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A higher order evolution inequality with a gradient term in the exterior of the half-ball

  • We study the existence and nonexistence of weak solutions to a semilinear higher order (in time) evolution inequality involving a convection term in the exterior of the half-ball, under Dirichlet-type boundary conditions. A weight function of the form |x|a is allowed in front of the power nonlinearity. When a>2, we show that the dividing line with respect to existence or nonexistence is given by a critical exponent (Fujita critical exponent), which depends on the parameters of the problem, but independent of the order of the time-derivative. Our study yields naturally optimal nonexistence results for the corresponding stationary problem.

    Citation: Ibtehal Alazman, Ibtisam Aldawish, Mohamed Jleli, Bessem Samet. A higher order evolution inequality with a gradient term in the exterior of the half-ball[J]. AIMS Mathematics, 2023, 8(4): 9230-9246. doi: 10.3934/math.2023463

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  • We study the existence and nonexistence of weak solutions to a semilinear higher order (in time) evolution inequality involving a convection term in the exterior of the half-ball, under Dirichlet-type boundary conditions. A weight function of the form |x|a is allowed in front of the power nonlinearity. When a>2, we show that the dividing line with respect to existence or nonexistence is given by a critical exponent (Fujita critical exponent), which depends on the parameters of the problem, but independent of the order of the time-derivative. Our study yields naturally optimal nonexistence results for the corresponding stationary problem.



    The nonsingular H-matrix and its subclass play an important role in a lot of fields of science such as computational mathematics, mathematical physics, and control theory, see [1,2,3,4]. Meanwhile, the infinity norm bounds of the inverse for nonsingular H-matrices can be used in convergence analysis of matrix splitting and matrix multi-splitting iterative methods for solving large sparse systems of linear equations [5], as well as bounding errors of linear complementarity problems [6,7]. In recent years, many scholars have developed a deep research interest in the infinity norm of the inverse for special nonsingular H-matrices, such as GSDD1 matrices [8], CKV-type matrices [9], S-SDDS matrices [10], S-Nekrasov matrices [11], S-SDD matrices [12], and so on, which depends only on the entries of the matrix.

    In this paper, we prove that M is a nonsingular H-matrix by constructing a scaling matrix D for SDD+1 matrices such that MD is a strictly diagonally dominant matrix. The use of the scaling matrix is important for some applications, for example, the infinity norm bound of the inverse [13], eigenvalue localization [3], and error bounds of the linear complementarity problem [14]. We consider the infinity norm bound of the inverse for the SDD+1 matrix by multiplying the scaling matrix, then we use the result to discuss the error bounds of the linear complementarity problem.

    For a positive integer n2, let N denote the set {1,2,,n} and Cn×n(Rn×n) denote the set of all complex (real) matrices. Successively, we review some special subclasses of nonsingular H-matrices and related lemmas.

    Definition 1. [5] A matrix M=(mij)Cn×n is called a strictly diagonally dominant (SDD) matrix if

    |mii|>ri(M),iN, (1.1)

    where ri(M)=nj=1,ji|mij|.

    Various generalizations of SDD matrices have been introduced and studied in the literatures, see [7,15,16,17,18].

    Definition 2. [8] A matrix M=(mij)Cn×n is called a generalized SDD1 (GSDD1) matrix if

    {ri(M)pN2i(M)>0,iN2,(ri(M)pN2i(M))(|mjj|pN1j(M))>pN1i(M)pN2j(M),iN2,jN1, (1.2)

    where N1={iN|0<|mii|ri(M)}, N2={iN||mii|>ri(M)}, pN2i(M)=jN2{i}|mij|rj(M)|mjj|, pN1i(M)=jN1{i}|mij|,iN.

    Definition 3. [9] A matrix M=(mij)Cn×n, with n2, is called a CKV-type matrix if for all iN the set Si(M) is not empty, where

    Si(M)={S(i):|mii|>rsi(M),andforallj¯S(|mii|rSi(M))(|mjj|r¯Sj(M))>r¯Si(M)rSj(M)},

    with (i)={SN:iS} and rSi(M):=jS{i}|mij|.

    Lemma 1. [8] Let M=(mij)Cn×n be a GSDD1 matrix. Then

    M1max{ε,maxiN2ri(M)|mii|}min{miniN2ϕi,miniN1ψi}, (1.3)

    where

    ϕi=ri(M)jN2{i}|mij|rj(M)|mjj|jN1|mij|ε,iN2, (1.4)
    ψi=|mii|εjN1{i}|mij|εjN2|mij|rj(M)|mjj|,iN1, (1.5)

    and

    ε{maxiN1pN2i(M)|mii|pN1i(M),minjN2rj(M)pN2j(M)pN1j(M)}. (1.6)

    Lemma 2. [8] Suppose that M=(mij)Rn×n is a GSDD1 matrix with positive diagonal entries, and D=diag(di) with di[0,1]. Then

    maxd[0,1]n(ID+DM)1max{max{ε,maxiN2ri(M)|mii|}min{miniN2ϕi,miniN1ψi},max{ε,maxiN2ri(M)|mii|}min{ε,miniN2ri(M)|mii|}}, (1.7)

    where ϕi,ψi,andε are shown in (1.4)–(1.6), respectively.

    Definition 4. [19] Matrix M=(mij)Cn×n is called an SDD1 matrix if

    |mii|>ri(M),foreachiN1, (1.8)

    where

    ri(M)=jN1{i}|mij|+jN2{i}|mij|rj(M)|mjj|,
    N1={iN|0<|mii|ri(M)},N2={iN||mii|>ri(M)}.

    The rest of this paper is organized as follows: In Section 2, we propose a new subclass of nonsingular H-matrices referred to SDD+1 matrices, discuss some of the properties of it, and consider the relationships among subclasses of nonsingular H-matrices by numerical examples, including SDD1 matrices, GSDD1 matrices, and CKV-type matrices. At the same time, a scaling matrix D is constructed to verify that the matrix M is a nonsingular H-matrix. In Section 3, two methods are utilized to derive two different upper bounds of infinity norm for the inverse of the matrix (one with parameter and one without parameter), and numerical examples are used to show the validity of the results. In Section 4, two error bounds of the linear complementarity problems for SDD+1 matrices are given by using the scaling matrix D, and numerical examples are used to illustrate the effectiveness of the obtained results. Finally, a summary of the paper is given in Section. 5.

    For the sake of the following description, some symbols are first explained:

    N=N1N2,N1=N(1)1N(1)2,ri(M)0,N1={iN|0<|mii|ri(M)},N2={iN||mii|>ri(M)}, (2.1)
    N(1)1={iN1|0<|mii|ri(M)},N(1)2={iN1||mii|>ri(M)}, (2.2)
    ri(M)=jN1{i}|mij|+jN2|mij|rj(M)|mjj|,iN1. (2.3)

    By the definitions of N(1)1 and N(1)2, for N1=N(1)1N(1)2, ri(M) in Definition 4 can be rewritten as

    ri(M)=jN1{i}|mij|+jN2{i}|mij|rj(M)|mjj|=jN(1)1{i}|mij|+jN(1)2{i}|mij|+jN2{i}|mij|rj(M)|mjj|. (2.4)

    According to (2.4), it is easy to get the following equivalent form for SDD1 matrices.

    A matrix M is called an SDD1 matrix, if

    {|mii|>jN(1)1{i}|mij|+jN(1)2|mij|+jN2rj(M)|mjj||mij|,iN(1)1,|mii|>jN(1)1|mij|+jN(1)2{i}|mij|+jN2rj(M)|mjj||mij|,iN(1)2. (2.5)

    By scaling conditions (2.5), we introduce a new class of matrices. As we will see, these matrices belong to a new subclass of nonsingular H-matrices.

