
Citation: Xinqi Dong, Ruijia Chen, Susan K. Roepke-Buehler. Characteristics Associated with Psychological, Physical, Sexual Abuse, Caregiver Neglect and Financial Exploitation in U.S. Chinese Older Adults: Findings from the Population-Based Cohort Study in the Greater Chicago Area[J]. AIMS Medical Science, 2014, 1(2): 103-124. doi: 10.3934/medsci.2014.2.103
[1] | Yali Zhao, Qixin Dong, Xiaoqing Huang . A self-adaptive viscosity-type inertial algorithm for common solutions of generalized split variational inclusion and paramonotone equilibrium problem. AIMS Mathematics, 2025, 10(2): 4504-4523. doi: 10.3934/math.2025208 |
[2] | Pongsakorn Yotkaew, Nopparat Wairojjana, Nuttapol Pakkaranang . Accelerated non-monotonic explicit proximal-type method for solving equilibrium programming with convex constraints and its applications. AIMS Mathematics, 2021, 6(10): 10707-10727. doi: 10.3934/math.2021622 |
[3] | Mohammad Dilshad, Fahad Maqbul Alamrani, Ahmed Alamer, Esmail Alshaban, Maryam G. Alshehri . Viscosity-type inertial iterative methods for variational inclusion and fixed point problems. AIMS Mathematics, 2024, 9(7): 18553-18573. doi: 10.3934/math.2024903 |
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[5] | Mohammad Dilshad, Aysha Khan, Mohammad Akram . Splitting type viscosity methods for inclusion and fixed point problems on Hadamard manifolds. AIMS Mathematics, 2021, 6(5): 5205-5221. doi: 10.3934/math.2021309 |
[6] | Saud Fahad Aldosary, Mohammad Farid . A viscosity-based iterative method for solving split generalized equilibrium and fixed point problems of strict pseudo-contractions. AIMS Mathematics, 2025, 10(4): 8753-8776. doi: 10.3934/math.2025401 |
[7] | Yu Zhang, Xiaojun Ma . An accelerated conjugate method for split variational inclusion problems with applications. AIMS Mathematics, 2025, 10(5): 11465-11487. doi: 10.3934/math.2025522 |
[8] | Bancha Panyanak, Chainarong Khunpanuk, Nattawut Pholasa, Nuttapol Pakkaranang . A novel class of forward-backward explicit iterative algorithms using inertial techniques to solve variational inequality problems with quasi-monotone operators. AIMS Mathematics, 2023, 8(4): 9692-9715. doi: 10.3934/math.2023489 |
[9] | Jun Yang, Prasit Cholamjiak, Pongsakorn Sunthrayuth . Modified Tseng's splitting algorithms for the sum of two monotone operators in Banach spaces. AIMS Mathematics, 2021, 6(5): 4873-4900. doi: 10.3934/math.2021286 |
[10] | Yasir Arfat, Supak Phiangsungnoen, Poom Kumam, Muhammad Aqeel Ahmad Khan, Jamshad Ahmad . Some variant of Tseng splitting method with accelerated Visco-Cesaro means for monotone inclusion problems. AIMS Mathematics, 2023, 8(10): 24590-24608. doi: 10.3934/math.20231254 |
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 $ \langle \cdot, \cdot \rangle $ and $ \| \cdot \|, $ respectively. Let $ C $ be a nonempty, closed and convex subset of $ H $ and $ A: C \rightarrow H $ be a nonlinear mapping. The variational inequality problem (VIP) associated with the set $ C $ and the mapping $ A $ is stated as follows:
$ findx∗∈Csuch that⟨Ax∗,x−x∗⟩≥0,∀x∈C. $ | (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 = I-S $, with $ S $ another nonexpansive self-mapping of $ C $. In other words, $ VIP $ is of the form
$ findx∗∈Fix(T)such that⟨x∗−Sx∗,x−x∗⟩≥0,∀x∈Fix(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)+(1−λn)(αnSxn+(1−αn)Txn), $ | (1.3) |
where $ f:C \rightarrow C $ and $ \{ \alpha_n \}, \{ \lambda_n \} \subset (0, 1) $. They also proved the strong convergence of the sequence $ \{ x_n \} $ 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 $ \{ x_n \} $ by the following explicit scheme:
$ {yn=βnSxn+(1−βn)xn,xn+1=αnf(xn)+(1−αn)Tyn,n≥1. $ | (1.4) |
Under appropriate conditions, the above iterative sequence $ \{x_n\} $ converges strongly to some fixed point of $ T $ where $ T $ is nonexpansive mapping and $ \{x_n\} $ 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 $ \{ x_n \} $ by the following iterative scheme:
$ {F(un,y)+h(un,y)+1rn⟨y−un,un−xn⟩,∀y∈C,yn,1=βn,1S1un+(1−βn,1)un,yn,i=βn,iSiun+(1−βn,i)yn,i−1,i=2,…,N,xn+1=αnf(xn)+(1−αn)Tyn,N],n≥1. $ | (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:C \rightarrow H $ be $ k $-Lipschitzian and $ \eta $-strongly monotone on the nonempty, closed and convex subset $ C $ of $ H $, where $ k $ and $ \eta $ are positive constants, that is,
$ ‖Fx−Fy‖≤k‖x−y‖and⟨Fx−Fy,x−y⟩≥η‖x−y‖2,∀x,y∈C. $ | (1.6) |
Let $ f:C \rightarrow H $ be a $ \rho $-contraction with a coefficient $ \rho \in [0, 1) $ and $ S, T: C \rightarrow C $ be two nonexpansive mappings with $ Fix(T) \neq \emptyset. $ Let $ 0 < \mu < \frac{2\eta}{k^2} $ and $ 0 < \gamma \leq \tau $, where $ \tau = 1- \sqrt{1-\mu (2 \eta -\mu k^2)}. $ Consider the following triple hierarchical variational inequality problem (THVI): find $ x^* \in \Xi $ such that
$ ⟨(μF−γf)x∗,x−x∗⟩≥0,∀x∈Ξ, $ | (1.7) |
where $ \Xi $ denotes the solution set of the following hierarchical variational inequality problem (HVIP): find $ z^* \in Fix(T) $ such that
$ ⟨(μF−γS)z∗,z−z∗⟩≥0,∀z∈Fix(T), $ | (1.8) |
where the solution set $ \Xi $ 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 $ x^{*} \in H_1 $ such that
$ {0∈f1x∗+B1x∗,y∗=Ax∗∈H2:0∈f2y∗+B2y∗, $ | (1.9) |
where $ f_1 : H_1 \rightarrow H_1 $ and $ f_2: H_2 \rightarrow H_2 $ are two given single-valued mappings, $ A:H_1 \rightarrow H_2 $ is a bounded linear operator, and $ B_1 : H_1 \rightarrow 2^{H_1} $ and $ B_2 : H_2 \rightarrow 2^{H_2} $ are multivalued maximal monotone mappings.
