Processing math: 73%
Research article

Adaptive neural networks event-triggered fault-tolerant consensus control for a class of nonlinear multi-agent systems

  • Received: 24 December 2019 Accepted: 05 March 2020 Published: 17 March 2020
  • MSC : 93B52, 93C95, 93D05

  • This paper studies the event-triggered fault-tolerant control problem of nonlinear multi-agent systems. The goal is to ensure the stability of event-based sampling multi-agent systems when the actuator faults occurs. The neural networks approximate property is used to approximate unknown ideal control parameters, which can reduce the exact requirements of control parameters. Based on the states information of neighboring agents, a distributed fault-tolerant consensus controller is designed for the leaderless multi-agent systems. Moreover, an event-triggered mechanism with special definition of event-triggered error is applied to reduce the amount of communications. In addition, the Zeno behaviour is avoided. By using Lyapunov stability theory, it is proved that all signals are bounded in the closed-loop systems. Finally, a numerical simulation result is presented to prove the effectiveness of the proposed method.

    Citation: Zuo Wang, Hong Xue, Yingnan Pan, Hongjing Liang. Adaptive neural networks event-triggered fault-tolerant consensus control for a class of nonlinear multi-agent systems[J]. AIMS Mathematics, 2020, 5(3): 2780-2800. doi: 10.3934/math.2020179

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  • This paper studies the event-triggered fault-tolerant control problem of nonlinear multi-agent systems. The goal is to ensure the stability of event-based sampling multi-agent systems when the actuator faults occurs. The neural networks approximate property is used to approximate unknown ideal control parameters, which can reduce the exact requirements of control parameters. Based on the states information of neighboring agents, a distributed fault-tolerant consensus controller is designed for the leaderless multi-agent systems. Moreover, an event-triggered mechanism with special definition of event-triggered error is applied to reduce the amount of communications. In addition, the Zeno behaviour is avoided. By using Lyapunov stability theory, it is proved that all signals are bounded in the closed-loop systems. Finally, a numerical simulation result is presented to prove the effectiveness of the proposed method.


    It is known that variational inequality, as a very important tool, has already been studied for a wide class of unilateral, obstacle, and equilibrium problems arising in several branches of pure and applied sciences in a unified and general framework. Many numerical methods have been developed for solving variational inequalities and some related optimization problems; see [1,2,3,4,5,6] and the references therein.

    Let H be a real Hilbert space whose inner product and norm are denoted by , and , respectively. Let C be a nonempty, closed and convex subset of H and A:CH be a nonlinear mapping. The variational inequality problem (VIP) associated with the set C and the mapping A is stated as follows:

    findxCsuch thatAx,xx0,xC. (1.1)

    In particular, the VIP(1.1) in the case C is the set Fix(T) of fixed points of a nonexpansive self-mapping T of C and A is of the form A=IS, with S another nonexpansive self-mapping of C. In other words, VIP is of the form

    findxFix(T)such thatxSx,xx0,xFix(T). (1.2)

    This problem, introduced by Mainge and Moudafi [8], is called hierarchical fixed point problem (HFPP).

    Subsequently, Moudafi and Mainge [7] studied the explicit scheme for computing a solution of VIP(1.2) by introducing the following iterative algorithm:

    xn+1=λnf(xn)+(1λn)(αnSxn+(1αn)Txn), (1.3)

    where f:CC and {αn},{λn}(0,1). They also proved the strong convergence of the sequence {xn} generalized by (1.3) to a solution of VIP(1.2).

    Yao et al. [9] introduced and analyzed the following two-step iterative algorithm that generates a sequence {xn} by the following explicit scheme:

    {yn=βnSxn+(1βn)xn,xn+1=αnf(xn)+(1αn)Tyn,n1. (1.4)

    Under appropriate conditions, the above iterative sequence {xn} converges strongly to some fixed point of T where T is nonexpansive mapping and {xn} is solves VIP(1.2).

    Marino et al. [10] introduced a multistep iterative method that generalizes the two-step method studied in [9] from two nonexpansive mappings to a finite family of nonexpansive mappings that generates a sequence {xn} by the following iterative scheme:

    {F(un,y)+h(un,y)+1rnyun,unxn,yC,yn,1=βn,1S1un+(1βn,1)un,yn,i=βn,iSiun+(1βn,i)yn,i1,i=2,,N,xn+1=αnf(xn)+(1αn)Tyn,N],n1. (1.5)

    They prove that strong convergence of the method to a common fixed point of a finite number of nonexpansive mappings that also solves a suitable equilibrium problem.

    On the other hand, by combining the regularization method, the hybrid steepest descent method, and the projection method, Ceng et al. [11] proposed an iterative algorithm that generates a sequence via the explicit scheme and proved that this sequence converges strongly to a unique solution of the following problem.

    Problem 1.1 Let F:CH be k-Lipschitzian and η-strongly monotone on the nonempty, closed and convex subset C of H, where k and η are positive constants, that is,

    FxFykxyandFxFy,xyηxy2,x,yC. (1.6)

    Let f:CH be a ρ-contraction with a coefficient ρ[0,1) and S,T:CC be two nonexpansive mappings with Fix(T). Let 0<μ<2ηk2 and 0<γτ, where τ=11μ(2ημk2). Consider the following triple hierarchical variational inequality problem (THVI): find xΞ such that

    (μFγf)x,xx0,xΞ, (1.7)

    where Ξ denotes the solution set of the following hierarchical variational inequality problem (HVIP): find zFix(T) such that

    (μFγS)z,zz0,zFix(T), (1.8)

    where the solution set Ξ is assumed to be nonempty.

    Since Problem 1.1 has a triple hierarchical structure, in contrast to bilevel programming problems [12,13], that is, a variational inequality problem with a variational inequality constraint over the fixed point set Fix(T), we also call (1.8) a triple hierarchical variational inequality problem (THVIP), which is a generalization of the triple hierarchical constrained optimization problem (THCOP) considered by [14,15].

    Recently, many authors introduced the split monotone variational inequality inclusion problem, which is the core of the modeling of many inverse problems arising in phase retrieval and other real-world problems. It has been widely studied in sensor networks, intensity-modulated radiation therapy treatment planning, data compression, and computerized tomography in recent years; see, e.g., [18,19,21,26,27] and the references therein.

    The split monotone variational inclusion problem (SMVIP) was first introduced by Moudafi [20] as follows: find xH1 such that

    {0f1x+B1x,y=AxH2:0f2y+B2y, (1.9)

    where f1:H1H1 and f2:H2H2 are two given single-valued mappings, A:H1H2 is a bounded linear operator, and B1:H12H1 and B2:H22H2 are multivalued maximal monotone mappings.

    If f1=f20, then (1.9) reduces to the following split variational inclusion problem (SVIP): find xH1 such that

    {0B1x,y=AxH2:0B2y. (1.10)

    Additionally, if f10, then (1.9) reduces to the following split monotone variational inclusion problem (SMVIP): find xH1 such that

    {0B1x,y=AxH2:0fy+B2y. (1.11)

    We denote the solution sets of variational inclusion 0B1x and 0fy+B2y by SOLVIP(B1) and SOLVIP(f+B2), respectively. Thus, the solution set of (1.11) can denoted by Γ={xH1:xSOLVIP(B1),AxSOLVIP(f+B2)}.

