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Research article

Error bounds for generalized vector inverse quasi-variational inequality Problems with point to set mappings

  • Received: 11 September 2020 Accepted: 08 November 2020 Published: 30 November 2020
  • MSC : 49J40, 47H09, 47J20, 54H25

  • The goal of this paper is further to study a kind of generalized vector inverse quasi-variational inequality problems and to obtain error bounds in terms of the residual gap function, the regularized gap function, and the global gap function by utilizing the relaxed monotonicity and Hausdorff Lipschitz continuity. These error bounds provide effective estimated distances between an arbitrary feasible point and the solution set of generalized vector inverse quasi-variational inequality problems.

    Citation: S. S. Chang, Salahuddin, M. Liu, X. R. Wang, J. F. Tang. Error bounds for generalized vector inverse quasi-variational inequality Problems with point to set mappings[J]. AIMS Mathematics, 2021, 6(2): 1800-1815. doi: 10.3934/math.2021108

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  • The goal of this paper is further to study a kind of generalized vector inverse quasi-variational inequality problems and to obtain error bounds in terms of the residual gap function, the regularized gap function, and the global gap function by utilizing the relaxed monotonicity and Hausdorff Lipschitz continuity. These error bounds provide effective estimated distances between an arbitrary feasible point and the solution set of generalized vector inverse quasi-variational inequality problems.


    In 2014, Li et al. [1] suggested a new class of inverse mixed variational inequality in Hilbert spaces that has simple problem of traffic network equilibrium control, market equilibrium issues as applications in economics and telecommunication network problems. The concept of gap function plays an important role in the development of iterative algorithms, an evaluation of their convergence properties and useful stopping rules for iterative algorithms, see [2,3,4,5]. Error bounds are very important and useful because they provide a measure of the distance between a solution set and a feasible arbitrary point. Solodov [6] developed some merit functions associated with a generalized mixed variational inequality, and used those functions to achieve mixed variational error limits. Aussel et al. [7] introduced a new inverse quasi-variational inequality (IQVI), obtained local (global) error bounds for IQVI in terms of certain gap functions to demonstrate the applicability of IQVI, and provided an example of road pricing problems, also see [8,9]. Sun and Chai [10] introduced regularized gap functions for generalized vector variation inequalities (GVVI) and obtained GVVI error bounds for regularized gap functions. Wu and Huang [11] implemented generalized f-projection operators to deal with mixed variational inequality. Using the generalized f-projection operator, Li and Li [12] investigated a restricted mixed set-valued variational inequality in Hilbert spaces and proposed four merit functions for the restricted mixed set valued variational inequality and obtained error bounds through these functions.

    Our goal in this paper is to present a problem of generalized vector inverse quasi-variational inequality problems. They propose three gap functions, the residual gap function, the regularized gap function, and the global gap function, and obtain error bounds for generalized vector inverse quasi-variational inequality problem using these gap functions and generalized f-projection operator under the monotonicity and Lipschitz continuity of underlying mappings.

    Throughout this article, R+ denotes the set of non-negative real numbers, 0 denotes the origins of all finite dimensional spaces, and , denotes the norms and the inner products in finite dimensional spaces, respectively. Let Ω,F,P:RnRn be the set-valued mappings with nonempty closed convex values, Ni:Rn×RnRn(i=1,2,,m) be the bi-mappings, B:RnRn be the single-valued mappings, and fi:RnR(i=1,2,,m) be real-valued convex functions. We put

    f=(f1,f2,,fm),N(,)=(N1(,),N2(,),,Nm(,)),

    and for any x,wRn,

    N(x,x),w=(N1(x,x),w,N2(x,x),w,,Nn(x,x),w).

    In this paper, we consider the following generalized vector inverse quasi-variational inequality for finding ˉxΩ(ˉx), ˉuF(ˉx) and ˉvP(ˉx) such that

    N(ˉu,ˉv),yB(ˉx)+f(y)f(B(ˉx))intRm+,yΩ(ˉx), (2.1)

    and solution set is denoted by .