If $ f_1 = f_2 \equiv 0 $, then (1.9) reduces to the following split variational inclusion problem (SVIP): find $ x^{*} \in H_1 $ such that
$ {0∈B1x∗,y∗=Ax∗∈H2:0∈B2y∗. $ | (1.10) |
Additionally, if $ f_1 \equiv 0 $, then (1.9) reduces to the following split monotone variational inclusion problem ($ SMVIP $): find $ x^* \in H_1 $ such that
$ {0∈B1x∗,y∗=Ax∗∈H2:0∈fy∗+B2y∗. $ | (1.11) |
We denote the solution sets of variational inclusion $ 0\in B_1 x^{*} $ and $ 0\in f y^{*} + B_2 y^{*} $ by $ SOLVIP(B_1) $ and $ SOLVIP(f+B_2) $, respectively. Thus, the solution set of $ (1.11) $ can denoted by $ \Gamma = \{x^*\in H_1 : x^{*} \in SOLVIP(B_1), Ax^{*} \in SOLVIP(f+B_2) \} $.
In 2012, Byrne et al. [21] studied the following iterative scheme for $ SVIP \; (1.10) $: for a given $ x_0 \in H_1 $ and $ \lambda > 0 $,
$ xn+1=JB1λ[xn+ϵA∗(JB2λ−I)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∗(JB2λ−I)Axn],xn+1=αnf(xn)+(1−αn)Tun, $ | (1.13) |
where $ A $ is a bounded linear operator, $ A^{*} $ is the adjoint of $ A $, $ f $ is a contraction on $ H_1 $, and $ T $ is a nonexpansive mapping of $ H_1 $. 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∗(JB2λ−I)Axn],yn=βnun+(1−βn)∞∑n=1η(n)iTiun,xn+1=αnτf(xn)+(I−αnD)yn,n≥1, $ | (1.14) |
where $ A $ is a bounded linear operator, $ A^{*} $ is the adjoint of $ A $, $ \{ T_i \}_{i = 1}^{N} $ is a family of $ k_i $-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 $ \{ \alpha_{n } \}, \{ \beta_{n } \} $ and $ \{ \eta_{i}^{(n)} \}_{i = 1}^{N} $ that $ \{x_n\} $, defined by (1.14), converges strongly to a common solution of $ SVIP \; (1.10) $ and a fixed point of a finite family of $ k_i $-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:C \rightarrow H $ be $ k $-Lipschitzian and $ \eta $-strongly monotone on the nonempty closed and convex subset $ C $ of $ H $, $ \psi:C \rightarrow H $ be a $ \rho $-contraction with coefficient $ \rho \in [0, 1) $ and $ S_i, S, T: C \rightarrow C $ be nonexpansive mappings for all $ i\in \{ 1, \ldots, N \} $. Let $ 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 objective is to find $ x^* \in \Omega $ such that
$ {⟨(μF−ξψ)x∗,x−x∗⟩≥0,∀x∈Ω,⟨(μF−ξS)x∗,y−x∗⟩≥0,∀y∈Ω, $ | (1.15) |
where $ \Omega = Fix(T)\cap(\bigcap_i Fix (S_i))\cap \Gamma \neq \emptyset. $
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 $ \{ x_n \} $ 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 $ \{ x_n \} $ to a point $ x \in H $ by $ x_n \rightarrow x $ and $ x_n \rightharpoonup x $, respectively. It is also well known [24] that the Hilbert space $ H $ satisfies $ Opail's \; condition $, that is, for any sequence $ \{ x_n \} $ with $ x_n \rightharpoonup x $, the inequality
$ lim supn→∞‖xn−x‖<lim supn→∞‖xn−y‖ $ | (2.1) |
holds for every $ y\in H $ with $ y \neq x $.
In the sequel, given a sequence $ \{ z_n \} $, we denote with $ \omega_w (z_n) $ the set of cluster points of $ \{ z_n \} $ with respect to the weak topology, that is,
$ ωw(zn)={z∈H:there existsnk→∞for whichznk⇀z}. $ |
Analogously, we denote by $ \omega_s (z_n) $ the set of cluster points of $ \{ z_n \} $ with respect to the norm topology, that is,
$ ωs(zn)={z∈H:there existsnk→∞for whichznk→z}. $ |
Lemma 2.1. In a real Hilbert space $ H $, the following inequalities hold:
(1) $ \|x-y\|^2 = \|x\|^2-\|y\|^2-2 \langle x-y, y\rangle, \forall x, y \in H $;
(2) $ \|x+y\|^2 \leq \|x\|^2+2 \langle y, x+y\rangle, \forall x, y \in H $;
(3) $ \|\lambda x+(1-\lambda) y\|^2 = \lambda \|x\|^2+(1-\lambda)\|y\|^2-\lambda(1-\lambda)\|x-y\|^2, \forall \lambda \in [0, 1], \forall x, y \in H $;
An element $ x \in C $ is called a $ fixed \; point $ of $ S $ if $ x\in Sx $. The set of all fixed point of $ S $ is denoted by $ Fix(S) $, that is, $ Fix(S) = \{ x\in C: x\in S x \} $.
Recall the following definitions. Moreover, $ S:H_1 \rightarrow H_1 $ is called
$ (1) $ a nonexpansive mapping if
$ ‖Sx−Sy‖≤‖x−y‖,∀x,y∈H1. $ | (2.2) |
A nonexpansive mapping with $ k = 1 $ can be strengthened to a firmly nonexpansive mapping in $ H_1 $ if the following holds:
$ ‖Sx−Sy‖2≤⟨x−y,Sx−Sy⟩,∀x,y∈H1. $ | (2.3) |
We note that every nonexpansive operator $ S:H_1\rightarrow H_1 $ satisfies, for all $ (x, y)\in H_1 \times H_1 $, the inequality
$ ⟨(x−Sx)−(y−Sy),Sy−Sx)⟩≤12‖(Sx−x)−(Sy−y)‖2, $ | (2.4) |
and therefore, we obtain, for all $ (x, y)\in H_1\times Fix(S) $,
$ ⟨x−Sx,y−Sx⟩≤12‖Sx−x‖2 $ | (2.5) |
(see, e.g., Theorem 3 in [16] and Theorem 1 in [17]).
