18,34,35,39]}. Taking $u_i(z) = u_i\in\mathbb{R}^{n}, 1\leq i\leq s$ , the VNCP reduces to the vertical linear complementarity problem (abbreviated as VLCP) ^[9,20,40]. Further, taking $s = 1$ , the VLCP reduces to the linear complementarity problem (abbreviated as LCP) ^[10].

In earlier literatures, there were some iterative methods for such problems. In ^[39], the Mangasarian's general iterative algorithm was constructed to solve the VLCP. Solving an approximation equation of the nonlinear complementarity problem by a continuation method can be extended to solve the VNCP; see ^[8]. Furthermore, in ^[37] the authors approximated the equivalent minimum equation of the VNCP by a sequence of aggregation functions, and found the zero solution of the approximated problems. Moreover, some interior-point methods were applied to solve complementarity problems, such as LCP ^[24,29], nonlinear complementarity problems (abbreviated as NCP) ^[36] and horizontal LCP ^[46]. In recent years, modulus-based matrix splitting (abbreviated as MMS) iteration methods have gained popularity for numerical solutions to various complementarity problems. References ^[2,12,15,16] detail their application to the LCP, while ^[13,32,50] focus on the horizontal LCP. The second-order cone LCP is discussed in ^[27], implicit complementarity problems in ^[7,14,25], quasi-complementarity problem in ^[38], NCP in ^[41], and the circular cone NCP is addressed in ^[28]. Numerical examples have demonstrated that MMS iteration methods often outperform state-of-the-art smooth Newton methods in practical applications. Specifically, for the VLCP, a special case of the VNCP, the MMS iteration method was introduced in ^[31]. Alternatively, an MMS iteration method without auxiliary variables based on a non-auxiliary-variable equivalent modulus equation was presented in ^[23] and shown to be more efficient than the method in ^[31]. Accelerated techniques and improved results for MMS iteration methods in the VLCP are further detailed in ^{[21,48,49,51]}. On the other hand, Projected type methods were also used to solve the VLCP; see ^[6,33]. For the VNCP, the only literature on the MMS iteration method currently is ^[42].

To improve the convergence rate of the MMS iteration method for solving the VNCP, in this work, we aim to construct a two-step MMS iteration method. The two-step splitting technique had been successfully used in other complementarity problems, e.g., see ^{[43,44,45,52]}, where the main idea is to change the iteration in the MMS to two iterations based on two matrix splittings of the system matrices, which can make full use of the information of the matrices for acceleration.

In the following, after presenting some required preliminaries, the new two-step MMS iteration method is established in Section 2. The convergence analysis of the proposed method is given in Section 3, which can generalize and improve the results in ^[42]. Some numerical examples are presented to illustrate the efficiency of the proposed method in Section 4, and concluding remarks of the whole work are presented in Section 5.

2. New method

First, some notations, definitions, and existing results needed in the following discussion are introduced.

Let $M = (m_{ij})\in\mathbb{R}^{n\times{n}}$ and $M = D_M-B_M$ , where $D_M$ and $-B_M$ are the diagonal and the nondiagonal matrices of $M$ , respectively. For two matrices $M$ and $N$ , the order in inequality $M\geq(>) N$ means $m_{ij}\geq(>)n_{ij}$ for every $i, j$ . Let $\lvert M\rvert = (\lvert m_{ij}\rvert)$ and the comparison matrix of $M$ be denoted by $\langle{M}\rangle = (\langle{m_{ij}}\rangle)$ , where $\langle{m_{ij}}\rangle = \lvert m_{ij}\rvert$ if $i = j$ and $\langle{m_{ij}}\rangle = -\lvert m_{ij}\rvert$ if $i\neq{j}$ . If $m_{ij}\leq0$ for any $i\neq{j}$ , $M$ is called a $Z$ -matrix. If $M$ is a nonsingular $Z$ -matrix and $M^{-1}\geq0$ , it is called a nonsingular $M$ -matrix. $M$ is called an $H$ -matrix if $\langle{M}\rangle$ is a nonsingular $M$ -matrix. If $\lvert m_{ii}\rvert > \sum\limits_{j\neq i}\lvert m_{ij}\rvert$ for all $1\leq i\leq n$ , $M$ is called a strictly diagonal dominant (abbreviated as s.d.d.) matrix (see ^[5]). If $M$ is an $H$ -matrix with all its diagonal entries positive (e.g., see ^[1]), it is called an $H_+$ -matrix. If $M$ is a nonsingular $M$ -matrix, it is well known that there exists a positive diagonal matrix $\Lambda$ , which can make $M\Lambda$ be an s.d.d. matrix with all the diagonal entries of $A\Lambda$ positive ^[5]. $M = F-G$ is called an $H$ -splitting if $\langle F\rangle-\lvert G\rvert$ is a nonsingular $M$ -matrix ^[19].

Let $\mathcal{Y} = \{Y_1, Y_2, \ldots, Y_s\}$ be a set of $n\times n$ real matrices (or $n\times 1$ real vectors). Denote a mapping $\varphi$ as follows:

$\varphi(\mathcal{Y}) = \sum\limits_{i = 1}^{s-1}2^{s-i-1}Y_i+Y_s.$

Let $\Omega\in\mathbb{R}^{n\times n}$ be a positive diagonal matrix, $\sigma > 0\in\mathbb{R}$ and $M_i = F_i-G_i$ $(1\leq i\leq s)$ be $s$ splittings of $M_i$ . Denote

$\mathcal{F} = \{F_1,F_2,\ldots,F_s\},\mathcal{G} = \{G_1,G_2,\ldots,G_s\},\mathcal{M} = \{M_1,M_2,\ldots,M_s\},$

$\mathcal{D_F} = \{D_{F_1},D_{F_2},\ldots,D_{F_s}\}, \mathcal{B_F} = \{B_{F_1},B_{F_2},\ldots,B_{F_s}\},$

and

$U(z) = \{u_1(z),u_2(z),\ldots,u_s(z)\}.$

By equivalently transforming the VNCP to a fixed-point equation, the MMS iteration method for the VNCP was given in ^[42].

Method 2.1. ^[42] (MMS) Given $x_1^{(0)}\in \mathbb{R}^{n}$ , for $k = 0, 1, 2, \ldots$ , compute $x_1^{(k+1)}\in \mathbb{R}^{n}$ by

$(2^{ s-1}\Omega+\varphi(\mathcal{F}))x_1^{(k+1)} = \varphi(\mathcal{G})x_1^{(k)}+(2^{ s-1}\Omega-\varphi(\mathcal{M}))\lvert x_1^{(k)}\rvert+\Omega\sum\limits_{i = 2}^{ s}2^{ s-i+1}\lvert x_i^{(k)}\rvert-\sigma \varphi(U(z^{(k)})),$

where

$z^{(k)} = \frac{1}{\sigma}(x_1^{(k)}+\lvert x_1^{(k)}\rvert)$

and $x_2^{(k)}, \ldots, x_ s^{(k)}$ are computed by

$\begin{equation} \left\{ \begin{array}{cll} x_ s^{(k)}& = &\frac{1}{2}\Omega^{-1}\big{[} (M_{ s-1}-M_ s)(\lvert x_1^{(k)}\rvert+x_1^{(k)})+\sigma u_{ s-1}(z^{(k)})-\sigma u_ s(z^{(k)}) \big{]},\\ x_j^{(k)}& = &\frac{1}{2}\Omega^{-1}\big{[}(M_{j-1}-M_j)(\lvert x_1^{(k)}\rvert+x_1^{(k)})+\Omega(\lvert x_{j+1}^{(k)}\rvert+x_{j+1}^{(k)})+\sigma u_{j-1}(z^{(k)})-\sigma u_j(z^{(k)}) \big{]},\\ &&j = s-1, s-2,\ldots,2, \end{array} \right. \end{equation}$

(2.1)

until the iteration is convergent.

