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On the convergence of two-step modulus-based matrix splitting iteration method

  • Ximing Fang , Shouzhong Fu and Ze Gu EMAIL logo
Published/Copyright: December 31, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

In this paper, based on the relationship between the linear complementarity problem and its reformulated fixed-point equation, we discuss the conditions of the modulus-based type iteration methods. Moreover, we present some convergence results on the two-step modulus-based matrix splitting iteration method with an H + -matrix. Finally, we give the numerical experiments.

Keywords: linear complementarity problem; modulus-based type iteration method; convergence condition
MSC 2010: 90C33

1 Introduction

The linear complementarity problem is to solve z ∈ R n such that

(1) z T r = 0 with z ≥ 0 , r = A z + q ≥ 0 ,

where A = ( a i j ) ∈ R n × n , q ∈ R n are given and the symbol “ T ” denotes the transpose operation. This problem is usually abbreviated as LCP( A , q ), which has many applications, such as the optimal stopping in Markov chain and the free boundary problems of journal bearings. For the detailed materials, see [1,2,3, 4,5,6, 7,8] and references therein.

In order to compute the numerical solution of the LCP( A , q ) with a large and sparse system matrix A , many kinds of modulus-based type iteration methods are presented recently, such as the modulus-based iteration method [8], the nonstationary extrapolated modulus method [9], the modified modulus-based iteration method [10], the modulus-based matrix splitting iteration method [3], the general modulus-based matrix splitting method [6], the two-step modulus-based matrix splitting iteration method [11], the two-sweep modulus-based matrix splitting iteration method [12], the preconditioned general modulus-based matrix splitting method [13], the accelerated modulus-based matrix splitting iteration methods [14,15], and so on. Most of these modulus-based type iteration methods are very practical and efficient. The common character of these methods is that the LCP( A , q ) is reformulated as a fixed-point equation through introducing the parameter matrices, and the original LCP( A , q ) is solved by solving such an equation with iteration approaches. Besides the modulus-based type iteration methods mentioned above, there are other iteration methods as well as the direct methods. For the detailed materials, readers can refer to [2,3,8, 9,10,16, 17,18,19, 20,21,22, 23,24,25, 26,27,28] and references therein.

We further study the modulus-based type iteration methods for solving the LCP( A , q ) in this paper. Although there are many modulus-based type iteration methods, the relationship between the linear complementarity problem and the reformulated equation is rarely discussed. Meanwhile, the involved system matrices in these modulus-based type iteration methods are mainly the positive definite matrix and the H + -matrix, which are two types of P -matrix, and these methods and the theories have not been extended to the general P -matrix. By studying the connection between the LCP( A , q ) and the fixed-point equation, we propose the conditions of the modulus-based type iteration method, which extends the application scope of this method. For the two-step modulus-based matrix splitting iteration method [11], the convergence has been further studied in [16]. Since this method is very effective, it has been applied to solve other complementarity problems. For the recent works on the two-step modulus-based methods for solving other complementarity problems, readers can refer to [29,30, 31,32,33, 34,35]. As the main part of this paper, we discuss the convergence problem of this method when the system matrix A is an H + -matrix and present some convergence conclusions from the matrix spectral radius. In the end, we provide the numerical experiments to verify the proposed results and illustrate the special cases of this method.

2 Preliminaries

In this section, we review some necessary preliminaries in brief, including the concepts, the notations, the reformulation of LCP( A , q ), and the two-step modulus-based matrix splitting iteration method. We first introduce the definition of P -matrix, and for this definition and its equivalent expressions as well as the properties of P -matrix, readers can refer to [8,36, 37,38].

Definition 2.1

A matrix A ∈ R n × n is called a P -matrix if all of its principal minors are positive.

A ∈ R n × n is called a Z -matrix if a i j ≤ 0 , i ≠ j , i , j = 1 , 2 , … , n , and A ∈ R n × n is called an M -matrix if A is a Z -matrix with A − 1 ≥ 0 . The comparison matrix of A is denoted by ⟨ A ⟩ = ( ⟨ a i j ⟩ ) with

⟨ a i j ⟩ = − ∣ a i j ∣ , i ≠ j , ∣ a i j ∣ , i = j ,

where “ ∣ ⋅ ∣ ” denotes the absolute value function. A ∈ R n × n is called an H -matrix if ⟨ A ⟩ is an M -matrix, and the H -matrix is called an H + -matrix if all of its diagonal elements are positive [3]. The absolute value matrix of a real matrix A ∈ R n × n is denoted by ∣ A ∣ = ( ∣ a i j ∣ ) . The positive vector and the nonnegative vector are denoted by v > 0 and v ≥ 0 , respectively. For details, readers can refer to [3,6] and references therein.

By setting

(2) z = ∣ x ∣ + x and r = Ω ( ∣ x ∣ − x ) ,

the LCP( A , q ) can be transformed into a fixed-point equation

(3) ( Ω + A ) x = ( Ω − A ) ∣ x ∣ − q ,

where Ω is a positive diagonal matrix. Equation (3) is a particular case of the original form (5) in [3], where γ is set to be 1. All the modulus-based type iteration methods introduced before are based on (3) or its changed forms.

From (2), we know that if z ∗ is a solution of (1), then we can obtain a solution x ∗ of (3), that is,

(4) x ∗ = z ∗ − Ω − 1 ( A z ∗ + q ) 2 .

On the contrary, if x ∗ is a solution of (3), then we can obtain a solution z ∗ of (1), that is,

(5) z ∗ = ∣ x ∗ ∣ + x ∗ .

In other words, we can solve (1) by solving (3) from [3].

For all of the modulus-based type iteration methods mentioned before, the involved initial iteration vector x ( 0 ) is arbitrary, which means that equation (3) should have a unique solution if the modulus-based type iteration methods are convergent. Thus, we first discuss the unique solution problem of equation (3) in Section 3.

The two-step modulus-based matrix splitting iteration method and its particular case, that is, the two-step modulus-based accelerated overrelaxition (MBAOR) iteration method are briefly reviewed in the following, and readers can refer to [11] for details.

2.1 The two-step modulus-based matrix splitting iteration method

Let A = M i − N i , i = 1 , 2 be two splittings of the matrix A . Given an initial vector x ( 0 ) , compute x ( k + 1 ) ∈ R n by solving the linear systems

(6) ( M 1 + Ω ) x k + 1 2 = N 1 x ( k ) + ( Ω − A ) ∣ x ( k ) ∣ − q , ( M 2 + Ω ) x ( k + 1 ) = N 2 x k + 1 2 + ( Ω − A ) ∣ x k + 1 2 ∣ − q .