    Definition 5. A matrix M=(mij)Cn×n is called an SDD+1 matrix if

    {|mii|>Fi(M)=jN(1)1{i}|mij|+jN(1)2|mij|rj(M)|mjj|+jN2|mij|rj(M)|mjj|,iN(1)1,|mii|>Fi(M)=jN(1)2{i}|mij|+jN2|mij|,iN(1)2, (2.6)

    where N1,N2,N(1)1,N(1)2,andri(M) are defined by (2.1)–(2.3), respectively.

    Proposition 1. If M=(mij)Cn×n is an SDD+1 matrix and N(1)1, then jN(1)2|mij|+jN2|mij|0 for iN(1)1.

    Proof. Assume that there exists iN(1)1 such that jN(1)2|mij|+jN2|mij|=0. We find that

    |mii|>jN(1)1{i}|mij|=jN(1)1{i}|mij|+jN(1)2{i}|mij|+jN2|mij|rj(M)|mjj|=jN(1)1{i}|mij|+jN2|mij|rj(M)|mjj|=ri(M),

    which contradicts iN(1)1. The proof is completed.

    Proposition 2. If M=(mij)Cn×n is an SDD+1 matrix with N(1)2=, then M is also an SDD1 matrix.

    Proof. From Definition 5, we have

    |mii|>jN(1)1{i}|mij|+jN2|mij|rj(M)|mjj|,iN(1)1.

    Because of N1=N(1)1, it holds that

    |mii|>jN1{i}|mij|+jN2|mij|rj(M)|mjj|=ri(M),iN1.

    The proof is completed.

    Example 1. Consider the following matrix:

    M1=(85233962429031210).

    In fact, N1={1,2} and N2={3,4}. Through calculations, we obtain that

    r1(M1)=10,r2(M1)=11,r3(M1)=6,r4(M1)=6,
    r1(M1)=|m12|+|m13|r3(M1)|m33|+|m14|r4(M1)|m44|8.1333,
    r2(M1)=|m21|+|m23|r3(M1)|m33|+|m24|r4(M1)|m44|=8.2000.

    Because of |m11|=8<8.1333=r1(M1), M1 is not an SDD1 matrix.

    Since

    r1(M1)8.1333>|m11|=8,
    r2(M1)=8.2000<|m22|=9,

    then N(1)1={1}, N(1)2={2}. As

    |m11|=8>|m12|r2(M1)|m22|+|m13|r3(M1)|m33|+|m14|r4(M1)|m44|7.6889,
    |m22|=9>|m23|+|m24|=8,

    M1 is an SDD+1 matrix by Definition 5.

    Example 2. Consider the following matrix:

    M2=(15487751216).

    In fact, N1={2} and N2={1,3}. By calculations, we get

    r1(M2)=12,r2(M2)=12,r3(M2)=3,
    r2(M2)=0+|m21|r1(M2)|m11|+|m23|r3(M2)|m33|=6.5375.

    Because of |m22|=7>6.5375=r2(M2), M2 is an SDD1 matrix.

    According to

    r2(M2)=6.5375<|m22|=7,

    we know that N(1)2={2}. In addition,

    |m22|=7<0+|m21|+|m23|=12,

    and M2 is not an SDD+1 matrix.

    As shown in Examples 1 and 2 and Proposition 2, it can be seen that SDD+1 matrices and SDD1 matrices have an intersecting relationship:

    {SDD1}{SDD+1} and {SDD+1}{SDD1}.

    The following examples will demonstrate the relationships between SDD+1 matrices and other subclasses of nonsingular H-matrices.

    Example 3. Consider the following matrix:

    M3=(4012120104.14620233480462023042040).

    In fact, N1={2,3} and N2={1,4,5}. Through calculations, we get that

    r1(M3)=6,r2(M3)=14.1,r3(M3)=34,r4(M3)=12,r5(M3)=36,
    r2(M3)=|m23|+|m21|r1(M3)|m11|+|m24|r4(M3)|m44|+|m25|r5(M3)|m55|=11.9>|m22|,
    r3(M3)=|m32|+|m31|r1(M3)|m11|+|m34|r4(M3)|m44|+|m35|r5(M3)|m55|=14.6<|m33|,

    and N(1)1={2}, N(1)2={3}. Because of

    |m22|>|m23|r3(M3)|m33|+|m21|r1(M3)|m11|+|m24|r4(M3)|m44|+|m25|r5(M3)|m55|9.61,
    |m33|=33>0+|m31|+|m34|+|m35|=32.

    So, M3 is an SDD+1 matrix. However, since |m22|=10<11.9=r2(M3), then M3 is not an SDD1 matrix. And we have

    pN11(M3)=3,pN12(M3)=4.1,pN13(M3)=2,pN14(M3)=10,pN15(M3)=6,
    pN21(M3)=2.4,pN22(M3)=7.8,pN23(M3)=12.6,pN24(M3)=1.8,pN25(M3)=4.5.

    Note that, taking i=1,j=2, we have that

    (r1(M3)pN21(M3))(|m22|pN12(M3))=21.24<pN11(M3)pN22(M3)=23.4.

    So, M3 is not a GSDD1 matrix. Moreover, it can be verified that M3 is not a CKV-type matrix.

    Example 4. Consider the following matrix:

    M4=(10.4000.510012.110.5120.311).

    It is easy to check that M4 is a CKV-type matrix and a GSDD1 matrix, but not an SDD+1 matrix.

    Example 5. Consider the following matrix:

    M5=(30.78666.755.256362.2533.78361.54.532.256633.036665.2530.752.25628.531.52.254.54.50.755.251.50.7529.28662.255.254.51.50.755.2535.2835.255.2534.5636.7530.033.754.50.753.757.55.250.750.7529.28).

    We know that M5 is not only an SDD+1 matrix, but also CKV-type matrix and GSDD1 matrix.

    According to Examples 3–5, we find that SDD+1 matrices have intersecting relationships with the CKV-type matrices and GSDD1 matrices, as shown in the Figure 1 below. In this paper, we take N1 and ri(M)0, so we will not discuss the relationships among SDD matrices, SDD+1 matrices, and GSDD1 matrices.

    Figure 1.  Relations between some subclasses of H-matrices.

    Example 6. Consider the tri-diagonal matrix MRn×n arising from the finite difference method for free boundary problems [8], where

    M6=(b+αsin(1n)c00ab+αsin(2n)c00ab+αsin(n1n)c00ab+αsin(1)).

    Take n=12000, a=5.5888, b=16.5150, c=10.9311, and α=14.3417. It is easy to verify that M6 is an SDD+1 matrix, but not an SDD matrix, a GSDD1 matrix, an SDD1 matrix, nor a CKV-type matrix.

    As is shown in [19] and [8], SDD1 matrix and GSDD1 matrix are both nonsingular H-matrices, and there exists an explicit construction of the diagonal matrix D, whose diagonal entries are all positive, such that MD is an SDD matrix. In the following, we construct a positive diagonal matrix D involved with a parameter that scales an SDD+1 matrix to transform it into an SDD matrix.

    Theorem 1. Let M=(mij)Cn×n be an SDD+1 matrix. Then, there exists a diagonal matrix D=diag(d1,d2,,dn) with

    di={1,iN(1)1,ε+ri(M)|mii|,iN(1)2,ε+ri(M)|mii|,iN2, (2.7)

    where

    0<ε<miniN(1)1pi, (2.8)

    and for all iN(1)1, we have

    pi=|mii|jN(1)1{i}|mij|jN(1)2|mij|rj(M)|mjj|jN2|mij|rj(M)|mjj|jN(1)2|mij|+jN2|mij|, (2.9)

    such that MD is an SDD matrix.

