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Research article

Pythagorean Cubic fuzzy Hamacher aggregation operators and their application in green supply selection problem

  • Received: 15 August 2021 Revised: 17 November 2021 Accepted: 25 November 2021 Published: 24 December 2021
  • MSC : 03E72, 47S40

  • The green chain supplier selection process plays a major role in the environmental decision for the efficient and effective supply chain management. Therefore, the aim of this paper is to develop a mechanism for decision making on green chain supplier problem. First, we define the Hamacher operational law for Pythagorean cubic fuzzy numbers (PCFNs) and study their fundamental properties. Based on the Hamacher operation law of PCFNs, we defined Pythagorean cubic fuzzy aggregation operators by using Hamacher t-norm and t-conorm. Further, we develop a series of Pythagorean cubic fuzzy Hamacher weighted averaging (PCFHWA), Pythagorean cubic fuzzy Hamacher order weighted averaging (PCFHOWA) Pythagorean Cubic fuzzy Hamacher hybrid averaging (PCFHHA), Pythagorean Cubic fuzzy Hamacher weighted Geometric (PCFHWG), Pythagorean Cubic fuzzy Hamacher order weighted Geometric (PCFHOWG), and Pythagorean Cubic fuzzy Hamacher hybrid geometric (PCFHHA) operators. Furthermore, we apply these aggregation operators of Pythagorean Cubic fuzzy numbers to the decision making problem for green supplier selection. We construct an algorithm for the group decision making by using aggregation operators and score function. The proposed decision making method applies to green chain supplier selection problem and find the best green supplier for green supply chain management. The proposed method compared with other group decision techniques under Pythagorean cubic fuzzy information. From the comparison and sensitivity analysis, we concluded that our proposed method is more generalized and effective method.

    Citation: Saleem Abdullah, Muhammad Qiyas, Muhammad Naeem, Mamona, Yi Liu. Pythagorean Cubic fuzzy Hamacher aggregation operators and their application in green supply selection problem[J]. AIMS Mathematics, 2022, 7(3): 4735-4766. doi: 10.3934/math.2022263

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  • The green chain supplier selection process plays a major role in the environmental decision for the efficient and effective supply chain management. Therefore, the aim of this paper is to develop a mechanism for decision making on green chain supplier problem. First, we define the Hamacher operational law for Pythagorean cubic fuzzy numbers (PCFNs) and study their fundamental properties. Based on the Hamacher operation law of PCFNs, we defined Pythagorean cubic fuzzy aggregation operators by using Hamacher t-norm and t-conorm. Further, we develop a series of Pythagorean cubic fuzzy Hamacher weighted averaging (PCFHWA), Pythagorean cubic fuzzy Hamacher order weighted averaging (PCFHOWA) Pythagorean Cubic fuzzy Hamacher hybrid averaging (PCFHHA), Pythagorean Cubic fuzzy Hamacher weighted Geometric (PCFHWG), Pythagorean Cubic fuzzy Hamacher order weighted Geometric (PCFHOWG), and Pythagorean Cubic fuzzy Hamacher hybrid geometric (PCFHHA) operators. Furthermore, we apply these aggregation operators of Pythagorean Cubic fuzzy numbers to the decision making problem for green supplier selection. We construct an algorithm for the group decision making by using aggregation operators and score function. The proposed decision making method applies to green chain supplier selection problem and find the best green supplier for green supply chain management. The proposed method compared with other group decision techniques under Pythagorean cubic fuzzy information. From the comparison and sensitivity analysis, we concluded that our proposed method is more generalized and effective method.



    The variational inequality problem (VIP) was introduced by Stampacchia [1] and provided a very useful tool for researching a large variety of interesting problems arising in physics, economics, finance, elasticity, optimization, network analysis, medical images, water resources, and structural analysis, see for example ([2,3,4,5,6,7,8,9,10,11,12,13,14,15]) and references therein.

    Let H be a real Hilbert space with inner product , and norm , respectively. Let C be a nonempty closed convex subset of H. Let B:CH be an operator.

    In this article, our study is related to a classical variational inequality problem (VIP) which aims to find an element xC such that

    Bx,xx0,  xC. (1.1)

    It is well known that xVI(B,C) if and only if x=PC(xζBx), where ζ>0, in other words, the VIP is equivalent to the fixed point problem (see [16]). Supposing that B is η-strongly monotone and L-Lipschitz continuous with 0<ζ<2ηL2, the following sequence {xn} of Picard iterates:

    xn+1=PC(xnζBxn), (1.2)

    converges strongly to a point xVI(B,C) due to the fact that PC(IζB) is a contraction on C. However, in general, the algorithm (1.2) fails when B is monotone and L-Lipschitz continuous (see [17]). In [7], Korpelevich put forward an extragradient method which provided an important idea for solving monotone variational inequality:

    yn=PC(xnλfxn),xn+1=PC(xnλfyn), (1.3)

    where f is monotone, L-Lipschitz continuous in the finite dimensional Euclidean space Rn and λ(0,1L).

    The another motivation of this article is the split common fixed point problem which aims to find a point uH1 such that

    uFix(T)   and   AuFix(S). (1.4)

    The split common fixed point problem can be regarded as a generalization of the split feasibility problem. Recall that the split feasibility problem is to find a point satisfying

    uC   and   AuQ, (1.5)

    where C and Q are two nonempty closed convex subsets of real Hilbert spaces H1 and H2, respectively and A:H1H2 is a bounded linear operator. Inverse problems in various disciplines can be expressed as the split feasibility problem and the split common fixed point problem. Problem (1.4) was firstly introduced by Censor and Segal [18]. Note that solving (1.4) can be translated to solve the fixed point equation:

    u=S(uτA(IT)Au), τ>0.

    Whereafter, Censor and Segal proposed an algorithm for directed operators. Since then, there has been growing interest in the split common fixed point problem (see [19,20,21,22]).

    Censor et al. [23] first proposed split variational inequality problems by combining the variational inequality problem and the split feasibility problem. Very recently, in 2017, Tian and Jiang [24] considered the following split variational inequality problem: finding an element u such that

    uVI(A,C) and BuFix(T), (1.6)

    where T:H2H2 is nonexpansive, B:H1H2 is a bounded linear operator with its adjoint B, and A:CH1 is a monotone and L-Lipschitz continuous mapping. Then they presented the following iteration method by combining the extragradient method with CQ algorithm for solving the (1.6):

    Algorithm 1.1. Choose an arbitrary initial value x1C. Assume xn has been constructed. Compute

    yn=PC(xnτnA(IT)Axn),zn=PC(ynςnFyn),xn+1=PC(ynςnFzn). (1.7)

    They proved that the iterative sequence {xn} defined by Eq (1.7) converges weakly to an element zΓ, where Γ is the set of solutions of the problem (1.6). However, Algorithm 1.1 fails, in general, to converge strongly in the setting of infinite-dimensional Hilbert spaces. We also notice that Algorithm 1.1 is involved with three metric projections in each iteration, which might seriously affect the efficiency of the method.