Then set

(7) z ( k + 1 ) = ∣ x ( k + 1 ) ∣ + x ( k + 1 )

for k = 0 , 1 , 2 , … until the iteration sequence { z ( k ) } k = 1 + ∞ ⊂ R n is convergent. Here Ω is a given positive diagonal matrix.

Suppose A = D − L − U , where D , − L , − U are the diagonal, strictly lower-triangular and strictly upper-triangular matrices of A , respectively. Set

(8) M 1 = 1 α 1 ( D − β 1 L ) , N 1 = 1 α 1 ( ( 1 − α 1 ) D + ( α 1 − β 1 ) L + α 1 U ) , M 2 = 1 α 2 ( D − β 2 U ) , N 2 = 1 α 2 ( ( 1 − α 2 ) D + ( α 2 − β 2 ) U + α 2 L ) ,

where α 1 > 0 , β 1 ≥ 0 , α 2 > 0 , β 2 ≥ 0 , then we have the particular two-step modulus-based matrix splitting iteration method based on the accelerate overrelaxation (AOR) splittings (8), that is, the two-step MBAOR iteration method below.

2.2 The two-step MBAOR iteration method

(9) 1 α 1 ( D − β 1 L ) + Ω x k + 1 2 = 1 α 1 ( ( 1 − α 1 ) D + ( α 1 − β 1 ) L + α 1 U ) x ( k ) + ( Ω − A ) ∣ x ( k ) ∣ − q , 1 α 2 ( D − β 2 U ) + Ω x ( k + 1 ) = 1 α 2 ( ( 1 − α 2 ) D + ( α 2 − β 2 ) U + α 2 L ) x k + 1 2 + ( Ω − A ) ∣ x k + 1 2 ∣ − q ,

with z ( k + 1 ) = ∣ x ( k + 1 ) ∣ + x ( k + 1 ) and α 1 > 0 , β 1 ≥ 0 , α 2 > 0 , β 2 ≥ 0 .

3 Main results

In this section, we discuss the conditions of the modulus-based type iteration methods and the convergence of the two-step modulus-based matrix splitting iteration method.

3.1 The conditions of the modulus-based type iteration method

From the relation between the linear complementarity problem (1) and equation (3), we have the following conclusion.

Theorem 3.1.1

If the linear complementarity problem (1) has a unique solution for any q ∈ R n , i.e., A is a P -matrix, then equation (3) has a unique solution for any positive diagonal matrix Ω .

Proof

From (4), we know that equation (3) has solutions for any q ∈ R n and any positive diagonal matrix Ω . Let x ∗ and x ∗ ∗ be two solutions of (3), that is,

(10) ( Ω + A ) x ∗ = ( Ω − A ) ∣ x ∗ ∣ − q , ( Ω + A ) x ∗ ∗ = ( Ω − A ) ∣ x ∗ ∗ ∣ − q .

On one hand, from (5), we know that both ∣ x ∗ ∣ + x ∗ and ∣ x ∗ ∗ ∣ + x ∗ ∗ are the solutions of (1). Since the linear complementarity problem (1) has a unique solution, we have

(11) ∣ x ∗ ∣ + x ∗ = ∣ x ∗ ∗ ∣ + x ∗ ∗ .

Then x ∗ and x ∗ ∗ have the same positive elements with the same positions.

On the other hand, from (10), we have

(12) ( Ω + A ) ( x ∗ − x ∗ ∗ ) = ( Ω − A ) ( ∣ x ∗ ∣ − ∣ x ∗ ∗ ∣ ) .

By reformulating, we have

(13) Ω ( x ∗ − x ∗ ∗ − ∣ x ∗ ∣ + ∣ x ∗ ∗ ∣ ) + A ( x ∗ − x ∗ ∗ + ∣ x ∗ ∣ − ∣ x ∗ ∗ ∣ ) = 0 .

Then, from (11),

(14) Ω ( x ∗ − x ∗ ∗ − ∣ x ∗ ∣ + ∣ x ∗ ∗ ∣ ) = 0

holds. So, we have

(15) ∣ x ∗ ∣ − x ∗ = ∣ x ∗ ∗ ∣ − x ∗ ∗ .

Thus, x ∗ and x ∗ ∗ have the same negative elements with the same positions. Therefore, we have

x ∗ = x ∗ ∗ . □

In Theorem 2.1 of [3], the author has proved the equivalence between the LCP( A , q ) and the fixed-point equation in terms of solutions. Here, we emphasize the case that the linear complementarity problem has a unique solution, which is the basis of exploring the convergence of modulus-based type iteration methods for solving the linear complementarity problems.

Corollary 3.1.1

If (1) has a unique solution for some q ∈ R n , then (3) has a unique solution for any positive diagonal matrix Ω .

Proof

Suppose z ∗ is the unique solution of (1). From (4), we know that (3) has a solution x Ω ∗ for any positive diagonal matrix Ω . Let x Ω ∗ ∗ be another solution of (3), similar to the proof of Theorem 3.1.1, we can prove that

(16) ∣ x Ω ∗ ∣ + x Ω ∗ = ∣ x Ω ∗ ∗ ∣ + x Ω ∗ ∗ , ∣ x Ω ∗ ∣ − x Ω ∗ = ∣ x Ω ∗ ∗ ∣ − x Ω ∗ ∗ .

Thus, x Ω ∗ = x Ω ∗ ∗ holds and the conclusion is proved.□

3.2 The convergence of the two-step modulus-based matrix splitting iteration method

Now, we discuss a concrete modulus-based type iteration method, that is, the two-step modulus-based matrix splitting iteration method.

Theorem 3.2.1

Let A ∈ R n × n be an H + -matrix and A = M i − N i , i = 1 , 2 , be two splittings of A with ⟨ M 1 ⟩ and ⟨ M 2 ⟩ being two M -matrices. If

(17) ρ ( ( ⟨ M 2 ⟩ + Ω ) − 1 ( ∣ N 2 ∣ + ∣ Ω − A ∣ ) ( ⟨ M 1 ⟩ + Ω ) − 1 ( ∣ N 1 ∣ + ∣ Ω − A ∣ ) ) < 1

for some positive diagonal matrix Ω , then { z ( k ) } k = 1 + ∞ ⊂ R + n generated by (6) with (7) converges to the unique solution of the LCP( A , q ) for any initial vector x ( 0 ) ∈ R n .