Based on Method 2.1, to fully utilize the information of the entries of the matrix set $\mathcal{M}$ , for $1\leq i\leq s$ , consider two matrix splittings of $M_i$ , e.g., $M_i = F_i^{(1)}-G_i^{(1)} = F_i^{(2)}-G_i^{(2)}$ . Then, the two-step MMS (abbreviated as TMMS) iteration method can be established as follows:

Method 2.2. (TMMS) Given $x_1^{(0)}\in \mathbb{R}^{n}$ , for $k = 0, 1, 2, \ldots$ , compute $x_1^{(k+1)}\in \mathbb{R}^{n}$ by

$\begin{equation} \left\{ \begin{array}{cll} (2^{ s-1}\Omega+\varphi(\mathcal{F}^{(1)}))x_1^{(k+\frac{1}{2})}& = &\varphi(\mathcal{G}^{(1)})x_1^{(k)}+(2^{ s-1}\Omega-\varphi(\mathcal{M}))\lvert x_1^{(k)}\rvert +\Omega\sum\limits_{i = 2}^{ s}2^{ s-i+1}\lvert x_i^{(k)}\rvert\\ &&-\sigma \varphi(U(z^{(k)})),\\ (2^{ s-1}\Omega+\varphi(\mathcal{F}^{(2)}))x_1^{(k+1)}& = &\varphi(\mathcal{G}^{(2)})x_1^{(k+\frac{1}{2})}+(2^{ s-1}\Omega-\varphi(\mathcal{M}))\lvert x_1^{(k+\frac{1}{2})}\rvert+\Omega\sum\limits_{i = 2}^{ s}2^{ s-i+1}\lvert x_i^{(k+\frac{1}{2})}\rvert\\ &&-\sigma \varphi(U(z^{(k+\frac{1}{2})})), \end{array} \right. \end{equation}$

(2.2)

where

$z^{(k)} = \frac{1}{\sigma}(x_1^{(k)}+\lvert x_1^{(k)}\rvert), z^{(k+\frac{1}{2})} = \frac{1}{\sigma}(x_1^{(k+\frac{1}{2})}+\lvert x_1^{(k+\frac{1}{2})}\rvert),$

where $x_2^{(k)}, \ldots, x_ s^{(k)}$ are computed by (2.1) and

$\begin{equation} \left\{ \begin{array}{cll} x_ s^{(k+\frac{1}{2})}& = &\frac{1}{2}\Omega^{-1}\big{[} (M_{ s-1}-M_ s)(\lvert x_1^{(k+\frac{1}{2})}\rvert+x_1^{(k+\frac{1}{2})})+\sigma u_{ s-1}(z^{(k+\frac{1}{2})})-\sigma u_ s(z^{(k+\frac{1}{2})}) \big{]},\\ x_j^{(k+\frac{1}{2})}& = &\frac{1}{2}\Omega^{-1}\big{[}(M_{j-1}-M_j)(\lvert x_1^{(k+\frac{1}{2})}\rvert+x_1^{(k+\frac{1}{2})})+\Omega(\lvert x_{j+1}^{(k+\frac{1}{2})}\rvert+x_{j+1}^{(k+\frac{1}{2})})\\ &&+\sigma u_{j-1}(z^{(k)})-\sigma u_j(z^{(k+\frac{1}{2})}) \big{]}, \; \; j = s-1, s-2,\ldots,2, \end{array} \right. \end{equation}$

(2.3)

until the iteration is convergent.

Clearly, if we take $F_i^{(1)} = F_i^{(2)}$ , Method 2.2 reduces to Method 2.1 immediately. Furthermore, we can obtain a class of relaxation methods from Method 2.2 by specially choosing the two matrix splittings, similar to those in ^{[43,44,45,49,50,52]}. For example, for $i = 1, 2, \ldots, s$ , taking

$\begin{equation} \left\{ \begin{array}{l} F_{i}^{(1)} = \frac{1}{\alpha}(D_{A_i}^{(1)}-\beta{L_{A_i}^{(1)}}), G_{i}^{(1)} = F_{i}^{(1)}-A_i,\\ F_{i}^{(2)} = \frac{1}{\alpha}(D_{A_i}^{(2)}-\beta{U_{A_i}^{(2)}}), G_{i}^{(2)} = F_{i}^{(2)}-A_i,\\ \end{array} \right. \end{equation}$

(2.4)

one can get the two-step modulus-based accelerated overrelaxation (abbreviated as TMAOR) iteration method, which can reduce to the two-step modulus-based successive overrelaxation (abbreviated as TMSOR) and Gauss-Seidel (abbreviated as TMGS) methods, when $(\alpha, \beta) = (\alpha, \alpha)$ and $(\alpha, \beta) = (1, 1)$ , respectively.

3. Convergence analysis

In this section, the convergence conditions of Method 2.2 are given under the assumption that the VNCP has a unique solution $z^*$ , the same as that in ^[42]. Furthermore, for $1\leq i\leq s$ , $u_i(z)$ is assumed to satisfy the locally Lipschitz smoothness conditions: Let

$u_i(z) = \big{(}u_i(z_1),u_i(z_2),\ldots,u_i(z_n)\big{)}^T$

be differentiable with

$\begin{equation} 0\leq\frac{\partial u_i(z)}{\partial z}\leq{U}_i,\; \; \; i = 1,2,\ldots,s. \end{equation}$

(3.1)

Then, by Lagrange mean value theorem, there exists $\xi_j$ between $z_j^{(k)}$ and $z_j^*$ such that

$\begin{equation} u_i(z^{(k)})-u_i(z^*) = {U}_i^{(k)}(z^{(k)}-z^*) = \frac{1}{\sigma}{U}_i^{(k)}\big{[}(x_1^{(k)}-x_1^*)+(\lvert x_1^{(k)}\rvert-\lvert x_1^*\rvert)\big{]}, \end{equation}$

(3.2)

where ${U}_i^{(k)}$ is a nonnegative diagonal matrix with diagonal entries $\frac{\partial u_i(z_j)}{\partial z_j}\bigg{|}_{z_j = \xi_j}, 1\leq j\leq n$ . Furthermore, by (3.1), we have

$\begin{equation} 0\leq{U}_i^{(k)}\leq{U}_i. \end{equation}$

(3.3)