    Special cases:

    (i) If P is a zero mapping and N(,)=N(), then (2.1) reduces to the following problem for finding ˉxΩ(ˉx) and ˉuF(ˉx) such that

    N(ˉu),yB(ˉx)+f(y)f(B(ˉx))intRm+,yΩ(ˉx), (2.2)

    studied in[13] and solution set is denoted by 1.

    (ii) If F is single valued mapping, then (2.2) reduces to the following vector inverse mixed quasi-variational inequality for finding ˉxΩ(ˉx) such that

    N(ˉx),yB(ˉx)+f(y)f(B(ˉx))intRm+,yΩ(ˉx), (2.3)

    studied in [14] and solution set is denoted by 2.

    (iii) If CRn is a nonempty closed and convex subset, B(x)=x and Ω(x)=C for all xRn, then (2.3) collapses to the following generalized vector variational inequality for finding ˉxC such that

    N(ˉx),yx+f(y)f(ˉx)intRm+,yC, (2.4)

    which is considered in [10].

    (iv) If f(x)=0 for all xRn, then (2.4) reduces to vector variational inequality for finding ˉxC such that

    N(ˉx),yxintRm+,yC, (2.5)

    studied in [15].

    (v) If Rm+=R+ then (2.5) reduces to variational inequality for finding ˉxC such that

    N(ˉx),yx0,yC, (2.6)

    studied in [16].

    Definition 2.1 [7] Let G:RnRn and g:RnRn be two maps.

    (i) (G,g) is said to be a strongly monotone if there exists a constant μg>0 such that

    G(y)G(x),g(y)g(x)μgyx2,x,yRn;

    (ii) g is said to be Lg-Lipschitz continuous if there exists a constant Lg>0 such that

    g(x)g(y)Lgxy,x,yRn.

    For any fixed γ>0, let G:RnטΩ(,+] be a function defined as follows:

    G(φ,x)=x22φ,x+φ2+2γf(x),φRn,x˜Ω, (2.7)

    where ˜ΩRn is a nonempty closed and convex subset, and f:RnR is convex.

    Definition 2.2 [11] We say that f˜Ω:Rn2˜Ω is a generalized f-projection operator if

    f˜Ωφ={w˜Ω:G(φ,w)=infy˜ΩG(φ,y)},φRn.

    Remark 2.3 If f(x)=0 for all x˜Ω, then the generalized f-projection operator f˜Ω is equivalent to the following metric projection operator:

    P˜Ω(φ)={w˜Ω:wφ=infy˜Ωyφ},φRn.

    Lemma 2.4 [1,11] The following statements hold:

    (i) For any given φRn,f˜Ωφ is nonempty and single-valued;

    (ii) For any given φRn,x=f˜Ωφ if and only if

    xφ,yx+γf(y)γf(x)0,y˜Ω;

    (iii) f˜Ω:RnΩ is nonexpansive, that is,

    f˜Ωxf˜Ωyxy,x,yRn.

    Lemma 2.5 [17] Let m be a positive number, BRn be a nonempty subset such that

    dmfor alldB.

    Let Ω:RnRn be a set-valued mapping such that, for each xRn, Ω(x) is a closed convex set, and let f:RnR be a convex function on Rn. Assume that

    (i) there exists a constant τ>0 such that

    D(Ω(x),Ω(y))τxy,x,yRn,

    where D(,) is a Hausdorff metric defined on Rn;

    (ii) 0wRnΩ(w);

    (iii) f is -Lipschitz continuous on Rn. Then there exists a constant κ=6τ(m+γ) such that

    fΩ(x)zfΩ(x)zκxy,x,yRn,zB.

    Lemma 2.6 A function r:RnR is said to be a gap function for a generalized vector inverse quasi-variational inequality on a set ˜SRn if it satisfies the following properties:

    (i) r(x)0for anyx˜S; (ii) r(ˉx)=0,ˉx˜S if and only if ˉx is a solution of (2.1).