    Proof. By (2.6), we have

    |mii|jN(1)1{i}|mij|jN(1)2|mij|rj(M)|mjj|jN2|mij|rj(M)|mjj|>0. (2.10)

    From Proposition 1, for all iN(1)1, it is easy to know that

    pi=|mii|jN(1)1{i}|mij|jN(1)2|mij|rj(M)|mjj|jN2|mij|rj(M)|mjj|jN(1)2|mij|+jN2|mij|>0. (2.11)

    Immediately, there exists a positive number ε such that

    0<ε<miniN(1)1pi. (2.12)

    Now, we construct a diagonal matrix D=diag(d1,d2,,dn) with

    di={1,iN(1)1,ε+ri(M)|mii|,iN(1)2,ε+ri(M)|mii|,iN2, (2.13)

    where ε is given by (2.12). It is easy to find that all the elements in the diagonal matrix D are positive. Next, we will prove that MD is strictly diagonally dominant.

    Case 1. For each iN(1)1, it is not difficult to find that |(MD)ii|=|mii|. By (2.11) and (2.13), we have

    ri(MD)=jN(1)1{i}dj|mij|+jN(1)2dj|mij|+jN2dj|mij|=jN(1)1{i}|mij|+jN(1)2(ε+rj(M)|mjj|)|mij|+jN2(ε+rj(M)|mjj|)|mij|=jN(1)1{i}|mij|+jN(1)2|mij|rj(M)|mjj|+jN2|mij|rj(M)|mjj|+ε(jN(1)2|mij|+jN2|mij|)<|mii|=|(MD)ii|.

    Case 2. For each iN(1)2, we obtain

    |(MD)ii|=|mii|(ε+ri(M)|mii|)=ε|mii|+ri(M). (2.14)

    From (2.3), (2.13), and (2.14), we derive that

    ri(MD)=jN(1)1|mij|+jN(1)2{i}|mij|rj(M)|mjj|+jN2|mij|rj(M)|mjj|+ε(jN(1)2{i}|mij|+jN2|mij|)ri(M)+ε(jN(1)2{i}|mij|+jN2|mij|)<ri(M)+ε|mii|=|(MD)ii|.

    The first inequality holds because of |mii|>ri(M) for any iN(1)2, and

    ri(M)=jN1{i}|mij|+jN2|mij|rj(M)|mjj|=jN(1)1|mij|+jN(1)2{i}|mij|+jN2|mij|rj(M)|mjj|jN(1)1|mij|+jN(1)2{i}|mij|rj(M)|mjj|+jN2|mij|rj(M)|mjj|.

    Case 3. For each iN2, we have

    ri(M)=jN1{i}|mij|+jN2{i}|mij|=jN(1)1|mij|+jN(1)2|mij|+jN2{i}|mij|jN(1)1|mij|+jN(1)2|mij|rj(M)|mjj|+jN2{i}|mij|rj(M)|mjj|. (2.15)

    Meanwhile, for each iN2, it is easy to get

    |mii|>ri(M)jN(1)2|mij|+jN2{i}|mij|, (2.16)

    and

    |(MD)ii|=|mii|(ε+ri(M)|mii|)=ε|mii|+ri(M). (2.17)

    From (2.13), (2.15), and (2.16), it can be deduced that

    ri(MD)=jN(1)1|mij|+jN(1)2(ε+rj(M)|mjj|)|mij|+jN2{i}(ε+rj(M)|mjj|)|mij|=jN(1)1|mij|+jN(1)2|mij|rj(M)|mjj|+jN2{i}|mij|rj(M)|mjj|+ε(jN(1)2|mij|+jN2{i}|mij|)<ri(M)+ε|mii|=|(MD)ii|.

    So, |(MD)ii|>ri(MD) for iN. Thus, MD is an SDD matrix. The proof is completed.

    It is well known that the H-matrix M is nonsingular if there exists a diagonal matrix D such that MD is an SDD matrix (see [1,19]). Therefore, from Theorem 1, SDD+1 matrices are nonsingular H-matrices.

    Corollary 1. Let M=(mij)Cn×n be an SDD+1 matrix. Then, M is also an H-matrix. If, in addition, M has positive diagonal entries, then det(M)>0.

    Proof. We see from Theorem 1 that there is a positive diagonal matrix D such that MD is an SDD matrix (cf. (M35) of Theorem 2.3 of Chapter 6 of [1]). Thus, M is a nonsingular H-matrix. Since the diagonal entries of M and D are positive, MD has positive diagonal entries. From the fact that MD is an SDD matrix, it is well known that 0<det(MD)=det(M)det(D), which means det(M)>0.

    In this section, we start to consider two infinity norm bounds of the inverse of SDD+1 matrices. Before that, some notations are defined:

    Mi=|mii|jN(1)1{i}|mij|jN(1)2|mij|rj(M)|mjj|jN2|mij|rj(M)|mjj|ε(jN(1)2|mij|+jN2|mij|),iN(1)1, (3.1)
    Ni=ri(M)jN(1)1|mij|jN(1)2{i}|mij|rj(M)|mjj|jN2|mij|rj(M)|mjj|+ε(|mii|jN(1)2{i}|mij|jN2|mij|),iN(1)2, (3.2)
    Zi=ri(M)jN(1)1|mij|jN(1)2|mij|rj(M)|mjj|jN(1)2{i}|mij|rj(M)|mjj|+ε(|mii|jN(1)2|mij|jN2{i}|mij|),iN2. (3.3)

    Next, let us review an important result proposed by Varah (1975).

    Theorem 2. [20] If M=(mij)Cn×n is an SDD matrix, then

    M11miniN{|mii|ri(M)}. (3.4)

    Theorem 2 can be used to bound the infinity norm of the inverse of an SDD matrix. This theorem together with the scaling matrix D=diag(d1,d2,,dn) allows us to gain the following Theorem 3.

    Theorem 3. Let M=(mij)Cn×n be an SDD+1 matrix. Then,

    M1max{1,maxiN(1)2(ε+ri(M)|mii|),maxiN2(ε+ri(M)|mii|)}min{miniN(1)1Mi,miniN(1)2Ni,miniN2Zi}, (3.5)

    where ε, Mi,Ni,andZi are defined in (2.8), (2.9), and (3.1)–(3.3), respectively.

    Proof. By Theorem 1, there exists a positive diagonal matrix D such that MD is an SDD matrix, where D is defined as (2.13). Hence, we have the following result:

    M1=D(D1M1)=D(MD)1D(MD)1 (3.6)

    and

    D=max1indi=max{1,maxiN(1)2(ε+ri(M)|mii|),maxiN2(ε+ri(M)|mii|)},

    where ε is given by (2.8). Note that MD is an SDD matrix, by Theorem 2, we have

    (MD)11miniN{|(MD)ii|ri(MD)}.

    However, there are three scenarios to solve |(MD)ii|ri(MD). For iN(1)1, we get

    |(MD)ii|ri(MD)=|mii|jN(1)1{i}|mij|jN(1)2(ε+rj(M)|mjj|)|mij|jN2(ε+rj(M)|mjj|)|mij|=|mii|jN(1)1{i}|mij|jN(1)2|mij|rj(M)|mjj|jN2|mij|rj(M)|mjj|ε(jN(1)2|mij|+jN2|mij|)=Mi.

    For iN(1)2, we have

    |(MD)ii|ri(MD)=(ε+ri(M)|mii|)|mii|jN(1)1|mij|jN(1)2{i}(ε+rj(M)|mjj|)|mij|jN2(ε+rj(M)|mjj|)|mij|=ri(M)jN(1)1|mij|jN(1)2{i}|mij|rj(M)|mjj|jN2|mij|rj(M)|mjj|+ε(|mii|jN(1)2{i}|mij|jN2|mij|)=Ni.

    For iN2, we obtain

    |(MD)ii|ri(MD)=(ε+ri(M)|mii|)|mii|jN(1)1|mij|jN(1)2(ε+rj(M)|mjj|)|mij|jN2{i}(ε+rj(M)|mjj|)|mij|=ri(M)jN(1)1|mij|jN(1)2|mij|rj(M)|mjj|jN2{i}|mij|rj(M)|mjj|+ε(|mii|jN(1)2|mij|jN2{i}|mij|)=Zi.