    In 2012, Byrne et al. [21] studied the following iterative scheme for SVIP(1.10): for a given x0H1 and λ>0,

    xn+1=JB1λ[xn+ϵA(JB2λI)Axn]. (1.12)

    In 2014, Kazmi and Rizvi [22] introduced a new iterative scheme for SVIP(1.10) and the fixed point problem of a nonexpansive mapping:

    {un=JB1λ[xn+ϵA(JB2λI)Axn],xn+1=αnf(xn)+(1αn)Tun, (1.13)

    where A is a bounded linear operator, A is the adjoint of A, f is a contraction on H1, and T is a nonexpansive mapping of H1. They obtained a strong convergence theorem under some mild restrictions on the parameters.

    Jitsupa et al. [1] modified algorithm (1.13) for SVIP(1.10) and the fixed point problem of a family of strict pseudo-contractions:

    {un=JB1λ[xn+γA(JB2λI)Axn],yn=βnun+(1βn)n=1η(n)iTiun,xn+1=αnτf(xn)+(IαnD)yn,n1, (1.14)

    where A is a bounded linear operator, A is the adjoint of A, {Ti}Ni=1 is a family of ki-strictly pseudo-contractions, f is a contraction, and D is a strong positive linear bounded operator. In [1], they prove under certain appropriate assumptions on the sequences {αn},{βn} and {η(n)i}Ni=1 that {xn}, defined by (1.14), converges strongly to a common solution of SVIP(1.10) and a fixed point of a finite family of ki-strictly pseudo-contractions, which solve a variational inequality problem (1.1).

    In this paper, we consider the following system of variational inequalities defined over a set consisting of the set of solutions of split monotone variational inclusion, the set of common fixed points of nonexpansive mappings, and the set of fixed points of a mapping.

    Problem 1.2 Let F:CH be k-Lipschitzian and η-strongly monotone on the nonempty closed and convex subset C of H, ψ:CH be a ρ-contraction with coefficient ρ[0,1) and Si,S,T:CC be nonexpansive mappings for all i{1,,N}. Let 0<μ<2ηk2 and 0<ξτ, where τ=11μ(2ημk2). Then, the objective is to find xΩ such that

    {(μFξψ)x,xx0,xΩ,(μFξS)x,yx0,yΩ, (1.15)

    where Ω=Fix(T)(iFix(Si))Γ.

    Motivated and inspired by the Moudafi and Mainge [7], Marino et al. [10], Ceng et al. [11] and Kazmi and Rizvi [22], in this paper, we consider a multistep which difference from (1.5). It is proven that under appropriate assumptions the proposed iterative method, the sequence {xn} converges strongly to a unique solution to Problem 1.2 and which is solve THVI(1.7). Finally, we give some example and numerical results to illustrate our main results.

    In this section, we collect some notations and lemmas. Let C be a nonempty closed convex subset of a real Hilbert space H. We denote the strong convergence and the weak convergence of the sequence {xn} to a point xH by xnx and xnx, respectively. It is also well known [24] that the Hilbert space H satisfies Opailscondition, that is, for any sequence {xn} with xnx, the inequality

    lim supnxnx<lim supnxny (2.1)

    holds for every yH with yx.

    In the sequel, given a sequence {zn}, we denote with ωw(zn) the set of cluster points of {zn} with respect to the weak topology, that is,

    ωw(zn)={zH:there existsnkfor whichznkz}.

    Analogously, we denote by ωs(zn) the set of cluster points of {zn} with respect to the norm topology, that is,

    ωs(zn)={zH:there existsnkfor whichznkz}.

    Lemma 2.1. In a real Hilbert space H, the following inequalities hold:

    (1) xy2=x2y22xy,y,x,yH;

    (2) x+y2x2+2y,x+y,x,yH;

    (3) λx+(1λ)y2=λx2+(1λ)y2λ(1λ)xy2,λ[0,1],x,yH;

    An element xC is called a fixedpoint of S if xSx. The set of all fixed point of S is denoted by Fix(S), that is, Fix(S)={xC:xSx}.

    Recall the following definitions. Moreover, S:H1H1 is called

    (1) a nonexpansive mapping if

    SxSyxy,x,yH1. (2.2)

    A nonexpansive mapping with k=1 can be strengthened to a firmly nonexpansive mapping in H1 if the following holds:

    SxSy2xy,SxSy,x,yH1. (2.3)

    We note that every nonexpansive operator S:H1H1 satisfies, for all (x,y)H1×H1, the inequality

    (xSx)(ySy),SySx)12(Sxx)(Syy)2, (2.4)

    and therefore, we obtain, for all (x,y)H1×Fix(S),

    xSx,ySx12Sxx2 (2.5)

    (see, e.g., Theorem 3 in [16] and Theorem 1 in [17]).

    (2) a contractive if there exists a constant α(0,1) such that

    SxSyαxy,x,yH1. (2.6)

    (3) an L-Lipschitzian if there exists a positive constant L such that

    SxSyLxy,x,yH1. (2.7)

    (4) an η-strongly monotone if there exists a positive constant η such that

    SxSy,xyηxy2,x,yH1. (2.8)

    (5) an β-inverse strongly monotone (βism) if there exists a positive constant β such that

    SxSy,xyβSxSy2,x,yH1. (2.9)

    (6) averaged if it can be expressed as the average of the identity mapping and a nonexpansive mapping, i.e.,

    S:=(1α)I+αT, (2.10)

    where α(0,1),I is the identity operator on H1 and T:H1H1 is nonexpansive.

    It is easily seen that averaged mappings are nonexpansive. In the meantime, firmly nonexpansive mappings are averaged.

    (7) A linear operator D is said to be a strongly positive bounded linear operator on H1 if there exists a positive constant ˉτ>0 such that

    Dx,xˉτx2,xH1. (2.11)

    From the definition above, we easily find that a strongly positive bounded linear operator D is ˉτ-strongly monotone and D-Lipschitzian.

    (8) A multivalued mapping M:D(M)H12H1 is called monotone if for all x,yD(M),uMx and vMy,

    xy,uv0. (2.12)

    A monotone mapping M is maximal if the Graph(M) is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mapping M is maximal if and only if for xD(M),uH1,xy,uv0 for each (y,v)Graph(M), uMx.

    (9) Let M:D(M)H12H1 be a multivalued maximal monotone mapping. Then, the resolvent operator JMλ:H1D(M) is defined by

    JMλx:=(I+λM)1(x),xH1, (2.13)

    for λ>0, where I stands for the identity operator on H1. We observe that JMλ is single-valued, nonexpansive, and firmly nonexpansive.

    We recall some concepts and results that are needed in the sequel. A mapping PC is said to be a metric projection of H1 onto C if for every point xH1, there exists a unique nearest point in C denoted by PCx such that

    xPCxxy,yC. (2.14)

    It is well known that PC is a nonexpansive mapping and is characterized by the following property:

    PCxPCy2xy,PCxPCy,x,yH1. (2.15)

    Moreover, PCx is characterized by the following properties:

    xPCx,yPCx0,xH1,yC, (2.16)
    xy2xPCx2+yPCx2,xH1,yC, (2.17)

    and

    (xy)(PCxPCy)2xy2PCxPCy2,x,yH1. (2.18)

    Proposition 2.2. [20]

    (1) If T=(1α)S+αV, where S:H1H1 is averaged, V:H1H1 is nonexpansive, and if α[0,1], then T is averaged.

    (2) The composite of finitely many averaged mappings is averaged.

    (3) If the mappings {Ti}Ni=1 are averaged and have a nonempty common fixed point, then

    Ni=1F(Ti)=F(T1T2TN). (2.19)

    (4) If T is a vism, then for γ>0,γT is a vγism.

