Synchronal Algorithm and Cyclic Algorithm for Hierarchical Fixed Point Problems and Variational Inequalities

Abstract

We propose synchronal algorithm and cyclic algorithm based on the general iterative method for solving a hierarchical fixed point problem. Under suitable parameters, the iterative sequence converges strongly to a common fixed point of 𝑁 nonexpansive mappings and also the unique solution of a variational inequality. The results presented in this paper improve and extend the corresponding results reported recently by some authors. Furthermore, a numerical example is given to demonstrate the effectiveness of our iterative schemes.

1. Introduction

Let 𝐻 be a real Hilbert space with an inner product ⟨,⟩ and its induced norm ‖⋅‖. Let 𝐶 be a nonempty, closed, and convex subset of 𝐻.

Let 𝑇∶𝐶→𝐻 be a nonlinear mapping; we denote the set of fixed points of 𝑇 by Fix(𝑇) (i.e., Fix(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥}). A mapping 𝑇∶𝐶→𝐻 is called 𝑘-Lipschitzian continuous if there exists a constant 𝑘>0 such that 𝑇𝑥−𝑇𝑦≤𝑘𝑥−𝑦,∀𝑥,𝑦∈𝐶. (1) In particular, 𝑇 is said to be a nonexpansive mapping if 𝑘=1. A mapping 𝐵 is called 𝜂-strongly monotone on 𝐶, if there exists a constant 𝜂>0 such that ⟨𝐵𝑥−𝐵𝑦,𝑥−𝑦⟩≥𝜂𝑥−𝑦²,∀𝑥,𝑦∈𝐶. (2)

A variational inequality (short for VI) is formulated as finding a point 𝑥^∗∈𝐶 such that ⟨𝐵𝑥^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈𝐶. (3)

If 𝐵 is a monotone operator, then VI (3) is known as a monotone variational inequality. If the set 𝐶 is replaced by the set of Fix(𝑇) of fixed points of a mapping 𝑇, then the VI (3) is called a hierarchical variational inequality problem.

Many practical problems in applied sciences such as signal processing [1], beamforming [2], and power control [3] are formulated as the monotone variational inequality with a fixed point constrained. In recent years, several authors paid attention toward this kind of problem. Some methods have been proposed to solve the hierarchical fixed point problems and variational inequalities; see for instance [4–10] and the references therein.

In 2010, Tian [11] proposed a general iterative method and revealed the inner contact of the Yamada’s algorithm [12] and viscosity iterative algorithm; then he obtained the following result in a real Hilbert space.

Theorem 1. Let 𝑥_𝑛 be generated by algorithm 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓(𝑥_𝑛)+(𝐼−𝛼_𝑛𝜇𝐴)𝑇𝑥_𝑛 with the sequence {𝛼_𝑛} of parameters satisfying conditions (C1)–(C3):(C1)𝛼_𝑛→0, (C2)∑^∞_𝑛=0𝛼_𝑛=∞, (C3) either ∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞ or lim_𝑛→∞(𝛼_𝑛+1/𝛼_𝑛)=1.Then 𝑥_𝑛 converges strongly to a fixed point 𝑥 of 𝑇 which solves the variational inequality ⟨𝜇𝐴−𝛾𝑓𝑥,𝑥−𝑧⟩≤0,∀𝑧∈Fix(𝑇). (4)

Recently, Yao et al. [10] investigated an iterative method for solving a hierarchical fixed point problem by 𝑦_𝑛=𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛,𝑥_𝑛+1=𝑃_𝐶𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇𝑦_𝑛, (5) where 𝑆, 𝑇 are nonexpansive mapping with Fix(𝑇)≠∅ and 𝑓 is a contraction; the sequence converges strongly to the unique solution of the variational inequality 𝑥^∗∈Fix(𝑇),⟨𝐼−𝑓𝑥^∗,𝑝−𝑥^∗⟩≤0,∀𝑝∈Fix(𝑇). (6)

Very recently, on this basis, Wang and Xu [8] introduced a new modified iterative method for solving a hierarchical fixed point problem. To be more precise, they proposed the following algorithm: 𝑦_𝑛=𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛,𝑥_𝑛+1=𝑃_𝐶𝛼_𝑛𝜌𝑈𝑥_𝑛+𝐼−𝛼_𝑛𝜇𝐹𝑇𝑦_𝑛, (7) where 𝑆,𝑇 are nonexpansive mappings with Fix(𝑇)≠∅, 𝑈 is a Lipschitzian mapping, and 𝐹 is a Lipschitzian and strongly monotone operator. They proved the sequence generated by the above algorithm converges strongly to the unique solution of the variational inequality ⟨𝜌𝑈−𝜇𝐹𝑥^∗,𝑝−𝑥^∗⟩≤0,∀𝑝∈Fix(𝑇). (8)

On the other hand, Tian and Di [13] established a synchronal algorithm and a cyclic algorithm for fixed point problems and variational inequalities. In 2012, Ceng et al. [4] proposed an iterative method to solve a special form of VI (3), where the constraint set is the set of common fixed points of 𝑁 nonexpansive mappings 𝑇₁,𝑇₂,…,𝑇_𝑁.

Motivated and inspired by the above works, in this paper, we combine the hybrid steepest descent algorithm and hierarchical variational inequalities to propose a synchronal algorithm and a cyclic algorithm involving finite family of nonexpansive mappings. Under certain assumptions, we will prove that the sequences converge strongly. Further an example will be given to demonstrate the effectiveness of our iterative schemes.

2. Preliminaries

Recall that given a nonempty, closed and convex subset 𝐶 of a real Hilbert space 𝐻, for any 𝑥∈𝐻, there exists a unique nearest point in 𝐶, denoted by 𝑃_𝐶𝑥, such that 𝑥−𝑃_𝐶𝑥≤𝑥−𝑦 (9) for all 𝑦∈𝐶. Such a 𝑃_𝐶 is called the metric (or the nearest point) projection of 𝐻 onto 𝐶. As we all know, 𝑦=𝑃_𝐶𝑥 if and only if there holds the relation ⟨𝑥−𝑦,𝑦−𝑧⟩≥0,∀𝑧∈𝐶. (10)

In the sequel, we will make use of the following lemmas in a real Hilbert space 𝐻.

