1 Introduction

Consider the following weakly nonlinear complementarity problems, which is to find \(z\in R^{n}\) such that

$$\begin{aligned} z\geq0,\quad Az+\Psi(z)\geq0, \quad z^{T}\bigl(Az+\Psi(z) \bigr)=0, \end{aligned}$$
(1)

abbreviated as WNCP, where \(A=(a_{ij})\in R^{n\times n}\) is a given large, sparse and real matrix, \(\Psi(z)\): \(R^{n}\rightarrow R^{n}\) is a Lipschitz continuous nonlinear function, \(z\geq0\) means \(z_{i}\geq0\), \(i=1,2,\ldots,n\), and T always denotes the transpose of a vector.

As is well known, the classical nonlinear complementarity problems (NCP) are very important and fundamental topics in optimization theory and they have been developed into a well-established and fruitful principle. See [1–3] for details as regards the basic theories, effective algorithms and important applications of NCP. Problem (1) is a special case of NCP, but it extends from linear complementarity problems. When \(\Psi(z)=q\) is a constant vector, problem (1) reduces to a linear complementarity problem. Recently, lots of researchers [4–8] have paid close attention to feasible and efficient methods for solving linear complementarity problems. Especially, by reformulating linear complementarity problems as an implicit fixed point equation, Van Bokhoven [9] proposed a modulus iteration method, which is defined as the solution of system of linear equations at each iteration. In 2010, Bai [10] presented a modulus-based matrix splitting iteration method and showed the convergence when the system matrix is an \(H_{+}\)-matrix. Consequently, Zhang [11] proposed two-step modulus-based matrix splitting iteration methods and considered the convergence theory when the system matrix is an \(H_{+}\)-matrix. Based on the above work, for solving problem (1), in [12] Sun and Zeng proposed a modified semismooth Newton method with A being an M-matrix and \(\Psi(z)\) being a continuously differentiable monotone diagonal function on \(R^{n}\).

In this paper, for satisfying the requirements of the application, more details can be found in [13–16], we present an accelerated modulus-based matrix splitting algorithm for dealing with WNCP. The organization of this paper is as follows: Some necessary notations and definitions are introduced in Section 2. In Section 3, we establish a class of accelerated modulus-based matrix splitting iteration algorithms. In Section 4, the convergence conditions are considered.

2 Preliminaries

Some necessary notations, definitions and lemmas used in the sequel discussions are introduced in this section. For \(B\in R^{n\times n}\), we write \(B^{-1}\), \(B^{T}\), \(\rho(B)\) to denote the inverse, the transpose, the spectral radius of the matrix B, respectively. For \(x\in R^{n}\), we write \(\Vert x\Vert \), \(\vert x\vert \) to denote the norm of the vector x, \(\vert x\vert =(\vert x_{1}\vert ,\ldots, \vert x_{n}\vert )\), respectively. \(\Vert A\Vert \) denotes any norm of matrix A. Especially, we use \(\Vert \cdot \Vert _{2}\) to denote a spectral norm. \(\lambda\in\lambda(A)\) denotes the eigenvalue of matrix A where \(\lambda(A)\) is the set of all eigenvalues of matrix A.

Definition 1

[17]

If for any \(x:=(x_{1},x_{2},\ldots,x_{n})\neq0\), there exists an index k, such that \(x_{k}(Ax)_{k}=x_{k}(a_{k1}x_{1}+\cdots+a_{kn}x_{n})>0\), we call that matrix A is a P-matrix.

Definition 2

For the function \(f(x): R^{n}\rightarrow R^{n}\), if for any x, \(y\in R^{n}\), there exists a constant L such that

$$\bigl\Vert f(x)-f(y)\bigr\Vert \leq L\Vert x-y\Vert , $$

then we call f a Lipschitz continuous function on \(R^{n}\), and L is called a Lipschitz constant.