    Hence, according to (3.6) we have

    M1max{1,maxiN(1)2(ε+ri(M)|mii|),maxiN2(ε+ri(M)|mii|)}min{miniN(1)1Mi,miniN(1)2Ni,miniN2Zi}.

    The proof is completed.

    It is noted that the upper bound in Theorem 3 is related to the interval values of the parameter. Next, another upper bound of M1 is given, which depends only on the elements in the matrix.

    Theorem 4. Let M=(mij)Cn×n be an SDD+1 matrix. Then,

    M1max{1miniN(1)1{|mii|Fi(M)},1miniN(1)2{|mii|Fi(M)},1miniN2{|mii|ji|mij|}}, (3.7)

    where Fi(M) and Fi(M) are shown in (2.6).

    Proof. By the well-known fact (see[21,22]) that

    M11=infx0Mxx=minx=1Mx=Mx=maxiN|(Mx)i|, (3.8)

    for some x=[x1,x2,,xn]T, we have

    M11|(Mx)i|. (3.9)

    Assume that there is a unique kN such that x=1=|xk|, then

    mkkxk=(Mx)kjkmkjxj=(Mx)kjN(1)1{k}mkjxjjN(1)2{k}mkjxjjN2{k}mkjxj.

    When kN(1)1, let |xj|=rj(M)|mjj| (jN(1)2), and |xj|=rj(M)|mjj| (jN2). Then we have

    |mkk|=|mkkxk|=|(Mx)kjN(1)1{k}mkjxjjN(1)2mkjxjjN2mkjxj||(Mx)k|+|jN(1)1{k}mkjxj|+|jN(1)2mkjxj|+|jN2mkjxj||(Mx)k|+jN(1)1{k}|mkj||xj|+jN(1)2|mkj||xj|+jN2|mkj||xj||(Mx)k|+jN(1)1{k}|mkj|+jN(1)2|mkj|rj(M)|mjj|+jN2|mkj|rj(M)|mjj|M11+jN(1)1{k}|mkj|+jN(1)2|mkj|rj(M)|mjj|+jN2|mkj|rj(M)|mjj|=M11+Fk(M).

    Which implies that

    M11|mkk|Fk(M)1miniN(1)1{|mii|Fi(M)}.

    For kN(1)2, let jN(1)1|mkj|=0. It follows that

    |mkk||(Mx)k|+jN(1)1|mkj||xj|+jN(1)2{k}|mkj||xj|+jN2|mkj||xj|M11+0+jN(1)2{k}|mkj|rj(M)|mjj|+jN2|mkj|rj(M)|mjj|=M11+Fk(M).

    Hence, we obtain that

    M11|mkk|Fk(M)1miniN(1)2{|mii|Fi(M)}.

    For kN2, we get

    0<miniN2{|mii|ji|mij|}|mkk|jk|mkj|,

    and

    0<miniN2(|mii|ji|mij|)|xk||mkk||xk|jk|mkj||xj||mkk||xk||jkmkjxj||kN2,jNmkjxj|maxiN2|jNmijxj|M11,

    which implies that

    M11|mkk|Fk(M)1miniN2{|mii|ji|mij|}.

    To sum up, we obtain that

    \begin{equation*} \begin{aligned} \left\|M^{-1}\right\|_{\infty} \leq& \max \left\{ \frac{1}{\min\limits _{i \in N_1^{(1)}}\left\{\left|m_{i i}\right|-F_i(M)\right\}}, \right.\left.\frac{1}{\min \limits_{i \in N_2^{(1)}}\left\{\left|m_{i i}\right|-F_i^{\prime}(M)\right\}}, \frac{1}{\min\limits _{i \in N_2}\left\{\left|m_{i i}\right|-\sum\limits_{j \neq i}\left|m_{i j}\right|\right\} } \right\}. \end{aligned} \end{equation*}

    The proof is completed.

    Next, some numerical examples are given to illustrate the superiority of our results.

    Example 7. Consider the following matrix:

    M_7 = \left( {\begin{array}{*{20}{c}} 2 &-1&{0}&0&0&0\\ {-1}&2 &-1&{0}&0&0\\ 0&-1&2 &-1&{0}&0\\ 0&0&-1&2 &-1&0\\ 0&0&0&-1&2 &-1\\ 0&0&0&0&2 &-1 \end{array}} \right).

    It is easy to verify that M_7 is an SDD_1^{+} matrix. However, we know that M_7 is not an SDD matrix, a GSDD_1 matrix, an SDD_1 matrix and a CKV -type matrix. By the bound in Theorem 3, we have

    \min p_i = 0.25, \; \varepsilon\in \left(0, 0.25\right).

    When \varepsilon = 0.1225 , we get

    \left\|M_7^{-1}\right\|_{\infty} \leq 8.1633 .

    The range of parameter values in Theorem 3 is not empty set, and its optimal solution can be illustrated through examples. In Example 7, the range of values for the error bound and its optimal solution can be seen from Figure 2. The bound for Example 7 is (8.1633,100), and the optimal solution for Example 7 is 8.1633.

    Figure 2.  The bound of Theorem 3.

    However, according to Theorem 4, we obtain

    \left\|M_7^{-1}\right\|_{\infty} \leq\max\left\{ 4, 1, 1 \right\} = 4.

    Through this example, it can be found that the bound of Theorem 4 is better than Theorem 3 in some cases.

    Example 8. Consider the following matrix:

    M_8 = \left( {\begin{array}{*{20}{c}} b_1 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ a & b_2 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & a & b_3 & c & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & a & b_4 & c & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & a & b_5 & c & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & a & b_6 & c & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & a & b_7 & c & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & a & b_8 & c & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & a & b_9 & c \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a & b_{10} \end{array}} \right).

    Here, a = -2 , c = 2.99 , b_1 = 6.3304 , b_2 = 6.0833 , b_3 = 5.8412 , b_4 = 5.6065 , b_5 = 5.3814 , b_6 = 5.1684 , b_7 = 4.9695 , b_8 = 4.7866 , b_9 = 4.6217 , and b_{10} = 4.4763 . It is easy to verify that M_8 is an SDD_1^{+} matrix. However, we can get that M_8 is not an SDD matrix, a GSDD_1 matrix, an SDD_1 matrix, and a CKV -type matrix. By the bound in Theorem 3, we have

    \min p_i = 0.1298, \; \varepsilon\in \left(0, 0.1298\right).

    When \varepsilon = 0.01 , we have

    \left\|M_8^{-1}\right\|_{\infty} \leq \frac{\max \{1, 1.0002, 0.9755\}}{\min \{0.5981, 0.0163, 0.1765\}} = 61.3620 .

    If \varepsilon = 0.1 , we have

    \left\|M_8^{-1}\right\|_{\infty} \leq \frac{\max \{1, 1.0902, 1.0655\}}{\min \{0.1490, 0.1632, 0.1925\}} = 7.3168 .

    Taking \varepsilon = 0.11 , then it is easy to calculate

    \left\|M_8^{-1}\right\|_{\infty} \leq \frac{\max \{1, 1.1002, 1.0755\}}{\min \{0.0991, 0.1795, 0.1943\}} = 11.1019 .

    By the bound in Theorem 4, we have

    \left\|M_8^{-1}\right\|_{\infty} \leq\max\left\{ 1.5433, 0.6129, 5.6054 \right\} = 5.6054.

    Example 9. Consider the following matrix:

    M_9 = \left( {\begin{array}{*{20}{c}} 11.3 & -1.2 & -1.1 & 4.7 \\ -1.2 & 14.2 & 9.1 & -4 \\ 3.2 & -1.1 & 14.3 & 0.3 \\ 4.6 & 7.6 & 3.2 & 11.3 \\ \end{array}} \right).