Denote

$\mathcal{{U}}^{(k)} = \{{U}_1^{(k)},{U}_2^{(k)},\ldots,{U}_s^{(k)}\}, \mathcal{{U}} = \{{U}_1,{U}_2,\ldots,{U}_s\}.$

Lemma 3.1. Let $M_i, 1\leq i\leq s$ , be $H_+$ -matrices, $\Omega\in\mathbb{R}^{n\times n}$ be a positive diagonal matrix, and $\sigma > 0\in\mathbb{R}$ . For $t = 1, 2$ , assume that:

(I) $D_{F_i^{(t)} } > 0, i = 1, 2, \ldots, s-1$ , and $M_ s = F_ s^{(t)} -G_ s^{(t)}$ be an $H$ -splitting of $M_ s$ ;

(II)

$\begin{equation} \left\{ \begin{array}{l} \langle F_{ s-1}^{(t)} \rangle\geq\langle F_ s^{(t)} \rangle,\\ 2^{ s-j}\langle F_{j-1}^{(t)} \rangle\geq\langle \sum\limits_{i = j}^{ s-1}2^{ s-i-1}F_i^{(t)} +F_ s^{(t)} \rangle,\; \; 2\leq j\leq s-1; \end{array} \right. \end{equation}$

(3.4)

(III) There exists a positive diagonal matrix $\Lambda$ such that $(\langle F_ s^{(t)} \rangle-\lvert G_ s^{(t)} \rvert)\Lambda$ is an s.d.d. matrix.

Then, $2^{ s-1}\Omega+\varphi(\mathcal{F}^{(t)})$ is an $H$ -matrix.

Proof. Let $e = \in\mathbb{R}^{n}$ be a vector with all entries being 1. By reusing (3.4) $s-2$ times, we have

$\begin{eqnarray*} &&\langle 2^{ s-1}\Omega+\varphi(\mathcal{F}^{(t)})\rangle \Lambda e\\ & > &\langle\varphi(\mathcal{F}^{(t)})\rangle \Lambda e\\ &\geq& (\langle 2^{ s-2}F_1^{(t)} \rangle+\langle\sum\limits_{i = 2}^{ s-1}2^{ s-i-1}F_i+F_{ s}^{(t)}\rangle )\Lambda e\\ &\geq& 2\langle\sum\limits_{i = 2}^{ s-1}2^{ s-i-1}F_i^{(t)}+F_{ s}^{(t)}\rangle \Lambda e\\ &\geq& 2(\langle 2^{ s-3}F_2^{(t)} \rangle+\langle\sum\limits_{i = 3}^{ s-1}2^{ s-i-1}F_i+F_{ s}^{(t)}\rangle )\Lambda e\\ &\geq&\ldots\geq2^{ s-1}\langle F_{ s}^{(t)}\rangle \Lambda e\\ &\geq&2^{ s-1}(\langle F_{ s}^{(t)}\rangle-\lvert G_ s^{(t)}\rvert) \Lambda e\\ & > &0. \end{eqnarray*}$

Hence, $\langle 2^{ s-1}\Omega+\varphi(\mathcal{F}^{(t)})\rangle \Lambda$ is an s.d.d matrix, which implies that $2^{ s-1}\Omega+\varphi(\mathcal{F}^{(t)})$ is an $H$ -matrix. □

Lemma 3.2. With the same notations as those in Lemma 3.1, denote $x_i^*, i = 1, 2, \ldots, s$ , as the solution of the VNCP, and let

$\delta_i^{(k)} = x_i^{(k)}-x_i^*, \bar\delta_i^{(k)} = \lvert x_i^{(k)}\rvert-\lvert x_i^*\rvert.$

Then, we have

$\begin{equation} \sum\limits_{i = 2}^{ s}2^{ s-i+1}\Omega\lvert\bar\delta_i^{(k)}\rvert\leq[2(\sum\limits_{j = 1}^{ s-2}2^{ s-j-1}\Gamma_j+\Gamma_{s-1})+\Theta]\lvert\delta_1^{(k)}\rvert, \end{equation}$

(3.5)

where

$\begin{equation} \notag \Gamma_j = \left\{ \begin{array}{lll} \lvert M_{ s-1}-M_ s\rvert,& \mathit{\text{if}} \; \; \; j = 1,\\ \lvert 2^{j-1}M_{ s-j}-\sum\limits_{t = s-j+1}^{ s-1}2^{ s-t-1}M_t-M_ s\rvert,& \mathit{\text{if}} \; \; \; 2\leq j\leq s-1, \end{array} \right. \end{equation}$

and

$\Theta = \sum\limits_{j = 2}^{s-1}(2^s-2^{s-j})U_j+(2^s-2)U_s.$

Proof. By (2.1), we get

$\begin{equation} \left\{ \begin{array}{cll} \delta_ s^{(k)}& = &\Omega^{-1}\big{[} (M_{ s-1}-M_ s)+(U_{ s-1}^{(k)}-U_ s^{(k)})\big{]} (\bar{\delta}_1^{(k)}+\delta_1^{(k)}),\\ \delta_j^{(k)}& = &\frac{1}{2}\Omega^{-1}\big{[}(M_{j-1}-M_j)+(U_{ j-1}^{(k)}-U_ {j}^{(k)})(\bar{\delta}_1^{(k)}+\delta_1^{(k)})+\Omega(\bar{\delta}_{j+1}^{(k)}+\delta_{j+1}^{(k)})\big{]}, \\ &&\; \; j = s-1, s-2,\ldots,2. \end{array} \right. \end{equation}$

(3.6)

By the first equation of (3.6), we have

$\begin{eqnarray*} &&2^1\Omega \lvert\delta_ s^{(k)}\rvert\\ & = &\lvert[(M_{ s-1}-M_ s)+({U}_{s-1}^{(k)}-{U}_s^{(k)})](\bar\delta_1^{(k)}+\delta_1^{(k)})\rvert\\ &\leq&(\lvert M_{ s-1}-M_ s\rvert+U_{s-1}+U_s)\lvert\delta_1^{(k)}\rvert\\ & = &2(\Gamma_1+U_{s-1}+U_s)\lvert\delta_1^{(k)}\rvert. \end{eqnarray*}$

Then, when the subscript is $s-1$ , with the second equation of (3.6), we can get

$\begin{eqnarray*} &&2^2\Omega \lvert\delta_{ s-1}^{(k)}\rvert\\ & = & \lvert[2(M_{ s-2}-M_{ s-1})+2({U}_{s-2}^{(k)}-{U}_{s-1}^{(k)})](\bar\delta_1^{(k)}+\delta_1^{(k)})+2\Omega(\bar\delta_ s^{(k)}+\delta_ s^{(k)})\rvert\\ & = &\lvert[(2M_{ s-2}-M_{ s-1}-M_ s)+(2U_{ s-2}^{(k)}-U_{ s-1}^{(k)}-U_ s^{(k)})](\bar\delta_1^{(k)}+\delta_1^{(k)})+2\Omega\bar\delta_ s^{(k)}\rvert\\ &\leq&2(\Gamma_2+2U_{ s-2}+U_{ s-1}+U_ s)\lvert\delta_1^{(k)}\rvert+2\Omega\lvert\delta_ s^{(k)}\rvert\\ &\leq&2[\Gamma_2+\Gamma_1+2(U_{ s-2}+U_{ s-1}+U_ s)]\lvert\delta_ 1^{(k)}\rvert. \end{eqnarray*}$