Lemma 3

[18]

Let \(A=(a_{ij})\in R^{n\times n}\) be a P-matrix, for any nonnegative diagonal matrix Ω, the matrix \(A+\Omega\) is nonsingular.

3 Algorithm

Theorem 4

Let \(M_{1}-N_{1}=M_{2}-N_{2}=A\) be two splittings of the matrix \(A\in R^{n\times n}\) and Ω, Γ be \(n\times n\) positive diagonal matrices, \(\Omega_{1}\), \(\Omega_{2}\) be \(n\times n\) nonnegative diagonal matrices such that \(\Omega=\Omega_{1}+\Omega_{2}\), then the following statements hold.

  1. (i)

    If z is a solution of problem (1), then \(x=\frac {1}{2} (\Gamma^{-1}z-\Omega^{-1}(Az+\Psi(z)) )\) satisfies the implicit fixed point equation

    $$ (M_{1}\Gamma+\Omega_{1})x =(N_{1}\Gamma-\Omega_{2})x+(\Omega-M_{2} \Gamma)\vert x\vert +N_{2}\Gamma \vert x\vert -\Psi \bigl(\Gamma \bigl(\vert x\vert +x\bigr) \bigr). $$
    (2)
  2. (ii)

    If x satisfies the implicit fixed point equation (2), then

    $$z=\Gamma\bigl(\vert x\vert +x\bigr) $$

    is a solution of problem (1).

Proof

First we prove part (i). Since z is a solution of problem (1), we have \(z\geq0\). So there exists \(x\in R^{n}\) such that

$$z=\Gamma\bigl(\vert x\vert +x\bigr). $$

Define another nonnegative vector,

$$\upsilon=\Omega\bigl(\vert x\vert -x\bigr). $$

It is easy to see that \(\upsilon\geq0\), \(z^{T}\upsilon=0\), and \(\upsilon =Az+\Psi(z)\) if and only if

$$\Omega\bigl(\vert x\vert -x\bigr)=A\Gamma\bigl(\vert x\vert +x\bigr)+ \Psi \bigl(\Gamma\bigl(\vert x\vert +x\bigr) \bigr). $$

Replace A with \(M_{1}-N_{1}\), we have

$$(M_{1}\Gamma+\Omega)x=N_{1}\Gamma x+ \bigl( \Omega-(M_{1}-N_{1})\Gamma \bigr)\vert x\vert -\Psi \bigl(\Gamma\bigl(\vert x\vert +x\bigr) \bigr). $$

Taking \(M_{1}-N_{1}=M_{2}-N_{2}\) and \(\Omega=\Omega_{1}+\Omega_{2}\) into account, it follows that

$$(M_{1}\Gamma+\Omega_{1})x=(N_{1}\Gamma- \Omega_{2}) x+(\Omega-M_{2}\Gamma )\vert x\vert +N_{2}\Gamma \vert x\vert -\Psi \bigl(\Gamma\bigl(\vert x\vert +x\bigr) \bigr). $$

This shows that equation (2) holds.

We now turn to prove part (ii). By some simple calculations, the implicit fixed point equation (2) is equivalent to

$$A\Gamma\bigl(\vert x\vert +x\bigr)+\Psi \bigl(\Gamma\bigl(\vert x\vert +x\bigr) \bigr)=\Omega\bigl(\vert x\vert -x\bigr). $$

Set \(z=\Gamma(\vert x\vert +x)\) and \(\upsilon=\Omega(\vert x\vert -x)\). Evidently, it yields \(z\geq0\), \(\upsilon\geq0\), \(z^{T}\upsilon=0\), and \(\upsilon =Az+\Psi(z)\), which means that z is a solution of problem (1). This completes the proof. □