    It is easy to verify that the matrix M_9 is a GSDD_1 matrix and an SDD_1^{+} matrix. When M_9 is a GSDD_1 matrix, it can be calculated according to Lemma 1 that

    \varepsilon\in \left(1.0484, 1.1265\right).

    According to Figure 3, if we take \varepsilon = 1.0964 , we can obtain an optimal bound, namely

    \left\|M_9^{-1}\right\|_{\infty} \leq 6.1806.

    When M_9 is an SDD_1^{+} matrix, it can be calculated according to Theorem 3. We obtain

    Figure 3.  The bound of infinity norm.
    \varepsilon\in \left(0, 0.2153\right).

    According to Figure 3, if we take \varepsilon = 0.1707 , we can obtain an optimal bound, namely

    \left\|M_9^{-1}\right\|_{\infty} \leq 1.5021.

    However, according to Theorem 4, we get

    \left\|M_9^{-1}\right\|_{\infty} \leq\max\left\{ 0.3016, 0.2564, 0.2326 \right\} = 0.3016.

    Example 10. Consider the following matrix:

    M_{10} = \left( {\begin{array}{*{20}{c}} 7 &3&{1}&1\\ {1}&7&3&4\\ 2&2&9&3\\ 3&1&3&7 \end{array}} \right).

    It is easy to verify that the matrix {M}_{10} is a C K V -type matrix and an S D D_1^{+} matrix. When {M}_{10} is a CKV -type matrix, it can be calculated according to Theorem 21 in [9]

    \left\|M_{10}{ }^{-1}\right\|_{\infty} \leq 11 .

    When {M}_{10} is an S D D_1^{+} matrix, take \varepsilon = 0.0914 according to Theorem 3. We obtain an optimal bound, namely

    \left\|M_{10}{ }^{-1}\right\|_{\infty} \leq \frac{\max \{1, 0.8737, 0.8692\}}{\min \{0.0919, 0.0914, 3.5901\}} = 10.9409.

    When {M}_{10} is an S D D_1^{+} matrix, it can be calculated according to Theorem 4. We can obtain

    \left\|M_{10}{ }^{-1}\right\|_{\infty} \leq \max \{1, 2149, 1, 1\} = 1.2149 .

    From Examples 9 and 10, it is easy to know that the bound in Theorem 3 and Theorem 4 in our paper is better than available results in some cases.

    The P -matrix refers to a matrix in which all principal minors are positive [19], and it is widely used in optimization problems in economics, engineering, and other fields. In fact, the linear complementarity problem in the field of optimization has a unique solution if and only if the correlation matrix is a P -matrix, so the P -matrix has attracted extensive attention, see [23,24,25]. As we all know, the linear complementarity problem of matrix M , denoted by LCP(M, q) , is to find a vector x\in R^{n} such that

    \begin{equation} Mx+q\geq 0, \ \ \ \ \ \ (Mx+q)^{T}x = 0, \ \ \ \ \ \ x\geq 0, \end{equation} (4.1)

    or to prove that no such vector x exists, where M\in R^{n\times n} and q\in R^{n} . One of the essential problems in LCP(M, q) is to estimate

    \max\limits_{d\in [0, 1]^{n}} \|(I-D+DM)^{-1}\|_{\infty},

    where D = diag(d_{i}) , d = (d_{1}, d_{2}, \cdots, d_{n}) , 0\leq d_{i}\leq 1 , i = 1, 2, \cdots, n . It is well known that when M is a P -matrix, there is a unique solution to linear complementarity problems.

    In [2], Chen et al. gave the following error bound for LCP(M, q) ,

    \begin{equation} \|x-x^{*}\|_{\infty}\leq \max\limits_{d\in [0, 1]^{n}} \| (I-D+DM)^{-1}\|_{\infty} \| r(x)\|_{\infty}, \ \ \ \ \ \forall x\in R^n, \end{equation} (4.2)

    where x^{*} is the solution of LCP(M, q) , r(x) = \min\{x, Mx+q\} , and the min operator r(x) denotes the componentwise minimum of two vectors. However, for P-matrices that do not have a specific structure and have a large order, it is very difficult to calculate the error bound of \max\limits_{d\in [0, 1]^{n}} \| (I-D+DM)^{-1}\|_{\infty} . Nevertheless, the above problem is greatly alleviated when the proposed matrix has a specific structure [7,16,26,27,28].

    It is well known that a nonsingular H -matrix with positive diagonal entries is a P -matrix. In [29], when the matrix M is a nonsingular H -matrix with positive diagonal entries, and there is a diagonal matrix D so that MD is an SDD matrix, the authors propose a method to solve the error bounds of the linear complementarity problem of the matrix M . Now let us review it together.

    Theorem 5. [29] Assume that M = (m_{ij})\in R^{n\times n} is an H -matrix with positive diagonal entries. Let D = diag(d_i) , d_i > 0 , for all i \in N = \left\{1, \ldots, n\right\} , be a diagonal matrix such that MD is strictly diagonally dominant by rows. For any i \in N = \left\{1, \ldots, n\right\} , let \beta_i : = m_{ii}d_i-\sum\limits_{j\neq i}|m_{ij}|d_j . Then,

    \begin{equation} \max\limits_{d \in[0, 1]^n}\left\|(I-D+D M)^{-1}\right\|_{\infty} \leq \max \left\{\frac{\max _i\left\{d_i\right\}}{\min _i\left\{\beta_i\right\}}, \frac{\max _i\left\{d_i\right\}}{\min _i\left\{d_i\right\}}\right\} . \end{equation} (4.3)

    Next, the error bound of the linear complementarity problem of SDD_1^{+} matrices is given by using the positive diagonal matrix D in Theorem 1.

    Theorem 6. Suppose that M = (m_{ij})\in R^{n\times n}\; (n\geq 2) is an SDD_1^{+} matrix with positive diagonal entries, and for any i\in N_{1}^{(1)} , \sum\limits_{j \in N_2^{(1)} }\left|m_{i j}\right|+\sum\limits_{j \in N_2}\left|m_{i j}\right|\neq 0 . Then,

    \begin{align} &\max _{d \in[0, 1]^n}\left\|(I-D+D M)^{-1}\right\|_{\infty} \\ \leq &\max \left\{\frac{\max \left\{1, \max\limits _{i \in N_2^{(1)}} \left(\varepsilon+\frac{r_i^{\prime}(M)}{\left|m_{i i}\right|}\right), \max\limits _{i \in N_2} \left(\varepsilon+\frac{r_i(M)}{\left|m_{i i}\right|}\right)\right\}}{\min \left\{\min\limits _{i \in N_1^{(1)}} M_i, \min\limits _{i \in N_2^{(1)}} N_i, \min\limits _{i \in N_2} Z_i\right\}}, \right. \left.\frac{\max \left\{1, \max\limits _{i \in N_2^{(1)}} \left(\varepsilon+\frac{r_i^{\prime}(M)}{\left|m_{i i}\right|}\right), \max \limits_{i \in N_2} \left(\varepsilon+\frac{r_i(M)}{\left|m_{i i}\right|}\right)\right\}}{\min \left\{1, \min \limits_{i \in N_2^{(1)}} \left(\varepsilon+\frac{r_i^{\prime}(M)}{\left|m_{i i}\right|}\right), \min \limits_{i \in N_2} \left(\varepsilon+\frac{r_i(M)}{\left|m_{i i}\right|}\right)\right\}}\right\} , \end{align} (4.4)

    where \varepsilon , M_i, \; N_i, \; and\; Z_i are defined in (2.8), (2.9), and (3.1)–(3.3), respectively.

