A Unified Iterative Treatment for Solutions of Problems of Split Feasibility and Equilibrium in Hilbert Spaces

Abstract

We at first raise the so called split feasibility fixed point problem which covers the problems of split feasibility, convex feasibility, and equilibrium as special cases and then give two types of algorithms for finding solutions of this problem and establish the corresponding strong convergence theorems for the sequences generated by our algorithms. As a consequence, we apply them to study the split feasibility problem, the zero point problem of maximal monotone operators, and the equilibrium problem and to show that the unique minimum norm solutions of these problems can be obtained through our algorithms. Since the variational inequalities, convex differentiable optimization, and Nash equilibria in noncooperative games can be formulated as equilibrium problems, each type of our algorithms can be considered as a generalized methodology for solving the aforementioned problems.

1. Introduction

Throughout this paper, ℋ denotes a real Hilbert space with inner product ⟨⋅,⋅⟩ and the norm ‖⋅‖, 𝐼 the identity mapping on ℋ, ℕ the set of all natural numbers, and ℝ the set of all real numbers. For a self-mapping 𝑇 on ℋ, Fix(𝑇) denotes the set of all fixed points of 𝑇. If 𝑀∶ℋ→2^ℋ is a set-valued mapping; then 𝒟(𝑀) denotes its domain, that is, 𝒟(𝑀)={𝑥∈ℋ∶𝑀(𝑥)≠∅}.

Let 𝐶 and 𝑄 be nonempty closed convex subsets of two Hilbert spaces ℋ₁ and ℋ₂, respectively, and let 𝐴∶ℋ₁→ℋ₂ be a bounded linear mapping. The split feasibility problem (SFP) is the problem of finding a point with the property 𝑥^∗∈𝐶,𝐴𝑥^∗∈𝑄. (1) The SFP was first introduced by Censor and Elfving [1] for modeling inverse problems which arise from phase retrievals and medical image reconstruction. Recently, it has been found that the SFP can also be used to model the intensity-modulated radiation therapy. For details, the readers are referred to Xu [2] and the references therein.

Assume that the SFP has a solution. There are many iterative methods designed to approximate its solutions. The most popular algorithm is the 𝐶𝑄 algorithm introduced by Byrne [3, 4].

Start with any 𝑥₁∈ℋ₁ and generate a sequence {𝑥_𝑛} through the iteration 𝑥_𝑛+1=𝑃_𝐶𝐼−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛, (2)where 𝛾∈(0,2/‖𝐴‖²), 𝐴^∗ the adjoint of 𝐴, and 𝑃_𝐶 and 𝑃_𝑄 are the metric projections onto 𝐶 and 𝑄, respectively.

The sequence {𝑥_𝑛} generated by the 𝐶𝑄 algorithm (2) converges weakly to a solution of SFP(1), cf. [2–4]. Under the assumption that SFP(1) has a solution, it is known that a point 𝑥^∗∈ℋ₁ solves SFP(1) if and only if 𝑥^∗ is a fixed point of the operator 𝑃_𝐶𝐼−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴, (3) cf. [2], where Xu also proposed the regularized method, 𝑥_𝑛+1=𝑃_𝐶𝐼−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼_𝑛𝐼𝑥_𝑛, (4) and proved that the sequence {𝑥_𝑛} converges strongly to the minimum norm solution of SFP(1) provided that the parameters {𝛼_𝑛} and {𝛾_𝑛} verify some suitable conditions. This regularized method was further investigated by Yao et al. [5] and Yao et al. [6].

Putting 𝑆=𝑃_𝐶 and 𝑇=𝑃_𝑄, SFP(1) is of the forms: find𝑥^∗∈ℋ₁sothat𝑥^∗∈Fix(𝑆),𝐴𝑥^∗∈Fix(𝑇). (5) As a metric projection is firmly nonexpansive, it is reasonable to require the 𝑆 and 𝑇 in (5) to be firmly nonexpansive only and call it a split feasibility fixed point problem (SFFP). Many interesting problems in the literature can be described as SFFP.