Note that the implicit fixed point equation (2) includes many parameters that are quite complicated to be determined in a computation. For solving this problem, we will give the simple formulation as follows. Subsequently, a matrix splitting iteration algorithm will be given for solving WNCP. Let \(\Omega_{1}=\Omega\), \(\Omega_{2}=0, \Gamma=\frac{1}{\gamma}I\), where \(\gamma> 0\) is a real number, then the implicit fixed point equation reduces to

$$(M_{1}+\gamma\Omega)x=N_{1} x+(\gamma\Omega-M_{2}) \vert x\vert +N_{2}\vert x\vert -\gamma \Psi \biggl( \frac{(\vert x\vert +x)}{\gamma} \biggr). $$

In fact, γΩ in the above equation denotes a positive diagonal parameter matrix, which can be replaced by Ω for simplicity. That is, the above equation is essentially equivalent to

$$\begin{aligned} (M_{1}+\Omega)x=N_{1} x+( \Omega-M_{2})\vert x\vert +N_{2}\vert x\vert -\gamma \Psi \biggl(\frac {(\vert x\vert +x)}{\gamma} \biggr). \end{aligned}$$
(3)

In the rest of the paper, we will use the fixed point equation above to give the algorithm and convergence analysis for solving WNCP.

Algorithm 5

Step 1. Choose two splittings of the matrix \(A\in R^{n\times n}\) satisfying \(A=M_{1}-N_{1}=M_{2}-N_{2}\).

Step 2. Set \(k=0\). Give an initial vector \(x^{0}\in R^{n}\), compute \(z^{0}=\frac{1}{\gamma}(\vert x^{0}\vert +x^{0})\).

Step 3. Choose \(x^{k+1}\) such that

$$\begin{aligned} (M_{1}+\Omega)x^{k+1}=N_{1}x^{k}+( \Omega -M_{2})\bigl\vert x^{k}\bigr\vert +N_{2}\bigl\vert x^{k+1}\bigr\vert -\gamma\Psi \bigl(z^{k}\bigr), \end{aligned}$$
(4)

and set

$$z^{k+1}=\frac{1}{\gamma}\bigl(\bigl\vert x^{k+1}\bigr\vert +x^{k+1}\bigr),\quad k=k+1. $$

Here, \(\Omega, \Gamma\in R^{n\times n}\) are positive diagonal matrices and γ is a positive constant.

Step 4. If the sequence \(\{z^{k}\}_{0}^{+\infty}\) is convergent, stop. Otherwise, go to Step 3.

4 Convergence theorems

In this section, we will consider the conditions that ensure the convergence of \(\{z^{k}\}_{0}^{+\infty}\) obtained by Algorithm 5.

Theorem 6

Let \(A\in R^{n\times n}\) be a P-matrix, \(M_{1}-N_{1}=M_{2}-N_{2}=A\) be two splittings of the matrix A with \(M_{1}\in R^{n\times n}\) being a P-matrix. Assume that \(\Omega\in R ^{n\times n}\) is a positive diagonal matrix, γ is a positive constant and \(\Psi(z)\): \(R^{n}\rightarrow R^{n}\) is a Lipschitz continuous function with the Lipschitz constant L. Set

$$\rho(\Omega)=2g(\Omega)+2q(\Omega)+f(\Omega), $$

where \(g(\Omega)=\Vert (M_{1}+\Omega)^{-1}N_{2}\Vert \), \(q(\Omega)=\Vert (M_{1}+\Omega)^{-1}N_{1}\Vert +L\Vert (M_{1}+\Omega)^{-1}\Vert \), \(f(\Omega)=\Vert (M_{1}+\Omega)^{-1}(\Omega-M_{1})\Vert \). If the matrix Ω satisfies \(\rho(\Omega)<1\), then for any initial vector \(x^{0}\in R^{n}\), the iteration sequence \(\{z^{k}\}^{+\infty }_{k=0}\) generated by Algorithm 5 converges to a solution \(z^{*}\in R^{n}_{+}\) of problem (1).