By induction, for $2\leq i\leq s$ , we have

$\begin{equation} 2^{ s-i+1}\Omega\lvert\delta_i^{(k)}\rvert\leq(\sum\limits_{j = 1}^{ s-i}2^{ s-i-j+1}\Gamma_j+2\Gamma_{ s-i+1}+2^{ s-i+1}\sum\limits_{j = i-1}^{s}U_j)\lvert\delta_1^{(k)}\rvert. \end{equation}$

(3.7)

Then, by (3.7), we get

$\begin{eqnarray} &&\sum\limits_{i = 2}^{ s}2^{ s-i+1}\Omega\lvert\bar\delta_i^{(k)}\rvert\\ &\leq&\sum\limits_{i = 2}^{ s}2^{ s-i+1}\Omega\lvert\delta_i^{(k)}\rvert\\ &\leq&\sum\limits_{i = 2}^{ s}(\sum\limits_{j = 1}^{ s-i}2^{ s-i-j+1}\Gamma_j+2\Gamma_{ s-i+1}+2^{ s-i+1}\sum\limits_{j = i-1}^{s}U_j)\lvert\delta_1^{(k)}\rvert\\ & = &(\sum\limits_{j = 1}^{ s-2}\sum\limits_{i = 2}^{ s-j}2^{ s-i-j+1}\Gamma_j+2\sum\limits_{i = 1}^{ s-1}\Gamma_i+\sum\limits_{i = 2}^{ s}2^{ s-i+1}\sum\limits_{j = i-1}^{s}U_j)\lvert\delta_1^{(k)}\rvert\\ & = &[\sum\limits_{j = 1}^{ s-2}(2^{ s-j}-2)\Gamma_j+2\sum\limits_{j = 1}^{ s-1}\Gamma_j+\sum\limits_{j = 2}^{s-1}(2^s-2^{s-j})U_j+(2^s-2)U_s)]\lvert\delta_1^{(k)}\rvert\\ & = &[2(\sum\limits_{j = 1}^{ s-2}2^{ s-j-1}\Gamma_j+\Gamma_{s-1})+\Theta]\lvert\delta_1^{(k)}\rvert. \end{eqnarray}$

□

Theorem 3.1. With the same notations and assumptions as those in Lemmas 3.1 and 3.2, for $t = 1, 2$ , assume that

$\begin{equation} \left\{ \begin{array}{l} \lvert G_{ s-1}^{(t)} \rvert \leq\lvert G_ s^{(t)} \rvert,\\ 2^{ s-j}\lvert G_j^{(t)} \rvert\leq\lvert \sum\limits_{i = j}^{ s-1}2^{ s-i-1}G_i^{(t)} +G_ s^{(t)} \rvert,\; \; 2\leq j\leq s-1, \end{array} \right. \end{equation}$

(3.8)

Then, Method 2.1 converges for any initial vector $x_1^{(0)}$ provided that

$\begin{equation} \Omega \geq2^{1- s}(\varphi(\mathcal{D_M}^{(t)})+\varphi(\mathcal{U})+\Theta) \end{equation}$

(3.9)

$\begin{equation} \big{[}2^{1- s}(\varphi(\mathcal{D_M}^{(t)})+\varphi(\mathcal{U}))+2^{-s}\Theta-(\langle F_ s^{(t)}\rangle-\lvert G_ s^{(t)}\rvert)\big{]}\Lambda e < \Omega \Lambda e < 2^{1- s}(\varphi(\mathcal{D_M}^{(t)})+\Theta)\Lambda e. \end{equation}$

(3.10)

Proof.

By (3.2), we get

$u_i(z^{(k)})-u_i(z^*) = \frac{1}{\sigma}{U}_i^{(k)}(\delta_1^{(k)}+\bar{\delta}_1^{(k)}).$

By the definition of $\varphi$ , we have

$\begin{equation} \varphi(U(z^{(k)}))-\varphi(U(z^*)) = \varphi(U(z^{(k)})-U(z^*)) = \frac{1}{\sigma}\varphi(\mathcal{{U}}^{(k)})(\delta_1^{(k)}+\bar\delta_1^{(k)}). \end{equation}$

(3.11)

Combining the first equation of (2.2) and (3.11), we can get

$\begin{eqnarray*} &&(2^{ s-1}\Omega+\varphi(\mathcal{F}^{(1)}))\delta_1^{(k+\frac{1}{2})}\\ & = &\varphi(\mathcal{G}^{(1)})\delta_1^{(k)}+(2^{ s-1}\Omega-\varphi(\mathcal{M}))\bar\delta_1^{(k)} +\Omega\sum\limits_{i = 2}^{ s}2^{ s-i+1}\bar\delta_i^{(k)}-\sigma[\varphi(U(z^{(k)}))-\varphi(U(z^*))]\\ & = &\varphi(\mathcal{G}^{(1)})\delta_1^{(k)}+(2^{ s-1}\Omega-\varphi(\mathcal{M}))\bar\delta_1^{(k)} +\Omega\sum\limits_{i = 2}^{ s}2^{ s-i+1}\bar\delta_i^{(k)}-\varphi(\mathcal{{U}}^{(k)})(\delta_i^{(k)}+\bar\delta_i^{(k)})\\ & = &[\varphi(\mathcal{G}^{(1)})-\varphi(\mathcal{{U}}^{(k)})]\delta_1^{(k)}+[2^{ s-1}\Omega-\varphi(\mathcal{M})-\varphi(\mathcal{{U}}^{(k)})]\bar\delta_1^{(k)} +\Omega\sum\limits_{i = 2}^{ s}2^{ s-i+1}\bar\delta_i^{(k)}. \end{eqnarray*}$

Similarly, by the second equation of (2.2), we have

$\begin{eqnarray*} &&(2^{ s-1}\Omega+\varphi(\mathcal{F}^{(2)}))\delta_1^{(k+1)}\\ & = &[\varphi(\mathcal{G}^{(2)})-\varphi(\mathcal{{U}}^{(k+\frac{1}{2})})]\delta_1^{(k+\frac{1}{2})}+[2^{ s-1}\Omega-\varphi(\mathcal{M})-\varphi(\mathcal{{U}}^{(k+\frac{1}{2})})]\bar\delta_1^{(k+\frac{1}{2})} +\Omega\sum\limits_{i = 2}^{ s}2^{ s-i+1}\bar\delta_i^{(k+\frac{1}{2})}. \end{eqnarray*}$