(i) A set-valued map 𝑀∶ℋ→2^ℋ is called monotone if 𝑥−𝑦,𝑢−𝑣≥0 (6) for all 𝑥,𝑦∈𝒟(𝑀) and for any 𝑢∈𝑀(𝑥), 𝑣∈𝑀(𝑦). 𝑀 is said to be maximal monotone if its graph {(𝑥,𝑢)∶𝑥∈ℋ,𝑢∈𝑀(𝑥)} is not properly contained in the graph of any other monotone operator. A point 𝑣∈ℋ is called a zero point of a maximal monotone operator 𝑀 if 0∈𝑀(𝑣). The set of all zero points of 𝑀 is denoted by 𝑀⁻¹0, which is equal to Fix(𝐽^𝑀_𝛼) for any 𝛼>0, where 𝐽^𝑀_𝛼 denotes the resolvent of a monotone operator 𝑀; that is, 𝐽^𝑀_𝛼(𝑥)=(𝐼+𝛼𝑀)⁻¹(𝑥) for any 𝑥∈ℋ. It is known that for any 𝛼>0, 𝐽^𝑀_𝛼 is firmly nonexpansive. Now, let 𝑀 and 𝑁 be two maximal monotone operators on ℋ₁ and ℋ₂, respectively. Replacing 𝐶 and 𝑄 with 𝑀⁻¹0=Fix(𝐽^𝑀_𝛼) and 𝑁⁻¹0=Fix(𝐽^𝑁_𝛽), respectively, in (1), the SFP becomes a SFFP: find𝑥^∗∈ℋ₁sothat𝑥^∗∈Fix𝐽^𝑀_𝛼,𝐴𝑥^∗∈Fix𝐽^𝑁_𝛽. (7) Putting 𝐴=𝐼, the previous SFFP is reduced to the common zero point problem of two maximal monotone operators: find𝑥^∗∈ℋ₁sothat𝑥^∗∈𝑀⁻¹0∩𝑁⁻¹0. (8)

(ii) Let 𝑓∶𝐶×𝐶→ℝ. An equilibrium problem is the problem of finding 𝑥∈𝐶 such that 𝑓𝑥,𝑦≥0,∀𝑦∈𝐶, (9) whose solution set is denoted by EP(𝑓). For solving an equilibrium problem, we usually assume that the function 𝑓 satisfies the following conditions:(A1)  𝑓(𝑥,𝑥)=0,  for all  𝑥∈𝐶;(A2)  𝑓  is monotone, that is,  𝑓(𝑥,𝑦)+𝑓(𝑦,𝑥)≤0,  for all  𝑥∈𝐶;(A3)  for all  𝑥,𝑦,𝑧∈𝐶,  limsup_𝑡↓0𝑓((1−𝑡)𝑥+𝑡𝑧,𝑦)≤𝑓(𝑥,𝑦);(A4) for all𝑥∈𝐶,𝑓(𝑥,⋅)  is convex and lower semicontinuous.Blum and Oettli [7] and Aoyama et al. [8] showed that there exists a unique 𝑧∈𝐶 such that 𝑓+1𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶. (10) Moreover, For 𝑟>0, define 𝐽^𝑓_𝑟∶ℋ→𝐶 by 𝐽^𝑓_𝑟+1𝑥=𝑧∈𝐶∶𝑓𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶 (11) for all 𝑥∈ℋ. Combettes and Hirstoaga [9] showed that there hold(a)𝐽^𝑓_𝑟  is single-valued;(b)𝐽^𝑓_𝑟  is firmly nonexpansive;(c)Fix(𝐽^𝑓_𝑟)=EP(𝑓);(d)EP(𝑓)  is closed and convex.