Proof

Suppose that \(z^{*}\in R^{n}_{+}\) is a solution of problem (1). Note that \(\Gamma=\frac{1}{\gamma}I\), by Theorem 4 and (3), we see that \(x^{*}=\frac{1}{2} (\gamma z^{*}-\Omega ^{-1} (A(z^{*})+\Psi(z^{*}) ) )\) is a solution of the equation

$$\begin{aligned} (M_{1}+\Omega)x^{*}=N_{1}x^{*}+( \Omega-M_{2})\bigl\vert x^{*}\bigr\vert +N_{2}\bigl\vert x^{*}\bigr\vert -\gamma \Psi \bigl(z^{*}\bigr) \end{aligned}$$
(5)

with \(z^{*}=\frac{1}{\gamma}(\vert x^{*}\vert +x^{*})\). Let us consider (4) minus (5); we have

$$\begin{aligned} &(M_{1}+\Omega) \bigl(x^{k+1}-x^{*}\bigr) \\ ={}&N_{1}\bigl(x^{k}-x^{*}\bigr)+( \Omega-M_{2}) \bigl(\bigl\vert x^{k}\bigr\vert -\bigl\vert x^{*}\bigr\vert \bigr) \\ &{}+N_{2}\bigl(\bigl\vert x^{k+1}\bigr\vert -\bigl\vert x^{*}\bigr\vert \bigr)-\gamma\bigl(\Psi\bigl(z^{k} \bigr)-\Psi\bigl(z^{*}\bigr)\bigr). \end{aligned}$$

Note that \(M_{1}-N_{1}=M_{2}-N_{2}\), we have

$$\begin{aligned} &{}(\Omega-M_{2}) \bigl(\bigl\vert x^{k}\bigr\vert -\bigl\vert x^{*}\bigr\vert \bigr) \\ ={}&(\Omega-M_{2}+N_{2}-N_{2}) \bigl(\bigl\vert x^{k}\bigr\vert -\bigl\vert x^{*}\bigr\vert \bigr) \\ ={}&(\Omega-M_{1}+N_{1}-N_{2}) \bigl(\bigl\vert x^{k}\bigr\vert -\bigl\vert x^{*}\bigr\vert \bigr) \\ ={}&(\Omega-M_{1}) \bigl(\bigl\vert x^{k}\bigr\vert -\bigl\vert x^{*}\bigr\vert \bigr)+N_{1} \bigl(\bigl\vert x^{k}\bigr\vert -\bigl\vert x^{*}\bigr\vert \bigr)-N_{2} \bigl(\bigl\vert x^{k}\bigr\vert -\bigl\vert x^{*}\bigr\vert \bigr). \end{aligned}$$
(6)

Since \(M_{1}\) is a P-matrix and Ω is a positive diagonal matrix, it follows from Lemma 3 that \(M_{1}+\Omega\) is a nonsingular matrix. Hence, by (5) and (6), we have

$$\begin{aligned} &{}x^{k+1}-x^{*} \\ ={}&(M_{1}+\Omega)^{-1}N_{1} \bigl(x^{k}-x^{*}\bigr)+(M_{1}+ \Omega)^{-1}(\Omega -M_{1}) \bigl(\bigl\vert x^{k} \bigr\vert -\bigl\vert x^{*}\bigr\vert \bigr) \\ &{}+(M_{1}+\Omega)^{-1}N_{1} \bigl(\bigl\vert x^{k}\bigr\vert -\bigl\vert x^{*}\bigr\vert \bigr)-(M_{1}+\Omega )^{-1}N_{2} \bigl(\bigl\vert x^{k}\bigr\vert -\bigl\vert x^{*}\bigr\vert \bigr) \\ &{}+(M_{1}+\Omega)^{-1}N_{2} \bigl(\bigl\vert x^{k+1}\bigr\vert -\bigl\vert x^{*}\bigr\vert \bigr)-(M_{1}+\Omega )^{-1}\gamma \bigl(\Psi \bigl(z^{k}\bigr)-\Psi\bigl(z^{*}\bigr) \bigr). \end{aligned}$$
(7)