Lemma 2. Let 𝐻 be a real Hilbert space; the following inequalities hold:(i)‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑥+𝑦⟩, ∀𝑥,𝑦∈𝐻, (ii)‖𝑡𝑥+(1−𝑡)𝑦‖²≤𝑡‖𝑥‖²+(1−𝑡)‖𝑦‖², ∀𝑡∈[0,1], ∀𝑥,𝑦∈𝐻.

Lemma 3 (see [13]). Let 𝐵∶𝐻→𝐻 be a 𝑘-Lipschitzian and 𝜂-strongly monotone operator on a Hilbert space 𝐻 with 𝑘>0, 𝜂>0, 0<𝜇<2𝜂/𝑘², and 0<𝑡<1. Then 𝑆=(𝐼−𝑡𝜇𝐵)∶𝐻→𝐻 is a contraction with contractive coefficient 1−𝑡𝜏 and 𝜏=(1/2)𝜇(2𝜂−𝜇𝑘²).

Lemma 4 (see [5]). Let 𝑉∶𝐶→𝐻 be an 𝑙-Lipschitz mapping with coefficient 𝑙≥0 and 𝐵∶𝐶→𝐻 a 𝑘-Lipschitzian continuous operator and 𝜂-strongly monotone operator with 𝑘>0, 𝜂>0. Then for 0<𝛾<𝜇𝜂/𝑙, 𝑦≥𝑥−𝑦,𝜇𝐵−𝛾𝑉𝑥−𝜇𝐵−𝛾𝑉𝜇𝜂−𝛾𝑙𝑥−𝑦²,𝑥,𝑦∈𝐻. (11) That is, 𝜇𝐵−𝛾𝑉 is strongly monotone with coefficient 𝜇𝜂−𝛾𝑙.

Lemma 5 (see [9]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛿_𝑛, (12) where {𝛾_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence such that (i)∑^∞_𝑛=1𝛾_𝑛=∞,(ii)limsup_𝑛→∞(𝛿_𝑛/𝛾_𝑛)≤0 or ∑^∞_𝑛=1|𝛿_𝑛|<∞.Then, lim_𝑛→∞𝑎_𝑛=0.

Lemma 6 (see [14]). Let 𝐻 be a real Hilbert, and let 𝑇_𝑖∶𝐻→𝐻(𝑖=1,2,…) be all nonexpansive mappings with ∩^∞_𝑖=1Fix(𝑇_𝑖)≠∅. Let 𝑇=∑^∞_𝑖=1𝜔_𝑖𝑇_𝑖(𝑖=1,2,…), where {𝜔_𝑖}⊂(0,1) such that ∑^∞_𝑖=1𝜔_𝑖=1. Then 𝑇 is a nonexpansive mapping with Fix(𝑇)=∩^∞_𝑖=1Fix(𝑇_𝑖).

Lemma 7 (see [13]). Let 𝐻 be a Hilbert space, and let 𝐶 be a nonempty closed convex subset of 𝐻 and 𝑇∶𝐶→𝐶 a nonexpansive mapping with Fix(𝑇)≠∅. If {𝑥_𝑛} is a sequence in 𝐶 weakly converging to 𝑥 and if {(𝐼−𝑇)𝑥_𝑛} converges strongly to 𝑦, then (𝐼−𝑇)𝑥=𝑦.

We adopt the following notations:(1)𝑥_𝑛⇀𝑥 stands for the weak convergence of {𝑥_𝑛} to 𝑥,(2)𝑥_𝑛→𝑥 stands for the strong convergence of {𝑥_𝑛} to 𝑥.

3. Synchronal Algorithm

Throughout the rest of this paper, we always assume that 𝑉∶𝐶→𝐻 is an 𝑙-Lipschitzian mapping with coefficient 𝑙≥0 and 𝐵∶𝐶→𝐻 is a 𝑘-Lipschitzian continuous operator and 𝜂-strongly monotone with 𝑘>0, 𝜂>0. Let 𝑁≥1 be an integer. Let, for each 1≤𝑖≤𝑁, 𝑇_𝑖∶𝐶→𝐶 be a nonexpansive mapping and 𝑆∶𝐶→𝐶 also nonexpansive. Assume that 0<𝜇<2𝜂/𝑘² and 0<𝛾<𝜇(𝜂−𝜇𝑘²/2)/𝛼=𝜏/𝑙.

Define a mapping 𝑈_𝑛=𝛽_𝑛𝑆+(1−𝛽_𝑛)𝐼. Since 𝑆 is nonexpansive, it is easy to get that 𝑈_𝑛 is also nonexpansive. Consider the following mapping 𝐺_𝑛 on 𝐶 defined by 𝐺_𝑛𝑥=𝑃_𝐶𝛼_𝑛𝛾𝑉(𝑥)+𝐼−𝛼_𝑛𝜇𝐵𝑇𝑈_𝑛𝑥,∀𝑥∈𝐶,𝑛∈ℕ, (13) where 𝛼_𝑛∈(0,1), 𝑇=∑^𝑁_𝑖=1𝜔_𝑖𝑇_𝑖 with 𝜔_𝑖>0, and ∑^𝑁_𝑖=1𝜔_𝑖=1. By Lemmas 2 and 3, we obtain 𝐺_𝑛𝑥−𝐺_𝑛𝑦≤𝛼_𝑛𝛾+𝑉𝑥−𝑉𝑦1−𝛼_𝑛𝜏𝑇𝑈_𝑛𝑥−𝑇𝑈_𝑛𝑦≤𝛼_𝑛𝛾𝑙𝑥−𝑦+1−𝛼_𝑛𝜏=𝑥−𝑦1−𝛼_𝑛𝜏−𝛾𝑙𝑥−𝑦. (14)

Since 0<1−𝛼_𝑛(𝜏−𝛾𝑙)<1, it follows that 𝐺_𝑛 is a contraction. Therefore, by the Banach contraction principle, 𝐺_𝑛 has a unique fixed point 𝑥^𝑉_𝑛∈𝐶 such that 𝑥^𝑉_𝑛=𝑃_𝐶𝛼_𝑛𝛾𝑉𝑥^𝑉_𝑛+𝐼−𝛼_𝑛𝜇𝐵𝑇𝑈_𝑛𝑥^𝑉_𝑛. (15)

For simplicity, we will write 𝑥_𝑛 for 𝑥^𝑉_𝑛 provided that no confusion occurs. Next we prove that the sequence {𝑥_𝑛} converges strongly to a point 𝑥^∗∈Ω=∩^𝑁_𝑖=1Fix(𝑇_𝑖) which solves the variational inequality ⟨𝛾𝑉−𝜇𝐵𝑥^∗,𝑝−𝑥^∗⟩≤0,∀𝑝∈Ω. (16) By the property of the projection, we can get 𝑥^∗=𝑃_Ω(𝐼−𝜇𝐵+𝛾𝑉)𝑥^∗ equivalently.