Now, let 𝑓∶𝐶×𝐶→ℝ and 𝑔∶𝑄×𝑄→ℝ be two functions satisfying conditions (A1)–(A4). Replacing 𝐶 and 𝑄 with EP(𝑓)=Fix(𝐽^𝑓_𝛼) and EP(𝑔)=Fix(𝐽^𝑔_𝛽), respectively, in (1), the SFP becomes a SFFP: find𝑥^∗∈ℋ₁sothat𝑥^∗∈Fix𝐽^𝑓_𝛼,𝐴𝑥^∗∈Fix𝐽^𝑔_𝛽. (12) Putting ℋ₁=ℋ₂, 𝐶=𝑄, and 𝐴=𝐼, the previous SFFP is reduced to the common equilibrium problem: find𝑥^∗𝑥∈𝐶sothat𝑓^∗,𝑦≥0,𝑔𝑥^∗,𝑦≥0,∀𝑦∈𝐶. (13)

(iii) When ℋ₁=ℋ₂=ℋ, and the bounded linear operator 𝐴 is the identity mapping, SFP(1) is reduced to the convex feasibility problem (CFP): find𝑥^∗∈ℋsothat𝑥^∗∈𝐶∩𝑄, (14) which in turn can be described as a SFFP: find𝑥^∗∈ℋsothat𝑥^∗∈Fix(𝑆),𝐴𝑥^∗∈Fix(𝑇), (15) where 𝑆=𝑃_𝐶 and 𝑇=𝑃_𝑄. Although SFFP(5) contains CFP as a special case, it cannot cover the multiple-set split convex feasibility problem described in [10].

In this paper, we are concerned with iterative methods for SFFP(5). We derive some weak convergence theorems for SFFP(5) in Section 3. In Section 4, we describe SFFP(5) in a more general form.

Let  {𝑆_𝑛}  and  {𝑇_𝑛}  be two families of firmly nonexpansive self-mappings on  ℋ₁  and ℋ₂,  respectively, so that  ∩^∞_𝑛=1Fix(𝑆_𝑛)≠∅  and  ∩^∞_𝑛=1Fix(𝑇_𝑛)≠∅𝑓𝑖𝑛𝑑𝑥^∗∈ℋ₁𝑠𝑜𝑡ℎ𝑎𝑡𝑥^∗∈∩^∞_𝑛=1Fix𝑆_𝑛,𝐴𝑥^∗∈∩^∞_𝑛=1Fix𝑇_𝑛 (16)and obtain the following main result.

Let  {𝑎_𝑛},  {𝑏_𝑛},  {𝑐_𝑛},  and  {𝑑_𝑛}  be sequences in  [0,1]  with  𝑎_𝑛+𝑏_𝑛+𝑐_𝑛+𝑑_𝑛=1  and  𝑎_𝑛∈(0,1)  for all  𝑛∈ℕ. Let {𝛾_𝑛} be a sequence in (0,2/‖𝐴‖²) and let {𝑒_𝑛} be a bounded sequence in ℋ₁. Suppose that the solution set Ω of  SFFP(16)  is nonempty. For any 𝑢∈ℋ₁, start with an arbitrary 𝑥₁∈ℋ₁ and define a sequence {𝑥_𝑛} by𝑥_𝑛+1=𝑎_𝑛𝑢+𝑏_𝑛𝑥_𝑛+𝑐_𝑛𝑆_𝑛𝐼−𝛾_𝑛𝐴^∗𝐼−𝑇_𝑛𝐴𝑥_𝑛+𝑑_𝑛𝑒_𝑛. (17)Then, the sequence {𝑥_𝑛} converges strongly to 𝑃_Ω𝑢 provided that the following conditions are satisfied: (i)lim_𝑛→∞𝑎_𝑛=lim_𝑛→∞(𝑑_𝑛/𝑎_𝑛)=0,  ∑^∞_𝑛=1𝑎_𝑛=∞,  ∑^∞_𝑛=1𝑑_𝑛<∞;(ii)lim_𝑛→∞inf𝑐_𝑛(2𝛾_𝑛−𝛾²_𝑛‖𝐴‖²)>0,  lim_𝑛→∞inf𝑏_𝑛𝑐_𝑛>0;(iii)there are two nonnegative real-valued functions  𝜅₁  and  𝜅₂  on  ℕ  with𝑆_𝑚𝑥−𝑥≤𝜅₁(𝑚)𝑆_𝑛𝑥−𝑥,∀𝑚∈ℕ,∀𝑛≥𝑚,∀𝑥∈ℋ₁,𝑇_𝑚𝑦−𝑦≤𝜅₂(𝑚)𝑇_𝑛𝑦−𝑦,∀𝑚∈ℕ,∀𝑛≥𝑚,∀𝑦∈ℋ₂. (18)