Then we have

$\begin{equation} \left\{ \begin{array}{lll} \lvert \delta_1^{(k+\frac{1}{2})}\rvert &\leq&\lvert 2^{ s-1}\Omega+\varphi(\mathcal{F}^{(1)})\rvert^{-1}\Big{(}\lvert\varphi(\mathcal{G}^{(1)})-\varphi(\mathcal{{U}}^{(k)})\rvert\lvert\delta_1^{(k)}\rvert\\ &&+\lvert2^{ s-1}\Omega-\varphi(\mathcal{M})-\varphi(\mathcal{{U}}^{(k)})\rvert\lvert\bar\delta_1^{(k)}\rvert+\sum\limits_{i = 2}^{ s}2^{ s-i+1}\Omega\lvert\bar\delta_i^{(k)}\rvert\Big{)},\\ \lvert \delta_1^{(k+1)}\rvert &\leq&\lvert 2^{ s-1}\Omega+\varphi(\mathcal{F}^{(2)})\rvert^{-1}\Big{(}\lvert\varphi(\mathcal{G}^{(2)})-\varphi(\mathcal{{U}}^{(k+\frac{1}{2})})\rvert\lvert\delta_1^{(k+\frac{1}{2})}\rvert\\ &&+\lvert2^{ s-1}\Omega-\varphi(\mathcal{M})-\varphi(\mathcal{{U}}^{(k+\frac{1}{2})})\rvert\lvert\bar\delta_1^{(k+\frac{1}{2})}\rvert+\sum\limits_{i = 2}^{ s}2^{ s-i+1}\Omega\lvert\bar\delta_i^{(k+\frac{1}{2})}\rvert\Big{)}.\\ \end{array} \right. \end{equation}$

(3.12)

By Lemma 3.1, we have that $2^{ s-1}\Omega+\varphi(\mathcal{F})$ is an $H$ -matrix. By ^[17], with (3.5) and (3.12), we get

$\begin{equation} \lvert\delta_1^{(k+1)}\rvert\leq{\mathcal{P}^{(2)}}^{-1}\mathcal{Q}^{(2)}{\mathcal{P}^{(1)}}^{-1}\mathcal{Q}^{(1)}\lvert\delta_1^{(k)}\rvert, \end{equation}$

(3.13)

where

${\mathcal{P}^{(1)}} = \langle2^{ s-1}\Omega+\varphi(\mathcal{F}^{(1)})\rangle,$

${\mathcal{Q}^{(1)}} = \lvert\varphi(\mathcal{G}^{(1)})-\varphi(\mathcal{U}^{(k)})\rvert+\lvert2^{ s-1}\Omega-\varphi(\mathcal{M})-\varphi(\mathcal{U}^{(k)})\rvert+2(\sum\limits_{j = 1}^{ s-2}2^{ s-j-1}\Gamma_j+\Gamma_{s-1})+\Theta,$

${\mathcal{P}^{(2)}} = \langle2^{ s-1}\Omega+\varphi(\mathcal{F}^{(2)})\rangle,$

${\mathcal{Q}^{(2)}} = \lvert\varphi(\mathcal{G}^{(2)})-\varphi(\mathcal{U}^{(k)})\rvert+\lvert2^{ s-1}\Omega-\varphi(\mathcal{M})-\varphi(\mathcal{U}^{(k)})\rvert+2(\sum\limits_{j = 1}^{ s-2}2^{ s-j-1}\Gamma_j+\Gamma_{s-1})+\Theta.$

First, we estimate ${\mathcal{F}^{(1)}}^{-1}\mathcal{G}^{(1)}$ . By Lemma 3.1 and ^[26], we have

$\begin{equation} \|\Lambda^{-1}{\mathcal{F}^{(1)}}^{-1}\mathcal{G}^{(1)}\Lambda\|_\infty = \|(\mathcal{F}^{(1)}\Lambda)^{-1}(\mathcal{G}^{(1)}\Lambda)\|_\infty\leq \max\limits_{1\leq i\leq n}\frac{(\mathcal{G}^{(1)}\Lambda e)_i}{(\mathcal{F}^{(1)}\Lambda e)_i}. \end{equation}$

(3.14)

Let

$\begin{equation} \notag \Gamma_j^{(1)} = \left\{ \begin{array}{lll} \lvert D_{F_{ s-1}^{(1)}}-D_{F_ s^{(1)}}\rvert,& \text{if} \; \; \; j = 1,\\ \lvert 2^{j-1}D_{F_{ s-j}^{(1)}}-\sum\limits_{t = s-j+1}^{ s-1}2^{ s-t-1}D_{F_t^{(1)}}-D_{F_ s^{(1)}}\rvert,& \text{if} \; \; \; 2\leq j\leq s-1, \end{array} \right. \end{equation}$

$\begin{equation} \notag \Gamma_j^{(2)} = \left\{ \begin{array}{lll} \lvert B_{F_{ s-1}^{(1)}}-B_{F_ s^{(1)}}\rvert,& \text{if} \; \; \; j = 1,\\ \lvert2^{j-1}B_{F_{ s-j}^{(1)}}-\sum\limits_{t = s-j+1}^{ s-1}2^{ s-t-1}B_{F_t^{(1)}}-B_{F_ s^{(1)}}\rvert,& \text{if} \; \; \; 2\leq j\leq s-1, \end{array} \right. \end{equation}$

and

$\begin{equation} \notag \Gamma_j^{(3)} = \left\{ \begin{array}{lll} \lvert G_{ s-1}^{(1)}-G_ s^{(1)}\rvert,& \text{if} \; \; \; j = 1,\\ \lvert2^{j-1}G_{ s-j}^{(1)}-\sum\limits_{t = s-j+1}^{ s-1}2^{ s-t-1}G_t^{(1)}-G_ s^{(1)}\rvert,& \text{if} \; \; \; 2\leq j\leq s-1. \end{array} \right. \end{equation}$

Clearly, we have $\Gamma_j\leq \Gamma_j^{(1)}+\Gamma_j^{(2)}+\Gamma_j^{(3)}$ . By (3.4), we can get

$\begin{equation} \notag \Gamma_j^{(1)} = \left\{ \begin{array}{lll} D_{F_{ s-1}}-D_{F_ s},& \text{if} \; \; \; j = 1,\\ 2^{j-1}D_{F_{ s-j}}-\sum\limits_{t = s-j+1}^{ s-1}2^{ s-t-1}D_{F_t}-D_{F_ s},& \text{if} \; \; \; 2\leq j\leq s-1. \end{array} \right. \end{equation}$

Then, by direct computation, we can obtain

$\begin{eqnarray} \varphi(\mathcal{D}_{M}^{(1)})-2(\sum\limits_{j = 1}^{ s-2}2^{ s-j-1}\Gamma_j^{(1)}+\Gamma_{s-1}^{(1)})& = &\sum\limits_{i = 1}^{ s-1}2^{ s-i-1}D_{F_i^{(1)}}+D_{F_ s^{(1)}}-2(\sum\limits_{j = 1}^{ s-2}2^{ s-j-1}\Gamma_j^{(1)}+\Gamma_{s-1}^{(1)})\\ & = &(2^{ s}-1)D_{F_ s^{(1)}}-\sum\limits_{i = 1}^{ s-1}2^{ s-i-1}D_{F_i^{(1)}}. \end{eqnarray}$

(3.15)