    Definition 2.7 Let B:RnRn be the single-valued mapping and N:Rn×RnRn be a bi-mapping.

    (i) (N,B) is said to be a strongly monotone with respect to the first argument of N and B, if there exists a constant μB>0 such that

    N(y,)N(x,),B(y)B(x)μByx2,x,yRn;

    (ii) (N,B) is said to be a relaxed monotone with respect to the second argument of N and B, if there exists a constant ζB>0 such that

    N(,y)N(,x),B(y)B(x)ζByx2,x,yRn;

    (iii) N is said to be σ-Lipschitz continuous with respect to the first argument with constant σ>0 and -Lipschitz continuous with respect to the second argument with constant >0 such that

    N(x,ˉx)N(y,ˉy)σxy+ˉxˉy,x,ˉx,y,ˉyRn.

    (iv) B is said to be -Lipschitz continuous if there exists a constant >0 such that

    B(x)B(y)xy,x,yRn.

    Example 2.8 The variational inequality (2.6) can be solved by transforming it into an equivalent optimization problem for the so-called merit function r(;τ):X=RnR{+} defined by

    r(x;τ)=sup{N(ˉx),yxXτˉxx2X|xC} for ˉxC,

    where τ is a nonnegative parameter. If X is finite dimensional, the function r(;0) is usually called the gap function for τ=0, and the function r(;τ) for τ>0 is called the regularized gap function.

    Example 2.9 Assume that N:RnRn be a given mapping and C a closed convex set in Rn. Let and be given scalar satisfying >>0 then (2.6) has a D-gap function if

    N(x)=N(x)N(x),xRn

    where D stands for difference.

    In this section, we discuss the residual gap function for generalized vector inverse quasi-variational inequality problem by using the strong monotonicity, relaxed monotonicity, Hausdorff Lipschitz continuity and prove error bounds related to the residual gap function. We define the residual gap function for (2.1) as follows:

    rγ(x)=min1im{B(x)fiΩ(x)[B(x)γNi(u,v)]},xRn,uF(x),vP(x),γ>0. (3.1)

    Theorem 3.1 Suppose that F,P:RnRn are set-valued mappings and Ni:Rn×RnRn(i=1,2,,m) are the bi-mappings. Assume that B:RnRn is single-valued mapping, then for any γ>0,rγ(x) is a gap function for (2.1) on Rn.

    Proof. For any xRn,

    rγ(x)0.

    On the other side, if

    rγ(ˉx)=0,

    then there exists 0i0m such that

    B(ˉx)=fi0Ω(ˉx)[B(ˉx)γNi0(ˉu,ˉv)],ˉuF(ˉx),ˉvP(ˉx).

    From Lemma 2.4, we have

    B(ˉx)[B(ˉx)γNi0(ˉu,ˉv)],yB(ˉx)+γf(y)γf(B(ˉx))0,yΩ(ˉx),ˉuF(ˉx),ˉvP(ˉx)

    and

    Ni0(ˉu,ˉv),yB(ˉx)+f(y)f(B(ˉx))0,yΩ(ˉx),ˉuF(ˉx),ˉvP(ˉx).

    It gives that

    N(ˉu,ˉv),yB(ˉx)+f(y)f(B(ˉx))intRm+,yΩ(ˉx),ˉuF(ˉx),ˉvP(ˉx).

    Thus, ˉx is a solution of (2.1).

    Conversely, if ˉx is a solution of (2.1), there exists 1i0m such that

    Ni0(ˉu,ˉv),yB(ˉx)+fi0(y)fi0(B(ˉx))0,yΩ(ˉx),ˉuF(ˉx),ˉvP(ˉx).

    By using the Lemma 2.4, we have

    B(ˉx)=fi0Ω(ˉx)[B(ˉx)γNi0(ˉu,ˉv)],ˉuF(ˉx),ˉvP(ˉx).