Based on the concept of using contractions to approximate nonexpansive mappings, another type of algorithms for SFFP(5) is also introduced, and the corresponding strong convergence theorem for the sequence generated by such algorithm is given too.

In Section 5, since resolvents of monotone operators are firmly nonexpansive, we replace the sequences {𝑆_𝑛} and {𝑇_𝑛} of firmly nonexpansive mappings in the previous condition (iii) by two sequences of resolvents of maximal monotone operators. Then, the proposed algorithm becomes a scheme to approach the minimum norm solution of zero point problem of maximal monotone operators and the equilibrium problem. It is worth noting that as Blum and Oettli [7] showed that the variational inequalities, convex differentiable optimization, and Nash equilibria in noncooperative games can be formulated as equilibrium problems, the proposed algorithm can be considered as a generalized methodology for solving all aforementioned problems.

2. Preliminaries

In order to facilitate our investigation in this paper, we recall some basic facts. A mapping 𝑆∶ℋ→ℋ is said to be(i)nonexpansive if 𝑆𝑥−𝑆𝑦≤𝑥−𝑦,∀𝑥,𝑦∈ℋ; (19)(ii)firmly nonexpansive if 𝑆𝑥−𝑆𝑦²≤𝑥−𝑦,𝑆𝑥−𝑆𝑦,∀𝑥,𝑦∈ℋ; (20)(iii)𝜆-averaged by 𝐺 if 𝑆=(1−𝜆)𝐼+𝜆𝐺, (21) for some 𝜆∈(0,1) and some nonexpansive mapping 𝐺;(iv)𝜈-inverse strongly monotone (𝜈-ism), with 𝜈>0, if 𝑥−𝑦,𝑆𝑥−𝑆𝑦≥𝜈𝑆𝑥−𝑆𝑦²,∀𝑥,𝑦∈ℋ. (22)If 𝑆 is nonexpansive, then the fixed point set Fix(𝑆) of 𝑆 is closed and convex, cf. [11]. If 𝑆=(1−𝜆)𝐼+𝜆𝐺 is averaged, then 𝑆 is nonexpansive with Fix(𝑆)=Fix(𝐺). It is well known that 𝑆 is firmly nonexpansive if and only if it is 1/2-averaged, cf. [11], and so is 𝐼−𝑆. Here we would like to mention that the term “averaged mapping” originated in [12, 13]. In [12], Baillon el at. showed that if 𝑆 is a 𝜆-averaged mapping by 𝐺 on a nonempty closed convex subset 𝐶 of a uniformly convex Banach space, then Fix(𝐺)=𝜙 if and only if lim_𝑛→∞‖𝑆^𝑛𝑥‖=∞ for all 𝑥 in 𝐶. Moreover, in [13], Bruck and Reich showed that if the above 𝐶 satisfies 𝐶=−𝐶 and 𝑆 is odd, then {𝑆^𝑛𝑥} converges strongly to a fixed point of 𝐺.

Let 𝐶 be a nonempty closed convex subset of ℋ. The metric projection 𝑃_𝐶 from ℋ onto 𝐶 is the mapping that assigns each 𝑥∈ℋ the unique point 𝑃_𝐶𝑥 in 𝐶 with the property 𝑥−𝑃_𝐶𝑥=min_𝑦∈𝐶𝑦−𝑥. (23) It is known that 𝑃_𝐶 is firmly nonexpansive and characterized by the inequality: for any 𝑥∈ℋ, 𝑥−𝑃_𝐶𝑥,𝑦−𝑃_𝐶𝑥≤0,∀𝑦∈𝐶. (24)

We need some lemmas that will be quoted in the sequel.