$ (2) $ a contractive if there exists a constant $ \alpha \in (0, 1) $ such that
$ ‖Sx−Sy‖≤α‖x−y‖,∀x,y∈H1. $ | (2.6) |
$ (3) $ an $ L $-Lipschitzian if there exists a positive constant $ L $ such that
$ ‖Sx−Sy‖≤L‖x−y‖,∀x,y∈H1. $ | (2.7) |
$ (4) $ an $ \eta $-strongly monotone if there exists a positive constant $ \eta $ such that
$ ⟨Sx−Sy,x−y⟩≥η‖x−y‖2,∀x,y∈H1. $ | (2.8) |
$ (5) $ an $ \beta $-inverse strongly monotone ($ \beta-ism $) if there exists a positive constant $ \beta $ such that
$ ⟨Sx−Sy,x−y⟩≥β‖Sx−Sy‖2,∀x,y∈H1. $ | (2.9) |
$ (6) $ averaged if it can be expressed as the average of the identity mapping and a nonexpansive mapping, i.e.,
$ S:=(1−α)I+αT, $ | (2.10) |
where $ \alpha \in (0, 1), I $ is the identity operator on $ H_1 $ and $ T:H_1 \rightarrow H_1 $ 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 $ H_1 $ if there exists a positive constant $ \bar{\tau} > 0 $ such that
$ ⟨Dx,x⟩≥ˉτ‖x‖2,∀x∈H1. $ | (2.11) |
From the definition above, we easily find that a strongly positive bounded linear operator $ D $ is $ \bar{\tau} $-strongly monotone and $ \|D\| $-Lipschitzian.
$ (8) $ A multivalued mapping $ M : D(M) \subseteq H_1 \rightarrow 2^{H_1} $ is called monotone if for all $ x, y \in D(M), u\in Mx $ and $ v \in My $,
$ ⟨x−y,u−v⟩≥0. $ | (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 $ x\in D(M), u\in H_1, \langle x-y, u-v \rangle \geq 0 $ for each $ (y, v) \in Graph (M) $, $ u\in Mx $.
$ (9) $ Let $ M:D(M) \subseteq H_1 \rightarrow 2^{H_1} $ be a multivalued maximal monotone mapping. Then, the resolvent operator $ J_{\lambda}^{M}: H_1 \rightarrow D(M) $ is defined by
$ JMλx:=(I+λM)−1(x),∀x∈H1, $ | (2.13) |
for $ \forall \lambda > 0 $, where $ I $ stands for the identity operator on $ H_1 $. We observe that $ J_{\lambda}^{M} $ is single-valued, nonexpansive, and firmly nonexpansive.
We recall some concepts and results that are needed in the sequel. A mapping $ P_C $ is said to be a metric projection of $ H_1 $ onto $ C $ if for every point $ x\in H_1 $, there exists a unique nearest point in $ C $ denoted by $ P_C x $ such that
$ ‖x−PCx‖≤‖x−y‖,∀y∈C. $ | (2.14) |
It is well known that $ P_C $ is a nonexpansive mapping and is characterized by the following property:
$ ‖PCx−PCy‖2≤⟨x−y,PCx−PCy⟩,∀x,y∈H1. $ | (2.15) |
Moreover, $ P_C x $ is characterized by the following properties:
$ ⟨x−PCx,y−PCx⟩≤0,∀x∈H1,y∈C, $ | (2.16) |
$ ‖x−y‖2≥‖x−PCx‖2+‖y−PCx‖2,∀x∈H1,y∈C, $ | (2.17) |
and
$ ‖(x−y)−(PCx−PCy)‖2≥‖x−y‖2−‖PCx−PCy‖2,∀x,y∈H1. $ | (2.18) |
Proposition 2.2. [20]
(1) If $ T = (1-\alpha) S + \alpha V, $ where $ S: H_1 \rightarrow H_1 $ is averaged, $ V: H_1 \rightarrow H_1 $ is nonexpansive, and if $ \alpha \in [0, 1], $ then $ T $ is averaged.
(2) The composite of finitely many averaged mappings is averaged.
(3) If the mappings $ \{ T_i \}_{i = 1}^{N} $ are averaged and have a nonempty common fixed point, then
$ N⋂i=1F(Ti)=F(T1∘T2∘…∘TN). $ | (2.19) |
(4) If $ T $ is a $ v-ism $, then for $ \gamma > 0, \gamma T $ is a $ \frac{v}{\gamma}-ism. $
(5) $ T $ is averaged if and only if its complement $ I-T $ is a $ v-ism $ for some $ v > \frac{1}{2} $.
Proposition 2.3. [20] Let $ \lambda > 0, h $ be an $ \alpha-ism $ operator, and $ B $ be a maximal monotone operator. If $ \lambda \in (0, 2\alpha) $, then it is easy to see that the operator $ J_{\lambda}^{B}(I-\lambda h) $ is averaged.
Proposition 2.4. [20] Let $ \lambda > 0 $ and $ B_1 $ be a maximal monotone operator. Then,
$ x∗solves(1.9)⇔x∗=JB1λ(I−λf1)x∗andAx∗=JB2λ(I−λf2)Ax∗. $ | (2.20) |
Lemma 2.5. [23] Let $ \{s_n\} $ be a sequence of nonnegative numbers satisfying the condition
$ sn+1≤(1−γn)sn+γnδn,n≥1, $ |
where $ \{ \gamma_n \}, \{ \delta_n \} $ are the sequences of real numbers such that
(i) $ \{ \gamma_n \} \subset [0, 1] $ and $ \mathop {\mathop \sum \limits^\infty }\limits_{n = 1} \gamma_n = \infty $, or equivalently,
$ ∞Πn=1(1−γn):=limn→∞∞Πk=1(1−γk)=0; $ |
(ii) $ \underset{n \rightarrow \infty}{\limsup} \delta_n \leq 0 $, or
(iii) $ \mathop {\mathop \sum \limits^\infty }\limits_{n = 1} \gamma_n \delta_n $ is convergent.
Then, $ \underset{n \rightarrow \infty}{\lim} s_n = 0 $.
Lemma 2.6. [23] Let $ \lambda $ be a number $ (0, 1] $, and let $ \mu > 0. $ Let $ F:C \rightarrow H $ be an operator on $ C $ such that for some constant $ k, \eta > 0, F $ is $ k $-Lipschitzian and $ \eta $-strongly monotone. Associating with a nonexpansive mapping $ T:C \rightarrow C $, we define the following the mapping $ T^{\lambda}:C \rightarrow H $ by
$ Tλx:=Tx−λμF(Tx),∀x∈C. $ | (2.21) |
Then, $ T^{\lambda} $ is a contraction provided $ \mu < \frac{2\eta }{k^2} $, that is,
$ ‖Tλx−Tλy‖≤(1−λτ)‖x−y‖,∀x,y∈C, $ | (2.22) |
where $ \tau = 1- \sqrt{1-\mu (2\eta - \mu k^2)} \in (0, 1] $.
Lemma 2.7. [25] Let $ \{ \alpha_n \} $ be a sequence of nonnegative real numbers with $ \underset{n \rightarrow \infty}{\limsup} \alpha_n < \infty $ and $ \{ \beta_n \} $ be a sequence of real numbers with $ \underset{n \rightarrow \infty}{\limsup} \beta_n \leq 0 $. Then, $ \underset{n \rightarrow \infty}{\limsup} \alpha_n \beta_n \leq 0 $.