Moreover, by reusing (3.4) $s-2$ times, we get

$\begin{eqnarray} &&2(\sum\limits_{j = 1}^{ s-2}2^{ s-j-1}\Gamma_j^{(2)}+\Gamma_{s-1}^{(2)})+2\lvert\sum\limits_{j = 1}^{ s-1}2^{ s-j-1}B_{F_j^{(1)}}+B_{F_ s^{(1)}}\rvert\\ & = &2\sum\limits_{j = 1}^{ s-2}2^{ s-j-1}\Gamma_j^{(2)}+{\Big{(}}2\lvert2^{ s-2}B_{F_{1}^{(1)}}-\sum\limits_{j = 2}^{ s-1}2^{ s-j-1}B_{F_j^{(1)}}-B_{F_ s^{(1)}}\rvert\\ &&+2\lvert2^{ s-2}B_{F_{1}^{(1)}}+\sum\limits_{j = 2}^{ s-1}2^{ s-j-1}B_{F_j^{(1)}}+B_{F_ s^{(1)}}\rvert{\Big{)}}\\ & = &2\sum\limits_{j = 1}^{ s-2}2^{ s-j-1}\Gamma_j^{(2)}+2^2\lvert\sum\limits_{j = 2}^{ s-1}2^{ s-j-1}B_{F_j^{(1)}}+B_{F_ s^{(1)}}\rvert\\ & = &2\sum\limits_{j = 1}^{ s-3}2^{ s-j-1}\Gamma_j^{(2)}+{\Big{(}}2^2\lvert2^{ s-3}B_{F_{2}^{(1)}}-\sum\limits_{j = 3}^{ s-1}2^{ s-j-1}B_{F_j^{(1)}}-B_{F_ s^{(1)}}\rvert\\ &&+2^2\lvert2^{ s-3}B_{F_{2}}+\sum\limits_{j = 3}^{ s-1}2^{ s-j-1}B_{F_j^{(1)}}+B_{F_ s^{(1)}}\rvert{\Big{)}}\\ & = &2\sum\limits_{j = 1}^{ s-3}2^{ s-j-1}\Gamma_j^{(2)}+2^3\lvert\sum\limits_{j = 3}^{ s-1}2^{ s-j-1}B_{F_j^{(1)}}+B_{F_ s^{(1)}}\rvert\\ & = &\ldots = 2^ s\lvert B_{F_ s}\rvert. \end{eqnarray}$

(3.16)

Analogously, by reusing (3.8) $s-2$ times, we get

$\begin{equation} 2(\sum\limits_{j = 1}^{ s-2}2^{ s-j-1}\Gamma_j^{(3)}+\Gamma_{s-1}^{(3)})+2\lvert\varphi(\mathcal{G}^{(1)})\rvert = 2^ s\lvert{G_ s^{(1)}}\rvert. \end{equation}$

(3.17)

By (3.15)–(3.17), we get

$\begin{eqnarray} &&\mathcal{P}^{(1)}\Lambda e-\mathcal{Q}^{(1)}\Lambda e\\ & = &\Big{[}\langle2^{ s-1}\Omega+\varphi(\mathcal{F}^{(1)})\rangle-\lvert\varphi(\mathcal{G}^{(1)})-\varphi(\mathcal{U}^{(k)})\rvert-\lvert2^{ s-1}\Omega-\varphi(\mathcal{M})-\varphi(\mathcal{U}^{(k)})\rvert\\ &&-2(\sum\limits_{j = 1}^{ s-2}2^{ s-j-1}\Gamma_j+\Gamma_{s-1})-\Theta\Big{]}\Lambda e\\ &\geq&\Big{[}2^{ s-1}\Omega+\varphi(\mathcal{D_{M}}^{(1)}) - \lvert\varphi(\mathcal{B_{M}}^{(1)})\rvert-\lvert\varphi(\mathcal{G}^{(1)})-\varphi(\mathcal{U}^{(k)})\rvert-\lvert2^{ s-1}\Omega-\varphi(\mathcal{D_M})-\varphi(\mathcal{U}^{(k)})\rvert\\ &&-\lvert\varphi(\mathcal{B_{M}}^{(1)})\rvert-\lvert\varphi(\mathcal{G}^{(1)})\rvert-2(\sum\limits_{j = 1}^{ s-2}2^{ s-j-1}\Gamma_j^{(1)}+\Gamma_{s-1}^{(1)})-2(\sum\limits_{j = 1}^{ s-2}2^{ s-j-1}\Gamma_j^{(2)}+\Gamma_{s-1}^{(2)})\\ &&-2(\sum\limits_{j = 1}^{ s-2}2^{ s-j-1}\Gamma_j^{(3)}+\Gamma_{s-1}^{(3)})-\Theta\Big{]}\Lambda e\\ &\geq&\bigg{\{}2^{ s-1}\Omega+\Big{[}\varphi(\mathcal{D_{M}}^{(1)})-2(\sum\limits_{j = 1}^{ s-2}2^{ s-j-1}\Gamma_j^{(1)}+\Gamma_{s-1}^{(1)})\Big{]} -\Big{[}2\lvert\varphi(\mathcal{B_{M}}^{(1)})\rvert+2(\sum\limits_{j = 1}^{ s-2}2^{ s-j-1}\Gamma_j^{(2)}+\Gamma_{s-1}^{(2)})\Big{]}\\ &&-\Big{[}2\lvert\varphi(\mathcal{G})\rvert+2(\sum\limits_{j = 1}^{ s-2}2^{ s-j-1}\Gamma_j^{(3)}+\Gamma_{s-1}^{(3)})\Big{]}-\varphi(\mathcal{U}^{(k)})-\lvert2^{ s-1}\Omega-\varphi(\mathcal{D_{M}}^{(1)})-\varphi(\mathcal{U}^{(k)})-\Theta\rvert\bigg{\}}\Lambda e\\ & = &\Big{[}2^{ s-1}\Omega+(2^{ s}-1)D_{F_ s^{(1)}}-\sum\limits_{i = 1}^{ s-1}2^{ s-i-1}D_{F_i^{(1)}}-2^ s \lvert B_{F_ s^{(1)}}\rvert-2^ s \lvert G_ s^{(1)}\rvert-\varphi(\mathcal{U}^{(k)})\\ &&-\lvert2^{ s-1}\Omega-\varphi(\mathcal{D_{M}}^{(1)})-\varphi(\mathcal{U}^{(k)})-\Theta\rvert\Big{]}\Lambda e. \end{eqnarray}$

(3.18)

Next, consider the following two cases.