    Motivated and inspired by the above works, in the present paper, we consider variational inequality problems and split common fixed point problems for finding an element u such that

    ˆxVI(A,C) and Bˆxn=1Fix(Tn), (1.8)

    where {Tn}n=1:H2H2 is an infinite family of nonexpansive mappings, B:H1H2 is a bounded linear operator with its adjoint B, and A:H1H1 is a monotone and L-Lipschitz continuous mapping. In contrast to Tian and Jiang [24], we consider the common fixed points of an infinite family of nonexpansive mappings instead of only the fixed points of a nonexpansive mapping. The efficiency of the algorithm is also improved by removing the projection operator in the first iteration which might affect the efficiency of the method to a certain extent. Finally, we present a very simple modification to extragradient method, which makes our algorithm have the strong convergence. It is well known that the strong convergence theorem is always more convenient to use.

    This paper is organized as follows: In Section 2, we give some definitions and key lemmas which are used in this paper. Section 3 consists of our algorithms and provides the strong convergence theorems. In Section 4, numerical examples are provided for illustration. Finally, this paper is concluded in Section 5.

    Let H be a real Hilbert space with inner product , and norm , respectively. Let C be a nonempty closed convex subset of H. Let T:CC be an operator. We use Fix(T) to denote the set of fixed points of T, that is, Fix(T)={x|x=Tx,xC}. First, we give some definitions and lemmas related to the involved operators.

    Definition 2.1. An operator T:CC is said to be nonexpansive if TuTvuv for all u,vC.

    Definition 2.2. An operator A:CH is said to be monotone if AxAy,xy0 for all x,yC.

    A monotone operator R:H2H is called maximal monotone if the graph of R is a maximal monotone set.

    Definition 2.3. An operator T:CH is said to be L-Lipschitzian if there exists L>0 such that TxTyLxy for all x,yC.

    Usually, the convergence of fixed point algorithms requires some additional smoothness properties of the mapping T such as demi-closedness.

    Definition 2.4. An operator T is said to be demiclosed if, for any sequence {un} which weakly converges to u, and if Tunw, then Tu=w.

    Recall that the (nearest point or metric) projection from H onto C, denoted by PC, assigns to each uH, the unique point PCuC with the property:

    uPCu=inf{uv:vC}.

    The metric projection \mathcal{P}_{\mathcal{C}} of \mathcal{H} onto \mathcal{C} is characterized by

    \begin{eqnarray} \begin{split} &\langle u\;-\mathcal{P}_{\mathcal{C}}u,v-\mathcal{P}_{\mathcal{C}}u\rangle\leq0\\ \; \mathrm{or}&\ \|u-v\|^{2}\geq\|u-\mathcal{P}_{\mathcal{C}}u\|^{2}+\|v-\mathcal{P}_{\mathcal{C}}u\|^{2} \end{split} \end{eqnarray} (2.1)

    for all u\in \mathcal{H}, v\in \mathcal{C} . It is well known that the metric projection \mathcal{P}_{\mathcal{C}}:\mathcal{H}\rightarrow \mathcal{C} is firmly nonexpansive, that is,

    \begin{eqnarray} \begin{split} &\langle u\;-v,\mathcal{P}_{\mathcal{C}}u-\mathcal{P}_{\mathcal{C}}v\rangle\geq\|\mathcal{P}_{\mathcal{C}}u-\mathcal{P}_{\mathcal{C}}v\|^{2}\\ \; \mathrm{or}&\ \|\mathcal{P}_{\mathcal{C}}u-\mathcal{P}_{\mathcal{C}}v\|^{2}\leq\|u-v\|^{2}-\|(I-\mathcal{P}_{\mathcal{C}})u-(I-\mathcal{P}_{\mathcal{C}})v\|^{2} \end{split} \end{eqnarray} (2.2)

    for all u, v\in \mathcal{H} . More information on the metric projection can be found, for example, in Section 3 of the book by Goebel et al. (see [25]).

    For all u, v \in H , the following conclusions hold:

    \begin{eqnarray} \|tu+(1-t)v\|^{2} = t\|u\|^{2}+(1-t)\|v\|^{2}-t(1-t)\|u-v\|^{2},\ \ t\in[0,1], \end{eqnarray} (2.3)
    \begin{eqnarray} \|u+v\|^{2} = \|u\|^{2}+2\langle u,v\rangle+\|v\|^{2} \end{eqnarray} (2.4)

    and

    \begin{eqnarray} \|u+v\|^{2}\leq\|u\|^{2}+2\langle v,u+v\rangle. \end{eqnarray} (2.5)

    Let \{\mathcal{T}_{n}\}^{\infty}_{n = 1}:\mathcal{H}\rightarrow \mathcal{H} be an infinite family of nonexpansive mappings and \lambda_{1}, \lambda_{2}, ... be real numbers such that 0\leq\lambda_{i}\leq1 for each i\in\mathbb{N} . For any n\in\mathbb{N} , define a mapping \mathcal{W}_{n} of \mathcal{C} into \mathcal{H} as follows:

    \begin{eqnarray} \begin{split} \;&\mathcal{U}_{n,n+1} = \mathcal{I},\\ \;&\mathcal{U}_{n,n} = \lambda_{n}\mathcal{T}_{n}\mathcal{U}_{n,n+1}+ (1-\lambda_{n})\mathcal{I},\\ \;&\mathcal{U}_{n,n-1} = \lambda_{n-1}\mathcal{T}_{n-1}\mathcal{U}_{n,n}+ (1-\lambda_{n-1})\mathcal{I},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \;&\ldots\\ \;&\mathcal{U}_{n,k} = \lambda_{k}\mathcal{T}_{k}\mathcal{U}_{n,k+1}+ (1-\lambda_{k})\mathcal{I},\\ \;&\mathcal{U}_{n,k-1} = \lambda_{k-1}\mathcal{T}_{k-1}\mathcal{U}_{n,k}+ (1-\lambda_{k-1})\mathcal{I},\\ \;&\ldots\\ \;&\mathcal{U}_{n,2} = \lambda_{2}\mathcal{T}_{2}\mathcal{U}_{n,3}+ (1-\lambda_{2})\mathcal{I},\\ \;&\mathcal{W}_{n} = \mathcal{U}_{n,1} = \lambda_{1}\mathcal{T}_{1}\mathcal{U}_{n,2}+ (1-\lambda_{1})\mathcal{I}.\\ \end{split} \end{eqnarray} (2.6)

    Such a mapping \mathcal{W}_{n} is called the \mathcal{W} -mapping generated by \mathcal{T}_{1}, \mathcal{T}_{2}, ..., \mathcal{T}_{n} and \lambda_{1}, \lambda_{2}, ..., \lambda_{n} . We have the following crucial Lemma concerning \mathcal{W}_{n} :

    Lemma 2.1. [26] Let \mathcal{H} be a real Hilbert space. Let \{\mathcal{T}_{n}\}^{\infty}_{n = 1}:\mathcal{H}\rightarrow \mathcal{H} be an infinite family of nonexpansive mappings such that \bigcap^{\infty}_{n = 1}\mathrm{Fix} (\mathcal{T}_{n})\neq\emptyset . Let \lambda_{1}, \lambda_{2}, ... be real numbers such that 0\leq\lambda_{i}\leq b < 1 for each i\geq1 . Then we have the following:

    (1) For any x\in\mathcal{H} and k\geq1 , the limit \lim_{n\rightarrow \infty}\mathcal{U}_{n, k}x exists;

    (2) \mathrm{Fix} (\mathcal{W}) = \bigcap^{\infty}_{n = 1}\mathrm{Fix} (\mathcal{T}_{n}) , where \mathcal{W}x = \lim_{n\rightarrow \infty}\mathcal{W}_{n}x = \lim_{n\rightarrow \infty}\mathcal{U}_{n, 1}x, \ \forall x\in\mathcal{C};

    (3) For any bounded sequence \{x_{n}\}\subset \mathcal{H} , \lim_{n\rightarrow \infty}\mathcal{W}x_{n} = \lim_{n\rightarrow \infty}\mathcal{W}_{n}x_{n} .