Proof

From the assumption that A is an H + -matrix, we know that the LCP( A , q ) has a unique solution z ∗ and equation (3) also has a unique solution x ∗ for any positive diagonal matrix Ω from Theorem 3.1.1. Since ⟨ M 1 ⟩ , ⟨ M 2 ⟩ are two M -matrices, we have ⟨ A ⟩ , ⟨ M 1 ⟩ , ⟨ M 1 ⟩ + Ω , ⟨ M 2 ⟩ , and ⟨ M 2 ⟩ + Ω are M -matrices. Then

(18) ( M 1 + Ω ) x ∗ = N 1 x ∗ + ( Ω − A ) ∣ x ∗ ∣ − q , ( M 2 + Ω ) x ∗ = N 2 x ∗ + ( Ω − A ) ∣ x ∗ ∣ − q .

Therefore, from (6) and (18), we have

(19) x k + 1 2 − x ∗ = ( M 1 + Ω ) − 1 ( N 1 ( x ( k ) − x ∗ ) + ( Ω − A ) ( ∣ x ( k ) ∣ − ∣ x ∗ ∣ ) ) , x ( k + 1 ) − x ∗ = ( M 2 + Ω ) − 1 N 2 x k + 1 2 − x ∗ + ( Ω − A ) ∣ x k + 1 2 ∣ − ∣ x ∗ ∣ .

By taking absolute value on both sides of each equation in (19), combining with the property of M -matrix, we have

(20) ∣ x ( k + 1 ) − x ∗ ∣ ≤ ( ⟨ M 2 ⟩ + Ω ) − 1 ( ∣ N 2 ∣ + ∣ Ω − A ∣ ) ( ⟨ M 1 ⟩ + Ω ) − 1 ( ∣ N 1 ∣ + ∣ Ω − A ∣ ) ( ∣ x ( k ) − x ∗ ∣ ) .

Then, if the spectral radius

ρ ( ( ⟨ M 2 ⟩ + Ω ) − 1 ( ∣ N 2 ∣ + ∣ Ω − A ∣ ) ( ⟨ M 1 ⟩ + Ω ) − 1 ( ∣ N 1 ∣ + ∣ Ω − A ∣ ) ) < 1 ,

the two-step modulus-based matrix splitting iteration method is convergent.□

For an H + -matrix, both the H -compatible splitting and the H -splitting satisfy the first condition in Theorem 3.2.1, that is, ⟨ M i ⟩ , i = 1 , 2 , are M -matrices. Therefore, for these two special matrix splittings, we can adopt the two-step modulus-based matrix splitting iteration method by checking condition (17).

For the H -compatible splitting, there is a convergence condition in [11], that is,

(21) ρ ( ( ⟨ M 2 ⟩ + Ω ) − 1 ( 2 ∣ N 2 ∣ + ∣ Ω − M 2 ∣ ) ( ⟨ M 1 ⟩ + Ω ) − 1 ( 2 ∣ N 1 ∣ + ∣ Ω − M 1 ∣ ) ) < 1 .

Since

0 ≤ ( ⟨ M 2 ⟩ + Ω ) − 1 ( ∣ N 2 ∣ + ∣ Ω − A ∣ ) ( ⟨ M 1 ⟩ + Ω ) − 1 ( ∣ N 1 ∣ + ∣ Ω − A ∣ ) ≤ ( ⟨ M 2 ⟩ + Ω ) − 1 ( 2 ∣ N 2 ∣ + ∣ Ω − M 2 ∣ ) ( ⟨ M 1 ⟩ + Ω ) − 1 ( 2 ∣ N 1 ∣ + ∣ Ω − M 1 ∣ ) ,

we have

ρ ( ( ⟨ M 2 ⟩ + Ω ) − 1 ( ∣ N 2 ∣ + ∣ Ω − A ∣ ) ( ⟨ M 2 ⟩ + Ω ) − 1 ( ∣ N 1 ∣ + ∣ Ω − A ∣ ) ) ≤ ρ ( ( ⟨ M 2 ⟩ + Ω ) − 1 ( 2 ∣ N 2 ∣ + ∣ Ω − M 2 ∣ ) ( ⟨ M 1 ⟩ + Ω ) − 1 ( 2 ∣ N 1 ∣ + ∣ Ω − M 1 ∣ ) )

holds. Therefore, (17) in Theorem 3.2.1 is better.

In the following, we mainly discuss the H -compatible splitting for the two-step modulus-based matrix splitting iteration method and obtain the concrete convergence results.

Theorem 3.2.2

Let A ∈ R n × n be an H + -matrix and A = M i − N i , i = 1 , 2 , be two H -compatible splittings of A , i.e., ⟨ A ⟩ = ⟨ M i ⟩ − ∣ N i ∣ , i = 1 , 2 . If

(22) Ω ≥ diag ( A ) ,

then { z ( k ) } k = 1 + ∞ ⊂ R + n generated by (6) with (7) is convergent for any initial vector x ( 0 ) ∈ R n .

Proof

From Theorem 3.2.1, we only need to prove that inequality (17) holds when Ω ≥ diag ( A ) .

Since A is an H + -matrix, ⟨ A ⟩ is an M -matrix. Thus, there exists a positive vector v > 0 such that

(23) ⟨ A ⟩ v > 0 .

If Ω ≥ diag ( A ) , then both Ω + ⟨ M 1 ⟩ and Ω + ⟨ M 2 ⟩ are M -matrices. For the matrix appeared in (17), we have

(24) 0 ≤ ( Ω + ⟨ M 1 ⟩ ) − 1 ( ∣ N 1 ∣ + ∣ Ω − A ∣ ) = I − ( Ω + ⟨ M 1 ⟩ ) − 1 ( Ω − ∣ Ω − A ∣ + ⟨ A ⟩ ) = I − ( Ω + ⟨ M 1 ⟩ ) − 1 2 ⟨ A ⟩ , 0 ≤ ( Ω + ⟨ M 2 ⟩ ) − 1 ( ∣ N 2 ∣ + ∣ Ω − A ∣ ) = I − ( Ω + ⟨ M 2 ⟩ ) − 1 ( Ω − ∣ Ω − A ∣ + ⟨ A ⟩ ) = I − ( Ω + ⟨ M 2 ⟩ ) − 1 2 ⟨ A ⟩ .