Case 1. When

$\Omega\geq2^{1- s}(\varphi(\mathcal{D_{M}}^{(1)})+\varphi(\mathcal{U})+\Theta),$

by (3.18), we have

$\begin{eqnarray} &&\mathcal{P}^{(1)}\Lambda e-\mathcal{Q}^{(1)}\Lambda e\\ &\geq&\Big{[}2^{ s-1}\Omega+(2^{ s}-1)D_{F_ s^{(1)}}-\sum\limits_{i = 1}^{ s-1}2^{ s-i-1}D_{F_i^{(1)}}-2^ s \lvert B_{F_ s^{(1)}}\rvert-2^ s\lvert G_ s^{(1)}\rvert-\varphi(\mathcal{U}^{(k)})\\ &&-(2^{ s-1}\Omega-\varphi(\mathcal{D_M})-\varphi(\mathcal{U}^{(k)})-\Theta)\Big{]}\Lambda e\\ & = &\big{[}2^ s(D_{F_ s^{(1)}}-\lvert G_ s^{(1)}\rvert-\lvert B_{F_ s^{(1)}}\rvert)+\Theta\big{]}\Lambda e\\ & = &\big{[}2^ s(\langle F_ s\rangle-\lvert G_ s^{(1)}\rvert)+\Theta\big{]}\Lambda e\\ & > &0. \end{eqnarray}$

Case 2. When

$\big{[}2^{1- s}(\varphi(\mathcal{D_{M}}^{(1)})+\varphi(\mathcal{U}))+2^{-s}\Theta-(\langle F_ s^{(1)}\rangle-\lvert G_ s^{(1)}\rvert)\big{]}\Lambda e < \Omega \Lambda e < 2^{1- s}(\varphi(\mathcal{D_{M}}^{(1)})+\Theta)\Lambda e,$

by (3.18), we have

$\begin{eqnarray} &&\mathcal{P}^{(1)}\Lambda e-\mathcal{Q}^{(1)}\Lambda e\\ &\geq&\Big{[}2^{ s-1}\Omega+(2^{ s}-1)D_{F_ s^{(1)}}-\sum\limits_{i = 1}^{ s-1}2^{ s-i-1}D_{F_i^{(1)}}-2^ s \lvert B_{F_ s^{(1)}}\rvert-2^ s \lvert G_ s^{(1)}\rvert-\varphi(\mathcal{U}^{(k)})\\ &&-(\varphi(\mathcal{D_{M}}^{(1)})+\varphi(\mathcal{U}^{(k)})+\Theta-2^{ s-1}\Omega)\Big{]}\Lambda e\\ &\geq&\Big{[}2^{ s}\Omega-2\varphi(\mathcal{D_{M}}^{(1)})-2\varphi(\mathcal{U})-\Theta+2^ s (D_{F_ s}-\lvert G_ s^{(1)}\rvert-\lvert B_{F_ s^{(1)}}\rvert)\Big{]}\Lambda e\\ & = &\Big{[}2^{ s}\Omega-2\varphi(\mathcal{D_{M}}^{(1)})-2\varphi(\mathcal{U})-\Theta+2^{ s}(\langle F_ s^{(1)}\rangle-\lvert G_ s^{(1)}\rvert)\Big{]}\Lambda e\\ & > &0. \end{eqnarray}$

Summarizing Cases 1 and 2, we immediately get $\mathcal{P}^{(1)}\Lambda e-\mathcal{Q}^{(1)}\Lambda e > 0$ , provided that (3.9) or (3.10) holds, which implies that

$\|{\Lambda}^{-1}{\mathcal{P}^{(1)}}^{-1}\mathcal{Q}^{(1)}\Lambda\|_\infty < 1$

by (3.14).

By similar deductions, we can also get

$\|{\Lambda}^{-1}{\mathcal{P}^{(2)}}^{-1}\mathcal{Q}^{(2)}\Lambda\|_\infty < 1,$

when (3.9) or (3.10) is satisfied.

In summary, the spectral radius of the iteration matrix given by (3.13) can be estimated as follows:

$\begin{eqnarray*} &&\rho({\mathcal{P}^{(2)}}^{-1}\mathcal{Q}^{(2)}{\mathcal{F}^{(1)}}^{-1}\mathcal{Q}^{(1)})\\ & = &\rho({\Lambda}^{-1}{\mathcal{P}^{(2)}}^{-1}\mathcal{Q}^{(2)}{\Lambda}{\Lambda}^{-1}{\mathcal{P}^{(1)}}^{-1}\mathcal{Q}^{(1)}\Lambda)\\ &\leq&\|{\Lambda}^{-1}{\mathcal{P}^{(2)}}^{-1}\mathcal{Q}^{(2)}{\Lambda}\|_\infty\|{\Lambda}^{-1}{\mathcal{P}^{(1)}}^{-1}\mathcal{Q}^{(1)}\Lambda\|_\infty\\ & < &1. \end{eqnarray*}$

Hence, we can see that $\{x_1^{(k)}\}_{k = 0}^{\infty}$ converges by (3.13). □

Based on Theorem 3.1, we have the next theorem for the TMAOR method.

Theorem 3.2. With the same notations and assumptions as those in Theorem 3.1, for $t = 1, 2$ , let $F_i^{(t)}$ and $G_i^{(t)}$ be given by (2.4), $1\leq i\leq {s}$ . Assume that

$\begin{equation} 0 < \beta\leq\alpha < \frac{2}{1+\rho(D_{M_s}^{-1}{\mid}B_{M_s}{\mid})}. \end{equation}$

(3.19)

Then, the TMAOR converges to the solution of the VNCP.

Proof. By (2.4), with direct computation, we can get

$\langle F_s^{(1)}\rangle-{\mid}G_s^{(1)}{\mid} = \langle F_s^{(2)}\rangle-{\mid}G_s^{(2)}{\mid} = \frac{1-{\mid}1-\alpha{\mid}}{\alpha}D_{M_s}-{\mid}B_{M_s}{\mid},$

if $0 < \beta\leq\alpha$ . Since $M_s$ is an $H_+$ -matrix, by ^[5] we have $\rho(D_{M_s}^{-1}{\mid}B_{M_s}{\mid}) < 1$ . Since (3.19) holds, we can easily have that $\frac{1-{\mid}1-\alpha{\mid}}{\alpha}D_{M_s}-{\mid}B_{M_s}{\mid}$ is a nonsingular $M$ -matrix. Note that all the assumptions of Theorem 3.1 hold. Hence, the TMAOR is convergent by Theorem 3.1. □

4. Numerical examples

Next, some numerical examples are given to shown the efficiency of Method 2.2 compared to Method 2.1.

Example 4.1. ^[42] Consider a VNCP ( $s = 2$ ), where the two system matrices are given by

$M_1 = \left( \begin{array}{ccccc} R & -I_m & && \\ -I_m &R & -I_m && \\ &\ddots&\ddots&\ddots&\\ && -I_m & R & -I_m\\ && & -I_m & R \end{array} \right)\in \mathbb{R}^{n\times n} \; \; {and}\; \; M_2 = \left( \begin{array}{ccccc} R & & && \\ &R & && \\ &&\ddots&&\\ && & R & \\ && & & R \end{array} \right)\in \mathbb{R}^{n\times n},$

where

$R = \left( \begin{array}{ccccc} 4 & -1 & & \\ -1 & 4 & \ddots & \\ & \ddots & \ddots & -1\\ & & -1 & 4 \end{array} \right)\in \mathbb{R}^{m\times m}.$

Example 4.2. ^[42] Consider a VNCP ( $s = 2$ ), where the two system matrices are given by