    (5) T is averaged if and only if its complement IT is a vism for some v>12.

    Proposition 2.3. [20] Let λ>0,h be an αism operator, and B be a maximal monotone operator. If λ(0,2α), then it is easy to see that the operator JBλ(Iλh) is averaged.

    Proposition 2.4. [20] Let λ>0 and B1 be a maximal monotone operator. Then,

    xsolves(1.9)x=JB1λ(Iλf1)xandAx=JB2λ(Iλf2)Ax. (2.20)

    Lemma 2.5. [23] Let {sn} be a sequence of nonnegative numbers satisfying the condition

    sn+1(1γn)sn+γnδn,n1,

    where {γn},{δn} are the sequences of real numbers such that

    (i) {γn}[0,1] and n=1γn=, or equivalently,

    Πn=1(1γn):=limnΠk=1(1γk)=0;

    (ii) lim supnδn0, or

    (iii) n=1γnδn is convergent.

    Then, limnsn=0.

    Lemma 2.6. [23] Let λ be a number (0,1], and let μ>0. Let F:CH be an operator on C such that for some constant k,η>0,F is k-Lipschitzian and η-strongly monotone. Associating with a nonexpansive mapping T:CC, we define the following the mapping Tλ:CH by

    Tλx:=TxλμF(Tx),xC. (2.21)

    Then, Tλ is a contraction provided μ<2ηk2, that is,

    TλxTλy(1λτ)xy,x,yC, (2.22)

    where τ=11μ(2ημk2)(0,1].

    Lemma 2.7. [25] Let {αn} be a sequence of nonnegative real numbers with lim supnαn< and {βn} be a sequence of real numbers with lim supnβn0. Then, lim supnαnβn0.

    Lemma 2.8. [28] Assume that T is nonexpansive self-mapping of a closed convex subset C of a Hilbert space H1. If T has a fixed point, then IT is demiclosed, i.e., whenever {xn} weakly converges to some x and {(IT)xn} converges strongly to y, it follows that (IT)x=y. Here, I is the identity mapping on H1.

    Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H1 and Q be a nonempty closed convex subset of a real Hilbert space H2. Let A:H1H2 be a bounded linear operator, A be the adjoint of A, and r be the spectral radius of the operator AA. Let f:H2H2 be a ς-inverse strongly monotone operator, B1:C2H1,B2:H22H2 be two multivalued maximal monotone operators, and F:CH1 be k-Lipschitzian and η-strongly monotone. Let ψ:CH1 be a ρ-contraction with a coefficient ρ[0,1) and Si,S,T:CC be nonexpansive mappings for all i{1,,N}. Let {λn},{αn},{βn,i},i=1,,N be sequences in (0,1) such that βn,iβi(0,1) as n for all i{1,,N}, 0<μ<2ηk2 and 0<ξτ, where τ=11μ(2ημk2). Then, the sequence {xn} is generated from an arbitrary initial point x1C by the following:

    {un=JB1λ1[xn+γA(JB2λ2(Iλ2f)I)Axn],yn,1=βn,1S1un+(1βn,1)un,yn,i=βn,iSiun+(1βn,i)yn,i1,i=2,,N,xn+1=PC[λnξ(αnψ(xn)+(1αn)Sxn)+(IλnμF)Tyn,N],n1. (3.1)

    Assume that Problem 1.2 has a solution. Suppose that the following conditions are satisfied:

    (C1) 0<lim infnαnlim supnαn<1;

    (C2) limnλn=0 and n=1λn=;

    (C3) n=2|αnλnαn1λn1|< or limn|αnλnαn1λn1|λn=0;

    (C4) n=2|λnλn1|< or limn|λnλn1|λn=0;

    (C5) n=2|βn,iβn1,i|< or limn|βn,iβn1,i|λn=0 for all i{1,,N};

    (C6) λ1>0,0<λ2<2ς,0<γ<1r.

    Then, {xn} converges strongly to a unique solution xΩ of Problem 1.2.

    Proof. Let {xn} be a sequence generated by scheme (4.1). First, note that 0<ξτ and

    μητμη11μ(2ημk2)1μ(2ημk2)1μη12μη+μ2k212μη+μ2η2k2η2kη.

    Then, it follows from the ρ-contractiveness of ψ that

    (μFξψ)x(μFξψ)y,xy(μηξρ)xy2,x,yC.

    Hence, from ξρ<ξτμη, we deduce that μFξψ is (μηξρ)-strongly monotone. Since it is clear that μFξψ is Lipshitz continuous, there exists a unique solution to the VIP:

    findxΩsuch that(μFξψ)x,xx0,xΩ.

    Additionally, since Problem 1.2 has a solution, it is easy to see that Problem 1.2 has a unique solution. In addition, taking into account condition (C1), without loss of generality, we may assume that {αn}[a,b] for some a,b(0,1).

    Let U:=JB2λ2(Iλ2f); the iterative scheme (4.1) can be rewritten as

    {un=JB1λ1[xn+γA(UI)Axn],yn,1=βn,1S1un+(1βn,1)un,yn,i=βn,iSiun+(1βn,i)yn,i1,i=2,,N,xn+1=PC[λnξ(αnψ(xn)+(1αn)Sxn)+(IλnμF)Tyn,N],n1. (3.2)

    The rest of the proof is divided into several steps.

    Step 1. We show that the sequences {xn},{yn,i} for all i,{un} are bounded.

    Indeed, take a point pΩ arbitrarily. Then, JB1λ1p=p,U(Ap)=Ap, and it is easily seen that Wp=p, where W:=I+γA(UI)A. From the definition of firm nonexpansion and Proposition 2.3, we have that JB1λ1 and U are averaged. Likewise, W is also averaged because it is a vrism for some v>12. Actually, by Proposition 2.2 (5), we know that IU is a vism with v>12. Hence, we have

    A(IU)AxA(IU)Ay,xy=(IU)Ax(IU)Ay,AxAyv(IU)Ax(IU)Ay2vrA(IU)AxA(IU)Ay2.

    Thus, γA(IU)A is a vγrism. Due to the condition 0<γ<1r, the complement IγA(IU)A is averaged, as well as M:=JB1λ1[I+γA(UI)A]. Therefore, JB1λ1,U,W, and M are nonexpansive mappings.

    From (3.2), we estimate

    unp2=JB1λ1[xn+γA(UI)Axn]JB1λ1p2xn+γA(UI)Axnp2=xnp2+γ2A(UI)Axn2+2γxnp,A(UI)Axn. (3.3)

    Thus, we obtain

    unp2xnp2+γ2(UI)Axn,AA(UI)Axn+2γxnp,A(UI)Axn. (3.4)

    Next, setting ϑ1:=γ2(UI)Axn,AA(UI)Axn, we estimate

    ϑ1=γ2(UI)Axn,AA(UI)Axnrγ2(UI)Axn,(UI)Axn=rγ2(UI)Axn2. (3.5)

    Setting ϑ2:=2γxnp,A(UI)Axn, we obtain from (2.5) the following:

    ϑ2=2γxnp,A(UI)Axn=2γA(xnp),(UI)Axn=2γA(xnp)+(UI)Axn(UI)Axn,(UI)Axn=2γ(UAxnAp,(UI)Axn(UI)Axn2)2γ(12(UI)Axn2(UI)Axn2)γ(UI)Axn2. (3.6)

    In view of (3.4)-(3.6), we have

    unp2xnp2+γ(rγ1)(UI)Axn2. (3.7)