It follows that

(25) ( ⟨ M 2 ⟩ + Ω ) − 1 ( ∣ N 2 ∣ + ∣ Ω − A ∣ ) ( ⟨ M 1 ⟩ + Ω ) − 1 ( ∣ N 1 ∣ + ∣ Ω − A ∣ ) v = ( I − 2 ( Ω + ⟨ M 2 ⟩ ) − 1 ⟨ A ⟩ ) ( I − 2 ( Ω + ⟨ M 1 ⟩ ) − 1 ⟨ A ⟩ ) v < ( I − 2 ( Ω + ⟨ M 2 ⟩ ) − 1 ⟨ A ⟩ ) v < v .

Therefore, according to the properties of the nonnegative matrix, we have

ρ ( ( ⟨ M 2 ⟩ + Ω ) − 1 ( ∣ N 2 ∣ + ∣ Ω − A ∣ ) ( ⟨ M 1 ⟩ + Ω ) − 1 ( ∣ N 1 ∣ + ∣ Ω − A ∣ ) ) < 1 ,

and then this theorem is proved.□

Theorem 3.2.3

Let A ∈ R n × n be an H + -matrix with ρ ( D − 1 ( ∣ L ∣ + ∣ U ∣ ) ) < 1 2 , where D , − L , − U are the diagonal, the strictly lower-triangular, and the strictly upper-triangular matrices of A , respectively. Let A = M i − N i , i = 1 , 2 , be two H -compatible splittings of A . If

(26) Ω ≥ 1 2 D ,

then { z ( k ) } k = 1 + ∞ ⊂ R + n generated by (6) with (7) is convergent for any initial vector x ( 0 ) ∈ R n .

Proof

Similar to the proof of Theorem 3.2.2, we only prove (17) holds here. Since A is an H + matrix, the equalities (24) can be concretely reformulated as

(27) 0 ≤ ( Ω + ⟨ M 1 ⟩ ) − 1 ( ∣ N 1 ∣ + ∣ Ω − A ∣ ) = I − ( Ω + ⟨ M 1 ⟩ ) − 1 ( Ω − ∣ Ω − A ∣ + ⟨ A ⟩ ) = I − ( Ω + ⟨ M 1 ⟩ ) − 1 ( Ω − ∣ Ω − D ∣ + D − 2 ( ∣ L ∣ + ∣ U ∣ ) ) = I − ( Ω + ⟨ M 1 ⟩ ) − 1 ( Ω − ∣ Ω − D ∣ + D ( I − 2 D − 1 ( ∣ L ∣ + ∣ U ∣ ) ) ) , 0 ≤ ( Ω + ⟨ M 2 ⟩ ) − 1 ( ∣ N 2 ∣ + ∣ Ω − A ∣ ) = I − ( Ω + ⟨ M 2 ⟩ ) − 1 ( Ω − ∣ Ω − A ∣ + ⟨ A ⟩ ) = I − ( Ω + ⟨ M 2 ⟩ ) − 1 ( Ω − ∣ Ω − D ∣ + D − 2 ( ∣ L ∣ + ∣ U ∣ ) ) = I − ( Ω + ⟨ M 2 ⟩ ) − 1 ( Ω − ∣ Ω − D ∣ + D ( I − 2 D − 1 ( ∣ L ∣ + ∣ U ∣ ) ) ) .

If Ω ≥ 1 2 , then Ω − ∣ Ω − D ∣ ≥ 0 . Moreover, I − 2 D − 1 ( ∣ L ∣ + ∣ U ∣ ) , D ( I − 2 D − 1 ( ∣ L ∣ + ∣ U ∣ ) ) , and Ω − ∣ Ω − D ∣ + D ( I − 2 D − 1 ( ∣ L ∣ + ∣ U ∣ ) ) are three M -matrices. Thus, there exists a positive vector v such that

( Ω − ∣ Ω − D ∣ + D ( I − 2 D − 1 ( ∣ L ∣ + ∣ U ∣ ) ) ) v > 0

and

(28) ( ⟨ M 2 ⟩ + Ω ) − 1 ( ∣ N 2 ∣ + ∣ Ω − A ∣ ) ( ⟨ M 1 ⟩ + Ω ) − 1 ( ∣ N 1 ∣ + ∣ Ω − A ∣ ) v = ( I − ( Ω + ⟨ M 2 ⟩ ) − 1 ( Ω − ∣ Ω − D ∣ + D ( I − 2 D − 1 ( ∣ L ∣ + ∣ U ∣ ) ) ) ) × ( I − ( Ω + ⟨ M 1 ⟩ ) − 1 ( Ω − ∣ Ω − D ∣ + D ( I − 2 D − 1 ( ∣ L ∣ + ∣ U ∣ ) ) ) ) v < ( I − ( Ω + ⟨ M 2 ⟩ ) − 1 ( Ω − ∣ Ω − D ∣ + D ( I − 2 D − 1 ( ∣ L ∣ + ∣ U ∣ ) ) ) ) v < v .

Thus, from the property of the nonnegative matrix, we have

ρ ( ( ⟨ M 2 ⟩ + Ω ) − 1 ( ∣ N 2 ∣ + ∣ Ω − A ∣ ) ( ⟨ M 1 ⟩ + Ω ) − 1 ( ∣ N 1 ∣ + ∣ Ω − A ∣ ) ) < 1 ,

and then the conclusion is proved.□

Lemma 3.2.1

Let A ∈ R n × n be an H + -matrix with A = D − L − U , where D , − L , − U are the same as that of Theorem 3.2.3. Let A = M i − N i , i = 1 , 2 , be two AOR splittings, that is, M i , N i , i = 1 , 2 , satisfy (8). Then the two splittings are H -compatible splittings if and only if

(29) 0 ≤ β i ≤ α i ≤ 1 , with α i ≠ 0 , i = 1 , 2 .

Proof

We only prove the case i = 1 .

The sufficiency: According to the definition of the H -compatible splitting, we only need to prove that the equality ⟨ A ⟩ = ⟨ M 1 ⟩ − ∣ N 1 ∣ holds when 0 ≤ β 1 ≤ α 1 ≤ 1 with α 1 ≠ 0 .

If 0 ≤ β 1 ≤ α 1 ≤ 1 with α 1 ≠ 0 , then

⟨ M 1 ⟩ − ∣ N 1 ∣ = 1 α 1 ( D − β 1 L ) − 1 α 1 ( ( 1 − α 1 ) D + ( α 1 − β 1 ) L + α 1 U ) = 1 α 1 D − β 1 α 1 ∣ L ∣ − 1 − α 1 α 1 D + α 1 − β 1 α 1 ∣ L ∣ + ∣ U ∣ = D − ∣ L ∣ − ∣ U ∣ = ⟨ A ⟩ .