Taking the facts

$$\begin{aligned} \bigl\Vert z^{k}-z^{*}\bigr\Vert =& \biggl\Vert \frac{\vert x^{k}\vert +x^{k}}{\gamma}-\frac {\vert x^{*}\vert +x^{*}}{\gamma} \biggr\Vert \leq \frac{2}{\gamma}\bigl\Vert x^{k}-x^{*}\bigr\Vert , \end{aligned}$$
(8)

\(\Psi(z)\) is a Lipschitz continuous function, and (8) into account, we have

$$\begin{aligned} \bigl\Vert \Psi\bigl(z^{k}\bigr)-\Psi \bigl(z^{*}\bigr)\bigr\Vert \leq L\bigl\Vert z^{k}-z^{*} \bigr\Vert \leq\frac{2L}{\gamma}\bigl\Vert x^{k}-x^{*} \bigr\Vert . \end{aligned}$$
(9)

Thereby, we derive from (7) and (9) that

$$\begin{aligned} &{} \bigl(1- \bigl\Vert (M_{1}+\Omega)^{-1}N_{2} \bigr\Vert \bigr)\bigl\Vert x^{k+1}-x^{*}\bigr\Vert \\ \leq{}& \bigl(2 \bigl(\bigl\Vert (M_{1}+\Omega)^{-1}N_{1} \bigr\Vert +L\bigl\Vert (M_{1}+\Omega)^{-1}\bigr\Vert \bigr) +\bigl\Vert (M_{1}+\Omega)^{-1}(\Omega-M_{1}) \bigr\Vert \\ &{}+\bigl\Vert (M_{1}+\Omega)^{-1}N_{2} \bigr\Vert \bigr)\bigl\Vert x^{k}-x^{*}\bigr\Vert , \end{aligned}$$

which is equivalent to

$$\bigl\Vert x^{k+1}-x^{*}\bigr\Vert \leq\frac{2q(\Omega)+f(\Omega)+g(\Omega)}{1-g(\Omega )} \bigl\Vert x^{k}-x^{*}\bigr\Vert $$

with \(g(\Omega)<1\). The condition

$$\frac{2q(\Omega)+f(\Omega)+g(\Omega)}{1-g(\Omega)}< 1 $$

with \(g(\Omega)<1\), which is equivalent to \(\rho(\Omega)=2g(\Omega )+2q(\Omega)+f(\Omega)<1\), ensures that the limit \(\lim_{k\rightarrow +\infty}x^{k}=x^{*}\) holds. These results complete the proof. □

Theorem 7

Let \(A\in R^{n\times n}\) be a P-matrix, \(M_{1}-N_{1}=M_{2}-N_{2}=A\) be two splittings of the matrix A with \(M_{1}\in R^{n\times n}\) being a symmetric P-matrix. Suppose that \(\Omega=\omega I\in R ^{n\times n}\) is a positive scalar matrix and ω is a positive constant. \(\Psi (z): R^{n}\rightarrow R^{n}\) is a Lipschitz continuous function with the Lipschitz constant L. \(\lambda_{\max}\) and \(\lambda_{\min}\) to denote the largest and smallest eigenvalue of the matrix \(M_{1}\), respectively. Let \(\tau_{1}=\Vert M_{1}^{-1}N_{1}\Vert _{2}\) and \(\tau_{2}=\Vert M_{1}^{-1}N_{2}\Vert _{2}\) satisfy \(\tau_{1}+\tau_{2}<1\). If \(\lambda _{\min}>L\), the choices of the parameters ω, \(M_{1}, N_{1}, M_{2}, N_{2}\) satisfy either of the following conditions, then the iteration sequence \(\{z_{k}\}_{k}^{+\infty}\subset R^{n}_{+}\) generated by Algorithm 5 converges to the unique solution \(z^{*}\in R^{n}_{+}\) of WNCP for any initial vector \(x^{0}\in R^{n}\).