    From 0<γ<1r, we obtain

    unpxnp. (3.8)

    Thus, we have from (3.2) and (3.8) that

    yn,1pβn,1S1unp+(1βn,1)unpunpxnp. (3.9)

    For all i from i=2 to i=N, by induction, one proves that

    yn,ipβn,iunp+(1βn,i)yn,i1punpxnp. (3.10)

    Hence, we obtain that for all i{1,,N},

    yn,ipunpxnp. (3.11)

    In addition, utilizing Lemma 2.6 and (3.2), we have

    xn+1p=PC[λnξ(αnψ(xn)+(1αn)Sxn)+(IλnμF)Tyn,N]PCpλnξ(αnψ(xn)+(1αn)Sxn)+(IλnμF)Tyn,Np=λnξ(αnψ(xn)+(1αn)Sxn)λnμFTp+(IλnμF)Tyn,N(IλnμF)Tpλnξ(αnψ(xn)+(1αn)Sxn)λnμFTp+(IλnμF)Tyn,N(IλnμF)Tp=λnαn(ξψ(xn)μFp)+(1αn)(ξSxnμFp)+(IλnμF)Tyn,N(IλnμF)Tpλn[αnξψ(xn)μFp+(1αn)ξSxnμFp]+(1λnτ)yn,Npλn[αn(ξψ(xn)ξψ(p)+ξψ(p)μFp)+(1αn)(ξSxnξSp+ξSpμFp)]+(1λnτ)yn,Npλn[αnξρxnp+αnξψ(p)μFp+(1αn)ξxnp+(1αn)ξSpμFp]+(1λnτ)xnpλn[ξ(1αn(1ρ))xnp+max{ξψ(p)μFp,ξSpμFp}]+(1λnτ)xnp(1λnξαn(1ρ))xnp+λnmax{ξψ(p)μFp,ξSpμFp}(1λnξa(1ρ))xnp+λnmax{ξψ(p)μFp,ξSpμFp}, (3.12)

    due to 0<ξτ. Thus, calling

    M=max{x1p,ξψ(p)μFpξa(1ρ),ξSpμFpξa(1ρ)},

    by induction, we derive xnpM for all n1. We thus obtain the claim.

    Step 2. We show that limnxn+1xn=0.

    Indeed, for each n1, we set

    zn=λnξ(αnψ(xn)+(1αn)Sxn)+(IλnμF)Tyn,N.

    Then, we observe that

    znzn1=αnλnξ[ψ(xn)ψ(xn1)]+λn(1αn)ξ(SxnSxn1)+[(IλnμF)Tyn,N(IλnμF)Tyn1,N]+(αnλnαn1λn1)ξ[ψ(xn1)Sxn1]+(λnλn1)(ξSxn1μFTyn1,N). (3.13)

    Let M0>0 be a constant such that

    supn1{ξψ(xn)Sxn+ξSxnμFTyn,N}M0.

    It follows from (3.2) and (3.13) that

    xn+1xn=PCznPCzn1znzn1αnλnξψ(xn)ψ(xn1)+λn(1αn)ξSxnSxn1+(IλnμF)Tyn,N(IλnμF)Tyn1,N+|αnλnαn1λn1|ξψ(xn1)Sxn1+|λnλn1|ξSxn1μFTyn1,Nαnλnξρxnxn1+λn(1αn)ξxnxn1+(1λnτ)yn,Nyn1,N+|αnλnαn1λn1|M0+|λnλn1|M0=λn(1αn(1ρ))ξxnxn1+(1λnτ)yn,Nyn1,N+[|αnλnαn1λn1|+|λnλn1|]M0λnξ(1a(1ρ))xnxn1+(1λnτ)yn,Nyn1,N+[|αnλnαn1λn1|+|λnλn1|]M0. (3.14)

    By the definition of yn,i, we obtain that for all i=N,,2,

    yn,iyn1,iβn,iunun1+Siun1yn1,i1|βn,iβn1,i|+(1βn,i)yn,i1yn1,i1. (3.15)

    In this case i=1, we have

    yn,1yn1,1βn,1unun1+S1un1un1|βn,1βn1,1|+(1βn,1)unun1=unun1+S1un1un1|βn,1βn1,1|. (3.16)

    Substituting (3.16) in all (3.15)-type inequalities, we find that for i=2,,N,

    yn,iyn1,i unun1+Nk=2Skun1yn1,k1|βn,kβn1,k|+S1un1un1|βn,1βn1,1|.

    Thus, we conclude that

    xn+1xnλnξ(1a(1ρ))xnxn1+(1λnτ)yn,Nyn1,N+[|αnλnαn1λn1|+|λnλn1||]M0λnξ(1a(1ρ))xnxn1+[|αnλnαn1λn1|+|λnλn1|]M0+(1λnτ)unun1+Nk=2Skun1yn1,k1|βn,kβn1,k|+S1un1un1|βn,1βn1,1|. (3.17)

    Since JB1λ1[I+γA(UI)A] is nonexpansive, we obtain

    unun1=JB1λ1[I+γA(UI)A]xnJB1λ1[I+γA(UI)A]xn1xnxn1. (3.18)

    Substituting (3.18) into (3.17), we have

    xn+1xnλnξ(1a(1ρ))xnxn1+[|αnλnαn1λn1|+|λnλn1|]M0+(1λnτ)xnxn1+Nk=2Skun1yn1,k1|βn,kβn1,k|+S1un1un1|βn,1βn1,1|. (3.19)

    If we call M1:=max{M0,supn2,i=2,,NSiun1yn1,i1,supn2S1un1un1}, we have

    xn+1xn(1λnξa(1ρ))xnxn1+M1[|αnλnαn1λn1|+|λnλn1|+Nk=2|βn,kβn1,k|], (3.20)

    due to 0<ξ<τ. By condition (C2)(C5) and Lemma 2.5, we obtain that

    limnxn+1xn=0. (3.21)

    Step 3. We show that limnxnun=0.

    From (3.2) and (3.7), we have

    xn+1p2λnξ(αnψ(xn)+(1αn)Sxn)+(IλnμF)Tyn,Np2=λnξ(αnψ(xn)+(1αn)Sxn)λnμFTp+(IλnμF)Tyn,N(IλnμF)Tp2{λnξ(αnψ(xn)+(1αn)Sxn)λnμFTp+(IλnμF)Tyn,N(IλnμF)Tp}2{λnαn(ξψ(xn)μFp)+(1αn)(ξSxnμFp)+(1λnτ)yn,Np}2λn1τ[αnξψ(xn)μFp+(1αn)ξSxnμFp]2+(1λnτ)yn,Np2λn1τ[ξψ(xn)μFp+ξSxnμFp]2+(1λnτ)unp2λn1τ[ξψ(xn)μFp+ξSxnμFp]2+(1λnτ)[xnp2+γ(rγ1)(UI)Axn2]=λn1τ[ξψ(xn)μFp+ξSxnμFp]2+(1λnτ)xnp2γ(1rγ)(1λnτ)(UI)Axn2, (3.22)

    which implies that

    (1λnτ)γ(1rγ)(UI)Axn2λn1τ[ξψ(xn)μFp+ξSxnμFp]2+(1λnτ)xnp2xn+1p2λn1τ[ξψ(xn)μFp+ξSxnμFp]2+xnp2xn+1p2λn1τ[ξψ(xn)μFp+ξSxnμFp]2+xn+1xn(xnp+xn+1p). (3.23)