Lemma 2.8. [28] Assume that $ T $ is nonexpansive self-mapping of a closed convex subset $ C $ of a Hilbert space $ H_1 $. If $ T $ has a fixed point, then $ I-T $ is demiclosed, i.e., whenever $ \{ x_n \} $ weakly converges to some $ x $ and $ \{ (I-T) x_n \} $ converges strongly to $ y $, it follows that $ (I-T)x = y $. Here, $ I $ is the identity mapping on $ H_1. $
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, $ 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:
$ {un=JB1λ1[xn+γA∗(JB2λ2(I−λ2f)−I)Axn],yn,1=βn,1S1un+(1−βn,1)un,yn,i=βn,iSiun+(1−βn,i)yn,i−1,i=2,…,N,xn+1=PC[λnξ(αnψ(xn)+(1−αn)Sxn)+(I−λnμF)Tyn,N],n≥1. $ | (3.1) |
Assume that Problem 1.2 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 1.2.
Proof. Let $ \{ x_n \} $ be a sequence generated by scheme (4.1). First, note that $ 0 < \xi \leq \tau $ and
$ μη≥τ⇔μη≥1−√1−μ(2η−μk2)⇔√1−μ(2η−μk2)≥1−μη⇔1−2μη+μ2k2≥1−2μη+μ2η2⇔k2≥η2⇔k≥η. $ |
Then, it follows from the $ \rho $-contractiveness of $ \psi $ that
$ ⟨(μF−ξψ)x−(μF−ξψ)y,x−y⟩≥(μη−ξρ)‖x−y‖2,∀x,y∈C. $ |
Hence, from $ \xi \rho < \xi \leq \tau \leq \mu \eta $, we deduce that $ \mu F - \xi \psi $ is $ (\mu \eta - \xi \rho) $-strongly monotone. Since it is clear that $ \mu F - \xi \psi $ is Lipshitz continuous, there exists a unique solution to the VIP:
$ findx∗∈Ωsuch that⟨(μF−ξψ)x∗,x−x∗⟩≥0,∀x∈Ω. $ |
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 $ \{ \alpha_n \} \subset [a, b] $ for some $ a, b \in (0, 1). $
Let $ \mathcal{U} : = J_{\lambda_2}^{B_2}(I-\lambda_2 f) $; the iterative scheme (4.1) can be rewritten as
$ {un=JB1λ1[xn+γA∗(U−I)Axn],yn,1=βn,1S1un+(1−βn,1)un,yn,i=βn,iSiun+(1−βn,i)yn,i−1,i=2,…,N,xn+1=PC[λnξ(αnψ(xn)+(1−αn)Sxn)+(I−λnμF)Tyn,N],n≥1. $ | (3.2) |
The rest of the proof is divided into several steps.
Step 1. We show that the sequences $ \{ x_n\}, \{ y_{n, i} \} $ for all $ i, \{ u_n \} $ are bounded.
Indeed, take a point $ p\in \Omega $ arbitrarily. Then, $ J_{\lambda_1}^{B_1}p = p, \mathcal{U}(Ap) = Ap $, and it is easily seen that $ Wp = p $, where $ W: = I+\gamma A^{*}(\mathcal{U}-I)A. $ From the definition of firm nonexpansion and Proposition 2.3, we have that $ J_{\lambda_1}^{B_1} $ and $ \mathcal{U} $ are averaged. Likewise, $ W $ is also averaged because it is a $ \frac{v}{r}-ism $ for some $ v > \frac{1}{2} $. Actually, by Proposition 2.2 (5), we know that $ I-\mathcal{U} $ is a $ v-ism $ with $ v > \frac{1}{2} $. Hence, we have
$ ⟨A∗(I−U)Ax−A∗(I−U)Ay,x−y⟩=⟨(I−U)Ax−(I−U)Ay,Ax−Ay⟩≥v‖(I−U)Ax−(I−U)Ay‖2≥vr‖A∗(I−U)Ax−A∗(I−U)Ay‖2. $ |
Thus, $ \gamma A^{*} (I-\mathcal{U})A $ is a $ \frac{v}{\gamma r}-ism. $ Due to the condition $ 0 < \gamma < \frac{1}{r} $, the complement $ I-\gamma A^{*}(I-\mathcal{U})A $ is averaged, as well as $ M: = J_{\lambda_1}^{B_1}[I+\gamma A^{*}(\mathcal{U}-I)A] $. Therefore, $ J_{\lambda_1}^{B_1}, \mathcal{U}, W, $ and $ M $ are nonexpansive mappings.
From $ (3.2) $, we estimate
$ ‖un−p‖2=‖JB1λ1[xn+γA∗(U−I)Axn]−JB1λ1p‖2≤‖xn+γA∗(U−I)Axn−p‖2=‖xn−p‖2+γ2‖A∗(U−I)Axn‖2+2γ⟨xn−p,A∗(U−I)Axn⟩. $ | (3.3) |
Thus, we obtain
$ ‖un−p‖2≤‖xn−p‖2+γ2⟨(U−I)Axn,AA∗(U−I)Axn⟩+2γ⟨xn−p,A∗(U−I)Axn⟩. $ | (3.4) |
Next, setting $ \vartheta_1: = \gamma^2 \langle (\mathcal{U}-I)A x_n, A A^{*} (\mathcal{U} - I)A x_n \rangle $, we estimate
$ ϑ1=γ2⟨(U−I)Axn,AA∗(U−I)Axn⟩≤rγ2⟨(U−I)Axn,(U−I)Axn⟩=rγ2‖(U−I)Axn‖2. $ | (3.5) |
Setting $ \vartheta_2: = 2\gamma \langle x_n-p, A^{*}(\mathcal{U} - I) A x_n \rangle $, we obtain from (2.5) the following:
$ ϑ2=2γ⟨xn−p,A∗(U−I)Axn⟩=2γ⟨A(xn−p),(U−I)Axn⟩=2γ⟨A(xn−p)+(U−I)Axn−(U−I)Axn,(U−I)Axn⟩=2γ(⟨UAxn−Ap,(U−I)Axn⟩−‖(U−I)Axn‖2)≤2γ(12‖(U−I)Axn‖2−‖(U−I)Axn‖2)≤−γ‖(U−I)Axn‖2. $ | (3.6) |
In view of (3.4)-(3.6), we have
$ ‖un−p‖2≤‖xn−p‖2+γ(rγ−1)‖(U−I)Axn‖2. $ | (3.7) |
From $ 0 < \gamma < \frac{1}{r} $, we obtain
$ ‖un−p‖≤‖xn−p‖. $ | (3.8) |