    Proof. Since M is an SDD_1^{+} matrix with positive diagonal elements, the existence of a positive diagonal matrix D such that MD is a strictly diagonal dominance matrix can be seen. For i\in N , we can get

    \begin{aligned} \beta_i = &\left|(M D)_{i i}\right|-\sum\limits_{j \in N \backslash\{i\}}\left|(M D)_{i j}\right| \\ = &\left|(M D)_{i i}\right|-\left(\sum\limits_{j \in N_1^{(1)} \backslash\{i\}}\left|(M D)_{i j}\right|+\sum\limits_{j \in N_2^{(1)} \backslash\{i\}}\left|(M D)_{i j}\right|+\sum\limits_{j \in N_2 \backslash\{i\}}\left|(M D)_{i j}\right|\right) \\ = &\left|(M D)_{i i}\right|-\sum\limits_{j \in N_1^{(1)} \backslash\{i\}}\left|(M D)_{i j}\right|-\sum\limits_{j \in N_2^{(1)} \backslash\{i\}}\left|(M D)_{i j}\right|-\sum\limits_{j \in N_2 \backslash\{i\}}\left|(M D)_{i j}\right|. \end{aligned}

    By Theorem 5, for i\in N_1^{(1)} , we get

    \begin{equation*} \begin{array}{l}\begin{aligned} \beta_i = &m_{i i} d_i-\sum\limits_{j \neq i}\left|m_{i j}\right| d_j = \left|m_{i i}\right|-\left(\sum\limits_{j \in N_1^{(1)} \backslash\{i\}}\left|m_{i j}\right|+\sum\limits_{j \in N_2^{(1)}}\left(\varepsilon+\frac{r_j^{\prime}(M)}{\left|m_{j j}\right|}\right)\left|m_{i j}\right|+\sum\limits_{j \in N_2}\left(\varepsilon+\frac{r_j(M)}{\left|m_{j j}\right|}\right)\left|m_{i j}\right|\right) \\ = &\left|m_{i i}\right|-\sum\limits_{j \in N_1^{(1)} \backslash\{i\}}\left|m_{i j}\right|-\sum\limits_{j \in N_2^{(1)}}\left|m_{i j}\right| \frac{r_j^{\prime}(M)}{\left|m_{j j}\right|}-\sum\limits_{j \in N_2}\left|m_{i j}\right| \frac{r_j(M)}{\left|m_{j j}\right|}-\varepsilon\left(\sum\limits_{j \in N_2^{(1)}}\left|m_{i j}\right|+\sum\limits_{j \in N_2}\left|m_{i j}\right|\right). \end{aligned}\end{array} \end{equation*}

    For i\in N_2^{(1)} , we have

    \begin{array}{l}\begin{aligned} \beta_i = &m_{i i} d_i-\sum\limits_{j \neq i}\left|m_{i j}\right| d_j \\ = &\left(\varepsilon+\frac{r_i^{\prime}(M)}{\left|m_{i i}\right|}\right)\left|m_{i i}\right|-\left(\sum\limits_{j \in N_1^{(1)}}\left|m_{i j}\right|+\sum\limits_{j \in N_2^{(1)} \backslash\{i\}}\left(\varepsilon+\frac{r_j^{\prime}(M)}{\left|m_{j j}\right|}\right)\left|m_{i j}\right|+\sum\limits_{j \in N_2}\left(\varepsilon+\frac{r_j(M)}{\left|m_{j j}\right|}\right)\left|m_{i j}\right|\right) \\ = &\varepsilon\left|m_{i i}\right|+r_i^{\prime}(M)-\sum\limits_{j \in N_1^{(1)}}\left|m_{i j}\right|-\sum\limits_{j \in N_2^{(1)} \backslash\{i\}}\left|m_{i j}\right| \frac{r_j^{\prime}(M)}{\left|m_{j j}\right|}-\varepsilon \sum\limits_{j \in N_2^{(1)} \backslash\{i\}}\left|m_{i j}\right|-\sum\limits_{j \in N_2}\left|m_{i j}\right| \frac{r_j(M)}{\left|m_{j j}\right|}-\varepsilon \sum\limits_{j \in N_2}\left|m_{i j}\right| \\ = &r_i^{\prime}(M)-\sum\limits_{j \in N_1^{(1)}}\left|m_{i j}\right|-\sum\limits_{j \in N_2^{(1)} \backslash\{i\}}\left|m_{i j}\right| \frac{r_j^{\prime}(M)}{\left|m_{j j}\right|}-\sum\limits_{j \in N_2}\left|m_{i j}\right| \frac{r_j(M)}{\left|m_{j j}\right|}+\varepsilon\left(\left|m_{i i}\right|-\sum\limits_{j \in N_2^{(1)} \backslash\{i\}}\left|m_{i j}\right|-\sum\limits_{j \in N_2}\left|m_{i j}\right|\right). \end{aligned}\end{array}

    For i\in N_2 , we have

    \begin{aligned} \beta_i = &m_{i i} d_i-\sum\limits_{j \neq i}\left|m_{i j}\right| d_j\\ = &\left(\varepsilon+\frac{r_i(M)}{\left|m_{i i}\right|}\right)\left|m_{i i}\right|-\left(\sum\limits_{j \in N_1^{(1)}}\left|m_{i j}\right|+\sum\limits_{j \in N_2^{(1)}}\left(\varepsilon+\frac{r_j^{\prime}(M)}{\left|m_{j j}\right|}\right)\left|m_{i j}\right|+\sum\limits_{j \in N_2 \backslash\{i\}}\left(\varepsilon+\frac{r_j(M)}{\left|m_{j j}\right|}\right)\left|m_{i j}\right|\right) \\ = &\varepsilon\left|m_{i i}\right|+r_i(M)-\sum\limits_{j \in N_1^{(1)}}\left|m_{i j}\right|-\sum\limits_{j \in N_2^{(1)}}\left|m_{i j}\right| \frac{r_j^{\prime}(M)}{\left|m_{j j}\right|}-\varepsilon \sum\limits_{j \in N_2^{(1)}}\left|m_{i j}\right|-\sum\limits_{j \in N_2 \backslash\{i\}}\left|m_{i j}\right| \frac{r_j(M)}{\left|m_{j j}\right|}-\varepsilon \sum\limits_{j \in N_2 \backslash\{i\}}\left|m_{i j}\right| \\ = &r_i(M)-\sum\limits_{j \in N_1^{(1)}}\left|m_{i j}\right|-\sum\limits_{j \in N_2^{(1)}}\left|m_{i j}\right| \frac{r_j^{\prime}(M)}{\left|m_{j j}\right|}-\sum\limits_{j \in N_2 \backslash\{i\}}\left|m_{i j}\right| \frac{r_j(M)}{\left|m_{j j}\right|}+\varepsilon\left(\left|m_{i i}\right|-\sum\limits_{j \in N_2^{(1)}}\left|m_{i j}\right|-\sum\limits_{j \in N_2 \backslash\{i\}}\left|m_{i j}\right|\right). \end{aligned}

    To sum up, it can be seen that

    \beta_i = \left\{\begin{array}{l} M_i, \; \; i \in N_1^{(1)}, \\ N_i, \; \; \; i \in N_2^{(1)}, \\ Z_i, \; \; \; \; i \in N_2 . \end{array}\right.

    According to Theorems 1 and 5, it can be obtained that

    \begin{equation*} \begin{array}{l}\begin{aligned} &\max _{d \in[0, 1]^n}\left\|(I-D+D M)^{-1}\right\|_{\infty} \\ \leq& \max \left\{\frac{\max \left\{1, \max\limits _{i \in N_2^{(1)}} \left(\varepsilon+\frac{r_i^{\prime}(M)}{\left|m_{i i}\right|}\right), \max\limits _{i \in N_2} \left(\varepsilon+\frac{r_i(M)}{\left|m_{i i}\right|}\right)\right\}}{\min \left\{\min\limits _{i \in N_1^{(1)}} M_i, \min\limits _{i \in N_2^{(1)}} N_i, \min\limits _{i \in N_2} Z_i\right\}}, \right. \left.\frac{\max \left\{1, \max\limits _{i \in N_2^{(1)}} \left(\varepsilon+\frac{r_i^{\prime}(M)}{\left|m_{i i}\right|}\right), \max \limits_{i \in N_2} \left(\varepsilon+\frac{r_i(M)}{\left|m_{i i}\right|}\right)\right\}}{\min \left\{1, \min \limits_{i \in N_2^{(1)}} \left(\varepsilon+\frac{r_i^{\prime}(M)}{\left|m_{i i}\right|}\right), \min \limits_{i \in N_2} \left(\varepsilon+\frac{r_i(M)}{\left|m_{i i}\right|}\right)\right\}}\right\}.\\ \end{aligned}\end{array} \end{equation*}

    The proof is completed.