Lemma 1 (see [4]). If 𝑆 is a self-mapping on ℋ, then the following assertions hold.(a)𝑆 is nonexpansive if and only if the complement 𝐼−𝑆 is 1/2-averaged.(b)If 𝑆 is 𝜈-ism and 𝛾>0, then 𝛾𝑆 is (𝜈/𝛾)-ism.(c)𝑆 is 𝛼-averaged if and only if 𝐼−𝑆 is (1/2𝛼)-ism.(d)If 𝑆 is 𝛼₁-averaged and 𝑇 is 𝛼₂-averaged, then the composite 𝑆𝑇 is 𝛼₁+𝛼₂−𝛼₁𝛼₂-averaged.(e)If 𝑆 and 𝑈 are averaged on ℋ so that Fix(𝑆)∩Fix(𝑈)≠∅, then Fix(𝑆𝑈)=Fix(𝑆)∩Fix(𝑈). (25)

For 𝛼>0, the resolvent 𝐽^𝐴_𝛼 of a maximal monotone operator 𝐴 on ℋ has the following properties.

Lemma 2. Let 𝐴 be a maximal monotone operator on ℋ. Then,(a)𝐽^𝐴_𝛼 is single-valued and firmly nonexpansive;(b)𝒟(𝐽^𝐴_𝛼)=ℋ and Fix(𝐽^𝐴_𝛼)=𝐴⁻¹0;(c) (the resolvent identity) for 𝜇,𝜆>0, the following identity holds: 𝐽^𝐴_𝜇𝑥=𝐽^𝐴_𝜆𝜆𝜇𝜆𝑥+1−𝜇𝐽^𝐴_𝜇𝑥,∀𝑥∈ℋ. (26)

We referred readers to [14–24] for maximal monotone operators and their related algorithms.

Lemma 3. Let 𝑥,𝑦,𝑧∈ℋ. Then,(a)‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑥+𝑦⟩;(b)for any 𝜆∈ℝ, 𝜆𝑥+(1−𝜆)𝑦²=𝜆‖𝑥‖²+(1−𝜆)𝑦²−𝜆(1−𝜆)𝑥−𝑦²; (27)(c)for 𝑎,𝑏,𝑐∈[0,1] with 𝑎+𝑏+𝑐=1, 𝑎𝑥+𝑏𝑦+𝑐𝑧²=𝑎‖𝑥‖²+𝑏𝑦²+𝑐‖𝑧‖²−𝑎𝑏𝑥−𝑦²−𝑎𝑐‖𝑥−𝑧‖²−𝑏𝑐𝑦−𝑧². (28)

Lemma 4 (see [11] (demiclosedness principle)). Let 𝑆 be a nonexpansive self-mapping on ℋ and suppose that {𝑥_𝑛} is a sequence in ℋ such that {𝑥_𝑛} converges weakly to some 𝑧∈ℋ and lim_𝑛→∞‖𝑥_𝑛−𝑆𝑥_𝑛‖=0. Then, 𝑆𝑧=𝑧.

Lemma 5 (see [23]). Let {𝑠_𝑛} be a sequence of nonnegative real numbers satisfying 𝑠_𝑛+1≤1−𝛼_𝑛𝑠_𝑛+𝛼_𝑛𝜇_𝑛+𝜈_𝑛,𝑛∈ℕ, (29) where {𝛼_𝑛}, {𝜇_𝑛}, and {𝜈_𝑛} verify the following conditions:(i){𝛼_𝑛}⊆[0,1], ∑^∞_𝑛=1𝛼_𝑛=∞;(ii)limsup_𝑛→∞𝜇_𝑛≤0;(iii){𝜈_𝑛}⊆[0,∞) and ∑^∞_𝑛=1𝜈_𝑛<∞.Then, lim_𝑛→∞𝑠_𝑛=0.