Theorem 8. Let 𝐶 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻 and 𝑉∶𝐶→𝐻 an 𝑙-Lipschitzian mapping with 𝑙≥0. Let 𝑁≥1 be an integer. Let, for each 1≤𝑖≤𝑁, 𝑇_𝑖∶𝐶→𝐶 be a nonexpansive mapping and let 𝑆∶𝐶→𝐶 be also nonexpansive. Assume the set Ω=⋂^𝑁_𝑖=1Fix(𝑇_𝑖)≠∅. Let 𝐵∶𝐶→𝐻 be a 𝑘-Lipschitzian continuous operator and 𝜂-strongly monotone with 𝑘>0, 𝜂>0, 0<𝜇<2𝜂/𝑘² and 0<𝛾<𝜇(𝜂−𝜇𝑘²/2)/𝑙=𝜏/𝑙. Given 𝑥₁∈𝐶, let {𝑥_𝑛} be the sequence generated by the following algorithm: 𝑦_𝑛=𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛,𝑥_𝑛+1=𝑃_𝐶𝛼_𝑛𝛾𝑉𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐵𝑇𝑦_𝑛. (17) If {𝛼_𝑛} and {𝛽_𝑛} satisfy the following properties: (i){𝛼_𝑛}⊂(0,1), lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞,(ii){𝛽_𝑛}⊂[0,1), lim_𝑛→∞(𝛽_𝑛/𝛼_𝑛)=0, (iii)∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞ and ∑^∞_𝑛=1|𝛽_𝑛+1−𝛽_𝑛|<∞. Then, {𝑥_𝑛} converges strongly to 𝑥^∗∈Ω, which solves the variational inequality (16).