The necessity: From (8), we have the splitting A = M 1 − N 1 is an H -compatible splitting with α 1 > 0 , β 1 ≥ 0 and A is an H + -matrix. Then

⟨ A ⟩ = ⟨ M 1 ⟩ − ∣ N 1 ∣ = 1 α 1 ( D − β 1 L ) − 1 α 1 ( ( 1 − α 1 ) D + ( α 1 − β 1 ) L + α 1 U ) = 1 α 1 D − β 1 α 1 ∣ L ∣ − 1 − α 1 α 1 D + α 1 − β 1 α 1 ∣ L ∣ + ∣ U ∣ = 1 α 1 − 1 − α 1 α 1 D − β 1 α 1 + α 1 − β 1 α 1 ∣ L ∣ − ∣ U ∣ = D − ∣ L ∣ − ∣ U ∣ .

By comparing both sides of the last equality, we can obtain 0 ≤ β 1 ≤ α 1 ≤ 1 with α 1 ≠ 0 .□

Theorem 3.2.4

Let A be an H + -matrix with A = D − L − U , where D , − L , − U are the same as that of Theorem 3.2.3. Then the two-step MBAOR iteration method (9) is convergent if

(30) 0 ≤ β i ≤ α i ≤ 1 , with α i ≠ 0 , i = 1 , 2 , and Ω ≥ D .

Proof

From 0 ≤ β i ≤ α i ≤ 1 with α i ≠ 0 , i = 1 , 2 , and Lemma 3.2.1, we know that the splittings in (8) are two H -compatible splittings. It follows from Theorem 3.2.2 that if Ω ≥ D , the two-step MBAOR iteration method (9) is convergent.□

From Theorem 3.2.3 and Lemma 3.2.1, we have the following conclusion on the two-step MBAOR iteration method, whose proof is similar to that of Theorem 3.2.4 and is omitted here.

Theorem 3.2.5

Let A be an H + -matrix with A = D − L − U , where D , − L , − U are the same as that of Theorem 3.2.3. If ρ ( D − 1 ( ∣ L ∣ + ∣ U ∣ ) ) < 1 2 , then the two-step MBAOR iteration method (9) is convergent if

(31) 0 ≤ β i ≤ α i ≤ 1 , with α i ≠ 0 , i = 1 , 2 , and Ω ≥ 1 2 D .

Theorem 3.2.6

Let A be an H + -matrix with A = D − L − U , where D , − L , − U are the same as that of Theorem 3.2.3. Then the two-step MBAOR iteration method (9) is convergent if either of the following conditions holds:

(32) 0 ≤ β i ≤ α i ≤ 1 , α i ≠ 0 , with Ω ≥ D , 0 ≤ β i ≤ α i , 1 ≤ α i < 1 ρ ( D − 1 ( ∣ L ∣ + ∣ U ∣ ) ) , α 1 = α 2 ≠ 0 , with Ω ≥ D .

Moreover, if ρ ( D − 1 ( ∣ L ∣ + ∣ U ∣ ) ) < 1 2 , then the two-step MBAOR iteration method (9) is convergent if either of the following conditions holds:

(33) 0 ≤ β i ≤ α i ≤ 1 , α i ≠ 0 , with Ω ≥ 1 2 D , 0 ≤ β i ≤ α i , 1 ≤ α i ≤ 2 , α 1 = α 2 ≠ 0 , with 1 2 3 − 2 α i D ≤ Ω ≤ D .

Proof

From α i > 0 , β i ≥ 0 , i = 1 , 2 , we have ⟨ M i ⟩ , i = 1 , 2 , in (9) are M -matrices. From Theorem 3.2.1, we need to only prove that (17) holds. For the two-step MBAOR iteration method, (27) in Theorem 3.2.3 can turn to

(34) 0 ≤ ( Ω + ⟨ M 1 ⟩ ) − 1 ( ∣ N 1 ∣ + ∣ Ω − A ∣ ) = I − ( Ω + ⟨ M 1 ⟩ ) − 1 Ω + 1 − ∣ 1 − α 1 ∣ α 1 D − ∣ Ω − D ∣ − 2 ( ∣ L ∣ + ∣ U ∣ ) , 0 ≤ ( Ω + ⟨ M 2 ⟩ ) − 1 ( ∣ N 2 ∣ + ∣ Ω − A ∣ ) = I − ( Ω + ⟨ M 2 ⟩ ) − 1 Ω + 1 − ∣ 1 − α 2 ∣ α 2 D − ∣ Ω − D ∣ − 2 ( ∣ L ∣ + ∣ U ∣ ) .

Then

(35) Ω + 1 − ∣ 1 − α i ∣ α i D − ∣ Ω − D ∣ − 2 ( ∣ L ∣ + ∣ U ∣ ) = Ω + D − ∣ Ω − D ∣ − 2 ( ∣ L ∣ + ∣ U ∣ ) when α i ≤ 1 , Ω + 2 − α i α i D − ∣ Ω − D ∣ − 2 ( ∣ L ∣ + ∣ U ∣ ) when α i ≥ 1 .

Next we discuss the two cases of (35) separately.

  1. When 0 < α i ≤ 1 , we have

    (36) Ω + D − ∣ Ω − D ∣ − 2 ( ∣ L ∣ + ∣ U ∣ ) = 2 ⟨ A ⟩ when Ω ≥ D , Ω − ∣ Ω − D ∣ + D ( I − 2 D − 1 ( ∣ L ∣ + ∣ U ∣ ) ) when Ω ≥ 1 2 D .

If Ω ≥ D or Ω ≥ 1 2 D with ρ ( D − 1 ( ∣ L ∣ + ∣ U ∣ ) ) < 1 2 , then the matrix (36) is an M -matrix, and the two-step MBAOR iteration method is convergent from the proofs of Theorems 3.2.2 and 3.2.3, respectively.

  1. When α i ≥ 1 , we have

    (37) Ω + 2 − α i α i D − ∣ Ω − D ∣ − 2 ( ∣ L ∣ + ∣ U ∣ ) = 2 D 1 α i I − D − 1 ( ∣ L ∣ + ∣ U ∣ ) when Ω ≥ D , 2 Ω + 2 α i − 3 D + D ( I − 2 D − 1 ( ∣ L ∣ + ∣ U ∣ ) ) when Ω ≤ D .