    Since γ(1rγ)>0,xn+1xn0,λn0 as n, and by the boundedness of {xn}, we conclude that

    limn(UI)Axn=0. (3.24)

    In addition, by the firm nonexpansion of JB1λ1,(3.3),(3.7), and γ(0,1r), we estimate

    unp2=JB1λ1[xn+γA(UI)Axn]JB1λ1p2JB1λ1[xn+γA(UI)Axn]JB1λ1p,xn+γA(UI)Axnp=unp,xn+γA(UI)Axnp=12(unp2+xn+γA(UI)Axnp2(unp)[xn+γA(UI)Axnp]2)12[unp2+xnp2+γ(rγ1)(UI)Axn2unxnγA(UI)Axn2]12[unp2+xnp2unxnγA(UI)Axn2]=12[unp2+xnp2unxn2γ2A(UI)Axn2+2γunxn,A(UI)Axn]12[unp2+xnp2unxn2+2γunxn,A(UI)Axn]=12[unp2+xnp2unxn2+2γA(unxn),(UI)Axn]12[unp2+xnp2unxn2+2γA(unxn)(UI)Axn],

    and hence,

    unp2xnp2unxn2+2γA(unxn)(UI)Axn. (3.25)

    In view of (3.22) and (3.25),

    xn+1p2λn1τ[ξψ(xn)μFp+ξSxnμFp]2+(1λnτ)unp2λn1τ[ξψ(xn)μFp+ξSxnμFp]2+(1λnτ)[xnp2unxn2+2γA(unxn)(UI)Axn]=λn1τ[ξψ(xn)μFp+ξSxnμFp]2+(1λnτ)xnp2(1λnτ)unxn2+2γ(1λnτ)A(unxn)(UI)Axn, (3.26)

    which implies that

    (1λnτ)unxn2λn1τ[ξψ(xn)μFp+ξSxnμFp]2+(1λnτ)xnp2+2γ(1λnτ)A(unxn)(UI)Axnxn+1p2λn1τ[ξψ(xn)μFp+ξSxnμFp]2+2γ(1λnτ)A(unxn)(UI)Axn+xnp2xn+1p2λn1τ[ξψ(xn)μFp+ξSxnμFp]2+2γ(1λnτ)A(unxn)(UI)Axn+xn+1xn(xnp+xn+1p). (3.27)

    Since xn+1xn0,(UI)Axn0, and λn0 as n, and owing to the boundedness of {xn}, we conclude that

    limnxnun=0. (3.28)

    Step 4. We show that limnSiunun=0 for i{1,,N}.

    Take a point pΩ arbitrarily. When i=N, utilizing Lemma 2.8 and (3.2), we have

    xn+1p2λnξ(αnψ(xn)+(1αn)Sxn)+(IλnμF)Tyn,Np2=λnξ(αnψ(xn)+(1αn)Sxn)λnμFTp+(IλnμF)Tyn,N(IλnμF)Tp2{λnξ(αnψ(xn)+(1αn)Sxn)λnμFTp+(IλnμF)Tyn,N(IλnμF)Tp}2{λnαn(ξψ(xn)μFp)+(1αn)(ξSxnμFp)+(1λnτ)yn,Np}2λn1τ[ξψ(xn)μFp+ξSxnμFp]2+(1λnτ)yn,Np2=λn1τ[ξψ(xn)μFp+ξSxnμFp]2+(1λnτ)βn,NSNuNp2+(1λnτ)(1βn,N)yn,N1p2(1λnτ)(1βn,N)βn,NSNunyn,N12λn1τ[ξψ(xn)μFp+ξSxnμFp]2+(1λnτ)unp2(1λnτ)(1βn,N)βn,NSNunyn,N12λn1τ[ξψ(xn)μFp+ξSxnμFp]2+xnp2(1λnτ)(1βn,N)βn,NSNunyn,N12. (3.29)

    Thus, we have

    (1λnτ)(1βn,N)βn,NSNunyn,N12λn1τ[ξψ(xn)μFp+ξSxnμFp]2+xnp2xn+1p2λn1τ[ξψ(xn)μFp+ξSxnμFp]2+xn+1xn(xnp+xn+1p). (3.30)

    Since βn,NβN(0,1),xn+1xn0 and λn0 as n, by the boundedness of {xn}, we conclude that

    limnSNunyn,N1=0. (3.31)

    Take i{1,,N1} arbitrarily. Then, we obtain

    xn+1p2λn1τ[ξψ(xn)μFp+ξSxnμFp]2+(1λnτ)yn,Np2λn1τ[ξψ(xn)μFp+ξSxnμFp]2+(1λnτ)[βn,NSNunp2+(1βn,N)yn,N1p2]λn1τ[ξψ(xn)μFp+ξSxnμFp]2+(1λnτ)βn,Nxnp2+(1λnτ)(1βn,N)yn,N1p2λn1τ[ξψ(xn)μFp+ξSxnμFp]2+(1λnτ)βn,Nxnp2+(1λnτ)(1βn,N)[βn,N1SN1unp2+(1βn,N1)yn,N2p2]λn1τ[ξψ(xn)μFp+ξSxnμFp]2+(1λnτ)(βn,N+(1βn,N)βn,N1)xnp2+(1λnτ)Nk=N1(1βn,k)yn,N2p2. (3.32)

    Hence, after (Ni+1)-iterations,

    xn+1p2λn1τ[ξψ(xn)μFp+ξSxnμFp]2+(1λnτ)(βn,N+Nj=i+2(Np=j(1βn,p)βn,j1)×xnp2+(1λnτ)Nk=i+1(1βn,k)yn,ip2λn1τ[ξψ(xn)μFp+ξSxnμFp]2+(1λnτ)(βn,N+Nj=i+2(Np=j(1βn,p))βn,j1)×xnp2+(1λnτ)Nk=i+1(1βn,k)×[βn,iSiunp2+(1βn,i)yn,i1p2βn,i(1βn,i)Siunyn,i12]λn1τ[ξψ(xn)μFp+ξSxnμFp]2+(1λnτ)xnp2βn,i(1λnτ)Nk=i(1βn,k)Siunyn,i12. (3.33)

    Again, we obtain

    βn,i(1λnτ)Nk=i(1βn,k)Siunyn,i12λn1τ[ξψ(xn)μFp+ξSxnμFp]2+xnp2xn+1p2λn1τ[ξψ(xn)μFp+ξSxnμFp]2+xn+1xn(xnp+xn+1p). (3.34)

    Since for all k{1,,N},βn,kβk(0,1),xn+1xn0 and λn0 as n, by the boundedness of {xn}, we conclude that

    limnSiunyn,i1=0. (3.35)

    Obviously, for i=1, we have limnS1unun=0. To conclude, we have that

    S2ununS2unyn,1+yn,1un=S2unyn,1+βn,1S1unun, (3.36)

    which implies that limnS2unun=0. Consequently, by induction, we obtain

    limnSiunun=0for alli=2,,N. (3.37)

    It is enough to observe that

    SiununSiunyn,i1+yn,i1Si1un+Si1ununSiunyn,i1+(1βn,i1)Si1unyn,i2+Si1unun. (3.38)

    Step 5. We show that limnyn,Nxn=limnxnTxn=0 and ωw(xn)Ω.

    Indeed, since xnun0 as n, we have ωw(xn)=ωw(un) and ωs(xn)=ωs(un). Now, we observe that

    xnyn,1xnun+yn,1un=xnun+βn,1S1unun. (3.39)

    By Step 4, S1unun0 as n. Hence, we obtain

    limnxnyn,1=0. (3.40)

    This implies that \omega_w (x_n) = \omega_w (y_{n, 1}) and \omega_s (x_n) = \omega_s (y_{n, 1}) .