Proof. The proof is divided into several steps.
Step  1. Show first that {𝑥_𝑛} is bounded.
Take any 𝑝∈Ω; we have 𝑦_𝑛−𝑝=𝛽_𝑛𝑆𝑥_𝑛−𝑝+1−𝛽_𝑛𝑥_𝑛−𝑝≤𝛽_𝑛𝑆𝑥_𝑛−𝑆𝑝+𝛽_𝑛+𝑆𝑝−𝑝1−𝛽_𝑛𝑥_𝑛−𝑝≤𝑥_𝑛−𝑝+𝛽_𝑛𝑆𝑝−𝑝. (18)
Further we get 𝑥_𝑛+1=𝑃−𝑝_𝐶𝛼_𝑛𝛾𝑉𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐵𝑇𝑦_𝑛−𝑃_𝐶𝑝≤𝛼_𝑛𝛾𝑉𝑥_𝑛+−𝜇𝐵𝑝𝐼−𝜇𝛼_𝑛𝐵𝑇𝑦_𝑛−𝐼−𝜇𝛼_𝑛𝐵𝑝≤𝛼_𝑛𝛾𝑉𝑥_𝑛++−𝛾𝑉𝑝𝛾𝑉𝑝−𝜇𝐵𝑝1−𝛼_𝑛𝜏𝑇𝑦_𝑛−𝑝≤𝛼_𝑛𝑙𝛾𝑥_𝑛−𝑝+𝛼_𝑛+𝛾𝑉𝑝−𝜇𝐵𝑝1−𝛼_𝑛𝜏𝑦_𝑛=−𝑝1−𝛼_𝑛𝜏−𝑙𝛾𝑥_𝑛−𝑝+𝛼_𝑛+𝜏−𝑙𝛾𝛾𝑉𝑝−𝜇𝐵𝑝𝑆𝑝−𝑝𝑥𝜏−𝑙𝛾≤max_𝑛,+−𝑝𝛾𝑉𝑝−𝜇𝐵𝑝𝑆𝑝−𝑝.𝜏−𝑙𝛾 (19)
By induction, we obtain ‖𝑥_𝑛−𝑝‖≤max{‖𝑥₁−𝑝‖,(‖𝛾𝑉𝑝−𝜇𝐵𝑝‖+‖𝑆𝑝−𝑝‖)/(𝜏−𝑙𝛾)}, 𝑛≥1. Hence, {𝑥_𝑛} is bounded, so is {𝑦_𝑛}. It follows from the Lipschitz continuity of 𝐵 and 𝑉 that {𝐵𝑥_𝑛}, {𝐵𝑢_𝑛}, and {𝑉𝑥_𝑛} are also bounded. From the nonexpansivity of 𝑇 and 𝑆, it follows that {𝑇𝑥_𝑛}, {𝑆𝑥_𝑛}, and {𝐵𝑇𝑦_𝑛} are also bounded.
Step  2. Show that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (20)
By (17), we have 𝑥_𝑛+1−𝑥_𝑛=𝑃_𝐶𝛼_𝑛𝛾𝑉𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐵𝑇𝑦_𝑛−𝑃_𝐶𝛼_𝑛−1𝛾𝑉𝑥_𝑛−1+𝐼−𝜇𝛼_𝑛−1𝐵𝑇𝑦_𝑛−1≤𝛼_𝑛𝛾𝑉𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐵𝑇𝑦_𝑛−𝛼_𝑛−1𝛾𝑉𝑥_𝑛−1+𝐼−𝜇𝛼_𝑛−1𝐵𝑇𝑦_𝑛−1≤𝛼_𝑛𝛾𝑉𝑥_𝑛−𝑉𝑥_𝑛−1+𝐼−𝜇𝛼_𝑛𝐵𝑇𝑦_𝑛−𝐼−𝜇𝛼_𝑛𝐵𝑇𝑦_𝑛−1+𝐼−𝜇𝛼_𝑛𝐵𝑇𝑦_𝑛−1−𝐼−𝜇𝛼_𝑛−1𝐵𝑇𝑦_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝛾𝑉𝑥_𝑛−1≤𝛼_𝑛𝛾𝑙𝑥_𝑛−𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝛾𝑙𝑉𝑥_𝑛−1+𝜇𝐵𝑇𝑦_𝑛−1+1−𝛼_𝑛𝜏𝑦_𝑛−𝑦_𝑛−1. (21)
Observe that 𝑦_𝑛−𝑦_𝑛−1=𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛−𝛽_𝑛−1𝑆𝑥_𝑛−1−1−𝛽_𝑛−1𝑥_𝑛−1≤𝛽_𝑛𝑆𝑥_𝑛−𝛽_𝑛𝑆𝑥_𝑛−1+1−𝛽_𝑛𝑥_𝑛−1−𝛽_𝑛𝑥_𝑛−1+𝛽_𝑛𝑆𝑥_𝑛−1−𝛽_𝑛−1𝑆𝑥_𝑛−1+1−𝛽_𝑛𝑥_𝑛−1−1−𝛽_𝑛−1𝑥_𝑛−1≤𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+1−𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑆𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑥_𝑛−1=𝑥_𝑛−𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑆𝑥_𝑛−1+𝑥_𝑛−1. (22)
Together with (21) and (22), we get 𝑥_𝑛+1−𝑥_𝑛≤𝛼_𝑛𝛾𝑙𝑥_𝑛−𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝛾𝑙𝑉𝑥_𝑛−1+𝜇𝐵𝑇𝑦_𝑛−1+1−𝛼_𝑛𝜏𝑥_𝑛−𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑆𝑥_𝑛−1+𝑥_𝑛−1=1−𝛼_𝑛𝜏−𝛾𝑙𝑥_𝑛−𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝛾𝑙𝑉𝑥_𝑛−1+𝜇𝐵𝑇𝑦_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑆𝑥_𝑛−1+𝑥_𝑛−1≤1−𝛼_𝑛𝜏−𝛾𝑙𝑥_𝑛−𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑀₁, (23) where 𝑀₁=sup_𝑛{‖𝛾𝑙𝑉𝑥_𝑛−1‖+𝜇‖𝐵𝑇𝑦_𝑛−1‖+‖𝑆𝑥_𝑛−1‖+‖𝑥_𝑛−1‖}.
By Lemma 5, we obtain 𝑥_𝑛+1−𝑥_𝑛⟶0. (24)
Step  3. Show that 𝑥_𝑛−𝑇𝑥_𝑛⟶0. (25)
Observe that 𝑥_𝑛+1−𝑇𝑦_𝑛=𝑃_𝐶𝛼_𝑛𝛾𝑉𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐵𝑇𝑦_𝑛−𝑃_𝐶𝑇𝑦_𝑛≤𝛼_𝑛𝛾𝑉𝑥_𝑛−𝜇𝐵𝑇𝑦_𝑛⟶0. (26) From condition (i) and (ii), we obtain 𝑥_𝑛+1−𝑇𝑥_𝑛≤𝑥_𝑛+1−𝑇𝑦_𝑛+𝑇𝑦_𝑛−𝑇𝑥_𝑛≤𝑥_𝑛+1−𝑇𝑦_𝑛+𝑦_𝑛−𝑥_𝑛=𝑥_𝑛+1−𝑇𝑦_𝑛+𝛽_𝑛𝑆𝑥_𝑛−𝑥_𝑛⟶0. (27)
Step  4. Show that limsup_𝑛→∞⟨𝛾𝑉−𝜇𝐵𝑥^∗,𝑥_𝑛−𝑥^∗⟩≤0, (28) where 𝑥^∗=𝑃_Ω(𝐼−𝜇𝐵+𝛾𝑉)𝑥^∗ is a unique solution of the variational inequality (16).
Indeed, take a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} such that limsup_𝑛→∞𝛾𝑉−𝜇𝐵𝑥^∗,𝑥_𝑛−𝑥^∗=lim_𝑗→∞⟨𝛾𝑉−𝜇𝐵𝑥^∗,𝑥_𝑛𝑗−𝑥^∗⟩. (29)
Since {𝑥_𝑛𝑗} is bounded, there exists a subsequence {𝑥_𝑛𝑗𝑘} of {𝑥_𝑛𝑗} which converges weakly to 𝑥. Without loss of generality, we can assume 𝑥_𝑛𝑗⇀𝑥 and 𝑥_𝑛𝑗−𝑇𝑥_𝑛𝑗⟶0. (30)
By Lemma 7, we have 𝑥=𝑇𝑥. From Lemma 6, we get 𝑥∈Fix(𝑇)=^𝑁_𝑖=1Fix𝑇_𝑖. (31)
Since 𝑥^∗=𝑃_Ω(𝐼−𝜇𝐵+𝛾𝑉)𝑥^∗, it follows that limsup_𝑛→∞𝛾𝑉−𝜇𝐵𝑥^∗,𝑥_𝑛−𝑥^∗=lim_𝑗→∞𝛾𝑉−𝜇𝐵𝑥^∗,𝑥_𝑛𝑗−𝑥^∗=𝛾𝑉−𝜇𝐵𝑥^∗,𝑥−𝑥^∗≤0. (32)
Step  5. Show that 𝑥_𝑛⟶𝑥^∗. (33)
Denote 𝑧_𝑛=𝛼_𝑛𝛾𝑉𝑥_𝑛+(𝐼−𝜇𝛼_𝑛𝐵)𝑇𝑦_𝑛, then 𝑥_𝑛+1=𝑃_𝐶𝑧_𝑛. From (17), we have 𝑥_𝑛+1−𝑥^∗2=𝑃_𝐶𝑧_𝑛−𝑥^∗2=𝑃_𝐶𝑧_𝑛−𝑧_𝑛,𝑃_𝐶𝑧_𝑛−𝑥^∗+𝑧_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗≤𝑧_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗=𝐼−𝜇𝛼_𝑛𝐵𝑇𝑦_𝑛−𝐼−𝜇𝛼_𝑛𝐵𝑇𝑥^∗+𝛼_𝑛𝛾𝑉𝑥_𝑛−𝜇𝐵𝑥^∗,𝑥_𝑛+1−𝑥^∗≤1−𝛼_𝑛𝜏𝑦_𝑛−𝑥^∗𝑥_𝑛+1−𝑥^∗+𝛼_𝑛𝛾𝑉𝑥_𝑛−𝛾𝑉𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝛼_𝑛𝛾𝑉𝑥^∗−𝜇𝐵𝑥^∗,𝑥_𝑛+1−𝑥^∗≤1−𝛼_𝑛𝜏𝑥_𝑛−𝑥^∗+𝛽_𝑛𝑆𝑥^∗−𝑥^∗𝑥_𝑛+1−𝑥^∗+𝛼_𝑛𝛾𝑙𝑥_𝑛−𝑥^∗𝑥_𝑛+1−𝑥^∗+𝛼_𝑛𝛾𝑉𝑥^∗−𝜇𝐵𝑥^∗,𝑥_𝑛+1−𝑥^∗≤1−𝛼_𝑛𝜏−𝛾𝑙2𝑥_𝑛−𝑥^∗2+𝑥_𝑛+1−𝑥^∗2+𝛽_𝑛𝑥_𝑛+1−𝑥^∗𝑆𝑥^∗−𝑥^∗+𝛼_𝑛⟨𝛾𝑉𝑥^∗−𝜇𝐵𝑥^∗,𝑥_𝑛+1−𝑥^∗⟩. (34)
This implies that 𝑥_𝑛+1−𝑥^∗2≤1−𝛼_𝑛𝜏−𝛾𝑙1+𝛼_𝑛𝑥𝜏−𝑙𝛾^𝑛−𝑥^∗2+21+𝛼_𝑛×𝛽𝜏−𝛾𝑙_𝑛𝑥_𝑛+1−𝑥^∗𝑆𝑥^∗−𝑥^∗+𝛼_𝑛𝛾𝑉𝑥^∗−𝜇𝐵𝑥^∗,𝑥_𝑛+1−𝑥^∗≤1−𝛼_𝑛𝜏−𝛾𝑙𝑥_𝑛−𝑥^∗2+2𝛽_𝑛𝑀₂+2𝛼_𝑛⟨𝛾𝑉𝑥^∗−𝜇𝐵𝑥^∗,𝑥_𝑛+1−𝑥^∗⟩, (35) where 𝑀₂=sup_𝑛‖𝑥_𝑛+1−𝑥^∗‖‖𝑆𝑥^∗−𝑥^∗‖, 𝑛≥1. Put 𝛾_𝑛=𝛼_𝑛(𝜏−𝑙𝛾), 𝛿_𝑛=2𝛽_𝑛𝑀₂+2𝛼_𝑛⟨𝛾𝑉𝑥^∗−𝜇𝐵𝑥^∗,𝑥_𝑛+1−𝑥^∗⟩. It is easy to see that limsup_𝑛→∞𝛿_𝑛/𝛾_𝑛≤0. Hence by Lemma 5, the sequence {𝑥_𝑛} converges strongly to 𝑥^∗.