$M_1 = \left( \begin{array}{ccccc} T & -0.5I_m & && \\ -1.5I_m &T & -0.5I_m && \\ &\ddots&\ddots&\ddots&\\ && -1.5I_m & T & -0.5I_m\\ && & -1.5I_m & T \end{array} \right)\in \mathbb{R}^{n\times n} \; \; {and}\; \; M_2 = \left( \begin{array}{ccccc} T & & && \\ &T & && \\ &&\ddots&&\\ && & T & \\ && & & T \end{array} \right)\in \mathbb{R}^{n\times n},$

where

$T = \left( \begin{array}{ccccc} 4 & -0.5 & & \\ -1.5 & 4 & \ddots & \\ & \ddots & \ddots & -0.5\\ & & -1.5 & 4 \end{array} \right)\in \mathbb{R}^{m\times m}.$

Example 4.3. Consider the VNCP ( $s = 2$ ) whose system matrices $M_1, M_2\in\mathbb{R}^{m^2\times m^2}$ come from the discretization of Hamilton-Jacobi-Bellman (HJB) equation ^[4]

$\left\{ \begin{array}{l} \max\limits_{i = 1,2}\{L_i+f_i\} = 0\; \; \; in\; \; \; \Gamma,\\ z = 0\; \; \; on\; \; \; \partial\Gamma, \end{array} \right.$

with $\Gamma = \{(x, y)\mid0 < x < 2, 0 < y < 1\}$ ,

$\left\{ \begin{array}{l} L_1 = 0.002z_{xx}+0.001z_{yy}-20u_1(z),f_1 = 1,\\ L_2 = 0.001z_{xx}+0.001z_{yy}-10u_2(z),f_2 = 1. \end{array} \right.$

Same as those in ^[42], all the nonlinear functions in Examples 4.1–4.3 are set to $u_1(z) = \frac{z}{1+z}$ and $u_2(z) = arctan(z)$ .

The programming language is MATLAB, whose codes are run on a PC with a 12th Gen Intel(R) Core(TM) i7-12700 2.10 GHz and 32G memory. Consider the Gauss-Seidel (abbreviated as GS), SOR and AOR splittings, where the SOR splitting is

$F_s^{(t)} = \frac{1}{\alpha}(D_{M_s^{(t)}}-\alpha L_{M_s^{(t)}}), G_s^{(t)} = F_s^{(t)}-M_s^{(t)},\; \; t = 1,2,$

with $\alpha$ being the relaxation parameter and starting from 0.8 to 1.2 with a step size of 0.1, while the parameters of AOR splitting given by (2.4) satisfy $\alpha = \beta+0.1$ , and $\beta$ starts from 0.8 to 1.2 with a step size of 0.1.

All initial vectors are set to $x_1^{(0)} = e$ . The stopping criterion in the iteration of both Methods 2.1 and 2.2 is taken as

$\|\min\{z^{(k)},M_1z^{(k)}+u_1(z^{(k)}),M_2z^{(k)}+u_2(z^{(k)})\}\|_2 < 10^{-10},$

and $\Omega$ is chosen as

$\Omega = \frac{1}{2\alpha} (D_{M_1}+D_{M_{2}})+I.$

Let $m = 256,512, 1024$ for the three examples. For fair comparison, when the relaxation parameters are chosen as the experimentally optimal ones for each size of the three examples, the results of the MMS and TMMS methods are shown in Tables 1–3, where "IT" is the iteration steps, "CPU" is the elapsed CPU time, and

$SAVE = \frac{CPU_{MMS}-CPU_{TMMS}}{CPU_{MMS}}\times 100\%.$

Table 1. Numerical results of Example 4.1.

$m$	splitting	Method 2.1		Method 2.2
		IT	CPU	IT	CPU	SAVE
256	GS	98	0.6945	50	0.5976	14%
	SOR	94	0.6679	46	0.5652	15%
	AOR	97	0.7469	47	0.5902	21%
512	GS	100	3.7686	51	3.1709	16%
	SOR	94	3.6039	46	3.0237	16%
	AOR	101	4.0266	47	3.1373	22%
1024	GS	102	22.7605	52	18.9571	17%
	SOR	134	21.1283	47	17.7199	16%
	AOR	97	24.4213	47	18.8083	23%

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Table 2. Numerical results of Example 4.2.

$m$	splitting	Method 2.1		Method 2.2
		IT	CPU	IT	CPU	SAVE
256	GS	86	0.5939	34	0.3792	36%
	SOR	69	0.4937	31	0.3571	27%
	AOR	71	0.5252	32	0.4136	21%
512	GS	89	3.4883	35	2.1793	38%
	SOR	70	2.6886	31	1.9244	28%
	AOR	71	4.5891	32	3.3323	27%
1024	GS	91	19.1887	36	12.4029	35%
	SOR	70	15.3799	32	11.5026	25%
	AOR	72	17.7342	33	13.3755	24%

| Show Table

DownLoad: CSV

Table 3. Numerical results of Example 4.3.

$m$	splitting	Method 2.1		Method 2.2
		IT	CPU	IT	CPU	SAVE
256	GS	76	0.6216	39	0.5478	12%
	SOR	59	0.4853	30	0.4394	9%
	AOR	54	0.4732	28	0.4103	13%
512	GS	288	17.4591	145	14.2667	18%
	SOR	253	12.6084	101	11.6480	7%
	AOR	204	14.9972	103	12.6497	16%
1024	GS	1137	172.0573	569	144.5840	16%
	SOR	885	137.8101	443	116.6376	15%
	AOR	805	130.3913	403	111.1060	15%

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It is found from the numerical results that Method 2.2 always converges faster than Method 2.1 for all cases. Specifically, the iteration steps of Method 2.1 take almost twice as much time as those of Method 2.2 due to the fact that there are two equations needed to solve in each iteration in Method 2.2 versus only one in Method 2.1. Focusing on the CPU time, the cost of Method 2.2 is 14%–23% less than that of Method 2.1 for Example 4.1, while the percentages of Examples 4.2 and 4.3 are 21%–38% and 7%–18%, respectively. It is clear that the two-step splitting technique works for accelerating the convergence of the MMS iteration method.

5. Conclusions

By the two-step matrix splitting technique, we have constructed the TMMS iteration method for solving the VNCP. We also present the convergence theorems of the TMMS iteration method for an arbitrary $s$ , which can extend the research scope of the modulus-based methods for the VNCP.

Note that to show the effectiveness of the proposed two-step method, the choice of the two-step splittings in Section 4 is common in existing literatures. However, since the iteration matrix in (3.13) is essentially an extended bound containing absolute operation, it is very hard to minimize its spectral radius. How to choose an optimal two-step splitting in general or based some special structure is still an open question. On the other hand, it is noted that there were some other accelerated techniques for the MMS iteration method, such as relaxation, precondition, two-sweep and so on. One can also mix a certain technique with the two-step splittings for acceleration, e.g., ^[3,22,30,47]. More techniques to improve the convergence of the MMS iteration method are worth studying in the future.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgement

This work was supported by Basic and Applied Basic Research Foundation of Guangdong (No. 2024A1515011822), Scientific Computing Research Innovation Team of Guangdong Province (No. 2021KCXTD052), Science and Technology Development Fund, Macau SAR (No. 0096/2022/A), Guangdong Key Construction Discipline Research Capacity Enhancement Project (No. 2022ZDJS049) and Technology Planning Project of Shaoguan (No. 230330108034184).

Conflict of interest

The authors declare that they have no conflict of interest.

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