    Lemma 2.2. [27] Assume that \{\alpha_{n}\} is a sequence of nonnegative real numbers such that

    \alpha_{n+1}\leq(1-\gamma_{n})\alpha_{n} + \delta_{n}, \ \ n\in \mathbb{N},

    where \{\gamma_{n}\} is a sequence in (0, 1) and \{\delta_{n}\} is a sequence such that

    (1) \sum^{\infty}_{n = 1}\gamma_{n} = \infty ;

    (2) \limsup_{n\rightarrow \infty}\frac{\delta_{n}}{\gamma_{n}}\leq0 or \sum^{\infty}_{n = 1}|\delta_{n}| < \infty. Then \lim_{n\rightarrow \infty}\alpha_{n} = 0.

    Lemma 2.3. [28] Let \{\varpi_{n}\} be a sequence of real numbers. Assume there exists at least a subsequence \{\varpi_{n_{k}}\} of \{\varpi_{n}\} such that \varpi_{n_{k}}\leq\varpi_{n_{k+1}} for all k\geq0 . For every n\geq N_{0} , define an integer sequence \{\tau(n)\} as:

    \tau(n) = \max\{i\leq n:\varpi_{n_{i}} < \varpi_{n_{i+1}}\}.

    Then, \tau(n) \rightarrow \infty as n\rightarrow \infty and for all n\geq N_{0} , we have max\{ \varpi_{\tau(n)}, \varpi_{n}\}\leq\varpi_{\tau(n)+1} .

    In this section, we introduce our algorithm and prove its strong convergence. Some assumptions on the underlying spaces and involved operators are listed below.

    (R_{1}) \mathcal{H}_{1} and \mathcal{H}_{2} are two real Hilbert spaces and \mathcal{C}\subset\mathcal{H}_{1} is a nonempty closed convex subset.

    (R_{2}) \mathcal{B}:\mathcal{H}_{1}\rightarrow \mathcal{H}_{2} is a bounded linear operator with its adjoint \mathcal{B}^{\ast} .

    (R_{3}) \mathcal{A}:\mathcal{H}_{1}\rightarrow \mathcal{H}_{1} is a monotone and L -Lipschitz continuous mapping.

    (R_{4}) \Omega = \{\hat{x}|\hat{x}\in VI(\mathcal{A}, \mathcal{C}) and \mathcal{B}\hat{x}\in \bigcap^{\infty}_{n = 1} \mathrm{Fix}(\mathcal{T}_{n})\} , where \Omega is the set of solutions of the problem (1.8).

    Next, we present the following iterative algorithm to find a point \hat{x}\in\Omega .

    Algorithm 3.1. Choose an arbitrary initial value x_{1}\in \mathcal{H} . Assume x_{n} has been constructed. Compute

    \begin{eqnarray} \begin{split} \;& y_{n} = x_{n}-\tau_{n} \mathcal{B}^\ast(I-\mathcal{W}_{n})\mathcal{B}x_{n},\\ \;& z_{n} = \mathcal{P}_{\mathcal{C}}(y_{n}-\varsigma_{n}\mathcal{A}y_{n}),\\ \;&x_{n+1} = \mathcal{P}_{\mathcal{C}}((1-\alpha_{n})(y_{n}-\varsigma_{n}\mathcal{A}z_{n})), \\ \end{split} \end{eqnarray} (3.1)

    where \{\alpha_{n}\} is a sequence in (0, 1) , \varsigma_{n} is a sequence in (0, \frac{1}{L}) , and \tau_{n} is a sequence in (0, \frac{1}{\|\mathcal{B}\|^{ 2}}) .

    Theorem 3.1. If \Omega\neq\emptyset and the following conditions are satisfied:

    ( C_{1} ) \lim_{n\rightarrow \infty}\alpha_{n} = 0\ and\ \sum^{\infty}_{n = 0}\alpha_{n} = \infty ;

    ( C_{2} ) 0 < \liminf_{n\rightarrow \infty}\varsigma_{n}\leq \limsup_{n\rightarrow \infty}\varsigma_{n} < \frac{1}{L} ;

    ( C_{3} ) 0 < \liminf_{n\rightarrow \infty}\tau_{n}\leq \limsup_{n\rightarrow \infty}\tau_{n} < \frac{1}{\|\mathcal{B}\|^{ 2}} .

    Then, the iterative sequence \{x_{n}\} defined by Eq (3.1) strongly converges to the minimum-norm solution \hat{x}( = \mathcal{P}_{\Omega}\theta) .

    Proof. Set z = \mathcal{P}_{\Omega}\theta . We can obtain that

    \begin{eqnarray} \begin{split} &\|y_{n}-z\|^{2}\\ = \;&\|x_{n}-z-\tau_{n} \mathcal{B}^\ast(I-\mathcal{W}_{n})\mathcal{B}x_{n}\|^{2}\\ = \;&\|x_{n}-z\|^{2}-2\tau_{n} \langle x_{n}-z,\mathcal{B}^\ast(I-\mathcal{W}_{n})\mathcal{B}x_{n}\rangle+\|\tau_{n}\mathcal{B}^\ast(I-\mathcal{W}_{n})\mathcal{B}x_{n}\|^{2}\\ = \;&\|x_{n}-z\|^{2}-2\tau_{n} \langle \mathcal{B}x_{n}-\mathcal{B}z,(I-\mathcal{W}_{n})\mathcal{B}x_{n}\rangle+\|\tau_{n}\mathcal{B}^\ast(I-\mathcal{W}_{n})\mathcal{B}x_{n}\|^{2}\\ \leq\;&\|x_{n}-z\|^{2}-\tau_{n} \|(I-\mathcal{W}_{n})\mathcal{B}x_{n}\|^{2}+\tau_{n}^{2}\|\mathcal{B}\|^{2}\cdot\|(I-\mathcal{W}_{n})\mathcal{B}x_{n}\|^{2}\\ \leq\;&\|x_{n}-z\|^{2}-\tau_{n}(1-\tau_{n}\|\mathcal{B}\|^{ 2})\|(I-\mathcal{W}_{n})\mathcal{B}x_{n}\|^{2}\\ \leq\;&\|x_{n}-z\|^{2}. \end{split} \end{eqnarray} (3.2)