Remark 9. Let 𝑁=1 in Theorem 8; we can get Theorem 3.1 of [8].

Remark 10. Let 𝑁=1, 𝛾=1, 𝜇=1, 𝐵=𝐼 and 𝑉, be a contraction in Theorem 8; it is easy to get the theorem of  [10].

4. Cyclic Algorithm

In this section, we consider the cyclic algorithm of 𝑁 nonexpansive mappings 𝑇₁,𝑇₂,…,𝑇_𝑁. Similarly, we can get that the mapping 𝐺_𝑛 on 𝐶 defined by 𝐺_𝑛𝑥=𝑃_𝐶𝛼_𝑛𝛾𝑉(𝑥)+𝐼−𝛼_𝑛𝜇𝐵𝑇_[𝑛]𝑈_𝑛𝑥,∀𝑥∈𝐶,𝑛∈ℕ, (36) is a contraction, where 𝑇_[𝑛]=𝑇_𝑖 with 𝑖=𝑛(mod)𝑁 taking values in {1,2,…,𝑁}.

Theorem 11. Let 𝐶 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻, and let 𝑉∶𝐶→𝐻 be an 𝑙-Lipschitzian mapping with 𝑙≥0. Let 𝑁≥1 be an integer. Let, for each 1≤𝑖≤𝑁, 𝑇_𝑖∶𝐶→𝐶 be a nonexpansive mapping and let 𝑆∶𝐶→𝐶 be also nonexpansive. Assume the set Ω=⋂^𝑁_𝑖=1Fix(𝑇_𝑖)≠∅. Let 𝐵∶𝐶→𝐻 be a 𝑘-Lipschitzian continuous operator and 𝜂-strongly monotone with 𝑘>0, 𝜂>0, 0<𝜇<2𝜂/𝑘², and 0<𝛾<𝜇(𝜂−𝜇𝑘²/2)/𝑙=𝜏/𝑙. Given 𝑥₁∈𝐶, let {𝑥_𝑛} be the sequence generated by the following algorithm: 𝑦_𝑛=𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛,𝑥_𝑛+1=𝑃_𝐶𝛼_𝑛𝛾𝑉𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐵𝑇_[𝑛]𝑦_𝑛. (37) If {𝛼_𝑛} and {𝛽_𝑛} satisfy the following properties:(i){𝛼_𝑛}⊂(0,1), lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞, (ii){𝛽_𝑛}⊂[0,1), lim_𝑛→∞(𝛽_𝑛/𝛼_𝑛)=0,(iii)∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞ and ∑^∞_𝑛=1|𝛽_𝑛+1−𝛽_𝑛|<∞. Assume in addition that Fix𝑇^𝑁_𝑖=1𝑖=Fix𝑇_𝑁𝑇_𝑁−1⋯𝑇₂𝑇₁=Fix𝑇₁𝑇_𝑁𝑇_𝑁−1⋯𝑇₂=⋯=Fix𝑇₂𝑇₁𝑇_𝑁⋯𝑇₃. (38) Then, {𝑥_𝑛} converges strongly to 𝑥^∗∈Ω, which solves the variational inequality (16).