    Take a point q\in \omega_w (x_n) arbitrarily. Since q\in \omega_w (u_n) , by Step 4 and the demiclosedness principle, we have q\in Fix (S_i) for all i \in \{ 1, \ldots, N \} , that is, q\in \underset{i}{\cap} Fix (S_i) . Moreover, note that

    \begin{equation} \|y_{n,N} - x_n\| \leq \mathop {\mathop \sum \limits^N }\limits_{k = 2} \| y_{n,k} - y_{n,k-1}\|+\|y_{n,1}-x_n\| = \mathop {\mathop \sum \limits^N }\limits_{k = 2} \beta_{n,k}\|S_k u_n - y_{n,k-1}\|+\|y_{n,1}-x_n\|; \end{equation} (3.41)

    hence,

    \begin{eqnarray} &&\|x_n-T x_n\| \\ &\leq& \|x_n-x_{n+1}\|+\|x_{n+1}-T y_{n,N}\|+\|T y_{n,N}-T x_n\| \\ &\leq& \|x_n-x_{n+1}\|+\| \lambda_n \xi ( \alpha_n \psi (x_n) + (1-\alpha_n)S x_n) + (I-\lambda_n \mu F) T y_{n,N}-T y_{n,N} \| \\ &&+\|y_{n,N} - x_n\| \\ & = & \|x_n-x_{n+1}\|+\lambda_n \|\alpha_n ( \xi \psi (x_n) - \mu F T y_{n,N}) + (1-\alpha_n)(\xi S x_n - \mu F T y_{n,N}) \| \\ &&+\|y_{n,N} - x_n\| \\ &\leq& \|x_n-x_{n+1}\|+\lambda_n [\| \xi \psi (x_n) - \mu F T y_{n,N}\| + \| \xi S x_n - \mu F T y_{n,N} \|]+\|y_{n,N} - x_n\| \\ &\leq& \|x_n-x_{n+1}\|+\lambda_n [\| \xi \psi (x_n) - \mu F T y_{n,N}\| + \| \xi S x_n - \mu F T y_{n,N} \|] \\ &&+ \mathop {\mathop \sum \limits^N }\limits_{k = 2} \beta_{n,k}\|S_k u_n - y_{n,k-1}\|+\|y_{n,1}-x_n\|. \end{eqnarray} (3.42)

    Since \|x_n-x_{n+1}\| \rightarrow 0, \lambda_n \rightarrow 0, \|y_{n, 1}-x_n\| \rightarrow 0, \beta_{n, k} \rightarrow \beta_k and \|S_k u_n - y_{n, k-1}\| \rightarrow 0 for all k\in \{1, \ldots, N \} , we obtain

    \begin{equation} \underset{n\rightarrow \infty}{\lim} \| y_{n,N}-x_n\| = \underset{n\rightarrow \infty}{\lim} \| x_n - T x_n\| = 0. \end{equation} (3.43)

    Thus, by the demiclosedness principle, we have q\in Fix (T) .

    In addition, we rewrite u_{n_k} = J_{\lambda_1}^{B_1}[x_{n_k} + \gamma A^{*}(\mathcal{U}-I)Ax_{n_k}] as

    \begin{equation} \frac{x_{n_k}- u_{n_k} + \gamma A^{*}(\mathcal{U}-I)Ax_{n_k}}{\lambda_1} \in B_1 u_{n_k}. \end{equation} (3.44)

    Taking k \rightarrow \infty in (3.44) and using (3.24), (3.28) and the fact that the graph of a maximal monotone operator is weakly strongly closed, we have 0 \in B_{1} q, i.e., q\in SOLVIP(B_1) . Furthermore, since x_n and u_n have the same asymptotical behavior, A x_{n_k} weakly converges to Aq . It follows from (3.24), the nonexpansion of \mathcal{U} , and Lemma 2.8 that (I-\mathcal{U})A q = 0 . Thus, by Proposition 2.4, we have 0\in f(Aq) + B_2 (Aq), i.e., Aq \in SOLVIP(B_2). As a result, q\in \Gamma . This shows that q \in \Omega . Therefore, we obtain the claim.

    Step 6. We show that \{ x_n \} converges strongly to a unique solution x^* to Problem 1.2.

    Indeed, according to \|x_{n+1} - x_n\| \rightarrow 0, we can take a subsequence \{ x_{n_j}\} of \{ x_n \} satisfying

    \begin{eqnarray} \underset{n\rightarrow \infty}{\limsup} \langle (\xi \psi - \mu F)x^*, x_{n+1}-x^* \rangle & = & \underset{n\rightarrow \infty}{\limsup} \langle (\xi \psi - \mu F)x^*, x_{n}-x^* \rangle \\ & = & \underset{j\rightarrow \infty}{\lim}\langle (\xi \psi - \mu F)x^*, x_{n_j}-x^* \rangle. \end{eqnarray} (3.45)

    Without loss of generality, we may further assume that x_{n_j} \rightharpoonup \tilde{x}; then, \tilde{x} \in \Omega , as we have just proved. Since x^* is a solution to Problem 1.2, we obtain

    \begin{equation} \underset{n\rightarrow \infty}{\limsup} \langle (\xi \psi - \mu F)x^*, x_{n+1}-x^* \rangle = \langle (\xi \psi - \mu F)x^*, \tilde{x}-x^* \rangle \leq 0. \end{equation} (3.46)

    Repeating the same argument as that of (3.46), we have

    \begin{equation} \underset{n\rightarrow \infty}{\limsup} \langle (\xi S - \mu F)x^*, x_{n+1}-x^* \rangle \leq 0. \end{equation} (3.47)

    From (3.2) and (3.9), it follows (noticing that x_{n+1} = P_C z_n and 0 < \xi \leq \tau ) that

    \begin{eqnarray} && \|x_{n+1}-x^*\|^2 \\ & = & \langle z_n-x^*, x_{n+1}-x^* \rangle + \langle P_C z_n - z_n, P_C z_n - x^* \rangle \\ &\leq& \langle z_n-x^*, x_{n+1} - x^* \rangle \\ & = & \langle (I-\lambda_n \mu F) T y_{n,N} - (I-\lambda_n \mu F) x^*, x_{n+1}-x^* \rangle \\ &&+ \alpha_n \lambda_n \xi \langle \psi (x_n) - \psi (x^*), x_{n+1}-x^* \rangle + \lambda_n (1-\alpha_n) \xi \langle Sx_n-Sx^*, x_{n+} -x^*\rangle \\ &&+\alpha_n \lambda_n \langle (\xi \psi - \mu F)x^*, x_{n+1}-x^* \rangle + \lambda_n (1-\alpha_n)\langle (\xi S - \mu F)x^*, x_{n+1}-x^* \rangle \\ &\leq& [1-\lambda_n \tau+\alpha_n \lambda_n \xi \rho + \lambda_n(1-\alpha_n) \xi]\|x_n-x^*\| \|x_{n+1}-x^*\| \\ &&+\alpha_n \lambda_n \langle (\xi \psi - \mu F)x^*, x_{n+1}-x^* \rangle + \lambda_n (1-\alpha_n)\langle (\xi S - \mu F)x^*, x_{n+1}-x^* \rangle \\ &\leq& [1-\alpha_n \lambda_n \xi(1-\rho)]\|x_n-x^*\| \|x_{n+1}-x^*\| \\ &&+\alpha_n \lambda_n \langle (\xi \psi - \mu F)x^*, x_{n+1}-x^* \rangle + \lambda_n (1-\alpha_n)\langle (\xi S - \mu F)x^*, x_{n+1}-x^* \rangle \\ &\leq& [1-\alpha_n \lambda_n \xi (1-\rho)]\frac{1}{2}(\|x_n-x^*\|^2+\|x_{n+1}-x^*\|^2) \\ &&+\alpha_n \lambda_n \langle (\xi \psi - \mu F)x^*, x_{n+1}-x^* \rangle + \lambda_n (1-\alpha_n)\langle (\xi S - \mu F)x^*, x_{n+1}-x^* \rangle. \end{eqnarray} (3.48)