    This means that

    rγ(ˉx)=min1im{B(ˉx)fiΩ(ˉx)[B(ˉx)γNi(ˉu,ˉv)]}=0.

    The proof is completed.

    Next we will give the residual gap function rγ, error bounds for (2.1).

    Theorem 3.2 Let F,P:RnRn be D-ϑF-Lipschitz continuous and D-ϱP-Lipschitz continuous mappings, respectively. Let Ni:Rn×RnRn(i=1,2,,m) be σi-Lipschitz continuous with respect to the first argument and i-Lipschits continuous with respect to the second argument, and B:RnRn be -Lipschitz continuous, and (Ni,B) be strongly monotone with respect to the first argument of Ni and B with positive constant μBi, and relaxed monotone with respect to the second argument of Ni and B with positive constant ζBi. Let

    mi=1(i).

    Assume that there exists κi(0,μBiζBiσiϑF+ϱPi) such that

    fiΩ(x)zfiΩ(y)zκixy,x,yRn,uF(x),vP(x),z{ww=B(x)γNi(u,v)}. (3.2)

    Then, for any xRn and μBi>ζBi+κi(σiϑF+iϱP),

    γ>κiμBiζBiκi(σiϑF+iϱP),
    d(x,)γ(σiϑFiϱP)+γ(μBiζBiκi(σiϑF+iϱP))κirγ(x),

    where

    d(x,)=infˉxxˉx

    denotes the distance between the point x and the solution set .

    Proof. Since

    mi=1(i).

    Let ˉxΩ(ˉx) be the solution of (2.1) and thus for any i{1,,m}, we have

    Ni(ˉu,ˉv),yB(ˉx)+fi(y)fi(B(ˉx))0,yΩ(ˉx),ˉuF(ˉx),ˉvP(ˉx). (3.3)

    From the definition of fiΩ(ˉx)[B(x)γNi(u,v)], and Lemma 2.4, we have

    fiΩ(ˉx)[B(x)γNi(u,v)](B(x)γNi(u,v)),yfiΩ(ˉx)[B(x)γNi(u,v)]+γfi(y)γfi(fiΩ(ˉx)[B(x)γNi(u,v)])0,yΩ(ˉx),uF(x),vP(x). (3.4)

    Since

    ˉxmi=1(i),andB(ˉx)Ω(ˉx).

    Replacing y by B(ˉx) in (3.4), we get

    fiΩ(ˉx)[B(x)γNi(u,v)](B(x)γNi(u,v)),B(ˉx)fiΩ(ˉx)[B(x)γNi(u,v)]+γfi(B(ˉx))γfi(fiΩ(ˉx)[B(x)γNi(u,v)])0,uF(x),vP(x). (3.5)

    Since

    fiΩ(ˉx)[B(x)γNi(u,v)]Ω(ˉx),

    from (3.3), it follows that

    γNi(ˉu,ˉv),fiΩ(ˉx)[B(x)γNi(u,v)]B(ˉx)+γfi(fiΩ(ˉx)[B(x)γNi(u,v)])γfi(B(ˉx))0. (3.6)

    Utilizing (3.5) and (3.6), we have

    γNi(ˉu,ˉv)γNi(u,v)fiΩ(ˉx)[B(x)γNi(u,v)]+B(x),fiΩ(ˉx)[B(x)γNi(u,v)]B(ˉx)0,

    which implies that

    γNi(ˉu,ˉv)γNi(u,v),fiΩ(ˉx)[B(x)γNi(u,v)]B(x)γNi(ˉu,ˉv)γNi(u,v),B(ˉx)B(x)
    +B(x)fiΩ(ˉx)[B(x)γNi(u,v)],fiΩ(ˉx)[B(x)γNi(u,v)]B(x)
    +B(x)fiΩ(ˉx)[B(x)γNi(u,v)],B(x)B(ˉx)0.