    It follows from (2.1) that

    \begin{eqnarray} \begin{split} &\|x_{n+1}-z\|^{2}\\ = \;&\|\mathcal{P}_{\mathcal{C}}((1-\alpha_{n}) (y_{n}-\varsigma_{n}\mathcal{A}z_{n}))-z\|^{2}\\ \leq\;&\|(1-\alpha_{n})(y_{n}-\varsigma_{n}\mathcal{A}z_{n})-z\|^{2}- \|(1-\alpha_{n})(y_{n} -\varsigma_{n}\mathcal{A}z_{n})-x_{n+1}\|^{2}\\ \leq\;&\|(1-\alpha_{n})(y_{n}-\varsigma_{n}\mathcal{A}z_{n}-z)+\alpha_{n}(-z)\|^{2} \\ \;&-\|(1-\alpha_{n})(y_{n} -\varsigma_{n}\mathcal{A}z_{n}-x_{n+1})+\alpha_{n}(-x_{n+1})\|^{2}\\ \leq\;&(1-\alpha_{n})\|y_{n}-\varsigma_{n}\mathcal{A}z_{n}-z\|^{2}+\alpha_{n}\|-z\|^{2}\\ \;&-(1-\alpha_{n})\alpha_{n}\|y_{n}-\varsigma_{n}\mathcal{A}z_{n}\|^{2}\\ \;&-((1-\alpha_{n})\|y_{n}-\varsigma_{n}\mathcal{A}z_{n}-x_{n+1}\|^{2}+\alpha_{n}\|-x_{n+1}\|^{2}\\ \;&-(1-\alpha_{n})\alpha_{n}\|y_{n}-\varsigma_{n}\mathcal{A}z_{n}\|^{2})\\ = \;&(1-\alpha_{n})(\|y_{n}-\varsigma_{n}\mathcal{A}z_{n}-z\|^{2}-\|y_{n} -\varsigma_{n}\mathcal{A}z_{n}-x_{n+1}\|^{2})\\ \;&+\alpha_{n}(\|z\|^{2}-\|x_{n+1}\|^{2}). \end{split} \end{eqnarray} (3.3)

    We also observe that

    \begin{eqnarray} \begin{split} &\|y_{n}-\varsigma_{n}\mathcal{A}z_{n}-z\|^{2}-\|y_{n} -\varsigma_{n}\mathcal{A}z_{n}-x_{n+1}\|^{2}\\ = \;& \|y_{n}-z\|^{2}-\|y_{n} -x_{n+1}\|^{2}+2\varsigma_{n}\langle\mathcal{A}z_{n},z-x_{n+1}\rangle\\ = \;&\|y_{n}-z\|^{2}-\|y_{n} -x_{n+1}\|^{2}+2\varsigma_{n}\langle\mathcal{A}z_{n},z-z_{n}\rangle +2\varsigma_{n}\langle\mathcal{A}z_{n},z_{n}-x_{n+1}\rangle\\ = \;&\|y_{n}-z\|^{2}-\|y_{n} -x_{n+1}\|^{2}+2\varsigma_{n}\langle\mathcal{A}z_{n}-\mathcal{A}z,z-z_{n}\rangle\\ \;&+2\varsigma_{n}\langle\mathcal{A}z,z-z_{n}\rangle +2\varsigma_{n}\langle\mathcal{A}z_{n},z_{n}-x_{n+1}\rangle\\ \geq\;&\|y_{n}-z\|^{2}-\|y_{n} -x_{n+1}\|^{2} +2\varsigma_{n}\langle\mathcal{A}z_{n},z_{n}-x_{n+1}\rangle\\ = \;&\|y_{n}-z\|^{2}-\|y_{n} -z_{n}\|^{2}-\|z_{n} -x_{n+1}\|^{2}\\ \;&+2\langle y_{n}-\varsigma_{n} \mathcal{A}z_{n}-z_{n},x_{n+1}-z_{n}\rangle.\\ \end{split} \end{eqnarray} (3.4)

    On the other hand, we have that

    \begin{eqnarray} \begin{split} &\langle y_{n}-\varsigma_{n} \mathcal{A}z_{n}-z_{n},x_{n+1}-z_{n}\rangle\\ = \;& \langle y_{n}-\varsigma_{n} \mathcal{A}y_{n}-z_{n},x_{n+1}-z_{n}\rangle+ \varsigma_{n}\langle \mathcal{A}y_{n}-\mathcal{A}z_{n},x_{n+1}-z_{n}\rangle\\ \leq\;&\varsigma_{n}\langle \mathcal{A}y_{n}-\mathcal{A}z_{n},x_{n+1}-z_{n}\rangle\\ \leq\;&\varsigma_{n}\|\mathcal{A}y_{n}-\mathcal{A}z_{n}\|\times\|x_{n+1}-z_{n}\|\\ \leq\;&\varsigma_{n} L\|y_{n}-z_{n}\|\times\|x_{n+1}-z_{n}\|.\\ \end{split} \end{eqnarray} (3.5)

    Hence, we can derive that

    \begin{eqnarray} \begin{split} \|x_{n+1}-z\|^{2} = \;&(1-\alpha_{n})(\|y_{n}-\varsigma_{n}\mathcal{A}z_{n}-z\|^{2}-\|y_{n} -\varsigma_{n}\mathcal{A}z_{n}-x_{n+1}\|^{2})+\alpha_{n}(\|z\|^{2}-\|x_{n+1}\|^{2}),\\ (by(3.4))\leq\;&(1-\alpha_{n})(\|y_{n}-z\|^{2}-\|y_{n} -z_{n}\|^{2}-\|z_{n} -x_{n+1}\|^{2}\\ \;&+2\langle y_{n}-\varsigma_{n} \mathcal{A}z_{n}-z_{n},x_{n+1}-z_{n}\rangle)+\alpha_{n}(\|z\|^{2}-\|x_{n+1}\|^{2}),\\ (by(3.5))\leq\;&(1-\alpha_{n})(\|y_{n}-z\|^{2}-\|y_{n} -z_{n}\|^{2}-\|z_{n} -x_{n+1}\|^{2}\\ \;&+2\varsigma_{n} L\|y_{n}-z_{n}\|\times\|x_{n+1}-z_{n}\|)+\alpha_{n}(\|z\|^{2}-\|x_{n+1}\|^{2})\\ \leq\;&(1-\alpha_{n})(\|y_{n}-z\|^{2}-\|y_{n} -z_{n}\|^{2}-\|z_{n} -x_{n+1}\|^{2}\\ \;&+\varsigma_{n}^{2} L^{2}\|y_{n}-z_{n}\|^{2}+\|x_{n+1}-z_{n}\|^{2})+\alpha_{n}(\|z\|^{2}-\|x_{n+1}\|^{2})\\ \leq\;&(1-\alpha_{n})(\|y_{n}-z\|^{2}+(\varsigma_{n}^{2} L^{2}-1)\|y_{n}-z_{n}\|^{2})+\alpha_{n}(\|z\|^{2}-\|x_{n+1}\|^{2}),\\ (by(3.2))\leq\;&(1-\alpha_{n})(\|x_{n}-z\|^{2}+(\varsigma_{n}^{2} L^{2}-1)\|y_{n}-z_{n}\|^{2})+\alpha_{n}(\|z\|^{2}-\|x_{n+1}\|^{2}).\\ \end{split} \end{eqnarray} (3.6)