Proof. The proof is divided into several steps.
Step  1. Show first that {𝑥_𝑛} is bounded.
The proof of Step 1 is similar to that of Theorem 8.
Step  2. Show that lim_𝑛→∞𝑥_𝑛+𝑁−𝑥_𝑛=0. (39)
By (37), we have 𝑥_𝑛+𝑁+1−𝑥_𝑛+1≤𝛼_𝑛+𝑁𝛾𝑉𝑥_𝑛+𝑁+𝐼−𝜇𝛼_𝑛+𝑁𝐵𝑇_[𝑛]𝑦_𝑛+𝑁−𝛼_𝑛𝛾𝑉𝑥_𝑛−𝐼−𝜇𝛼_𝑛𝐵𝑇_[𝑛]𝑦_𝑛≤𝛼_𝑛+𝑁𝛾𝑉𝑥_𝑛+𝑁−𝑉𝑥_𝑛+𝐼−𝜇𝛼_𝑛+𝑁𝐵𝑇_[𝑛]𝑦_𝑛+𝑁−𝐼−𝜇𝛼_𝑛+𝑁𝐵𝑇_[𝑛]𝑦_𝑛+𝐼−𝜇𝛼_𝑛+𝑁𝐵𝑇_[𝑛]𝑦_𝑛−𝐼−𝜇𝛼_𝑛𝐵𝑇_[𝑛]𝑦_𝑛+𝛼_𝑛+𝑁−𝛼_𝑛𝛾𝑉𝑥_𝑛≤𝛼_𝑛+𝑁𝛾𝑙𝑥_𝑛+𝑁−𝑥_𝑛+1−𝛼_𝑛+𝑁𝜏𝑦_𝑛+𝑁−𝑦_𝑛+𝛼_𝑛+𝑁−𝛼_𝑛𝛾𝑙𝑉𝑥_𝑛+𝜇𝐵𝑇_[𝑛]𝑦_𝑛. (40)
Observe that 𝑦_𝑛+𝑁−𝑦_𝑛=𝛽_𝑛+𝑁𝑆𝑥_𝑛+𝑁+1−𝛽_𝑛+𝑁𝑥_𝑛+𝑁−𝛽_𝑛𝑆𝑥_𝑛−1−𝛽_𝑛𝑥_𝑛≤𝛽_𝑛+𝑁𝑆𝑥_𝑛+𝑁−𝛽_𝑛+𝑁𝑆𝑥_𝑛+1−𝛽_𝑛+𝑁𝑥_𝑛+𝑁−1−𝛽_𝑛+𝑁𝑥_𝑛+𝛽_𝑛+𝑁𝑆𝑥_𝑛−𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛+𝑁𝑥_𝑛−1−𝛽_𝑛𝑥_𝑛≤𝛽_𝑛+𝑁𝑥_𝑛+𝑁−𝑥_𝑛+1−𝛽_𝑛+𝑁𝑥_𝑛+𝑁−𝑥_𝑛+𝛽_𝑛+𝑁−𝛽_𝑛𝑆𝑥_𝑛+𝛽_𝑛+𝑁−𝛽_𝑛𝑥_𝑛=𝑥_𝑛+𝑁−𝑥_𝑛+𝛽_𝑛+𝑁−𝛽_𝑛𝑆𝑥_𝑛+𝑥_𝑛. (41)
Together with (40) and (41), we have 𝑥_𝑛+𝑁+1−𝑥_𝑛+1≤𝛼_𝑛+𝑁𝛾𝑙𝑥_𝑛+𝑁−𝑥_𝑛+𝛼_𝑛+𝑁−𝛼_𝑛𝛾𝑙𝑉𝑥_𝑛+𝜇𝐵𝑇_[𝑛]𝑦_𝑛+1−𝛼_𝑛+𝑁𝜏𝑥_𝑛+𝑁−𝑥_𝑛+𝛽_𝑛+𝑁−𝛽_𝑛𝑆𝑥_𝑛+𝑥_𝑛=1−𝛼_𝑛+𝑁𝜏−𝛾𝑙𝑥_𝑛+𝑁−𝑥_𝑛+𝛼_𝑛+𝑁−𝛼_𝑛𝛾𝑙𝑉𝑥_𝑛+𝜇𝐵𝑇_[𝑛]𝑦_𝑛+𝛽_𝑛+𝑁−𝛽_𝑛𝑆𝑥_𝑛+𝑥_𝑛≤1−𝛼_𝑛+𝑁𝜏−𝛾𝑙𝑥_𝑛+𝑁−𝑥_𝑛+𝛼_𝑛+𝑁−𝛼_𝑛+𝛽_𝑛+𝑁−𝛽_𝑛𝑀₃, (42) where 𝑀₃=sup_𝑛{‖𝛾𝑙𝑉𝑥_𝑛‖+𝜇‖𝐵𝑇_[𝑛]𝑦_𝑛‖+‖𝑆𝑥_𝑛‖+‖𝑥_𝑛‖}.
By Lemma 5, we get 𝑥_𝑛+𝑁−𝑥_𝑛⟶0. (43)
Step  3. Show that 𝑥_𝑛−𝑇_{[𝑛+𝑁−1]}𝑇_{[𝑛+𝑁−2]}⋯𝑇_[𝑛]𝑥_𝑛⟶0. (44)
Observe that 𝑥_𝑛+1−𝑇_[𝑛]𝑦_𝑛=𝑃_𝐶𝛼_𝑛𝛾𝑉𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐵𝑇_[𝑛]𝑦_𝑛−𝑃_𝐶𝑇_[𝑛]𝑦_𝑛≤𝛼_𝑛𝛾𝑉𝑥_𝑛−𝜇𝐵𝑇_[𝑛]𝑦_𝑛⟶0. (45)
From conditions (i) and (ii) of Theorem 11, we obtain 𝑥_𝑛+1−𝑇_[𝑛]𝑥_𝑛≤𝑥_𝑛+1−𝑇_[𝑛]𝑦_𝑛+𝑇_[𝑛]𝑦_𝑛−𝑇_[𝑛]𝑥_𝑛≤𝑥_𝑛+1−𝑇_[𝑛]𝑦_𝑛+𝑦_𝑛−𝑥_𝑛=𝑥_𝑛+1−𝑇_[𝑛]𝑦_𝑛+𝛽_𝑛𝑆𝑥_𝑛−𝑥_𝑛⟶0. (46)
Recursively, 𝑥_𝑛+𝑁−𝑇_{[𝑛+𝑁−1]}𝑥_{𝑛+𝑁−1}𝑥⟶0,_{𝑛+𝑁−1}−𝑇_{[𝑛+𝑁−2]}𝑥_{𝑛+𝑁−2}⟶0. (47)
Since every 𝑇_[𝑛] is nonexpansive, it is easy to get 𝑇_{[𝑛+𝑁−1]}𝑥_{𝑛+𝑁−1}−𝑇_{[𝑛+𝑁−1]}𝑇_{[𝑛+𝑁−2]}𝑥_{𝑛+𝑁−2}⟶0. (48)
Similarly, we obtain 𝑇_{[𝑛+𝑁−1]}𝑇_{[𝑛+𝑁−2]}𝑥_{𝑛+𝑁−2}−𝑇_{[𝑛+𝑁−1]}𝑇_{[𝑛+𝑁−2]}𝑇_{[𝑛+𝑁−3]}𝑥_{𝑛+𝑁−3}⋮𝑇⟶0_{[𝑛+𝑁−1]}𝑇_{[𝑛+𝑁−2]}⋯𝑇_[𝑛+1]𝑥_𝑛+1−𝑇_{[𝑛+𝑁−1]}𝑇_{[𝑛+𝑁−2]}⋯𝑇_[𝑛]𝑥_𝑛⟶0. (49)
Thus we get 𝑥_𝑛+𝑁−𝑇_{[𝑛+𝑁−1]}⋯𝑇_[𝑛]𝑥_𝑛≤𝑥_𝑛+𝑁−𝑇_{[𝑛+𝑁−1]}𝑥_{𝑛+𝑁−1}+𝑇_{[𝑛+𝑁−1]}𝑥_{𝑛+𝑁−1}−𝑇_{[𝑛+𝑁−1]}𝑇_{[𝑛+𝑁−2]}𝑥_{𝑛+𝑁−2}+⋯+𝑇_{[𝑛+𝑁−1]}𝑇_{[𝑛+𝑁−2]}⋯𝑇_[𝑛+1]𝑥_𝑛+1−𝑇_{[𝑛+𝑁−1]}⋯𝑇_[𝑛]𝑥_𝑛⟶0. (50)
Since 𝑥_𝑛−𝑇_{[𝑛+𝑁−1]}𝑇_{[𝑛+𝑁−2]}⋯𝑇_[𝑛]𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛+𝑁+𝑥_𝑛+𝑁−𝑇_{[𝑛+𝑁−1]}⋯𝑇_[𝑛]𝑥_𝑛, (51) we obtain (44).