    It is noted that the error bound of Theorem 6 is related to the interval of the parameter \varepsilon .

    Lemma 3. [16] Letting \gamma > 0 and \eta > 0 , for any x\in[0, 1] ,

    \begin{equation} \frac{1}{1-x+x \gamma} \leq \frac{1}{\min \{\gamma, 1\}}, \frac{\eta x}{1-x+x \gamma} \leq \frac{\eta}{\gamma} . \end{equation} (4.5)

    Theorem 7. Let M = ({{m_{ij}}}) \in {R^{n \times n}} be an SDD_{1}^+ matrix. Then, \overline {M} = (\overline {m}_{ij}) = I - D + DM is also an SDD_{1}^+ matrix, where D = diag\left({{d_i}} \right) with 0 \le {d_i} \le 1 , \forall i\in N .

    Proof. Since \overline {M} = I-D+DM = (\overline {m}_{ij}) , then

    \overline{m}_{ij} = \left\{\begin{array}{cc} 1-d_{i}+d_{i}m_{ii}, & i = j, \\ d_{i}m_{ij}, &i\neq j. \end{array} \right.

    By Lemma 3, for any i\in N_1^{(1)} , we have

    \begin{equation*} \begin{aligned} F_i(\overline{M}) = &\sum\limits_{j \in N_1^{(1)} \backslash\{i\}}\left|d_i m_{i j}\right|+\sum\limits_{j \in N_2^{(1)}}\left|d_i m_{i j}\right| \frac{d_j r_j^{\prime}(M)}{1-d_j+m_{j j} d_j}+\sum\limits_{j \in N_2}\left|d_i m_{i j}\right| \frac{d_j r_j(M)}{1-d_j+m_{j j} d_j} \\ = &d_i\left(\sum\limits_{j \in N_1^{(1)} \backslash\{i\}}\left|m_{i j}\right|+\sum\limits_{j \in N_2^{(1)}}\left|m_{i j}\right| \frac{d_j r_j^{\prime}(M)}{1-d_j+m_{j j} d_j}\right. \left.+\sum\limits_{j \in N_2}\left|m_{i j}\right| \frac{d_j r_j(M)}{1-d_j+m_{jj} d_j}\right) \\ \leq &d_i\left(\sum\limits_{j \in N_1^{(1)} \backslash\{i\}}\left|m_{i j}\right|+\sum\limits_{j \in N_2^{(1)}}\left|m_{i j}\right| \frac{r_j^{\prime}(M)}{\left|m_{j j}\right|}+\sum\limits_{j \in N_2}\left|m_{i j}\right| \frac{r_j(M)}{\left|m_{j j}\right|}\right) \\ = & d_i F_i(M). \end{aligned} \end{equation*}

    In addition, d_i F_i(M) < 1-d_i+d_i\left|m_{i i}\right| = \left|\overline{m}_{i i}\right| , that is, for each i \in N_1^{(1)}(\overline{M}) \subseteq N_1^{(1)}(M), \left|\overline{m}_{i i}\right| > F_i(\overline{M}) .

    For any i \in N_2^{(1)} , we have

    \begin{aligned} F_i(\overline{M})& = \sum\limits_{j \in N_2^{(1)} \backslash\{i\}}\left|d_i m_{i j}\right|+\sum\limits_{j \in N_2}\left|d_i m_{i j}\right| = d_i\left(\sum\limits_{j \in N_2^{(1)} \backslash\{i\}}\left|m_{i j}\right|+\sum\limits_{j \in N_2}\left|m_{i j}\right|\right) = d_i F_i^{\prime}(M).\end{aligned}

    So, d_i F_i^{\prime}(M) < 1-d_i+d_i\left|m_{i i}\right| = \left|\overline{m}_{i i}\right| , that is, for each i \in N_2^{(1)}(\overline{M}) \subseteq N_2^{(1)}(M), \left|\overline{m}_{i i}\right| > F_i^{\prime}(\overline{M}) . Therefore, \overline{M} = \left(\overline{m}_{i j}\right) = I-D+D M is an SDD_{1}^+ matrix.

    Next, another upper bound about \max\limits _{d \in[0, 1]^n}\left\|(I-D+D M)^{-1}\right\|_{\infty} is given, which depends on the result in Theorem 4.

    Theorem 8. Assume that M = (m_{ij})\in R^{n\times n}\; (n\geq 2) is an SDD_1^{+} matrix with positive diagonal entries, and {\overline M} = I-D+DM , D = diag\left({{d_i}} \right) with 0 \le {d_i} \le 1 . Then,

    \begin{eqnarray} \max _{d \in[0, 1]^n}\left\|\overline{M}^{-1}\right\|_{\infty} \leq \max\left\{\frac{1}{\min\limits _{i \in N_1^{(1)}}\left\{\left|m_{i i}\right|-F_i(M), 1\right\}}, \frac{1}{\min\limits _{i \in N_2^{(1)}}\left\{\left|m_{i i}\right|-F_i^{\prime}(M), 1\right\}}, \frac{1}{\min \limits _{i \in N_2}\left\{\left|m_{i i}\right|-\sum\limits _{j \neq i}\left|m_{i j}\right|, 1\right\}}\right\}, \end{eqnarray} (4.6)

    where N_2, \; N_1^{(1)}, \; N_2^{(1)}, \; F_i({M}), \; and\; F_i^{\prime}(M) are given by (2.1), (2.2), and (2.6), respectively.

    Proof. Because M is an SDD_1^{+} matrix, according to Theorem 7, {\overline M} = I-D+DM is also an SDD_1^{+} matrix, where

    {\overline M} = \left(\overline{m}_{ij}\right) = \left\{\begin{array}{cc} 1-d_{i}+d_{i}m_{ii}, & i = j, \\ d_{i}m_{ij}, &i\neq j. \end{array} \right.

    By (3.7), we can obtain that

    \begin{array}{l}\begin{aligned} \left\|\overline{M}^{-1}\right\|_{\infty} \leq& \max \left\{\frac{1}{\min\limits _{i \in N_1^{(1)}(\overline{M})}\left\{\left|\overline{m}_{i i}\right|-F_i(\overline{M})\right\}}, \right. \left.\frac{1}{\min \limits _{i \in N_2^{(1)}(\overline{M})}\left\{\left|\overline{m}_{i i}\right|-F_i^{\prime}(\overline{M})\right\}}, \frac{1}{\min \limits _{i \in N_2(\overline{M})}\left\{\left|\overline{m}_{i i}\right|-\sum\limits _{j \neq i}\left|\overline{m}_{i j}\right|\right\}}\right\} . \\ \end{aligned}\end{array}