    Owing to the assumption ( C_{2} ), it follows from (3.6) that

    \begin{eqnarray} \begin{split} \|x_{n+1}-z\|^{2} \leq\;&(1-\alpha_{n})\|y_{n}-z\|^{2}+\alpha_{n}(\|z\|^{2}-\|x_{n+1}\|^{2}),\\ (by(3.2))\leq\;&(1-\alpha_{n})\|x_{n}-z\|^{2}+\alpha_{n}(\|z\|^{2}-\|x_{n+1}\|^{2})\\ \leq\;&(1-\alpha_{n})\|x_{n}-z\|^{2}+\alpha_{n}\|z\|^{2}\\ \leq\;&\max\{\|x_{n}-z\|^{2},\|z\|^{2}\}\\ \end{split} \end{eqnarray} (3.7)

    and so

    \begin{eqnarray} \begin{split} \|x_{n}-z\|^{2} \leq\max\{\|x_{1}-z\|^{2},\|z\|^{2}\}, \end{split} \end{eqnarray} (3.8)

    which implies that the sequence \{x_{n}\} is bounded. In view of (3.2) and (3.7), we obtain that

    \begin{eqnarray} \begin{split} &\tau_{n}(1-\tau_{n}\|\mathcal{B}\|^{ 2})\|(I-\mathcal{W}_{n})\mathcal{B}x_{n}\|^{2}\\ \leq\;& \|x_{n}-z\|^{2}-\|y_{n}-z\|^{2}\\ \leq\;& \|x_{n}-z\|^{2}-\|x_{n+1}-z\|^{2}+\alpha_{n}(\|z\|^{2}-\|x_{n+1}\|^{2}-\|y_{n}-z\|^{2}).\\ \end{split} \end{eqnarray} (3.9)

    CASE \mathbb{I} . Suppose that there exists m > 0 such that the sequence \{\|x_{n}-z\|\} is decreasing when n\geq m . Then, \lim_{n\rightarrow \infty}\|x_{n}-z\| exists. Consequently, according to the assumptions ( C_{1} ) and ( C_{3} ), we deduce that

    \begin{eqnarray} \begin{split} \lim\limits_{n\rightarrow \infty}\|(I-\mathcal{W}_{n})\mathcal{B}x_{n}\| = 0. \end{split} \end{eqnarray} (3.10)

    In virtue of the boundedness of the sequence \{\mathcal{B}x_{n}\} and Lemma 2.1, we get that

    \begin{eqnarray} \begin{split} \lim\limits_{n\rightarrow \infty}\|W\mathcal{B}x_{n}- W_{n}\mathcal{B}x_{n}\| = 0. \end{split} \end{eqnarray} (3.11)

    This together with (3.24) implies that

    \begin{eqnarray} \begin{split} \lim\limits_{n\rightarrow \infty}\|(I-\mathcal{W})\mathcal{B}x_{n}\| = 0. \end{split} \end{eqnarray} (3.12)

    It follows from (3.6) that

    \begin{eqnarray} \begin{split} &(1-\alpha_{n})(1-\varsigma_{n}^{2} L^{2})\|y_{n}-z_{n}\|^{2}\\ \leq\;&(1-\alpha_{n})\|x_{n}-z\|^{2}-\|x_{n+1}-z\|^{2}\\ \;&+\alpha_{n}(\|z\|^{2}-\|x_{n+1}\|^{2})\\ \leq\;&\|x_{n}-z\|^{2}-\|x_{n+1}-z\|^{2}\\ \;&+\alpha_{n}(\|z\|^{2}-\|x_{n+1}\|^{2}-\|x_{n}-z\|^{2}).\\ \end{split} \end{eqnarray} (3.13)

    Thanks to the boundedness of the sequence \{x_{n}\} , we derive that

    \begin{eqnarray} \begin{split} \lim\limits_{n\rightarrow \infty}\|y_{n}-z_{n}\| = 0. \end{split} \end{eqnarray} (3.14)

    In view of (3.30), we can also get that

    \begin{eqnarray} \begin{split} \lim\limits_{n\rightarrow \infty}\|y_{n}-x_{n}\| = \lim\limits_{n\rightarrow \infty}\|\tau_{n} \mathcal{B}^\ast(I-\mathcal{W}_{n})\mathcal{B}x_{n}\| = 0(by(3.10)). \end{split} \end{eqnarray} (3.15)

    Combining (3.14) and (3.15), we obtain that

    \begin{eqnarray} \begin{split} \lim\limits_{n\rightarrow \infty}\|z_{n}-x_{n}\| = 0. \end{split} \end{eqnarray} (3.16)

    On the other hand, we get that

    \begin{eqnarray} \begin{split} \|x_{n+1}-z_{n}\| = \;&\|\mathcal{P}_{\mathcal{C}}((1-\alpha_{n})(y_{n}-\varsigma_{n}\mathcal{A}z_{n}))-\mathcal{P}_{\mathcal{C}}(y_{n}-\varsigma_{n}\mathcal{A}y_{n})\|\\ \leq\;&\|(1-\alpha_{n})(y_{n}-\varsigma_{n}\mathcal{A}z_{n})-(y_{n}-\varsigma_{n}\mathcal{A}y_{n})\|\\ \leq\;&\|(y_{n}-\varsigma_{n}\mathcal{A}z_{n})-(y_{n}-\varsigma_{n}\mathcal{A}y_{n})\|+\alpha_{n}\|y_{n}-\varsigma_{n}\mathcal{A}z_{n})\|\\ \leq\;&\|\varsigma_{n}\mathcal{A}z_{n}-\varsigma_{n}\mathcal{A}y_{n}\|+\alpha_{n}\|y_{n}-\varsigma_{n}\mathcal{A}z_{n}\|\\ \leq\;&\varsigma_{n}\|\mathcal{A}z_{n}-\mathcal{A}y_{n}\| +\alpha_{n}\|y_{n}-\varsigma_{n}\mathcal{A}z_{n}\|\\ \leq\;&\varsigma_{n}L\|z_{n}-y_{n}\| +\alpha_{n}\|y_{n}-\varsigma_{n}\mathcal{A}z_{n}\|.\\ \end{split} \end{eqnarray} (3.17)

    Hence, by (3.14), it turns out that

    \begin{eqnarray} \begin{split} \lim\limits_{n\rightarrow \infty}\|x_{n+1}-z_{n}\| = 0 \end{split} \end{eqnarray} (3.18)

    and consequently, according to (3.16), we have that

    \begin{eqnarray} \begin{split} \lim\limits_{n\rightarrow \infty}\|x_{n+1}-x_{n}\| = 0. \end{split} \end{eqnarray} (3.19)

    Next, we can take a subsequence \{n_{i}\} such that

    \begin{eqnarray} \begin{split} \limsup\limits_{n\rightarrow \infty}(\|z\|^{2}-\|x_{n+1}\|^{2}) = \;&\lim\limits_{i\rightarrow \infty}(\|z\|^{2}-\|x_{n_{i}+1}\|^{2}).\\ \end{split} \end{eqnarray} (3.20)

    By the boundedness of the real sequence \{x_{n_{i}+1}\}, we may assume that x_{n_{i}+1}\rightharpoonup x^{\dagger}. Since \mathcal{W} is nonexpansive, we can derive that \mathcal{B}x^{\dagger} = \mathcal{W}\mathcal{B}x^{\dagger} (see Corollary 4.28 in [29]), that is, \mathcal{B}x^{\dagger}\in Fix(\mathcal{W}) = \bigcap^{\infty}_{n = 1}Fix(\mathcal{T}_{n}) .