Thus, we have from $ (3.2) $ and $ (3.8) $ that
$ ‖yn,1−p‖≤βn,1‖S1un−p‖+(1−βn,1)‖un−p‖≤‖un−p‖≤‖xn−p‖. $ | (3.9) |
For all $ i $ from $ i = 2 $ to $ i = N $, by induction, one proves that
$ ‖yn,i−p‖≤βn,i‖un−p‖+(1−βn,i)‖yn,i−1−p‖≤‖un−p‖≤‖xn−p‖. $ | (3.10) |
Hence, we obtain that for all $ i\in \{ 1, \ldots, N \} $,
$ ‖yn,i−p‖≤‖un−p‖≤‖xn−p‖. $ | (3.11) |
In addition, utilizing Lemma 2.6 and (3.2), we have
$ ‖xn+1−p‖=‖PC[λnξ(αnψ(xn)+(1−αn)Sxn)+(I−λnμF)Tyn,N]−PCp‖≤‖λnξ(αnψ(xn)+(1−αn)Sxn)+(I−λnμF)Tyn,N−p‖=‖λnξ(αnψ(xn)+(1−αn)Sxn)−λnμFTp+(I−λnμF)Tyn,N−(I−λnμF)Tp‖≤‖λnξ(αnψ(xn)+(1−αn)Sxn)−λnμFTp‖+‖(I−λnμF)Tyn,N−(I−λnμF)Tp‖=λn‖αn(ξψ(xn)−μFp)+(1−αn)(ξSxn−μFp)‖+‖(I−λnμF)Tyn,N−(I−λnμF)Tp‖≤λn[αn‖ξψ(xn)−μFp‖+(1−αn)‖ξSxn−μFp‖]+(1−λnτ)‖yn,N−p‖≤λn[αn(‖ξψ(xn)−ξψ(p)‖+‖ξψ(p)−μFp‖)+(1−αn)(‖ξSxn−ξSp‖+‖ξSp−μFp‖)]+(1−λnτ)‖yn,N−p‖≤λn[αnξρ‖xn−p‖+αn‖ξψ(p)−μFp‖+(1−αn)ξ‖xn−p‖+(1−αn)‖ξSp−μFp‖]+(1−λnτ)‖xn−p‖≤λn[ξ(1−αn(1−ρ))‖xn−p‖+max{‖ξψ(p)−μFp‖,‖ξSp−μFp‖}]+(1−λnτ)‖xn−p‖≤(1−λnξαn(1−ρ))‖xn−p‖+λnmax{‖ξψ(p)−μFp‖,‖ξSp−μFp‖}≤(1−λnξa(1−ρ))‖xn−p‖+λnmax{‖ξψ(p)−μFp‖,‖ξSp−μFp‖}, $ | (3.12) |
due to $ 0 < \xi \leq \tau. $ Thus, calling
$ M=max{‖x1−p‖,‖ξψ(p)−μFp‖ξa(1−ρ),‖ξSp−μFp‖ξa(1−ρ)}, $ |
by induction, we derive $ \| x_n-p\| \leq M $ for all $ n \geq 1 $. We thus obtain the claim.
Step 2. We show that $ \underset{n \rightarrow \infty}{\lim} \| x_{n+1} - x_n \| = 0 $.
Indeed, for each $ n\geq 1, $ we set
$ zn=λnξ(αnψ(xn)+(1−αn)Sxn)+(I−λnμF)Tyn,N. $ |
Then, we observe that
$ zn−zn−1=αnλnξ[ψ(xn)−ψ(xn−1)]+λn(1−αn)ξ(Sxn−Sxn−1)+[(I−λnμF)Tyn,N−(I−λnμF)Tyn−1,N]+(αnλn−αn−1λn−1)ξ[ψ(xn−1)−Sxn−1]+(λn−λn−1)(ξSxn−1−μFTyn−1,N). $ | (3.13) |
Let $ M_{0} > 0 $ be a constant such that
$ supn≥1{ξ‖ψ(xn)−Sxn‖+‖ξSxn−μFTyn,N‖}≤M0. $ |
It follows from (3.2) and (3.13) that
$ ‖xn+1−xn‖=‖PCzn−PCzn−1‖≤‖zn−zn−1‖≤αnλnξ‖ψ(xn)−ψ(xn−1)‖+λn(1−αn)ξ‖Sxn−Sxn−1‖+‖(I−λnμF)Tyn,N−(I−λnμF)Tyn−1,N‖+|αnλn−αn−1λn−1|ξ‖ψ(xn−1)−Sxn−1‖+|λn−λn−1|‖ξSxn−1−μFTyn−1,N‖≤αnλnξρ‖xn−xn−1‖+λn(1−αn)ξ‖xn−xn−1‖+(1−λnτ)‖yn,N−yn−1,N‖+|αnλn−αn−1λn−1|M0+|λn−λn−1|M0=λn(1−αn(1−ρ))ξ‖xn−xn−1‖+(1−λnτ)‖yn,N−yn−1,N‖+[|αnλn−αn−1λn−1|+|λn−λn−1|]M0≤λnξ(1−a(1−ρ))‖xn−xn−1‖+(1−λnτ)‖yn,N−yn−1,N‖+[|αnλn−αn−1λn−1|+|λn−λn−1|]M0. $ | (3.14) |
By the definition of $ y_{n, i} $, we obtain that for all $ i = N, \ldots, 2 $,
$ ‖yn,i−yn−1,i‖≤βn,i‖un−un−1‖+‖Siun−1−yn−1,i−1‖|βn,i−βn−1,i|+(1−βn,i)‖yn,i−1−yn−1,i−1‖. $ | (3.15) |
In this case $ i = 1 $, we have
$ ‖yn,1−yn−1,1‖≤βn,1‖un−un−1‖+‖S1un−1−un−1‖|βn,1−βn−1,1|+(1−βn,1)‖un−un−1‖=‖un−un−1‖+‖S1un−1−un−1‖|βn,1−βn−1,1|. $ | (3.16) |
Substituting (3.16) in all (3.15)-type inequalities, we find that for $ i = 2, \ldots, N $,
$ ‖yn,i−yn−1,i‖≤ ‖un−un−1‖+N∑k=2‖Skun−1−yn−1,k−1‖|βn,k−βn−1,k|+‖S1un−1−un−1‖|βn,1−βn−1,1|. $ |
Thus, we conclude that
$ ‖xn+1−xn‖≤λnξ(1−a(1−ρ))‖xn−xn−1‖+(1−λnτ)‖yn,N−yn−1,N‖+[|αnλn−αn−1λn−1|+|λn−λn−1||]M0≤λnξ(1−a(1−ρ))‖xn−xn−1‖+[|αnλn−αn−1λn−1|+|λn−λn−1|]M0+(1−λnτ)‖un−un−1‖+N∑k=2‖Skun−1−yn−1,k−1‖|βn,k−βn−1,k|+‖S1un−1−un−1‖|βn,1−βn−1,1|. $ | (3.17) |
Since $ J_{\lambda_1}^{B_1}[I + \gamma A^{*}(\mathcal{U}-I)A] $ is nonexpansive, we obtain
$ ‖un−un−1‖=‖JB1λ1[I+γA∗(U−I)A]xn−JB1λ1[I+γA∗(U−I)A]xn−1‖≤‖xn−xn−1‖. $ | (3.18) |
Substituting (3.18) into (3.17), we have
$ ‖xn+1−xn‖≤λnξ(1−a(1−ρ))‖xn−xn−1‖+[|αnλn−αn−1λn−1|+|λn−λn−1|]M0+(1−λnτ)‖xn−xn−1‖+N∑k=2‖Skun−1−yn−1,k−1‖|βn,k−βn−1,k|+‖S1un−1−un−1‖|βn,1−βn−1,1|. $ | (3.19) |
If we call $ M_{1} : = \max \bigg \{ M_0, \underset{n \geq 2, i = 2, \ldots, N}\sup \| S_i u_{n-1} - y_{n-1, i-1}\|, \underset{n\geq 2} \sup \|S_1 u_{n-1}-u_{n-1} \| \bigg \} $, we have
$ ‖xn+1−xn‖≤(1−λnξa(1−ρ))‖xn−xn−1‖+M1[|αnλn−αn−1λn−1|+|λn−λn−1|+N∑k=2|βn,k−βn−1,k|], $ | (3.20) |
due to $ 0 < \xi < \tau $. By condition $ (C2)-(C5) $ and Lemma 2.5, we obtain that
$ limn→∞‖xn+1−xn‖=0. $ | (3.21) |
Step 3. We show that $ \underset{n \rightarrow \infty}{\lim} \| x_{n} - u_{n} \| = 0 $.