    According to Theorem 7, for i \in N_1^{(1)} , let

    \begin{aligned} \nabla(\overline{M}) = &\sum\limits_{j \in N_1^{(1)} \backslash\{i\}}\left|d_i m_{i j}\right|+\sum\limits_{j \in N_2^{(1)} \backslash\{i\}}\left|d_i m_{i j}\right| \frac{d_j r_j^{\prime}(M)}{1-d_j+d_j m_{j j}}+\sum\limits_{j \in N_2}\left|d_i m_{i j}\right| \frac{d_j r_j(M)}{1-d_j+d_j m_{j j}}, \end{aligned}

    we have

    \begin{aligned} \frac{1}{\left|\overline{m}_{i i}\right|-F_i(\overline{M})} = &\frac{1}{1-d_i+d_i m_{i i}-\nabla(\overline{M})} \\ \leq & \frac{1}{1-d_i+d_i m_{i i}-d_i\left(\sum\limits_{j \in N_1^{(1)} \backslash\{i\}}\left|m_{i j}\right|+\sum\limits_{j \in N_2^{(1)} \backslash\{i\}}\left|m_{i j}\right| \frac{r_j^{\prime}(M)}{|m_{j j}|}+\sum\limits_{j \in N_2}\left|m_{i j}\right| \frac{r_j(M)}{|m_{j j}|}\right)} \\ = & \frac{1}{1-d_i+d_i\left(\left|m_{i i}\right|-F_i(M)\right)} \\ \leq & \frac{1}{\min \left\{\left|m_{i i}\right|-F_i(M), 1\right\}} . \end{aligned}

    For i \in N_2^{(1)} , we get

    \begin{aligned} \frac{1}{\left|\overline{m}_{i i}\right|-F_i^{\prime}(\overline{M})} & = \frac{1}{1-d_i+d_i m_{i i}-\left(\sum\limits_{j \in N_2^{(1)} \backslash\{i\}}\left|d_i m_{i j}\right|+\sum\limits_{j \in N_2}\left|d_i m_{i j}\right|\right)} \\ & = \frac{1}{1-d_i+d_i m_{i i}-d_i\left(\sum\limits_{j \in N_2^{(1)} \backslash\{i\}}\left|m_{i j}\right|+\sum\limits_{j \in N_2}\left|m_{i j}\right|\right)} \\ & = \frac{1}{1-d_i+d_i\left(\left|m_{i i}\right|-F_i^{\prime}(M)\right)} \\ & \leq \frac{1}{\min \left\{\left|m_{i i}\right|-F_i^{\prime}(M), 1\right\}}. \end{aligned}

    For i \in N_2 , we can obtain that

    \begin{aligned} \frac{1}{\left|\overline{m}_{i i}\right|-\sum\limits_{j \neq i}\left|\overline{m}_{i j}\right|} & = \frac{1}{1-d_i+d_i m_{i i}-\sum\limits_{j \neq i}\left|d_i m_{i j}\right|} \\ & = \frac{1}{1-d_i+d_i\left(\left|m_{i i}\right|-\sum\limits_{j \neq i}\left|m_{i j}\right|\right)} \\ & \leq \frac{1}{\min \left\{\left|m_{i i}\right|-\sum\limits_{j \neq i}\left|m_{i j}\right|, 1\right\}} . \end{aligned}

    To sum up, it holds that

    \begin{equation*} \begin{aligned} \max _{d \in[0, 1]^n}&\left\|\overline{M}^{-1}\right\|_{\infty} \leq \max\left\{\frac{1}{\min\limits _{i \in N_1^{(1)}}\left\{\left|m_{i i}\right|-F_i(M), 1\right\}}, \right. \nonumber\left.\frac{1}{\min\limits _{i \in N_2^{(1)}}\left\{\left|m_{i i}\right|-F_i^{\prime}(M), 1\right\}}, \frac{1}{\min \limits _{i \in N_2}\left\{\left|m_{i i}\right|-\sum\limits _{j \neq i}\left|m_{i j}\right|, 1\right\}}\right\}. \end{aligned} \end{equation*}

    The proof is completed.

    The following examples show that the bound (4.6) in Theorem 8 is better than the bound (4.4) in some conditions.

    Example 11. Let us consider the matrix in Example 6. According to Theorem 6, by calculation, we obtain

    \varepsilon\in \left( 0, 0.0386 \right).

    Taking \varepsilon = 0.01 , then

    \max\limits _{d \in[0, 1]^{12000}}\left\|(I-D+D M_6)^{-1}\right\|_{\infty}\leq 589.4024.

    In addition, from Theorem 8, we get

    \max\limits _{d \in[0, 1]^{12000}}\left\|(I-D+D M_6)^{-1}\right\|_{\infty}\leq929.6202.

    Example 12. Let us consider the matrix in Example 9. Since M_9 is a GSDD_1 matrix, then, by Lemma 2, we get

    \varepsilon\in \left( 1.0484, 1.1265 \right).

    From Figure 4, when \varepsilon = 1.0964 , the optimal bound can be obtained as follows:

    \max\limits _{d \in[0, 1]^4}\left\|(I-D+D M_9)^{-1}\right\|_{\infty}\leqslant6.1806.

    Moreover, M_9 is an SDD_1^{+} matrix, and by Theorem 6, we get

    Figure 4.  The bound of LCP.
    \varepsilon\in \left( 0, 0.2153 \right).

    From Figure 4, when \varepsilon = 0.1794 , the optimal bound can be obtained as follows:

    \max\limits _{d \in[0, 1]^4}\left\|(I-D+D M_9)^{-1}\right\|_{\infty}\leq1.9957.

    However, according to Theorem 8, we obtain that

    \max\limits _{d \in[0, 1]^4}\left\|(I-D+D M_9)^{-1}\right\|_{\infty}\leq1.

    Example 13. Consider the following matrix:

    M_{11} = \left( {\begin{array}{*{20}{c}} 7 &-3&{-1}&-1\\ {-1}&7&-3&-4\\ -2&-2&9&-3\\ -3&-1&-3&7 \end{array}} \right).

    Obviously, {B}^{+} = {M}_{11} and {C} = 0 . By calculations, we know that the matrix {B}^{+} is a {CKV} -type matrix with positive diagonal entries, and thus {M}_{11} is a CKV -type B -matrix. It is easy to verify that the matrix {M}_{11} is an S D D_1^{+} matrix. By bound (4.4) in Theorem 6, we get

    \max\limits _{d \in[0, 1]^4}\left\|\left(I-D+D M_{11}\right)^{-1}\right\|_{\infty}\leq10.9890(\varepsilon = 0.091), \; \; \; \; \varepsilon \in(0, 0.1029).

    By the bound (4.6) in Theorem 8, we get

    \max\limits _{d \in[0, 1]^4}\left\|\left(I-D+D M_{11}\right)^{-1}\right\|_{\infty} \leq 1.2149,

    while by Theorem 3.1 in [18], it holds that

    \max\limits_{d \in[0, 1]^4}\left\|\left(I-D+D M_{11}\right)^{-1}\right\|_{\infty} \leq 147.

    From Examples 12 and 13, it is obvious to know that the bounds in Theorems 6 and 8 in our paper is better than available results in some cases.

    In this paper, a new subclass of nonsingular H -matrices, SDD_1^{+} matrices, have been introduced. Some properties of SDD_1^{+} matrices are discussed, and the relationships among SDD_1^{+} matrices and SDD_1 matrices, GSDD_1 matrices, and CKV -type matrices are analyzed through numerical examples. A scaling matrix D is used to transform the matrix M into a strictly diagonal dominant matrix, which proves the non-singularity of SDD_1^{+} matrices. Two upper bounds of the infinity norm of the inverse matrix are deduced by two methods. On this basis, two error bounds of the linear complementarity problem are given. Numerical examples show the validity of the obtained results.

    Lanlan Liu: Conceptualization, Methodology, Supervision, Validation, Writing-review; Yuxue Zhu: Investigation, Writing-original draft, Conceptualization, Writing-review and editing; Feng Wang: Formal analysis, Validation, Conceptualization, Writing-review; Yuanjie Geng: Conceptualization, Validation, Editing, Visualization. All authors have read and approved the final version of the manuscript for publication.

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

    This research is supported by Guizhou Minzu University Science and Technology Projects (GZMUZK[2023]YB10), the Natural Science Research Project of Department of Education of Guizhou Province (QJJ2022015), the Talent Growth Project of Education Department of Guizhou Province (2018143), and the Research Foundation of Guizhou Minzu University (2019YB08).

    The authors declare no conflicts of interest.



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