Step  4. Show that limsup_𝑛→∞⟨𝛾𝑉−𝜇𝐵𝑥^∗,𝑥_𝑛−𝑥^∗⟩≤0, (52) where 𝑥^∗=𝑃_Ω(𝐼−𝜇𝐵+𝛾𝑉)𝑥^∗ is a unique solution of the variational inequality (16).
Indeed, take a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} such that limsup_𝑛→∞𝛾𝑉−𝜇𝐵𝑥^∗,𝑥_𝑛−𝑥^∗=lim_𝑗→∞⟨𝛾𝑉−𝜇𝐵𝑥^∗,𝑥_𝑛𝑗−𝑥^∗⟩. (53)
Since {𝑥_𝑛𝑗} is bounded, there exists a subsequence {𝑥_𝑛𝑗𝑘} of {𝑥_𝑛𝑗} which converges weakly to 𝑥. Without loss of generality, we can assume 𝑥_𝑛𝑗⇀𝑥 and 𝑥_𝑛𝑗−𝑇_{[𝑛𝑗+𝑁−1]}𝑇_{[𝑛𝑗+𝑁−2]}⋯𝑇_[𝑛𝑗]𝑥_𝑛𝑗⟶0. (54)
Notice that, for each 𝑛_𝑗, 𝑇_{[𝑛𝑗+𝑁−1]}𝑇_{[𝑛𝑗+𝑁−2]}⋯𝑇_[𝑛𝑗] is some permutation of the mappings 𝑇₁𝑇₂⋯𝑇_𝑁. Since 𝑇₁𝑇₂⋯𝑇_𝑁 are finite, all the finite permutations are 𝑁!; there must be some permutation that appears infinite times. Without loss of generality, we can assume this permutation is 𝑇_𝑁𝑇_𝑁−1⋯𝑇₁. We obtain 𝑥_𝑛𝑗−𝑇_𝑁𝑇_𝑁−1⋯𝑇₁𝑥_𝑛𝑗⟶0. (55)
Obviously, 𝑇_𝑁𝑇_𝑁−1⋯𝑇₁ is nonexpansive. By Lemma 7, we have 𝑥=𝑇_𝑁𝑇_𝑁−1⋯𝑇₁𝑥. Further by the assumption in Theorem 11, we get 𝑥∈Fix𝑇_𝑁𝑇_𝑁−1⋯𝑇₁=^𝑁_𝑖=1Fix𝑇_𝑖. (56)
Since 𝑥^∗=𝑃_Ω(𝐼−𝜇𝐵+𝛾𝑉)𝑥^∗, it follows that limsup_𝑛→∞𝛾𝑉−𝜇𝐵𝑥^∗,𝑥_𝑛−𝑥^∗=lim_𝑗→∞𝛾𝑉−𝜇𝐵𝑥^∗,𝑥_𝑛𝑗−𝑥^∗=⟨𝛾𝑉−𝜇𝐵𝑥^∗,𝑥−𝑥^∗⟩≤0. (57)
Step  5. Show that 𝑥_𝑛⟶𝑥^∗. (58)
Denote 𝑧_𝑛=𝛼_𝑛𝛾𝑉𝑥_𝑛+(𝐼−𝜇𝛼_𝑛𝐵)𝑇_[𝑛]𝑦_𝑛; then 𝑥_𝑛+1=𝑃_𝐶𝑧_𝑛. From (37), we have 𝑥_𝑛+1−𝑥^∗2=𝑃_𝐶𝑧_𝑛−𝑥^∗2=𝑃_𝐶𝑧_𝑛−𝑧_𝑛,𝑃_𝐶𝑧_𝑛−𝑥^∗+𝑧_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗≤𝑧_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗=𝐼−𝜇𝛼_𝑛𝐵𝑇_[𝑛]𝑦_𝑛−𝐼−𝜇𝛼_𝑛𝐵𝑇_[𝑛]𝑥^∗+𝛼_𝑛𝛾𝑉𝑥_𝑛−𝜇𝐵𝑥^∗,𝑥_𝑛+1−𝑥^∗≤1−𝛼_𝑛𝜏𝑦_𝑛−𝑥^∗𝑥_𝑛+1−𝑥^∗+𝛼_𝑛𝛾𝑉𝑥_𝑛−𝛾𝑉𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝛼_𝑛𝛾𝑉𝑥^∗−𝜇𝐵𝑥^∗,𝑥_𝑛+1−𝑥^∗≤1−𝛼_𝑛𝜏𝑥_𝑛−𝑥^∗+𝛽_𝑛𝑆𝑥^∗−𝑥^∗×𝑥_𝑛+1−𝑥^∗+𝛼_𝑛𝛾𝑙𝑥_𝑛−𝑥^∗𝑥_𝑛+1−𝑥^∗+𝛼_𝑛𝛾𝑉𝑥^∗−𝜇𝐵𝑥^∗,𝑥_𝑛+1−𝑥^∗≤1−𝛼_𝑛𝜏−𝛾𝑙2𝑥_𝑛−𝑥^∗2+𝑥_𝑛+1−𝑥^∗2+𝛽_𝑛𝑥_𝑛−𝑥^∗𝑆𝑥^∗−𝑥^∗+𝛼_𝑛⟨𝛾𝑉𝑥^∗−𝜇𝐵𝑥^∗,𝑥_𝑛+1−𝑥^∗⟩. (59)
This implies that 𝑥_𝑛+1−𝑥^∗2≤1−𝛼_𝑛𝜏−𝛾𝑙1+𝛼_𝑛𝑥𝜏−𝛾𝑙^𝑛−𝑥^∗2+21+𝛼_𝑛×𝛽𝜏−𝛾𝑙_𝑛𝑥_𝑛−𝑥^∗𝑆𝑥^∗−𝑥^∗+𝛼_𝑛⟨𝛾𝑉𝑥^∗−𝜇𝐵𝑥^∗,𝑥_𝑛+1−𝑥^∗⟩≤1−𝛼_𝑛𝜏−𝛾𝑙𝑥_𝑛−𝑥^∗2+2𝛽_𝑛𝑀₁+2𝛼_𝑛⟨𝛾𝑉𝑥^∗−𝜇𝐵𝑥^∗,𝑥_𝑛+1−𝑥^∗⟩, (60) where 𝑀₁=sup_𝑛‖𝑥_𝑛−𝑥^∗‖‖𝑆𝑥^∗−𝑥^∗‖, 𝑛≥1. Put 𝛾_𝑛=𝛼_𝑛(𝜏−𝑙𝛾), 𝛿_𝑛=2𝛽_𝑛𝑀₁+2𝛼_𝑛⟨𝛾𝑉𝑥^∗−𝜇𝐵𝑥^∗,𝑥_𝑛+1−𝑥^∗⟩. It is easy to see that limsup_𝑛→∞𝛿_𝑛/𝛾_𝑛≤0. Hence, by Lemma 6, the sequence {𝑥_𝑛} converges strongly to 𝑥^∗.