From (3.2) and (3.7), we have
$ ‖xn+1−p‖2≤‖λnξ(αnψ(xn)+(1−αn)Sxn)+(I−λnμF)Tyn,N−p‖2=‖λnξ(αnψ(xn)+(1−αn)Sxn)−λnμFTp+(I−λnμF)Tyn,N−(I−λnμF)Tp‖2≤{‖λnξ(αnψ(xn)+(1−αn)Sxn)−λnμFTp‖+‖(I−λnμF)Tyn,N−(I−λnμF)Tp‖}2≤{λn‖αn(ξψ(xn)−μFp)+(1−αn)(ξSxn−μFp)‖+(1−λnτ)‖yn,N−p‖}2≤λn1τ[αn‖ξψ(xn)−μFp‖+(1−αn)‖ξSxn−μFp‖]2+(1−λnτ)‖yn,N−p‖2≤λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+(1−λnτ)‖un−p‖2≤λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+(1−λnτ)[‖xn−p‖2+γ(rγ−1)‖(U−I)Axn‖2]=λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+(1−λnτ)‖xn−p‖2−γ(1−rγ)(1−λnτ)‖(U−I)Axn‖2, $ | (3.22) |
which implies that
$ (1−λnτ)γ(1−rγ)‖(U−I)Axn‖2≤λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+(1−λnτ)‖xn−p‖2−‖xn+1−p‖2≤λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+‖xn−p‖2−‖xn+1−p‖2≤λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+‖xn+1−xn‖(‖xn−p‖+‖xn+1p‖). $ | (3.23) |
Since $ \gamma (1-r \gamma) > 0, \|x_{n+1}-x_n\| \rightarrow 0, \lambda_{n} \rightarrow 0 $ as $ n \rightarrow \infty $, and by the boundedness of $ \{ x_n \} $, we conclude that
$ limn→∞‖(U−I)Axn‖=0. $ | (3.24) |
In addition, by the firm nonexpansion of $ J_{\lambda_1}^{B_1}, (3.3), (3.7), $ and $ \gamma \in (0, \frac{1}{r}) $, we estimate
$ ‖un−p‖2=‖JB1λ1[xn+γA∗(U−I)Axn]−JB1λ1p‖2≤⟨JB1λ1[xn+γA∗(U−I)Axn]−JB1λ1p,xn+γA∗(U−I)Axn−p⟩=⟨un−p,xn+γA∗(U−I)Axn−p⟩=12(‖un−p‖2+‖xn+γA∗(U−I)Axn−p‖2−‖(un−p)−[xn+γA∗(U−I)Axn−p]‖2)≤12[‖un−p‖2+‖xn−p‖2+γ(rγ−1)‖(U−I)Axn‖2−‖un−xn−γA∗(U−I)Axn‖2]≤12[‖un−p‖2+‖xn−p‖2−‖un−xn−γA∗(U−I)Axn‖2]=12[‖un−p‖2+‖xn−p‖2−‖un−xn‖2−γ2‖A∗(U−I)Axn‖2+2γ⟨un−xn,A∗(U−I)Axn⟩]≤12[‖un−p‖2+‖xn−p‖2−‖un−xn‖2+2γ⟨un−xn,A∗(U−I)Axn⟩]=12[‖un−p‖2+‖xn−p‖2−‖un−xn‖2+2γ⟨A(un−xn),(U−I)Axn⟩]≤12[‖un−p‖2+‖xn−p‖2−‖un−xn‖2+2γ‖A(un−xn)‖‖(U−I)Axn‖], $ |
and hence,
$ ‖un−p‖2≤‖xn−p‖2−‖un−xn‖2+2γ‖A(un−xn)‖‖(U−I)Axn‖. $ | (3.25) |
In view of (3.22) and (3.25),
$ ‖xn+1−p‖2≤λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+(1−λnτ)‖un−p‖2≤λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+(1−λnτ)[‖xn−p‖2−‖un−xn‖2+2γ‖A(un−xn)‖‖(U−I)Axn‖]=λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+(1−λnτ)‖xn−p‖2−(1−λnτ)‖un−xn‖2+2γ(1−λnτ)‖A(un−xn)‖‖(U−I)Axn‖, $ | (3.26) |
which implies that
$ (1−λnτ)‖un−xn‖2≤λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+(1−λnτ)‖xn−p‖2+2γ(1−λnτ)‖A(un−xn)‖‖(U−I)Axn‖−‖xn+1−p‖2≤λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+2γ(1−λnτ)‖A(un−xn)‖‖(U−I)Axn‖+‖xn−p‖2−‖xn+1−p‖2≤λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+2γ(1−λnτ)‖A(un−xn)‖‖(U−I)Axn‖+‖xn+1−xn‖(‖xn−p‖+‖xn+1−p‖). $ | (3.27) |
Since $ \|x_{n+1}-x_n\| \rightarrow 0, \| (\mathcal{U}-I)Ax_n \| \rightarrow 0, $ and $ \lambda_n \rightarrow 0 $ as $ n \rightarrow \infty $, and owing to the boundedness of $ \{ x_n \} $, we conclude that
$ limn→∞‖xn−un‖=0. $ | (3.28) |
Step 4. We show that $ \underset{n\rightarrow \infty}{\lim} \| S_i u_n-u_{n}\| = 0 $ for $ i\in \{ 1, \ldots, N \} $.