    Now, we show that x^{\dagger}\in VI(\mathcal{A}, \mathcal{C}) . Let

    \begin{equation} \mathcal{R}(v) = \left\{ \begin{aligned} &\mathcal{A}v+\mathcal{N}_{\mathcal{C}}(v),\ &v\in \mathcal{C}, \\ &\emptyset\ \ &v\not\in \mathcal{ C}, \end{aligned} \right. \end{equation} (3.21)

    where \mathcal{N}_{\mathcal{C}}(v) is the normal cone to \mathcal{C} at v . According to Reference [30], we can easily derive that \mathcal{R} is maximal monotone. Let (v, w)\in G(\mathcal{R}) . Since w -Av \in\mathcal{N}_{\mathcal{C}}(v) and x_{n}\in \mathcal{C} , we have that

    \langle v-x_{n},w -Av\rangle\geq0.

    Noting that, due to v\in \mathcal{C} , we get

    \langle v-x_{n+1},x_{n+1}-(1-\alpha_{n})(y_{n}-\varsigma_{n}\mathcal{A}z_{n})) \rangle\geq0.

    It follows that

    \langle v-x_{n+1},\frac{x_{n+1}-y_{n}}{\varsigma_{n}}+\mathcal{A}z_{n}+ \frac{\alpha_{n}}{\varsigma_{n}}(y_{n}-\varsigma_{n}\mathcal{A}z_{n})\rangle\geq0.

    Thus, we can deduce that

    \begin{eqnarray} \begin{split} &\langle v-x_{n_{i}+1},w \rangle\\ \geq\;&\langle v-x_{n_{i}+1},Av\rangle\\ \geq\;&-\langle v-x_{n_{i}+1},\frac{x_{n_{i}+1}-y_{n_{i}}}{\varsigma_{n_{i}}}+\mathcal{A}z_{n_{i}}+ \frac{\alpha_{n_{i}}}{\varsigma_{n_{i}}}(y_{n_{i}}-\varsigma_{n_{i}}\mathcal{A}z_{n_{i}})\rangle\\ \;&+\langle v-x_{n_{i}+1},Av\rangle\\ \geq\;&\langle v-x_{n_{i}+1},\mathcal{A}v-\mathcal{A}z_{n_{i}}\rangle-\langle v-x_{n_{i}+1},\frac{x_{n_{i}+1}-y_{n_{i}}}{\varsigma_{n_{i}}} \rangle\\ \;&-\langle v-x_{n_{i}+1}, \frac{\alpha_{n_{i}}}{\varsigma_{n_{i}}}(y_{n_{i}}-\varsigma_{n_{i}}\mathcal{A}z_{n_{i}})\rangle\\ \geq\;&\langle v-x_{n_{i}+1},\mathcal{A}v-\mathcal{A}x_{n_{i}+1}\rangle+ \langle v-x_{n_{i}+1},\mathcal{A}x_{n_{i}+1}-\mathcal{A}z_{n_{i}}\rangle\\ \;&-\langle v-x_{n_{i}+1},\frac{x_{n_{i}+1}-y_{n_{i}}}{\varsigma_{n_{i}}} \rangle- \langle v-x_{n_{i}+1}, \frac{\alpha_{n_{i}}}{\varsigma_{n_{i}}}(y_{n_{i}}-\varsigma_{n_{i}}\mathcal{A}z_{n_{i}})\rangle\\ \geq \;&-\langle v-x_{n_{i}+1},\frac{x_{n_{i}+1}-y_{n_{i}}}{\varsigma_{n_{i}}} \rangle- \langle v-x_{n_{i}+1}, \frac{\alpha_{n_{i}}}{\varsigma_{n_{i}}}(y_{n_{i}}-\varsigma_{n_{i}}\mathcal{A}z_{n_{i}})\rangle\\ \;&+\langle v-x_{n_{i}+1},\mathcal{A}x_{n_{i}+1}-\mathcal{A}z_{n_{i}}\rangle.\\ \end{split} \end{eqnarray} (3.22)

    As i\rightarrow \infty , we obtain that

    \langle v-x^{\dagger},w \rangle\geq0.

    By the maximal monotonicity of \mathcal{R} , we derive that x^{\dagger}\in\mathcal{R}^{-1}0 . Hence, x^{\dagger}\in VI(\mathcal{A}, \mathcal{C}) . Therefore, x^{\dagger}\in\Omega . Since the norm of the Hilbert space \mathcal{H}_{1} is weakly lower semicontinuous(see Lemma 2.42 in [29]), we have the following inequality:

    \|x^{\dagger}\|\leq\liminf\limits_{i\rightarrow \infty}\|x_{n_{i}+1}\|

    and therefore

    -\|x^{\dagger}\|\geq\limsup\limits_{i\rightarrow \infty}(-\|x_{n_{i}+1}\|).

    From (3.7), we observe that

    \begin{eqnarray} \begin{split} \|x_{n+1}-z\|^{2} \leq\;&(1-\alpha_{n})\|x_{n}-z\|^{2}+\alpha_{n}(\|z\|^{2}-\|x_{n+1}\|^{2}).\\ \end{split} \end{eqnarray} (3.23)

    Thanks to z = \mathcal{P}_{\Omega}\theta and x^{\dagger}\in\Omega , we can deduce that

    \limsup\limits_{n\rightarrow \infty}(\|z\|^{2}-\|x_{n_{i}+1}\|^{2}) = \|z\|^{2}+\limsup\limits_{n\rightarrow \infty}(-\|x_{n_{i}+1}\|^{2})\leq\|z\|^{2}-\|x^{\dagger}\|^{2}\leq0.

    Applying Lemma 2.2 to (3.23), we derive that \lim_{n\rightarrow \infty}\|x_{n}-z\| = 0 , which implies that the sequence \{x_{n}\} converges strongly to z .

    CASE \mathbb{II} . For any n_{0} , there exists an integer m\geq n_{0} such that \|x_{m}-z\|\leq\|x_{m+1}-z\| . At this case, we set \varpi_{n} = \|x_{n}-z\| . For n\geq n_{0} , we define a sequence \{\tau_{n}\} by

    \tau(n) = \max\{l\in\mathbb{N}|n_{0}\leq l\leq n,\varpi_{l}\leq\varpi_{l+1}\}.

    It is easy to show that \tau(n) is a non-decreasing sequence such that

    \lim\limits_{n\rightarrow \infty}\tau(n) = +\infty

    and

    \varpi_{\tau(n)}\leq\varpi_{\tau(n)+1}.