  1. (i)

    When \(0<\tau_{1}+\tau_{2}<\frac{\lambda _{\min}-L}{\lambda_{\max}}\),

    $$\omega=\sqrt{\lambda_{\max}\lambda_{\min}}. $$
  2. (ii)

    When \(\frac{\lambda_{\min}-L}{\lambda_{\max}} <\tau _{1}+\tau_{2}<\frac{\lambda_{\min}-L}{\sqrt{\lambda_{\max}\lambda_{\min}}}\),

    $$\sqrt{\lambda_{\max}\lambda_{\min}}\leq\omega< \frac{[1-(\tau_{1}+\tau _{2})]\lambda_{\max}\lambda_{\min}-L\lambda_{\max}}{(\tau_{1}+\tau _{2})\lambda_{\max}+L-\lambda_{\min}}. $$
  3. (iii)

    When \(\tau_{1}+\tau_{2}=\frac{\lambda _{\min}-L}{\lambda_{\max}}\),

    $$\omega\geq\sqrt{\lambda_{\max}\lambda_{\min}}. $$

Proof

We first give some formulations, which will be used in the proof. Since M is a symmetric P-matrix and \(\tau_{1}+\tau_{2}<1\), by the definition of the spectral norm, we have

$$\begin{aligned} \bigl\Vert (M_{1}+\Omega)^{-1}N_{1} \bigr\Vert _{2} =&\bigl\Vert (M_{1}+\omega I)^{-1}M_{1}M_{1}^{-1}N_{1} \bigr\Vert _{2} \\ \leq&\bigl\Vert (M_{1}+\omega I)^{-1}M_{1} \bigr\Vert _{2}\bigl\Vert M_{1}^{-1}N_{1} \bigr\Vert _{2} \\ =&\max_{\lambda\in\lambda(M_{1})}\frac{\lambda\tau_{1}}{\omega+\lambda } \\ =&\frac{\lambda_{\max}\tau_{1}}{\omega+\lambda_{\max}}. \end{aligned}$$
(10)

Similarly, we have

$$\begin{aligned} \bigl\Vert (M_{1}+\Omega)^{-1}N_{2} \bigr\Vert _{2} =&\bigl\Vert (M_{1}+\omega I)^{-1}M_{1}M_{1}^{-1}N_{1} \bigr\Vert _{2} \\ \leq&\frac{\lambda_{\max}\tau_{2}}{\omega+\lambda_{\max}} \end{aligned}$$
(11)

and

$$\begin{aligned} \bigl\Vert (M_{1}+\Omega)^{-1}\bigr\Vert _{2} =&\bigl\Vert (M_{1}+\omega I)^{-1}\bigr\Vert _{2} \\ =&\max_{\lambda\in\lambda(M_{1})}\frac{1}{\omega+\lambda} \\ =&\frac{1}{\omega+\lambda_{\min}}. \end{aligned}$$
(12)

In addition, from a simple calculating process, we have

$$\begin{aligned} \bigl\Vert (M_{1}+\Omega)^{-1}( \Omega-M_{1})\bigr\Vert _{2} =&\bigl\Vert (M_{1}+\omega I)^{-1}(\omega I-M_{1})\bigr\Vert _{2} \\ =&\max_{\lambda\in\lambda(M_{1})}\frac{\vert \omega-\lambda \vert }{\omega+\lambda } =\max\biggl\{ \frac{\vert \omega-\lambda_{\max} \vert }{\omega+\lambda_{\max}},\frac {\vert \omega-\lambda_{\min} \vert }{\omega+\lambda_{\min}}\biggr\} \\ =&\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \frac{\lambda_{\max}-\omega}{\lambda_{\max}+\omega}, & \omega \leq\sqrt{\lambda_{\max}\lambda_{\min}},\\ \frac{\omega-\lambda_{\min}}{\omega+\lambda_{\min}}, & \omega \geq\sqrt{\lambda_{\max}\lambda_{\min}}. \end{array}\displaystyle \right . \end{aligned}$$
(13)