5. Numerical Result

In this section, we consider the following simple example to demonstrate the effectiveness, realization, and convergence of the algorithms in Theorems 8 and 11.

Example 12. Let 𝐻=ℝ, 𝐶=[1/4,+∞). Define 𝑇₁∶𝑥↦√𝑥, 𝑇₂∶𝑥↦𝑥+𝜋/4−arctan𝑥, 𝑆∶𝑥↦sin𝑥. Take 𝐵=𝐼 with Lipschitz constant 𝑘=1 and strongly monotone constant 𝜂=1, 𝑉𝑥=2𝑥, ∀𝑥∈𝐻, with Lipschitz coefficient 𝑙=2. Give the parameters 𝛼_𝑛=1/2𝑛; 𝛽_𝑛=1/𝑛² for every 𝑛≥1; fix 𝜇=1 and 𝛾=1/4. Then by Theorems 8 and 11, respectively, the sequence {𝑥_𝑛} is generated by 𝑦_𝑛=1𝑛²sin𝑥_𝑛+11−𝑛²𝑥_𝑛,𝑥_𝑛+1=𝑃_𝐶1𝑥4𝑛^𝑛+11−2𝑛𝑇𝑦_𝑛,𝑦_𝑛=1𝑛²sin𝑥_𝑛+11−𝑛²𝑥_𝑛,𝑥_𝑛+1=𝑃_𝐶1𝑥4𝑛^𝑛+11−𝑇2𝑛^[𝑛]𝑦_𝑛. (61)(62) As 𝑛→∞, we have {𝑥_𝑛}→𝑥^∗=1.

Let 𝜔_𝑖=1/2, 𝑖=1,2; then we have 𝑇𝑥=(1/2)(√𝑥+𝑥+𝜋/4−arctan𝑥), 𝑇_[𝑛]𝑥=√𝑥 if 𝑛 is odd and 𝑇_[𝑛]𝑥=𝑥+𝜋/4−arctan𝑥 if 𝑛 is even. Put 𝑧_𝑛=(1/4𝑛)𝑥_𝑛+(1−1/2𝑛)𝑇𝑦_𝑛; then (61) is equivalent to 𝑥_𝑛+1=𝑧_𝑛,if𝑧_𝑛1∈𝐶,4,if𝑧_𝑛∈𝐶. (63) Using the same method to treat (62), we can get similar equation as the above formula.

Now we turn to numerical simulation using the algorithms (17) and (37), respectively. Take the initial guess 𝑥₁=2; using software Matlab R2012, we obtain the numerical experiment results in Tables 1 and 2.

Table 1 

𝑥₁=2.

Table 2 

𝑥₁=2.

From the computer programming point of view, the algorithms are easier to implement in this paper.

Acknowledgments

The authors would like to thank the referee for valuable suggestions to improve the manuscript and the Fundamental Research Funds for the Central Universities (Grant 3122013k004).