Lemma 6 (see [25]). Let {𝑠_𝑛} be a sequence in ℝ that does not decrease at infinity in the sense that there exists a subsequence {𝑠_𝑛𝑖} such that 𝑠_𝑛𝑖<𝑠_𝑛𝑖+1,∀𝑖∈ℕ. (30) For any 𝑘∈ℕ, define 𝑚_𝑘=max{𝑗≤𝑘∶𝑠_𝑗<𝑠_𝑗+1}. Then, 𝑚_𝑘→∞ as 𝑘→∞ and max{𝑠_𝑚𝑘,𝑠_𝑘}≤𝑠_𝑚𝑘+1, for all 𝑘∈ℕ.

3. Weak Convergence Theorems

In this section, we at first transform SFFP(5) into a fixed point problem for the operator 𝑆(𝐼−𝛾𝐴^∗(𝐼−𝑇)𝐴), where 𝛾 is any positive real number. And then use fixed point algorithms to solve SFFP(5). From now on until the end of this paper, unless we state specifically, 𝑆 and 𝑆_𝑛, 𝑛∈ℕ (resp., 𝑇 and 𝑇_𝑛, 𝑛∈ℕ) are firmly nonexpansive self-mappings on ℋ₁ (resp., ℋ₂), and 𝐴 denotes a bounded linear operator from ℋ₁ into ℋ₂ with adjoint 𝐴^∗.

Lemma 7. Let 𝑆 be a firmly nonexpansive self-mapping on ℋ with Fix(𝑆)≠∅. Then, for any 𝑥∈ℋ, one has ⟨𝑥−𝑆𝑥,𝑣−𝑆𝑥⟩≤0,∀𝑣∈Fix(𝑆). (31)

Proof. Since 𝑣∈Fix(𝑆) and 𝑆 is firmly nonexpansive, we have ‖𝑆𝑥−𝑣‖²=‖𝑆𝑥−𝑆𝑣‖²≤⟨⟩=⟨⟩,𝑥−𝑣,𝑆𝑥−𝑆𝑣𝑥−𝑣,𝑆𝑥−𝑣 (32) and hence, ⟨⟩=⟨⟩+⟨⟩=‖‖𝑥−𝑆𝑥,𝑣−𝑆𝑥𝑥−𝑣,𝑣−𝑆𝑥𝑣−𝑆𝑥,𝑣−𝑆𝑥𝑣−𝑆𝑥²−⟨⟩≤‖‖𝑥−𝑣,𝑆𝑥−𝑣𝑣−𝑆𝑥²−‖𝑆𝑥−𝑣‖²=0. (33)

Although ⟨𝑥−𝑆𝑥,𝑣−𝑆𝑥⟩≤0, for all 𝑣∈Fix(𝑆) is similar to the characterization inequality (24) for the metric projection 𝑃_Fix(𝑆), as 𝑆𝑥 needs not to be in Fix(𝑆), it is in general different from 𝑃_Fix(𝑆)(𝑥). For example, let 𝑆𝑥=3𝑥/4 for all 𝑥∈ℝ, which is obviously firmly nonexpansive with Fix(𝑆)={0}. Thus, 𝑃_Fix(𝑆)(𝑥)=0 for all 𝑥∈ℝ, while 𝑆𝑥≠0 for all 𝑥≠0.

Proposition 8. For any 𝛾∈(0,2/‖𝐴‖²), the operator 𝐼−𝛾𝐴^∗(𝐼−𝑇)𝐴 is 𝛾‖𝐴‖²/2-averaged and 𝑆[𝐼−𝛾𝐴^∗(𝐼−𝑇)𝐴] is (2+𝛾‖𝐴‖²)/4-averaged.