As follows from (10)-(13), we have

$$\begin{aligned} \rho(\Omega) =&2g(\Omega)+2q(\Omega)+f(\Omega) \\ =&2 \bigl\Vert (M_{1}+\Omega)^{-1}N_{2} \bigr\Vert _{2}+2 \bigl(\bigl\Vert (M_{1}+\Omega )^{-1}N_{1}\bigr\Vert _{2} \\ &{}+L\bigl\Vert (M_{1}+\Omega)^{-1}\bigr\Vert _{2} \bigr)+\bigl\Vert (M_{1}+\Omega)^{-1}( \Omega -M_{1})\bigr\Vert _{2} \\ =&2\frac{\lambda_{\max}\tau_{2}}{\omega+\lambda_{\max}}+2\biggl(\frac{\lambda _{\max}\tau_{1}}{\omega+\lambda_{\max}} +\frac{L}{\omega+\lambda_{\min}}\biggr)+\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \frac{\lambda_{\max}-\omega}{\omega+\lambda_{\max}}, & \omega \leq\sqrt{\lambda_{\max}\lambda_{\min}},\\ \frac{\omega-\lambda_{\min}}{\omega+\lambda_{\min}}, & \omega \geq\sqrt{\lambda_{\max}\lambda_{\min}}. \end{array}\displaystyle \right . \end{aligned}$$
(14)

We then consider two cases.

(a) When \(\omega\leq\sqrt{\lambda_{\max}\lambda_{\min}}\), by a simple calculation on (14), we see that ω, \(\tau_{1}\), and \(\tau_{2}\) satisfy \(\rho(\Omega)<1\), which is equivalent to

$$\omega^{2}- \bigl((\tau_{1}+\tau_{2}) \lambda_{\max}+L-\lambda_{\min} \bigr)\omega- \bigl(( \tau_{1}+\tau_{2})\lambda_{\max} \lambda_{\min}+L\lambda _{\max} \bigr)>0. $$

Note that \(\omega>0\) and \(\omega\leq\sqrt{\lambda_{\max}\lambda_{\min}}\), then the solution of the above inequality is

$$\theta(\tau_{1},\tau_{2})< \omega\leq\sqrt{ \lambda_{\max}\lambda_{\min}}, $$

where

$$\begin{aligned} &{}\theta(\tau_{1},\tau_{2}) \\ =&\frac{\sqrt{ ((\tau_{1}+\tau_{2})\lambda_{\max}+L-\lambda_{\min} )^{2}+4(\tau_{1}+\tau_{2})\lambda_{\max}\lambda_{\min}+L\lambda _{\max}}}{2} \\ &{}+\frac{(\tau_{1}+\tau_{2})\lambda_{\max}+L-\lambda_{\min}}{2}. \end{aligned}$$

Certainly, we have

$$\begin{aligned} \theta(\tau_{1},\tau_{2})< \sqrt{ \lambda_{\max}\lambda_{\min}}. \end{aligned}$$
(15)

Since \(\lambda_{\min}>L\), by the definitions of \(\tau_{1}\), \(\tau_{2}\) and solving (15), we get

$$0< \tau_{1}+\tau_{2}< \frac{\lambda_{\min}-L}{\sqrt{\lambda_{\max}\lambda_{\min}}}. $$

(b) When \(\omega\geq\sqrt{\lambda_{\max}\lambda_{\min}}\), in a same way as (a), ω, \(\tau_{1}\), and \(\tau_{2}\) satisfying \(\rho(\Omega)<1\), which is equivalent to

$$\begin{aligned} \bigl[(\tau_{1}+\tau_{2}) \lambda_{\max}+L-\lambda_{\min} \bigr]\omega+(\tau _{1}+\tau_{2}-1)\lambda_{\max}\lambda_{\min}+L \lambda_{\max}< 0. \end{aligned}$$
(16)