If Ω ≥ D with 1 ≤ α i < 1 ρ ( D − 1 ( ∣ L ∣ + ∣ U ∣ ) ) or 1 2 3 − 2 α i D ≤ Ω ≤ D with 1 ≤ α i ≤ 2 and ρ ( D − 1 ( ∣ L ∣ + ∣ U ∣ ) ) < 1 2 , then the matrix (37) is an M -matrix. Set α 1 = α 2 , for the two cases, there exist positive vectors u , v satisfying

(38) 2 D 1 α i I − D − 1 ( ∣ L ∣ + ∣ U ∣ ) u > 0 , 2 Ω + 2 α i − 3 D + D ( I − 2 D − 1 ( ∣ L ∣ + ∣ U ∣ ) ) v > 0 , i = 1 , 2 ,

respectively. Thus, we have

(39) ( Ω + ⟨ M 2 ⟩ ) − 1 ( ∣ N 2 ∣ + ∣ Ω − A ∣ ) ( Ω + ⟨ M 1 ⟩ ) − 1 ( ∣ N 1 ∣ + ∣ Ω − A ∣ ) u = I − ( Ω + ⟨ M 2 ⟩ ) − 1 2 D 1 α 2 I − D − 1 ( ∣ L ∣ + ∣ U ∣ ) I − ( Ω + ⟨ M 1 ⟩ ) − 1 2 D 1 α 1 I − D − 1 ( ∣ L ∣ + ∣ U ∣ ) u < I − ( Ω + ⟨ M 2 ⟩ ) − 1 2 D 1 α 2 I − D − 1 ( ∣ L ∣ + ∣ U ∣ ) u < u

and

(40) ( Ω + ⟨ M 2 ⟩ ) − 1 ( ∣ N 2 ∣ + ∣ Ω − A ∣ ) ( Ω + ⟨ M 1 ⟩ ) − 1 ( ∣ N 1 ∣ + ∣ Ω − A ∣ ) v = I − ( Ω + ⟨ M 2 ⟩ ) − 1 2 Ω + 2 α 2 − 3 D + D ( I − 2 D − 1 ( ∣ L ∣ + ∣ U ∣ ) ) × I − ( Ω + ⟨ M 1 ⟩ ) − 1 2 Ω + 2 α 1 − 3 D + D ( I − 2 D − 1 ( ∣ L ∣ + ∣ U ∣ ) ) v < I − ( Ω + ⟨ M 2 ⟩ ) − 1 2 Ω + 2 α 2 − 3 D + D ( I − 2 D − 1 ( ∣ L ∣ + ∣ U ∣ ) ) v < v .

From the properties of nonnegative matrix, we always have

ρ ( ( Ω + ⟨ M 2 ⟩ ) − 1 ( ∣ N 2 ∣ + ∣ Ω − A ∣ ) ( Ω + ⟨ M 1 ⟩ ) − 1 ( ∣ N 1 ∣ + ∣ Ω − A ∣ ) ) < 1 .

Therefore, the two-step MBAOR iteration is convergent for the case α i ≥ 1 with α 1 = α 2 .

Collecting (i) and (ii), that is, 0 < α i ≤ 1 and α i ≥ 1 , (32) and (33) can be proved.□

From (35), (37), and the proof of Theorem 3.2.6, we can find that if the nonnegative matrix D − 1 ( ∣ L ∣ + ∣ U ∣ ) has a positive eigenvector w > 0 , which satisfies

D − 1 ( ∣ L ∣ + ∣ U ∣ ) w = ρ ( D − 1 ( ∣ L ∣ + ∣ U ∣ ) ) w ,

then replacing u and v as w in (38) we can prove (17) holds even if α 1 ≠ α 2 . Thus, the condition α 1 = α 2 can be deleted in Theorem 3.2.6. We know that when A is an irreducible H + -matrix, such positive eigenvector w exists. Hence, the conclusion of Theorem 3.2.6 can be further improved in this case and the result is omitted here.

It is well known that the AOR splitting (8) includes some particular cases, for example: (I) Jacobian(J) splitting when α i = 1 , β i = 0 ; (II) Gauss-Seidel splitting when α i = β i = 1 ; (III) Successive overrelaxation splitting when α i = β i > 0 . For these particular matrix splittings, the conclusions in Theorems 3.2.5 and 3.2.6 also hold.

4 Numerical experiments

In this section, we supply some experiments. We consider the two-step MBAOR iteration method and its particular case, that is, the two-step modulus-based Gauss-Seidel (MBGS) iteration method. The iteration steps, the elapsed time, the norm of the residual vector, and the spectral radius function in (17) are denoted by IT, CPU, RES, and ρ ( α , β ) , respectively. RES and ρ ( α , β , ω ) are defined as:

RES ( z ( k ) ) = ∣ ∣ min ( z ( k ) , A z ( k ) + q ) ∣ ∣ 2 , ρ ( α , β , ω ) = ρ ( ℒ 2 , α , β , ω ℒ 1 , α , β , ω ) .

Here, ‖ ⋅ ‖ is a vector norm, z ( k ) is the k th approximate solution, and

ℒ 1 , α 1 , β 1 , ω = 1 α 1 ( D − β 1 L ) + ω D − 1 1 α 1 ( ( 1 − α 1 ) D + ( α 1 − β 1 ) L + α 1 U ) + ∣ ω D − A ∣ , ℒ 2 , α 2 , β 2 , ω = 1 α 2 ( D − β 2 U ) + ω D − 1 1 α 2 ( ( 1 − α 2 ) D + ( α 2 − β 2 ) U + α 2 L ) + ∣ ω D − A ∣ .

In addition, we denote

ρ = ρ ( D − 1 ( ∣ L ∣ + ∣ U ∣ ) )

and

ρ ( ω ) = ρ ( ℒ 2 , 1 , 1 , ω ℒ 1 , 1 , 1 , ω ) ,

respectively. The matrix A in LCP( A , q ) is generated by A ( μ , η , ζ ) = A ^ + μ I + η B + ζ C , where μ , η , and ζ are three given constants, which guarantee the matrix A ( μ , η , ζ ) to be a P -matrix. Specifically,

A ^ = Tridiag ( − I , S , − I ) ∈ R n × n

is a block-tridiagonal matrix,

B = Tridiag ( 0 , 0 , 1 ) ∈ R n × n and S = Tridiag ( − 1 , 4 , − 1 ) ∈ R m × m

are two tridiagonal matrices, and

C = diag ( [ 1 , 2 , 1 , 2 , … ] ) ∈ R n × n

is a diagonal matrix, where m and n are two positive integers satisfying n = m 2 . For convenience, we set

q = ( 1 , − 1 , 1 , − 1 , … , 1 , − 1 , … ) T ∈ R n ,

then the LCP( A ( μ , η , ζ ) , q ) has a unique solution when A ( μ , η , ζ ) is a P -matrix. All initial vectors in our experiments are selected to be

x ( 0 ) = ( 1 , 0 , 1 , 0 , … , 1 , 0 , … ) T ∈ R n .