    Since F is D-ϑF-Lipschitz continuous, P is D-ϱP-Lipschits continuous and Ni is σi-Lipschitz continuous with respect to the first argument and i-Lipschitz continuous with respect to the second argument, we have

    ˉuuD(F(ˉx),F(x))ϑFˉxx;ˉvvD(P(ˉx),P(x))ϱPˉxx;Ni(ˉu,ˉv)Ni(u,v)σiˉuu+iˉvv. (3.7)

    Again, for i=1,2,,m, (Ni,B) are strongly monotone with respect to the first argument of Ni and B with a positive constant μBi,, and relaxed monotone with respect to the second argument of Ni and B with a positive constant ζBi, we have

    γNi(ˉu,ˉv)γNi(u,v),fiΩ(ˉx)[B(x)γNi(u,v)]B(x)B(x)fiΩ(ˉu)[B(x)γNi(u,v)]2
    +B(x)fiΩ(ˉx)[B(x)γNi(u,v)],B(x)B(ˉx)γμBixˉx2γζBixˉx2.

    By adding fiΩ(x)[B(x)γNi(u,v)] and using the Cauchy-Schwarz inequality along with the triangular inequality, we have

    γNi(ˉu,ˉv)γNi(u,v){fiΩ(ˉx)[B(x)γNi(u,v)]fiΩ(x)[B(x)γNi(u,v)]
    +fiΩ(x)[B(x)γNi(u,v)]B(x)}
    +B(x)B(ˉx){B(x)fiΩ(x)[B(x)γNi(u,v)]+fiΩ(x)[B(x)γNi(u,v)]
    fiΩ(ˉx)[B(x)γNi(u,v)]}γμBixˉx2γζBixˉx2.

    Using the (3.7) and condition (3.2), we have

    (σiϑF+iϱP)γˉxx{κiˉxx+fiΩ(x)[B(x)γNi(u,v)]B(x)}
    +xˉx{B(x)fiΩ(x)[B(x)γNi(u,v)]+κixˉx}γ(μBiζBi)xˉx2.

    Hence, for any xRn and i{1,2,,m}, μBi>ζBi+κi(σiϑF+iϱP),

    γ>κiμBiζBiκi(σiϑF+iϱP),

    we have

    xˉxγ(σiϑF+iϱP)+γ(μBiζBiκi(σiϑF+iϱP))κiB(x)fiΩ(x)[B(x)γNi(u,v)],uF(x),vP(x).

    This implies

    xˉxγ(σiϑF+iϱP)+γ(μBiζBiκi(σiϑF+iϱP))κimin1im{B(x)fiΩ(x)[B(x)γNi(u,v)]}

    which means that

    d(x,)xˉxγ(σiϑF+iϱP)+γ(μBiζBiκi(σiϑF+iϱP))κirγ(x).

    The proof is completed.

    The regularized gap function for (2.1) is defined for all xRn as follows:

    ϕγ(x)=min1imsupyΩ(x),uF(x),vP(x){Ni(u,v),B(x)y+fi(B(x))fi(y)12γB(x)y2}

    where γ>0 is a parameter.

    Lemma 4.1 We have

    ϕγ(x)=min1im{Ni(u,v),Riγ(x)+fi(B(x))fi(B(x)Riγ(x))12γRiγ(x)2}, (4.1)

    where

    Riγ(x)=B(x)fiΩ(x)[B(x)γNi(u,v)],xRn,uF(x),vP(x)

    and if

    xB1(Ω)

    and

    B1(Ω)={ξRn|B(ξ)Ω(ξ)},

    then

    ϕγ(x)12γrγ(x)2. (4.2)

    Proof. For given xRn,uF(x),vP(x) and i{1,2,,m}, set

    ψi(x,y)=Ni(u,v),B(x)y+fi(B(x))fi(y)12γB(x)y2,yRn.

    Consider the following problem:

    gi(x)=maxyΩ(x)ψi(x,y).