Take a point $ p\in \Omega $ arbitrarily. When $ i = N, $ utilizing Lemma 2.8 and (3.2), we have
$ ‖xn+1−p‖2≤‖λnξ(αnψ(xn)+(1−αn)Sxn)+(I−λnμF)Tyn,N−p‖2=‖λnξ(αnψ(xn)+(1−αn)Sxn)−λnμFTp+(I−λnμF)Tyn,N−(I−λnμF)Tp‖2≤{‖λnξ(αnψ(xn)+(1−αn)Sxn)−λnμFTp‖+‖(I−λnμF)Tyn,N−(I−λnμF)Tp‖}2≤{λn‖αn(ξψ(xn)−μFp)+(1−αn)(ξSxn−μFp)‖+(1−λnτ)‖yn,N−p‖}2≤λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+(1−λnτ)‖yn,N−p‖2=λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+(1−λnτ)βn,N‖SNuN−p‖2+(1−λnτ)(1−βn,N)‖yn,N−1−p‖2−(1−λnτ)(1−βn,N)βn,N‖SNun−yn,N−1‖2≤λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+(1−λnτ)‖un−p‖2−(1−λnτ)(1−βn,N)βn,N‖SNun−yn,N−1‖2≤λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+‖xn−p‖2−(1−λnτ)(1−βn,N)βn,N‖SNun−yn,N−1‖2. $ | (3.29) |
Thus, we have
$ (1−λnτ)(1−βn,N)βn,N‖SNun−yn,N−1‖2≤λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+‖xn−p‖2−‖xn+1−p‖2≤λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+‖xn+1−xn‖(‖xn−p‖+‖xn+1−p‖). $ | (3.30) |
Since $ \beta_{n, N} \rightarrow \beta_N \in (0, 1), \|x_{n+1} - x_n\| \rightarrow 0 $ and $ \lambda_n \rightarrow 0 $ as $ n \rightarrow \infty $, by the boundedness of $ \{ x_n \} $, we conclude that
$ limn→∞‖SNun−yn,N−1‖=0. $ | (3.31) |
Take $ i\in \{ 1, \ldots, N-1 \} $ arbitrarily. Then, we obtain
$ ‖xn+1−p‖2≤λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+(1−λnτ)‖yn,N−p‖2≤λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+(1−λnτ)[βn,N‖SNun−p‖2+(1−βn,N)‖yn,N−1−p‖2]≤λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+(1−λnτ)βn,N‖xn−p‖2+(1−λnτ)(1−βn,N)‖yn,N−1−p‖2≤λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+(1−λnτ)βn,N‖xn−p‖2+(1−λnτ)(1−βn,N)[βn,N−1‖SN−1un−p‖2+(1−βn,N−1)‖yn,N−2−p‖2]≤λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+(1−λnτ)(βn,N+(1−βn,N)βn,N−1)‖xn−p‖2+(1−λnτ)N∏k=N−1(1−βn,k)‖yn,N−2−p‖2. $ | (3.32) |
Hence, after $ (N-i+1) $-iterations,
$ ‖xn+1−p‖2≤λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+(1−λnτ)(βn,N+N∑j=i+2(N∏p=j(1−βn,p)βn,j−1)×‖xn−p‖2+(1−λnτ)N∏k=i+1(1−βn,k)‖yn,i−p‖2≤λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+(1−λnτ)(βn,N+N∑j=i+2(N∏p=j(1−βn,p))βn,j−1)×‖xn−p‖2+(1−λnτ)N∏k=i+1(1−βn,k)×[βn,i‖Siun−p‖2+(1−βn,i)‖yn,i−1−p‖2−βn,i(1−βn,i)‖Siun−yn,i−1‖2]≤λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+(1−λnτ)‖xn−p‖2−βn,i(1−λnτ)N∏k=i(1−βn,k)‖Siun−yn,i−1‖2. $ | (3.33) |
Again, we obtain
$ βn,i(1−λnτ)N∏k=i(1−βn,k)‖Siun−yn,i−1‖2≤λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+‖xn−p‖2−‖xn+1−p‖2≤λn1τ[‖ξψ(xn)−μFp‖+‖ξSxn−μFp‖]2+‖xn+1−xn‖(‖xn−p‖+‖xn+1−p‖). $ | (3.34) |
Since for all $ k\in \{ 1, \ldots, N \}, \beta_{n, k} \rightarrow \beta_k \in (0, 1), \|x_{n+1} - x_n\| \rightarrow 0 $ and $ \lambda_n \rightarrow 0 $ as $ n \rightarrow \infty $, by the boundedness of $ \{ x_n\} $, we conclude that
$ limn→∞‖Siun−yn,i−1‖=0. $ | (3.35) |
Obviously, for $ i = 1 $, we have $ \underset{n\rightarrow \infty}{\lim} \| S_1 u_n-u_{n}\| = 0 $. To conclude, we have that
$ ‖S2un−un‖≤‖S2un−yn,1‖+‖yn,1−un‖=‖S2un−yn,1‖+βn,1‖S1un−un‖, $ | (3.36) |
which implies that $ \underset{n\rightarrow \infty}{\lim} \| S_2 u_n-u_{n}\| = 0 $. Consequently, by induction, we obtain
$ limn→∞‖Siun−un‖=0for alli=2,…,N. $ | (3.37) |
It is enough to observe that
$ ‖Siun−un‖≤‖Siun−yn,i−1‖+‖yn,i−1−Si−1un‖+‖Si−1un−un‖≤‖Siun−yn,i−1‖+(1−βn,i−1)‖Si−1un−yn,i−2‖+‖Si−1un−un‖. $ | (3.38) |
Step 5. We show that $ \underset{n\rightarrow \infty}{\lim} \| y_{n, N}-x_n\| = \underset{n\rightarrow \infty}{\lim} \|x_n-T x_n\| = 0 $ and $ \omega_w (x_n) \subset \Omega $.