If \((\tau_{1}+\tau_{2})\lambda_{\max}+L-\lambda_{\min}>0\), that is, \(\tau _{1}+\tau_{2}>\frac{\lambda_{\min}-L}{\lambda_{\max}}\), then

$$\omega< \frac{[1-(\tau_{1}+\tau_{2})]\lambda_{\max}\lambda_{\min}-L\lambda _{\max}}{(\tau_{1}+\tau_{2})\lambda_{\max}+L-\lambda_{\min}}. $$

Combined with \(\omega\geq\sqrt{\lambda_{\max}\lambda_{\min}}\), we get

$$\sqrt{\lambda_{\max}\lambda_{\min}}\leq\omega< \frac{[1-(\tau_{1}+\tau _{2})]\lambda_{\max}\lambda_{\min}-L\lambda_{\max}}{(\tau_{1}+\tau _{2})\lambda_{\max}+L-\lambda_{\min}}. $$

Naturally, we have

$$\sqrt{\lambda_{\max}\lambda_{\min}}< \frac{[1-(\tau_{1}+\tau_{2})]\lambda _{\max}\lambda_{\min}-L\lambda_{\max}}{(\tau_{1}+\tau_{2})\lambda _{\max}+L-\lambda_{\min}}. $$

That is,

$$\tau_{1}+\tau_{2}< \frac{\lambda_{\min}-L}{\sqrt{\lambda_{\max}\lambda_{\min}}}. $$

This, together with \(\tau_{1}+\tau_{2}>\frac{\lambda_{\min}-L}{\lambda _{\max}}\), shows that we have

$$\frac{\lambda_{\min}-L}{\lambda_{\max}}< \tau_{1}+\tau_{2}< \frac{\lambda _{\min}-L}{\sqrt{\lambda_{\max}\lambda_{\min}}}. $$

If \((\tau_{1}+\tau_{2})\lambda_{\max}+L-\lambda_{\min}\leq0\), that is, \(\tau_{1}+\tau_{2}\leq\frac{\lambda_{\min}-L}{\lambda_{\max}}\), then for any \(\omega>0\) (16) holds. So

$$\omega\geq{\sqrt{\lambda_{\max}\lambda_{\min}}}. $$

Hence, from (a) and (b), we see that when \(0<\tau_{1}+\tau_{2}<\frac {\lambda_{\min}-L}{\lambda_{\max}}, \omega=\sqrt{\lambda_{\max}\lambda _{\min}}\); when \(\frac{\lambda_{\min}-L}{\lambda_{\max}} <\tau_{1}+\tau _{2}<\frac{\lambda_{\min}-L}{\sqrt{\lambda_{\max}\lambda_{\min}}}\), \(\sqrt{\lambda_{\max}\lambda_{\min}}\leq\omega<\frac{[1-(\tau_{1}+\tau _{2})]\lambda_{\max}\lambda_{\min}-L\lambda_{\max}}{(\tau_{1}+\tau _{2})\lambda_{\max}+L-\lambda_{\min}}\); when \(\tau_{1}+\tau_{2}=\frac{\lambda_{\min}-L}{\lambda_{\max}}\), \(\omega \geq\sqrt{\lambda_{\max}\lambda_{\min}}\). The proof is completed. □

5 Results and discussion

This study focused on the weakly nonlinear complementarity problems with a large sparse matrix. We proposed an algorithm that is not only computationally more convenient to use but also faster than the modulus-based matrix splitting iteration methods and the convergence conditions are presented when the system matrix is a P-matrix.

Some scholars had already stressed the accelerated modulus-based matrix splitting iteration methods for linear complementarity problems and pointed out that the system matrix is either a positive definite matrix or an \(H_{+}\)-matrix. However, we suggest that the system matrix is a P-matrix, this is more adaptable but also a limitation. Notwithstanding its limitation, this study does suggest that WNCP can be solved faster.

6 Conclusions

In this paper, by reformulating the complementarity problem (1) as an implicit fixed point equation based on splittings of the system matrix A, we establish an accelerated modulus-based matrix splitting iteration algorithm and show the convergence analysis when the involved matrix of the WNCP is a P-matrix.