    It turns out that

    \begin{eqnarray} && \|x_{n+1} - x^*\|^2 \\ &\leq& \frac{1-\alpha_n \lambda_n \xi (1-\rho)}{1+\alpha_n \lambda_n \xi (1-\rho)}\|x_n-x^*\|^2 +\frac{2}{1+\alpha_n \lambda_n \xi (1-\rho)}[ \alpha_n \lambda_n \langle (\xi \psi - \mu F)x^*, x_{n+1} - x^* \rangle \\ &&+\lambda_n (1-\alpha_n) \langle (\xi S - \mu F)x^*, x_{n+1}-x^* \rangle ] \\ &\leq& [1-\alpha_n \lambda_n \xi (1-\rho)] \|x_n - x^*\|^2 +\frac{2}{1+\alpha_n \lambda_n \xi (1-\rho)}[ \alpha_n \lambda_n \langle (\xi \psi - \mu F)x^*, x_{n+1} - x^* \rangle \\ &&+\lambda_n (1-\alpha_n) \langle (\xi S - \mu F)x^*, x_{n+1}-x^* \rangle ] \\ & = & [1- \alpha_n \lambda_n \xi (1-\rho) ]\|x_n-x^*\|^2 + \alpha_n \lambda_n \xi (1-\rho) \bigg\{ \frac{2}{\xi (1-\rho)[1+\alpha_n \lambda_n \xi (1-\rho)]} \\ &&\times \langle (\xi \psi - \mu F)x^*, x_{n+1} - x^* \rangle \\ &&+\frac{2(1-\alpha_n)}{\alpha_n \xi (1-\rho)[1+\alpha_n \lambda_n \xi (1-\rho)]}\langle (\xi S - \mu F)x^*, x_{n+1}-x^* \rangle \bigg\}. \end{eqnarray} (3.49)

    Put s_{n} = \|x_n-x^*\|^2, \xi_n = \alpha_n \lambda_n \xi (1-\rho) and

    \begin{eqnarray*} \delta_n & = & \frac{2}{\xi (1-\rho)[1+\alpha_n \lambda_n \xi (1-\rho)]} \langle (\xi \psi - \mu F)x^*, x_{n+1} - x^* \rangle \nonumber \\ &&+\frac{2(1-\alpha_n)}{\alpha_n \xi (1-\rho)[1+\alpha_n \lambda_n \xi (1-\rho)]}\langle (\xi S - \mu F)x^*, x_{n+1}-x^* \rangle. \end{eqnarray*}

    Then, (3.49) can be rewritten as

    \begin{equation*} s_{n+1} \leq (1-\gamma_n) s_n +\xi_n \delta_n. \end{equation*}

    From conditions (C1) and (C2) , we conclude from 0 < 1-\rho \leq 1 that

    \begin{equation*} \{ \xi_n \} \subset [0,1] \; \text{and}\; \mathop {\mathop \sum \limits^\infty }\limits_{n = 1} \xi_n = \infty. \end{equation*}

    Note that

    \begin{equation*} \frac{2}{\xi (1-\rho) [1+\alpha_n \lambda_{n} \xi (1-\rho)]} \leq \frac{2}{\xi (1-\rho)} \end{equation*}

    and

    \begin{equation*} \frac{2(1-\alpha_n)}{\alpha_n \xi (1-\rho) [1+\alpha_n \lambda_{n} \xi (1-\rho)]} \leq \frac{2}{a \xi (1-\rho)}. \end{equation*}

    Consequently, utilizing Lemma 2.5, we find that

    \begin{eqnarray*} \underset{n \rightarrow \infty}{\limsup} \delta_n &\leq& \underset{n \rightarrow \infty}{\limsup} \frac{2}{\xi (1-\rho) [1+\alpha_n \lambda_{n} \xi (1-\rho)]} \langle (\xi \psi - \mu F)x^*, x_{n+1} - x^* \rangle \\ &&+\underset{n \rightarrow \infty}{\limsup} \frac{2(1-\alpha_n)}{\alpha_n \xi (1-\rho) [1+\alpha_n \lambda_{n} \xi (1-\rho)]} \langle (\xi \psi - \mu F)x^*, x_{n+1} - x^* \rangle \\ &\leq& 0. \end{eqnarray*}

    Thus, this, together with Lemma 2.5, leads to \underset{n \rightarrow \infty}{\lim} \|x_n-x^*\| = 0 . The proof is complete.

    In Theorem 3.1, if \lambda_1 = \lambda_2 = \lambda and f = 0 , the we obtain the following corollary immediately.

    Corollary 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H_1 and Q be a nonempty closed convex subset of a real Hilbert space H_2 . Let A:H_1\rightarrow H_2 be a bounded linear operator, A^* be the adjoint of A , and r be the spectral radius of the operator A^* A . Let B_1:C \rightarrow 2^{H_1}, B_2: H_2 \rightarrow 2^{H_2} be two multivalued maximal monotone operators, and F:C \rightarrow H_1 be k -Lipschitzian and \eta -strongly monotone. Let \psi: C \rightarrow H_1 be a \rho -contraction with a coefficient \rho \in [0, 1) and S_i, S, T:C \rightarrow C be nonexpansive mappings for all i\in \{ 1, \ldots, N \} . Let \{ \lambda_n \}, \{ \alpha_n \}, \{ \beta_{n, i} \}, i = 1, \ldots, N be sequences in (0, 1) such that \beta_{n, i} \rightarrow \beta_i \in (0, 1) as n \rightarrow \infty for all i\in \{ 1, \ldots, N \} , 0 < \mu < \frac{2 \eta}{k^2} and 0 < \xi \leq \tau , where \tau = 1- \sqrt{1-\mu(2\eta - \mu k^2)} . Then, the sequence \{ x_n \} is generated from an arbitrary initial point x_1 \in C by the following:

    \begin{equation} \left \{ \begin{array}{l} u_n = J_{\lambda}^{B_1}[x_n + \gamma A^{*}( J_{\lambda}^{B_2}-I)Ax_n],\\ y_{n,1} = \beta_{n,1} S_1 u_n + (1-\beta_{n,1})u_n, \\ y_{n,i} = \beta_{n,i} S_i u_n + (1-\beta_{n,i})y_{n,i-1}, i = 2,\ldots,N,\\ x_{n+1} = P_C [\lambda_n \xi ( \alpha_n \psi (x_n) + (1-\alpha_n)S x_n)+(I-\lambda_n \mu F)T y_{n,N} ] ,\; n\geq 1. \end{array}\right. \end{equation} (3.50)

    Suppose that the following conditions are satisfied:

    (C1) 0 < \underset{n \rightarrow \infty}{\liminf}\alpha_n \leq \underset{n \rightarrow \infty}{\limsup}\alpha_n < 1 ;