    Since ψi(x,) is a strongly concave function and Ω(x) is nonempty closed convex, the above optimization problem has a unique solution zΩ(x). Evoking the condition of optimality at z, we get

    0Ni(u,v)+fi(z)+1γ(zB(x))+NΩ(x)(z),

    where NΩ(x)(z) is the normal cone at z to Ω(x) and fi(z) denotes the subdifferential of fi at z. Therefore,

    z(B(x)γNi(u,v)),yz+γfi(y)γfi(z)0,yΩ(x),uF(x),vP(x)

    and so

    z=fiΩ(x)[B(x)γNi(u,v)],uF(x),vP(x).

    Hence gi(x) can be rewritten as

    gi(x)=Ni(u,v),B(x)fiΩ(x)[B(x)γNi(u,v)]+fi(B(x))fi(fiΩ(x)[B(x)γNi(u,v)])
    12γB(x)fiΩ(x)[B(x)γNi(u,v)]2,uF(x),vP(x).

    Letting

    Riγ(x)=B(x)fiΩ(x)[B(x)γNi(u,v)],uF(x),vP(x),

    we get

    gi(x)=Ni(u,v),Riγ(x)+fi(B(x))fi(B(x)Riγ(x))12γRiγ(x)2,uF(x),vP(x), (4.3)
    (4.4)

    and so

    ϕγ(x)=min1im{Ni(u,v),Riγ(x)+fi(B(x))fi(B(x)Riγ(x))12γRiγ(x)2}.

    From the definition of projection fiΩ(x)[B(x)γNi(u,v)], we have

    fiΩ(x)[B(x)γNi(u,v)]B(x)+γNi(u,v),yfiΩ(x)[B(x)γNi(u,v)]+γfi(y)γfi(fiΩ(x)[B(x)γNi(u,v)])0,uF(x),vP(x). (4.5)

    For any xB1(Ω), we have

    B(x)Ω(x).

    Therefore, putting y=B(x) in (4.5), we get

    γNi(u,v)Riγ(x),Riγ(x)+γfi(B(x))γfi(B(x)Riγ(x))0,uF(x),vP(x),

    that is,

    Ni(u,v),Riγ(x)+fi(B(x))fi(B(x)Riγ(x))1γRiγ(x),Riγ(x)=1γRiγ(x)2. (4.6)

    From the definition of rγ(x) and (4.1), we get

    ϕγ(x)12γrγ(x)2.

    The proof is completed.

    Theorem 4.2 For γ>0,ϕγ is a gap function for (2.1) on the set

    B1(Ω)={ξRn|B(ξ)Ω(ξ)}.

    Proof. From the definition of ϕγ, we have

    ϕγ(x)min1im{Ni(u,v),B(x)y+fi(B(x))fi(y)12γB(x)y2},for allyΩ(x),uF(x),vP(x). (4.7)

    Therefore, for any xB1(Ω), putting y=B(x) in (4.7), we have

    ϕγ(x)0.

    Suppose that ˉxB1(ξ) with ϕγ(ˉx)=0. From (4.2), it follows that

    rγ(ˉx)=0,

    which implies that ˉx is the solution of (2.1).

    Conversely, if ˉx is a solution of (2.1), there exists 1i0m such that

    Ni0(ˉu,ˉv),B(ˉx)y+fi0(B(ˉx))fi0(y)0,yΩ(ˉx),ˉuF(ˉx),ˉvP(ˉx),

    which means that

    min1im{supyΩ(ˉx),ˉuF(ˉx),ˉvP(ˉx){Ni(ˉu,ˉv),B(ˉx)y+fi(B(ˉx))fi(y)12γB(ˉx)y2}}0.

    Thus,

    ϕγ(ˉx)0.

    The preceding claim leads to

    ϕγ(ˉx)0

    and it implies that

    ϕγ(ˉx)=0.

    The proof is completed.

    Since ϕγ can act as a gap function for (2.1), according to Theorem 4.2, investigating the error bound properties that can be obtained with ϕγ is interesting. The following corollary is obtained directly by Theorem 3.2 and (3.5).