Indeed, since $ \|x_n-u_n\| \rightarrow 0 $ as $ n\rightarrow \infty $, we have $ \omega_w(x_n) = \omega_w(u_n) $ and $ \omega_s(x_n) = \omega_s(u_n) $. Now, we observe that
$ ‖xn−yn,1‖≤‖xn−un‖+‖yn,1−un‖=‖xn−un‖+βn,1‖S1un−un‖. $ | (3.39) |
By Step 4, $ \| S_1 u_n - u_n\| \rightarrow 0 $ as $ n\rightarrow \infty $. Hence, we obtain
$ limn→∞‖xn−yn,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
$ ‖yn,N−xn‖≤N∑k=2‖yn,k−yn,k−1‖+‖yn,1−xn‖=N∑k=2βn,k‖Skun−yn,k−1‖+‖yn,1−xn‖; $ | (3.41) |
hence,
$ ‖xn−Txn‖≤‖xn−xn+1‖+‖xn+1−Tyn,N‖+‖Tyn,N−Txn‖≤‖xn−xn+1‖+‖λnξ(αnψ(xn)+(1−αn)Sxn)+(I−λnμF)Tyn,N−Tyn,N‖+‖yn,N−xn‖=‖xn−xn+1‖+λn‖αn(ξψ(xn)−μFTyn,N)+(1−αn)(ξSxn−μFTyn,N)‖+‖yn,N−xn‖≤‖xn−xn+1‖+λn[‖ξψ(xn)−μFTyn,N‖+‖ξSxn−μFTyn,N‖]+‖yn,N−xn‖≤‖xn−xn+1‖+λn[‖ξψ(xn)−μFTyn,N‖+‖ξSxn−μFTyn,N‖]+N∑k=2βn,k‖Skun−yn,k−1‖+‖yn,1−xn‖. $ | (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
$ limn→∞‖yn,N−xn‖=limn→∞‖xn−Txn‖=0. $ | (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
$ xnk−unk+γA∗(U−I)Axnkλ1∈B1unk. $ | (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
$ lim supn→∞⟨(ξψ−μF)x∗,xn+1−x∗⟩=lim supn→∞⟨(ξψ−μF)x∗,xn−x∗⟩=limj→∞⟨(ξψ−μF)x∗,xnj−x∗⟩. $ | (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
$ lim supn→∞⟨(ξψ−μF)x∗,xn+1−x∗⟩=⟨(ξψ−μF)x∗,˜x−x∗⟩≤0. $ | (3.46) |
Repeating the same argument as that of (3.46), we have
$ lim supn→∞⟨(ξS−μF)x∗,xn+1−x∗⟩≤0. $ | (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
$ ‖xn+1−x∗‖2=⟨zn−x∗,xn+1−x∗⟩+⟨PCzn−zn,PCzn−x∗⟩≤⟨zn−x∗,xn+1−x∗⟩=⟨(I−λnμF)Tyn,N−(I−λnμF)x∗,xn+1−x∗⟩+αnλnξ⟨ψ(xn)−ψ(x∗),xn+1−x∗⟩+λn(1−αn)ξ⟨Sxn−Sx∗,xn+−x∗⟩+αnλn⟨(ξψ−μF)x∗,xn+1−x∗⟩+λn(1−αn)⟨(ξS−μF)x∗,xn+1−x∗⟩≤[1−λnτ+αnλnξρ+λn(1−αn)ξ]‖xn−x∗‖‖xn+1−x∗‖+αnλn⟨(ξψ−μF)x∗,xn+1−x∗⟩+λn(1−αn)⟨(ξS−μF)x∗,xn+1−x∗⟩≤[1−αnλnξ(1−ρ)]‖xn−x∗‖‖xn+1−x∗‖+αnλn⟨(ξψ−μF)x∗,xn+1−x∗⟩+λn(1−αn)⟨(ξS−μF)x∗,xn+1−x∗⟩≤[1−αnλnξ(1−ρ)]12(‖xn−x∗‖2+‖xn+1−x∗‖2)+αnλn⟨(ξψ−μF)x∗,xn+1−x∗⟩+λn(1−αn)⟨(ξS−μF)x∗,xn+1−x∗⟩. $ | (3.48) |
It turns out that
$ ‖xn+1−x∗‖2≤1−αnλnξ(1−ρ)1+αnλnξ(1−ρ)‖xn−x∗‖2+21+αnλnξ(1−ρ)[αnλn⟨(ξψ−μF)x∗,xn+1−x∗⟩+λn(1−αn)⟨(ξS−μF)x∗,xn+1−x∗⟩]≤[1−αnλnξ(1−ρ)]‖xn−x∗‖2+21+αnλnξ(1−ρ)[αnλn⟨(ξψ−μF)x∗,xn+1−x∗⟩+λn(1−αn)⟨(ξS−μF)x∗,xn+1−x∗⟩]=[1−αnλnξ(1−ρ)]‖xn−x∗‖2+αnλnξ(1−ρ){2ξ(1−ρ)[1+αnλnξ(1−ρ)]×⟨(ξψ−μF)x∗,xn+1−x∗⟩+2(1−αn)αnξ(1−ρ)[1+αnλnξ(1−ρ)]⟨(ξS−μF)x∗,xn+1−x∗⟩}. $ | (3.49) |
Put $ s_{n} = \|x_n-x^*\|^2, \xi_n = \alpha_n \lambda_n \xi (1-\rho) $ and
$ δn=2ξ(1−ρ)[1+αnλnξ(1−ρ)]⟨(ξψ−μF)x∗,xn+1−x∗⟩+2(1−αn)αnξ(1−ρ)[1+αnλnξ(1−ρ)]⟨(ξS−μF)x∗,xn+1−x∗⟩. $ |
Then, (3.49) can be rewritten as
$ sn+1≤(1−γn)sn+ξnδn. $ |
From conditions $ (C1) $ and $ (C2) $, we conclude from $ 0 < 1-\rho \leq 1 $ that
$ {ξn}⊂[0,1]and∞∑n=1ξn=∞. $ |
Note that
$ 2ξ(1−ρ)[1+αnλnξ(1−ρ)]≤2ξ(1−ρ) $ |
and
$ 2(1−αn)αnξ(1−ρ)[1+αnλnξ(1−ρ)]≤2aξ(1−ρ). $ |
Consequently, utilizing Lemma 2.5, we find that
$ lim supn→∞δn≤lim supn→∞2ξ(1−ρ)[1+αnλnξ(1−ρ)]⟨(ξψ−μF)x∗,xn+1−x∗⟩+lim supn→∞2(1−αn)αnξ(1−ρ)[1+αnλnξ(1−ρ)]⟨(ξψ−μF)x∗,xn+1−x∗⟩≤0. $ |
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:
$ {un=JB1λ[xn+γA∗(JB2λ−I)Axn],yn,1=βn,1S1un+(1−βn,1)un,yn,i=βn,iSiun+(1−βn,i)yn,i−1,i=2,…,N,xn+1=PC[λnξ(αnψ(xn)+(1−αn)Sxn)+(I−λnμF)Tyn,N],n≥1. $ | (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:
$ {un=xn+γA∗(I−λ2f)Axn,yn,1=βn,1S1un+(1−βn,1)un,yn,i=βn,iSiun+(1−βn,i)yn,i−1,i=2,…,N,xn+1=PC[λnξ(αnψ(xn)+(1−αn)Sxn)+(I−λnμF)Tyn,N],n≥1. $ | (4.1) |
Assume that the problem
$ ⟨(μF−ξψ)x∗,x−x∗⟩≥0,∀x∈Ω, $ | (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
$ Ti(x)={x,x∈(−∞,0),2x,x∈[0,∞). $ | (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\}. $
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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