The iteration process stops when RES ( z ( k ) ) < 1 0 − 5 or the number of iteration steps reaches 10 m . We consider six cases in our experiments, that is, A ( 0 , 0 , 1 ) , A ( 0 , 1 , 1 ) , A ( 1 , 0 , 0 ) , A ( 1 , 1 , − 1 ) , A ( 1 , 0 , 1 ) , and A ( 1 , 1 , 1 ) , all of which are H + -matrices. Under the same conditions, we also test other cases, that is, A ^ = Tridiag ( − 0.5 I , S , − 0.5 I ) ∈ R n × n and A ^ = Tridiag ( − S , S , − S ) ∈ R n × n . Since the numerical results are similar, we omit them in this paper. All computations are run by using Matlab version 2016 on a Dell Laptop (Intel(R) Core(TM) i5-7200U CPU @ 2.50 GHz 4.00 GB RAM).

Example 4.1

In this example, we test the convergence domain and the parameter ω for the two-step MBGS iteration method with Ω = ω D , ω > 0 . Set m = 30 , then we obtain Table 1 as follows.

From Table 1, we have the following observations:

  1. For A ( 0 , 0 , 1 ) , A ( 1 , 0 , 0 ) , and A ( 1 , 1 , − 1 ) , all of which satisfy ρ = 0.7296 > 1 2 . Thus, for the three cases, the convergence domain satisfies Ω ≥ D from Theorem 3.2.4. Although the two-step MBGS iteration method is convergent for the three cases when Ω = 1 2 D and the initial vector is x ( 0 ) , the convergence domain cannot be expanded to Ω ≥ 1 2 D , which cannot be guaranteed by Theorem 3.2.5. However, for A ( 0 , 1 , 1 ) , since ρ = 0.4325 < 1 2 , we have a larger convergence domain Ω ≥ 1 2 D from Theorem 3.2.5.

  2. The numerical results also show that the two-step MBGS iteration method with Ω = D is the best case in our experiments.

Example 4.2

In this example, we test the parameters α and β in the two-step MBAOR iteration method. In the two-step MBAOR iteration method (9), we set Ω to be a fixed positive diagonal matrix, that is, Ω = D , and set α 1 = α 2 = α , β 1 = β 2 = β ∈ [ 0 , 1 ] with α ≠ 0 . We divide the interval [ 0 , 1 ] into ten equal parts and set α , β to be i 10 , i = 1 , 2 , … , 10 , and m = 30 . The numerical results on the spectral radius ρ ( α , β ) and the iteration steps IT are shown in Tables 2 and 3 and Figure 1 as follows.

Figure 1 corresponds to Tables 2 and 3. We can see:

  1. For Ω = D , when 0 ≤ β ≤ α ≤ 1 with α ≠ 0 , the two splittings in (8) are H -compatible splittings of an H + -matrix. When α = β = 1 , that is, the two-step MBGS iteration method is the best case in the experiments.

  2. For Ω = D , when 0 < α ≤ β ≤ 1 , the splittings in (8) are not H -compatible splittings. However, the two-step MBAOR iteration method is also convergent based on Theorem 3.2.2. In addition, the two-step MBGS iteration method is still the best case in the larger convergence domain.

Figure 1 The spectral radius function ρ ( α , β , 1 ) \rho \left(\alpha ,\beta ,1) (for (a) A(1,0,1) and (c) A(1,1,1)) and the iteration steps IT (for (b) A(1,0,1) and (d) A(1,1,1)) for the two-step MBAOR iteration method.
Figure 1

The spectral radius function ρ ( α , β , 1 ) (for (a) A(1,0,1) and (c) A(1,1,1)) and the iteration steps IT (for (b) A(1,0,1) and (d) A(1,1,1)) for the two-step MBAOR iteration method.

Table 1

Numerical results of the two-step MBGS iteration method

A ( 0 , 0 , 1 ) A ( 0 , 1 , 1 )
ρ 0.7296 0.4255
Region Ω ≥ D Ω ≥ 1 2 D
ρ ( ω ) IT CPU RES ρ ( ω ) IT CPU RES
Ω = 1 2 D 1.9720 11 0.051 3.5726 × 1 0 − 6 0.7759 11 0.049 5.4198 × 1 0 − 6
Ω = D 0.4480 7 0.040 6.4079 × 1 0 − 7 0.1262 6 0.027 8.3381 × 1 0 − 6
Ω = 3 2 D 0.5576 11 0.064 2.2155 × 1 0 − 6 0.2463 10 0.048 2.8405 × 1 0 − 6
Ω = 2 D 0.6317 14 0.065 6.0896 × 1 0 − 6 0.3458 13 0.067 2.9037 × 1 0 − 6
A ( 1 , 0 , 0 ) A ( 1 , 1 , − 1 )
ρ 0.7959 0.6951
Region Ω ≥ D Ω ≥ D
ρ ( ω ) IT CPU RES ρ ( ω ) IT CPU RES
Ω = 1 2 D 2.3615 12 0.054 5.3077 × 1 0 − 6 1.7950 24 0.124 8.9534 × 1 0 − 6
Ω = D 0.5555 8 0.035 2.6641 × 1 0 − 6 0.3961 32 0.151 5.6348 × 1 0 − 6
Ω = 3 2 D 0.6494 13 0.056 2.1278 × 1 0 − 6 0.5091 43 0.198 8.0561 × 1 0 − 6
Ω = 2 D 0.7109 16 0.065 9.0763 × 1 0 − 6 0.5941 54 0.256 9.3205 × 1 0 − 6
Table 2

Numerical results of the two-step MBAOR iteration method for A ( 1 , 0 , 1 )