    Corollary 4.3 Let F,P:RnRn be D-ϑF-Lipschitz continuous and D-ϱP-Lipschitz continuous mappings, respectively. Let Ni:Rn×RnRn(i=1,2,,m) be σi-Lipschitz continuous with respect to the first argument and i-Lipschitz continuous with respect to the second argument, B:RnRn be -Lipschitz continuous, and (Ni,B) be strongly monotone with respect to the first argument of N and B with respect to the constant μBi>0, and relaxed monotone with respect to the second argument of N and B with respect to the constant ζBi>0. Let

    mi=1(i).

    Assume that there exists κi(0,μBiζBiϑFσi+iϱP) such that

    fiΩ(x)zfiΩ(y)zκixy,x,yRn,uF(x),vP(x)z{ww=B(x)γNi(u,v)}.

    Then, for any xB1(Ω) and any

    γ>κiμBiζBiκi(ϑFσi+iϱP),
    d(x,)γ(ϑFσi+iϱP)+γ(μBiζBiκi(ϑFσi+iϱP))κi2γϕγ(x).

    The regularized gap function ϕγ does not provide global error bounds for (2.1) on Rn. In this section, we first discuss the D-gap function, see [6] for (2.1), which gives Rn the global error bound for (2.1).

    For (2.1) with >>0, the D-gap function is defined as follows:

    G(x)=min1im{supyΩ(x),uF(x),vP(x){Ni(u,v),B(x)y+fi(B(x))fi(y)12B(x)y2}supyΩ(x)uF(x),vP(x){Ni(u,v),B(x)y+fi(B(x))fi(y)12B(x)y2}}.

    From (4.1), we know G can be rewritten as

    G(x)=min1im{Ni(u,v),Ri(x)+fi(B(x))fi(B(x)Ri(x))12Ri(x)2(Ni(u,v),Ri(x)+fi(B(x))fi(B(x)Ri(x))12Ri(x)2)},

    where

    Ri(x)=B(x)fiΩ(x)[B(x)Ni(u,v)]

    and

    Ri(x)=B(x)fiΩ(x)[B(x)Ni(u,v)],xRn,uF(x),vP(x).

    Theorem 5.1 For any xRn,>>0, we have

    12(11)r2(x)G(x)12(11)r2(x). (5.1)

    Proof. From the definition of G(x), it follows that

    G(x)=min1im{Ni(u,v),Ri(x)Ri(x)fi(B(x)Ri(x))12Ri(x)2+fi(B(x)Ri(x))+12Ri(x)2},uF(x),vP(x).

    For any given i{1,2,,m}, we set

    gi(x)=Ni(u,v),Ri(x)Ri(x)fi(B(x)Ri(x))12Ri(x)2+fi(B(x)Ri(x))+12Ri(x)2,uF(x),vP(x). (5.2)

    From fiΩ(x)[B(x)Ni(u,v)]Ω(x), by Lemma 2.4, we know

    fiΩ(x)[B(x)Ni(u,v)](B(x)Ni(u,v)),fiΩ(x)[B(x)Ni(u,v)]fiΩ(x)[B(x)Ni(u,v)]
    +fi(fiΩ(x)[B(x)Ni(u,v)])fi(fiΩ(x)[B(x)Ni(u,v)])0,uF(x),vP(x)

    which means that

    Ni(u,v)Ri(x),Ri(x)Ri(x)+fi(B(x)Ri(x))fi(B(x)Ri(x))0. (5.3)

    Combining (5.2) and (5.3), we get

    gi(x)1Ri(x),Ri(x)Ri(x)12Ri(x)2+12Ri(x)2=12Ri(x)Ri(x)2+12(11)Ri(x)2. (5.4)

    Since

    fiΩ(x)[B(x)Ni(u,v)]Ω(x),

    from Lemma 2.4, we have

    fiΩ(x)[B(x)Ni(u,v)](B(x)Ni(u,v)),fiΩ(x)[B(x)Ni(u,v)]fiΩ(x)[B(x)Ni(u,v)]
    +fi(fiΩ(x)[B(x)Ni(u,v)])fi(fiΩ(x)[B(x)Ni(u,v)])0,uF(x),vP(x).