    (C2) \underset{n \rightarrow \infty}{\lim}\lambda_n = 0 and \mathop {\mathop \sum \limits^\infty }\limits_{n = 1} \lambda_n = \infty ;

    (C3) \mathop {\mathop \sum \limits^\infty }\limits_{n = 2} | \alpha_{n }\lambda_{n}-\alpha_{n-1}\lambda_{n-1}| < \infty or \underset{n \rightarrow \infty}{\lim} \frac{ |\alpha_{n }\lambda_{n}-\alpha_{n-1}\lambda_{n-1} |}{\lambda_n} = 0 ;

    (C4) \mathop {\mathop \sum \limits^\infty }\limits_{n = 2} | \lambda_{n}-\lambda_{n-1}| < \infty or \underset{n \rightarrow \infty}{\lim} \frac{ |\lambda_{n}-\lambda_{n-1} |}{\lambda_n} = 0 ;

    (C5) \mathop {\mathop \sum \limits^\infty }\limits_{n = 2} | \beta_{n, i}-\beta_{n-1, i}| < \infty or \underset{n \rightarrow \infty}{\lim} \frac{ |\beta_{n, i}-\beta_{n-1, i} |}{\lambda_n} = 0 for all i\in \{ 1, \ldots, N \} ;

    (C6) 0 < \gamma < \frac{1}{r} .

    Then, \{ x_n \} converges strongly to a unique solution x^{*} \in Fix(T)\cap(\bigcap_i Fix (S_i))\cap SVIP .

    Here as a numerical illustration, we consider a split common fixed points of a family of nonexpansive mappings, which is a particular case of problem 1.2. To that end, we have the following, which is an equivalent formulation of Theorem 3.1.

    Let C be a nonempty closed convex subset of a real Hilbert space H_1 and Q be a nonempty closed convex subset of a real Hilbert space H_2 . Let A:H_1\rightarrow H_2 be a bounded linear operator, A^* be the adjoint of A , and r be the spectral radius of the operator A^* A . Let f: H_2 \rightarrow H_2 be a \varsigma -inverse strongly monotone operator, and F:C \rightarrow H_1 be k -Lipschitzian and \eta -strongly monotone. Let \psi: C \rightarrow H_1 be a \rho -contraction with a coefficient \rho \in [0, 1) and S_i, S, T:C \rightarrow C be nonexpansive mappings for all i\in \{ 1, \ldots, N \} . Let \{ \lambda_n \}, \{ \alpha_n \}, \{ \beta_{n, i} \}, i = 1, \ldots, N be sequences in (0, 1) such that \beta_{n, i} \rightarrow \beta_i \in (0, 1) as n \rightarrow \infty for all i\in \{ 1, \ldots, N \} , 0 < \mu < \frac{2 \eta}{k^2} and 0 < \xi \leq \tau , where \tau = 1- \sqrt{1-\mu(2\eta - \mu k^2)} . Then, the sequence \{ x_n \} is generated from an arbitrary initial point x_1 \in C by the following:

    \begin{equation} \left \{ \begin{array}{l} u_n = x_n + \gamma A^{*}(I-\lambda_2 f)Ax_n,\\ y_{n,1} = \beta_{n,1} S_1 u_n + (1-\beta_{n,1})u_n, \\ y_{n,i} = \beta_{n,i} S_i u_n + (1-\beta_{n,i})y_{n,i-1}, i = 2,\ldots,N,\\ x_{n+1} = P_C [\lambda_n \xi ( \alpha_n \psi (x_n) + (1-\alpha_n)S x_n)+(I-\lambda_n \mu F)T y_{n,N} ] ,\; n\geq 1. \end{array}\right. \end{equation} (4.1)

    Assume that the problem

    \begin{equation} \langle (\mu F - \xi \psi)x^*, x-x^* \rangle \geq 0, \forall x \in \Omega, \end{equation} (4.2)

    has a solution. Suppose that the following conditions are satisfied:

    (C1) 0 < \underset{n \rightarrow \infty}{\liminf}\alpha_n \leq \underset{n \rightarrow \infty}{\limsup}\alpha_n < 1 ;

    (C2) \underset{n \rightarrow \infty}{\lim}\lambda_n = 0 and \mathop {\mathop \sum \limits^\infty }\limits_{n = 1} \lambda_n = \infty ;

    (C3) \mathop {\mathop \sum \limits^\infty }\limits_{n = 2} | \alpha_{n }\lambda_{n}-\alpha_{n-1}\lambda_{n-1}| < \infty or \underset{n \rightarrow \infty}{\lim} \frac{ |\alpha_{n }\lambda_{n}-\alpha_{n-1}\lambda_{n-1} |}{\lambda_n} = 0 ;

    (C4) \mathop {\mathop \sum \limits^\infty }\limits_{n = 2} | \lambda_{n}-\lambda_{n-1}| < \infty or \underset{n \rightarrow \infty}{\lim} \frac{ |\lambda_{n}-\lambda_{n-1} |}{\lambda_n} = 0 ;

    (C5) \mathop {\mathop \sum \limits^\infty }\limits_{n = 2} | \beta_{n, i}-\beta_{n-1, i}| < \infty or \underset{n \rightarrow \infty}{\lim} \frac{ |\beta_{n, i}-\beta_{n-1, i} |}{\lambda_n} = 0 for all i\in \{ 1, \ldots, N \} ;

    (C6) \lambda_1 > 0, 0 < \lambda_2 < 2 \varsigma, 0 < \gamma < \frac{1}{r} .

    Then, \{ x_n \} converges strongly to a unique solution x^{*} \in \Omega of Problem (4.2). Suppose H = C = \mathbb{R}, for each x \in \mathbb{R} the mappings S_i and T_i are defined as follows

    S_ix = \frac{i}{i+1}x

    and

    \begin{equation} T_i(x) = \begin{cases} x ,\quad x \in (- \infty, 0),\\ 2x ,\quad x \in [0, \infty). \end{cases} \end{equation} (4.3)

    Observe that S_i for i \ge 1 are nonexpansive and T is \frac{1}{3} -demicontractive mapping [29]. Take \beta_{n, i} = \frac{6}{n^2 i^2}, \alpha_n = \frac{3}{n^2} and \lambda_n = \frac{1}{n^2 + 2} . Also define \psi (x) = \frac{2x}{3} and Ax = 2x with \|A \| = 2 . Therefore it can be seen that the sequences satisfy the conditions in the (C1) - (C6).

    It can be observed from Figure 1, that the sequence \{x_n\} generated converges to 0 , which is the only element of the solution set, i.e \Omega = \{0\}.

    Figure 1.  Plot of the iterative sequence after 200 iterations.

    In this paper, we first propose triple hierarchical variational inequality problem (4.1) in Theorem 3.1 and then we prove some strong convergence of the sequence \{x_n\} generated by (4.1) to a common solution of variational inequality problem, split monotone variational inclusion problem and fixed point problems. We divide the proof into 6 steps and our theorem is extends and improves the corresponding results of Jitsupa et al. [1] and Kazmi and Rizvi [22].

    The authors thank the referees for their comments and suggestions regarding this manuscript. The last author would like to thank King Mongkut's University of Technology North Bangkok, Rayong Campus (KMUTNB-Rayong). This research was funded by King Mongkut's University of Technology North Bangkok. Contract no. KMUTNB-63-KNOW-016.

    The authors declare that they have no competing interests.



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