ρ ( α , β , 1 ) α 1 α 2 α 3 α 4 α 5 α 6 α 7 α 8 α 9 α 10 = 1
β 1 2157 2504 3781 5012 509 760 919 1528 877 1608 791 1586 198 431 112 263 289 728 1199 3224
β 2 595 694 481 643 492 743 194 327 1147 2136 809 1650 862 1911 8938 21397 677 1740 1115 3061
β 3 2140 2509 593 800 991 1515 1171 2003 1089 2062 1091 2266 956 2161 3489 8525 323 848 361 1013
β 4 4260 5023 957 1304 5195 8048 1026 1783 594 1145 348 737 601 1387 173 432 556 1493 1249 3587
β 5 729 865 1393 1919 1247 1960 1004 1775 1921 3775 935 2022 1535 3622 718 1835 589 1620 4116 12115
β 6 1708 2041 791 1103 397 634 1163 2095 603 1210 151 334 737 1781 671 1758 1133 3197 327 988
β 7 391 471 723 1022 961 1562 211 388 869 1784 956 2167 367 910 262 705 1031 2990 1150 3573
β 8 491 597 1724 2475 491 814 1104 2077 257 541 2045 4761 1486 3789 269 745 303 905 313 1002
β 9 1939 2384 873 1276 1051 1782 415 801 590 1277 1371 3287 800 2103 765 2186 3203 9876 389 1286
β 10 = 1 535 667 519 775 1105 1923 388 771 1318 2943 2556 6331 625 1699 579 1712 339 1082 149 510
IT α 1 α 2 α 3 α 4 α 5 α 6 α 7 α 8 α 9 α 10 = 1
β 1 59 31 21 17 14 12 10 9 8 8
β 2 58 30 21 16 13 11 10 9 8 7
β 3 56 29 20 16 13 11 10 9 8 7
β 4 54 28 20 15 13 11 9 8 8 7
β 5 51 27 19 15 12 10 9 8 7 7
β 6 49 26 18 14 12 10 9 8 7 7
β 7 46 25 18 14 11 10 9 8 7 6
β 8 42 23 17 13 11 9 8 7 7 6
β 9 44 22 16 13 11 9 8 7 7 6
β 10 = 1 45 22 15 12 10 9 8 7 6 6
Table 3

Numerical results of the two-step MBAOR iteration method for A ( 1 , 1 , 1 )

ρ ( α , β , 1 ) α 1 α 2 α 3 α 4 α 5 α 6 α 7 α 8 α 9 α 10 = 1
β 1 1807 2324 630 1027 621 1268 37 94 1201 3726 646 2457 418 1923 348 1951 641 4317 235 1951
β 2 477 616 1443 2374 435 896 1174 3009 613 1943 1583 6137 52 245 475 2707 3926 27125 665 5556
β 3 791 1025 355 587 233 485 1099 2860 285 917 746 2935 556 2691 467 2748 149 1060 992 8667
β 4 669 871 398 663 973 2049 991 2619 1093 3584 1072 4317 1683 8378 196 1181 243 1775 475 4251
β 5 811 1060 400 673 569 1212 9935 26748 427 1427 319 1319 298 1509 309 1916 853 6563 219 2026
β 6 763 1003 685 1162 770 1663 427 1160 600 2039 618 2599 577 3051 641 4134 946 7529 811 7818
β 7 1028 1361 591 1013 833 1835 237 661 515 1796 481 2070 662 3533 862 5689 226 1835 403 4064
β 8 745 989 947 1640 799 1776 884 2489 320 1147 610 2703 407 2245 2319 15628 91 762 377 4025
β 9 2931 3916 1523 2670 796 1799 551 1586 617 2247 1545 7036 193 1103 347 2432 496 4329 99 1066
β 10 = 1 1033 1391 1647 2918 2027 4651 813 2396 970 3603 213 1010 367 2143 119 874 485 4451 367 4197
IT α 1 α 2 α 3 α 4 α 5 α 6 α 7 α 8 α 9 α 10 = 1
β 1 55 29 20 15 13 11 10 9 8 7
β 2 54 28 20 15 12 11 9 8 8 7
β 3 52 27 19 15 12 10 9 8 7 7
β 4 51 27 19 14 12 10 9 8 7 7
β 5 49 26 18 14 12 10 9 8 7 7
β 6 46 25 18 14 11 10 9 8 7 6
β 7 43 24 17 13 11 10 8 8 7 6
β 8 43 22 16 13 11 9 8 7 7 6
β 9 45 21 16 13 11 9 8 7 7 6
β 10 = 1 45 22 15 12 10 9 8 7 6 6

From Examples 4.1 and 4.2, we can find that for the two-step MBAOR iteration method, when 0 ≤ β i ≤ α i ≤ 1 with α i ≠ 0 , if Ω = ω D with ω ∈ [ 1 , + ∞ ) or ω ∈ 1 2 , + ∞ and ρ ( D − 1 ( ∣ L ∣ + ∣ U ∣ ) ) < 1 2 , then the two-step MBGS iteration method with Ω = D , i.e., α i = β i = 1 , ω = 1 in (9) is the best case in our experiments.

5 Concluding remarks

In this paper, we discuss the condition of the modulus-based type iteration methods for solving the LCP( A , q ). Then, for the two-step modulus-based matrix splitting iteration method, we propose some convergence conditions when the system matrix A is an H + -matrix. In addition, we give the numerical experiments to show the presented results and investigate some particular cases of the two-step MBAOR iteration method.

Acknowledgements

The authors thank the anonymous referees for providing many valuable suggestions that made this paper a lot more readable.

  1. Funding information: This research was supported by Zhaoqing University Research Program (No. 611-612279), Zhaoqing Education and Development Project (No. ZQJYY2020093), the Characteristic Innovation Project of Department of Education of Guangdong Province (No. 2020KTSCX159), the Science and Technology Innovation Guidance Project of Zhaoqing City (No. 2021040315026), the Innovative Research Team Project of Zhaoqing University, and the Scientific Research Ability Enhancement Program for Excellent Young Teachers of Zhaoqing University.

  2. Conflict of interest: Authors state no conflict of interest.

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Received: 2021-05-19
Revised: 2021-09-14
Accepted: 2021-12-05
Published Online: 2021-12-31

© 2021 Ximing Fang et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  13. Pentagonal quasigroups, their translatability and parastrophes
  14. Counting certain quadratic partitions of zero modulo a prime number
  15. Global attractors for a class of semilinear degenerate parabolic equations
  16. A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation
  17. On sub-class sizes of mutually permutable products
  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
  19. Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity
  20. On kernels by rainbow paths in arc-coloured digraphs
  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
https://doi.org/10.1515/math-2021-0132
Keywords for this article
linear complementarity problem; modulus-based type iteration method; convergence condition
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  16. A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation
  17. On sub-class sizes of mutually permutable products
  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
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  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
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  26. Hodge-Deligne polynomials of character varieties of free abelian groups
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  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
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  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
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