    Hence

    Ni(u,v)Ri(x),Ri(x)Ri(x)+fi(B(x)Ri(x))
    fi(B(x)Ri(x))0,uF(x),vP(x)

    and so

    1Ri(x),Ri(x)Ri(x)Ni(u,v),Ri(x)Ri(x)fi(B(x)Ri(x))+fi(B(x)Ri(x)).

    It will require and (5.3),

    gi(x)1Ri(x),Ri(x)Ri(x)12Ri(x)2+12Ri(x)2=12Ri(x)Ri(x)2+12(11)Ri(x)2. (5.5)

    From (5.4) and (5.5), for any i{1,2,,m}, we get

    12(11)Ri(x)2gi(x)12(11)Ri(x)2.

    Hence

    12(11)min1im{Ri(x)2}min1im{gi(x)}12(11)min1im{Ri(x)2},

    and so

    12(11)r2(x)G(x)12(11)r2(x).

    The proof is completed.

    Now we are in position to prove that G in the set Rn is a global gap function for (2.1).

    Theorem 5.2 For 0<<, G is a gap function for (2.1) on Rn.

    Proof. From (5.2), we have

    G(x)0,xRn.

    Suppose that ˉxRn with

    G(ˉx)=0,

    then (5.2) implies that

    r(ˉx)=0.

    From Theorem 3.1, we know ˉx is a solution of (2.1).

    Conversely, if ˉx is a solution of (2.1), than from Theorem 3.1, it follows that

    r(ˉx)=0.

    Obviously, (5.2) shows that

    G(ˉx)=0.

    The proof is completed.

    Use Theorem 3.2 and (5.2), we immediately get a global error bound in the set Rn for (2.1).

    Corollary 5.3 Let F,P:RnRn be D-ϑF-Lipschitz continuous and D-ϱP-Lipschitz continuous mappings, respectively. Let Ni:Rn×RnRn(i=1,2,,m) be σi-Lipschitz continuous with respect to the first argument and i-Lipschitz continuous with respect to the second argument, and B:RnRn be -Lipschitz continuous. Let (Ni,B) be the strongly monotone with respect to the first argument of Ni and B with constant μBi and relaxed monotone with respect to the second argument of N and B with modulus ζBi. Let

    mi=1(i).

    Assume that there exists κi(0,μBiζBiϑFσi+iϱP) such that

    fiΩ(x)zfiΩ(y)zκixy,x,yRn,uF(x),vP(x),z{ww=B(x)Ni(u,v)}.

    Then, for any xRn and

    >κiμBiζBiκi(ϑFσi+iϱP),
    d(x,i)(ϑFσi+ϱPi)+(μBiζBiκi(ϑFσi+ϱPi))κi2G(x).

    One of the traditional approaches to evaluating a variational inequality (VI) and its variants is to turn into an analogous optimization problem by notion of a gap function. In addition, gap functions play a pivotal role in deriving the so-called error bounds that provide a measure of the distances between the solution set and feasible arbitrary point. Motivated and inspired by the researches going on in this direction, the main purpose of this paper is to further study the generalized vector inverse quasi-variational inequality problem (1.2) and to obtain error bounds in terms of the residual gap function, the regularized gap function, and the global gap function by utilizing the relaxed monotonicity and Hausdorff Lipschitz continuity. These error bounds provide effective estimated distances between an arbitrary feasible point and the solution set of (1.2).

    The authors are very grateful to the referees for their careful reading, comments and suggestions, which improved the presentation of this article.

    This work was supported by the Scientific Research Fund of Science and Technology Department of Sichuan Provincial (2018JY0340, 2018JY0334) and the Scientific Research Fund of SiChuan Provincial Education Department (16ZA0331).

    The authors declare that they have no competing interests.



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