Proof. Using the fact that 𝑇 is firmly nonexpansive, it is routine to show that 𝐴^∗(𝐼−𝑇)𝐴 is 1/‖𝐴‖²-ism, and so 𝛾𝐴^∗(𝐼−𝑇)𝐴 is 1/𝛾‖𝐴‖²-ism by Lemma 1(b). Thus, Lemma 1(c) shows that 𝐼−𝛾𝐴^∗(𝐼−𝑇)𝐴 is 𝛾‖𝐴‖²/2-averaged. As 𝑆 is firmly nonexpansive, it is 1/2-averaged. Therefore, 𝑆[𝐼−𝛾𝐴^∗(𝐼−𝑇)𝐴] is (1/2)+(𝛾‖𝐴‖²/2)−(𝛾‖𝐴‖²/4)=((2+𝛾‖𝐴‖²)/4)-averaged by Lemma 1(d).

Proposition 9. Let Ω be the solution set of SFFP(5); that is, Ω=Fix(𝑆)∩𝐴⁻¹(Fix(𝑇)). For any 𝛾∈(0,2/‖𝐴‖²), let 𝑈=𝐼−𝛾𝐴^∗(𝐼−𝑇)𝐴. Suppose that Ω≠∅. Then, Fix(𝑆𝑈)=Fix(𝑆)∩Fix(𝑈)=Ω.

Proof. If 𝑥^∗ solves SFFP(5), we have 𝑥^∗∈Fix(𝑆) and 𝐴𝑥^∗∈Fix(𝑇). Now, note that 𝐴𝑥^∗∈Fix(𝑇) implies (𝐼−𝑇)𝐴𝑥^∗=0, (34) and so 𝐼−𝛾𝐴^∗(𝐼−𝑇)𝐴𝑥^∗=𝑥^∗, (35) which means that 𝑥^∗∈Fix(𝑈). Consequently, 𝑥^∗∈Fix(𝑆)∩Fix(𝑈). This shows that Ω⊆Fix(𝑆)∩Fix(𝑈).
For the inverse inclusion, let 𝑥^∗ be any member of Fix(𝑆)∩Fix(𝑈) and pick 𝑣∈Ω. It is readily seen from 𝑥^∗∈Fix(𝑈) that 𝐴^∗(𝐼−𝑇)𝐴𝑥^∗=0. (36) Since 𝐴𝑣∈Fix(𝑇), an application of Lemma 7 yields (𝐼−𝑇)𝐴𝑥^∗,𝐴𝑣−𝑇𝐴𝑥^∗≤0, (37) which together with (36) implies that (𝐼−𝑇)𝐴𝑥^∗2=(𝐼−𝑇)𝐴𝑥^∗,𝐴𝑥^∗+)−𝐴𝑣(𝐼−𝑇𝐴𝑥^∗,𝐴𝑣−𝑇𝐴𝑥^∗≤(𝐼−𝑇)𝐴𝑥^∗,𝐴𝑥^∗=𝐴−𝐴𝑣_∗(𝐼−𝑇)𝐴𝑥,𝑥^∗−𝑣=0. (38) This comes to conclude that 𝐴𝑥^∗∈Fix(𝑇), and hence 𝑥^∗∈Ω once we note that 𝑥^∗∈Fix(𝑆). Finally, since Ω≠∅ by assumption, we have Fix(𝑆)∩Fix(𝑈)=Ω≠∅. Thus, Fix(𝑆𝑈)=Fix(𝑆)∩Fix(𝑈) follows from Lemma 1(e).

Proposition 10 (see [2, 4]). If 𝐺 is an averaged self-mapping on ℋ with Fix(𝐺)≠∅, then for any 𝑥∈ℋ, the sequence {𝐺^𝑛𝑥} converges weakly to a fixed point of 𝐺.

An immediate consequence of Propositions 8, 9, and 10 is the following convergence result.

Theorem 11. Assume that SFFP(5) has a solution. Then, for any 𝛾∈(0,2/‖𝐴‖²) and starting with any point 𝑥₁∈ℋ, the sequence {𝑥_𝑛} generated by 𝑥_𝑛+1=𝑆𝐼−𝛾𝐴^∗(𝐼−𝑇)𝐴𝑥_𝑛 (39) converges weakly to a solution of SFFP(5).