    This together with (3.9) implies that

    \begin{eqnarray} \begin{split} \lim\limits_{n\rightarrow \infty}\|(I-\mathcal{W}_{\tau(n)})\mathcal{B}x_{\tau(n)}\|^{2} = 0. \end{split} \end{eqnarray} (3.24)

    Employing techniques similar to CASE \mathbb{I} , we have

    \begin{eqnarray} \begin{split} \limsup\limits_{n\rightarrow \infty}(\|z\|^{2}-\|x_{\tau(n)+1}\|^{2})\leq0 \end{split} \end{eqnarray} (3.25)

    and

    \begin{eqnarray} \begin{split} \varpi_{\tau(n)+1}^{2} \leq\;&(1-\alpha_{\tau(n)})\varpi_{\tau(n)}^{2} +\alpha_{\tau(n)}(\|z\|^{2}-\|x_{\tau(n)+1}\|^{2}).\\ \end{split} \end{eqnarray} (3.26)

    Since \varpi_{\tau(n)}\leq\varpi_{\tau(n)+1}, we have

    \begin{eqnarray} \begin{split} \varpi^{2}_{\tau(n)} \leq\|z\|^{2}-\|x_{\tau(n)+1}\|^{2}. \end{split} \end{eqnarray} (3.27)

    By (3.25), we obtain that

    \limsup\limits_{n\rightarrow \infty}\varpi_{\tau(n)}\leq0

    and so

    \begin{eqnarray} \lim\limits_{n\rightarrow \infty}\varpi_{\tau(n)} = 0. \end{eqnarray} (3.28)

    By Eq (3.26), we also obtain

    \limsup\limits_{n\rightarrow \infty}\varpi_{\tau(n)+1}\leq \limsup\limits_{n\rightarrow \infty}\varpi_{\tau(n)}.

    In the light of the last inequality and Eq (3.28), we derive that

    \lim\limits_{n\rightarrow \infty}\varpi_{\tau(n)+1} = 0.

    Applying Lemma 2.3, we obtain

    \varpi_{n}\leq\varpi_{\tau(n)+1}.

    Therefore, we get that \varpi_{n}\rightarrow0 , that is, x_{n}\rightarrow z . This completes the proof.

    Algorithm 3.2. Choose an arbitrary initial value x_{1}\in \mathcal{C} . Assume x_{n} has been constructed. Compute

    \begin{eqnarray} \begin{split} \;& y_{n} = x_{n}-\tau_{n} \mathcal{B}^\ast(I-\mathcal{T})\mathcal{B}x_{n},\\ \;& z_{n} = \mathcal{P}_{\mathcal{C}}(y_{n}-\varsigma_{n}\mathcal{A}y_{n}),\\ \;&x_{n+1} = \mathcal{P}_{\mathcal{C}}((1-\alpha_{n})(y_{n}-\varsigma_{n}\mathcal{A}z_{n})), \\ \end{split} \end{eqnarray} (3.29)

    where \{\alpha_{n}\} is a sequence in (0, 1) , \varsigma_{n} is a sequence in (0, \frac{1}{L}) , and \tau_{n} is a sequence in (0, \frac{1}{\|\mathcal{B}\|^{ 2}}) .

    Theorem 3.2. If \hat{\Omega}\neq\emptyset and the following conditions are satisfied:

    ( C_{1} ) lim_{n\rightarrow \infty}\alpha_{n} = 0\ and\ \sum^{\infty}_{n = 0}\alpha_{n} = \infty ;

    ( C_{2} ) 0 < \liminf_{n\rightarrow \infty}\varsigma_{n}\leq \limsup_{n\rightarrow \infty}\varsigma_{n} < \frac{1}{L} ;

    ( C_{3} ) 0 < \liminf_{n\rightarrow \infty}\tau_{n}\leq \limsup_{n\rightarrow \infty}\tau_{n} < \frac{1}{\|\mathcal{B}\|^{ 2}} .

    Then, the iterative sequence \{x_{n}\} defined by Eq (3.29) strongly converges to the minimum-norm solution \hat{x}( = \mathcal{P}_{\hat{\Omega}}\theta) , where

    \hat{\Omega} = \{\hat{x}|\hat{x}\in VI(\mathcal{A},\mathcal{C})\ and \ \mathcal{B}\hat{x}\in Fix(\mathcal{T})\}\neq\emptyset.

    Algorithm 3.3. Choose an arbitrary initial value x_{1}\in \mathcal{C} . Assume x_{n} has been constructed. Compute

    \begin{eqnarray} \begin{split} \;& z_{n} = \mathcal{P}_{\mathcal{C}}(x_{n}-\varsigma_{n}\mathcal{A}x_{n}),\\ \;&x_{n+1} = \mathcal{P}_{\mathcal{C}}((1-\alpha_{n})(x_{n}-\varsigma_{n}\mathcal{A}z_{n})), \\ \end{split} \end{eqnarray} (3.30)

    where \{\alpha_{n}\} is a sequence in (0, 1) and \varsigma_{n} is a sequence in (0, \frac{1}{L}) .

    Theorem 3.3. If \hat{\Omega}\neq\emptyset and the following conditions are satisfied:

    ( C_{1} ) lim_{n\rightarrow \infty}\alpha_{n} = 0\ and\ \sum^{\infty}_{n = 0}\alpha_{n} = \infty ;

    ( C_{2} ) 0 < \liminf_{n\rightarrow \infty}\varsigma_{n}\leq \limsup_{n\rightarrow \infty}\varsigma_{n} < \frac{1}{L} ;

    Then, the iterative sequence \{x_{n}\} defined by Eq (3.30) strongly converges to the minimum-norm solution \hat{x}( = \mathcal{P}_{\Omega}\theta) , where \hat{\Omega} = \{\hat{x}|\hat{x}\in VI(\mathcal{A}, \mathcal{C})\}\neq\emptyset.

    In this section, we present some numerical examples to illustrate our main results. The MATLAB codes run in MATLAB version 9.5 (R2018b) on a PC Intel(R) Core(TM)i5-6200 CPU @ 2.30 GHz 2.40 GHz, RAM 8.00 GB. In all examples y -axes shows the value of \|x_{n+1}-x_{n}\| while the x -axis indicates to the number of iterations.

    Example 4.1. Let \mathcal{H}_{1} = \mathcal{H}_{2} = R^{n} . The feasible set is defined as:

    \mathcal{C}: = \{x\in R^{n}:\|x\|\leq1\}.

    Let G:R^{n}\rightarrow R^{n} is a linear operator defined by:

    \mathcal{A}x: = Gx

    for all x\in R^{n} , where G = (g_{ij})_{1\leq i, j \leq n} is a matrix in R^{n\times n} whose terms are given by:

    \begin{equation} g_{ij} = \left\{ \begin{aligned} &-1,\ & \mathrm{if}\ j = n+1-i\ \mathrm{and}\ j > i, \\ &1,\ & \mathrm{if}\ j = n+1-i\ \mathrm{and}\ j < i, \\ &0,\ & \mathrm{otherwise}. \\ \end{aligned} \right. \end{equation} (4.1)

    It is obvious that \mathcal{A} is \|G\| -Lipschitz continuous. By a direct calculation, we also have that \langle \mathcal{A}x, x\rangle = \langle Gx, x\rangle = 0 and so, \mathcal{A} is monotone. Let \mathcal{B} be a matrix in R^{n\times n} which is randomly generated.

    Taking cognizance of the difference of the problems handled by Algorithm 3.1 and Algorithm in Tian and Jiang [24], in order to comparing these two algorithms, we make a very small modification to the one in [24] such that it can also solve the problem (1.8). The modified algorithm can be written as follows:

    Algorithm 4.1.

    \begin{eqnarray} \begin{split} \;& y_{n} = x_{n}-\tau_{n} \mathcal{B}^\ast(I-\mathcal{W}_{n})\mathcal{B}x_{n},\\ \;& z_{n} = \mathcal{P}_{\mathcal{C}}(y_{n}-\varsigma_{n}\mathcal{A}y_{n}),\\ \;&x_{n+1} = \mathcal{P}_{\mathcal{C}}((1-\alpha_{n})(y_{n}-\varsigma_{n}\mathcal{A}z_{n})), \\ \end{split} \end{eqnarray} (4.2)

    According to the proof of Theorem 3.1, we can easily verify that this modified algorithm works for solving (1.8). The values of control parameters in these two Algorithms are \varsigma_{n} = \frac{1}{2\|G\|} , \tau_{n} = \frac{1}{2\|B\|^{2}} , \alpha_{1} = \frac{1}{2} , \alpha_{n} = \frac{1}{n} (for all n\geq2 ), \lambda_{n} = \frac{1}{n+1} and x_{1} = (1, \cdots, 1)^{T} , and the infinite family of nonexpansive mappings \{\mathcal{T}_{k}\}^{\infty}_{k = 1}:R^{n}\rightarrow R^{n} is defined by:

    \mathcal{T}_{k}x: = M_{k}x,

    for all x\in R^{n} , where \{M_{k}\} is a sequence of diagonal matrixes in R^{n\times n} :

    \begin{equation} M_{k} = \left[ \begin{array}{ccccc} 1-\frac{1}{k+2} &\ &\ &\ &\ \\ \ &1-\frac{1}{k+2} &\ &\ &\ \\ \ &\ &\ddots &\ &\ \\ \ &\ &\ &1-\frac{1}{k+2} &\ \\ \ &\ &\ &\ &1-\frac{1}{k+3}\\ \end{array} \right ]. \end{equation} (4.3)

    The numerical results of the Example 4.1 are reported in Table 1 and Figures 14 by using the stopping criterion \|x_{n+1}-x_{n}\|\leq10^{-10} .

    Table 1.  Example 4.1: Comparison of Algorithm 3.1 with Algorithm 4.1.
    No. of Iter. Time
    n Alg. 3.1 Alg. 4.1 Alg. 3.1 Alg. 4.1
    2 120 154 0.288s 1.133s
    10 156 201 0.691s 2.516s
    50 157 202 4.641s 13.853s
    100 157 203 12.333s 53.145s

     | Show Table
    DownLoad: CSV
    Figure 1.  Example 4.1: Comparison of Algorithm 3.1 with Algorithm 4.1 when n = 2 .
    Figure 2.  Example 4.1: Comparison of Algorithm 3.1 with Algorithm 4.1 when n = 10 .
    Figure 3.  Example 4.1: Comparison of Algorithm 3.1 with Algorithm 4.1 when n = 50 .
    Figure 4.  Example 4.1: Comparison of Algorithm 3.1 with Algorithm 4.1 when n = 100 .

    Example 4.2. Let \mathcal{H}_{1} = \mathcal{H}_{2} = L^{2}([0, 1]) with the inner product:

    \langle x,y\rangle = \int^{1}_{0}x(t)y(t)dt

    and the induced norm:

    \|x\|: = (\int^{1}_{0}x^{2}(t)dt)^{\frac{1}{2}}.

    The feasible set is defined as:

    \mathcal{C}: = \{x\in R^{n}:\|x\|\leq1\}.

    The mapping \mathcal{A}:L^{2}([0, 1])\rightarrow L^{2}([0, 1]) is defined by:

    \mathcal{A}x(t): = (1+t)\max\{0,x(t)\} = (1+t)\frac{x(t)+|x(t)|}{2},\ x\in L^{2}([0,1]).

    It is easy to see that

    \begin{eqnarray} \begin{split} \langle \mathcal{A}x\;-\mathcal{A}y,x-y\rangle& = \int^{1}_{0}(\mathcal{A}x(t)-\mathcal{A}y(t))(x(t)-y(t))dt \\ \;& = \int^{1}_{0}(1+t)\frac{x(t)-y(t)+|x(t)|-|y(t)|}{2}(x(t)-y(t))dt \\ \;& = \int^{1}_{0}\frac{1}{2}(1+t)((x(t)-y(t))^{2}+(|x(t)|-|y(t)|)(x(t)-y(t)))dt \\ \;&\geq0\\ \end{split} \end{eqnarray} (4.4)

    and

    \begin{eqnarray} \begin{split} \|\mathcal{A}x-\mathcal{A}y\|^{2} \;& = \int^{1}_{0}(\mathcal{A}x(t)-\mathcal{A}y(t))^{2}dt \\ \;& = \int^{1}_{0}(1+t)^{2}\frac{(x(t)-y(t)+|x(t)|-|y(t)|)^{2}}{4}dt \\ \;& = \int^{1}_{0}(1+t)^{2}(x(t)-y(t))^{2}dt \\ \;&\leq4\|x-y\|^{2}.\\ \end{split} \end{eqnarray} (4.5)

    Therefore, the operator \mathcal{A} is monotone and 2 -Lipschitz continuous. Let \mathcal{W}_{n} = \mathcal{I} (Identity mapping). The values of control parameters for Algorithm 4.1 and Algorithm 3.1 are \varsigma_{n} = \frac{1}{4} , \alpha_{1} = \frac{1}{2} , \alpha_{n} = \frac{1}{n} (for all n\geq2 ), \lambda_{n} = \frac{1}{n+1} and x_{1} = 8t^{2} . It can be seen easily that \{x_{n}\} strongly converges to the zero vector \theta(\in L^{2}([0, 1])). The numerical results of the Example 4.2 are reported in Table 2 and Figures 5 by using the stopping criterion \|x_{n+1}-x_{n}\|\leq\varepsilon = 0.01 .

    Table 2.  Example 4.2: Comparison of Algorithm 3.1 with Algorithm 4.1 when \varepsilon = 0.01 .
    No. of Iter. Time
    \varepsilon Alg. 3.1 Alg. 4.1 Alg. 3.1 Alg. 4.1
    0.01 8 13 0.678s 79.280s

     | Show Table
    DownLoad: CSV
    Figure 5.  Example 4.2: Comparison of Algorithm 3.1 with Algorithm 4.1 when \varepsilon = 0.01 .

    Remark 4.1. The numerical results of Example 4.1 and Example 4.2 show that the performance of Algorithm 3.1 is better than Algorithm 4.1 both in CPU time and the number of iterations. Algorithm 3.1 is more effective in both finite and infinite dimensional spaces and especially in conditions involving complex projection calculations, see Tables 1, 2 and Figures 15. In Example 4.1, we observe that the number of iterations tends to be stable, while the CPU time increases, as n increasing.

    In the present paper, we consider variational inequality problems and split common fixed point problems. We construct an iterative algorithm for solving Eq (1.8) which can be regard as a modification and generalization of Algorithm 1.1 with fewer metric projection operators. Under some mild restrictions, we demonstrate the strong convergence analysis of the presented algorithm. We also give some numerical examples to illustrate our main results. Noticeably, in our article, \mathcal{A} is assumed to a monotone and L -Lipschitz continuous mapping. A natural question arises: how to weaken this assumption?

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

    This work was supported by National Natural Science Foundation of China under Grant (No. 62103289), National Natural Science Foundation of China (No. 11761074), Project of Jilin Science and Technology Development for Leading Talent of Science and Technology Innovation in Middle and Young and Team Project (No. 20200301053RQ), and Natural Science Foundation of Jilin Province(No. 2020122336JC).

    The authors declare no conflict of interest.



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