Proposition 12 (see [2]). Let 𝐺 be a 𝛼-averaged self-mapping on ℋ with Fix(𝐺)≠∅ and assume that {𝛼_𝑛} is a sequence in [0,1/𝛼] such that ^∞_𝑛=1𝛼_𝑛1𝛼−𝛼_𝑛=∞. (40) Then, for any 𝑥₁∈ℋ, the sequence {𝑥_𝑛} generated by the Mann's algorithm 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝐺𝑥_𝑛 (41) converges weakly to a fixed point of 𝐺.

Applying Propositions 8, 9, and 12, we have the following result.

Theorem 13. Assume that SFFP(5) has a solution and 𝛾∈(0,2/‖𝐴‖²). Let {𝛼_𝑛} be a sequence in [0,4/(2+𝛾‖𝐴‖²)] with ^∞_𝑛=1𝛼_𝑛42+‖𝐴‖²−𝛼_𝑛=∞. (42) Then, for any 𝑥₁∈ℋ, the sequence {𝑥_𝑛} generated by the Mann's algorithm 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑆𝐼−𝛾𝐴^∗(𝐼−𝑇)𝐴𝑥_𝑛 (43) converges weakly to a solution of SFFP(5).

4. Strong Convergence Theorems

In this section, we devise two algorithms, one for SFFP(16) and the other for SFFP(5). We deal with SFFP(16) firstly. To begin with, we need a lemma.

Lemma 14. For any 𝛾∈(0,2/‖𝐴‖²) and all 𝑥,𝑦∈ℋ₁, one has [𝐼−𝛾𝐴^∗(𝐼−𝑇)𝐴]𝑥−[𝐼−𝛾𝐴^∗(𝐼−𝑇)𝐴]𝑦²≤𝑥−𝑦²−2𝛾−𝛾²‖𝐴‖²×(𝐼−𝑇)𝐴𝑥−(𝐼−𝑇)𝐴𝑦². (44)

Proof. Since 𝑇 is firmly nonexpansive, so is 𝐼−𝑇. Hence, for all 𝑥,𝑦∈ℋ₁, we have (𝐼−𝑇)𝐴𝑥−(𝐼−𝑇)𝐴𝑦²≤𝐴𝑥−𝐴𝑦,(𝐼−𝑇)𝐴𝑥−(𝐼−𝑇)𝐴𝑦. (45) Consequently, [𝐼−𝛾𝐴^∗(𝐼−𝑇)𝐴]𝑥−[𝐼−𝛾𝐴^∗(𝐼−𝑇)𝐴]𝑦²=𝑥−𝑦²×−2𝛾𝑥−𝑦,𝐴^∗(𝐼−𝑇)𝐴𝑥−𝐴^∗(𝐼−𝑇)𝐴𝑦+𝛾²𝐴^∗(𝐼−𝑇)𝐴𝑥−𝐴^∗(𝐼−𝑇)𝐴𝑦²=𝑥−𝑦²×()()−2𝛾𝐴𝑥−𝐴𝑦,𝐼−𝑇𝐴𝑥−𝐼−𝑇𝐴𝑦+𝛾²𝐴^∗(𝐼−𝑇)𝐴𝑥−𝐴^∗(𝐼−𝑇)𝐴𝑦²≤𝑥−𝑦²−2𝛾(𝐼−𝑇)𝐴𝑥−(𝐼−𝑇)𝐴𝑦²+𝛾²‖𝐴‖²(𝐼−𝑇)𝐴𝑥−(𝐼−𝑇)𝐴𝑦²=𝑥−𝑦²−2𝛾−𝛾²‖𝐴‖²×(𝐼−𝑇)𝐴𝑥−(𝐼−𝑇)𝐴𝑦². (46)

Theorem 15. Let {𝑎_𝑛}, {𝑏_𝑛}, {𝑐_𝑛}, and {𝑑_𝑛} be sequences in [0,1] with 𝑎_𝑛+𝑏_𝑛+𝑐_𝑛+𝑑_𝑛=1 and 𝑎_𝑛∈(0,1) for all 𝑛∈ℕ. Let {𝛾_𝑛} be a sequence in (0,2/‖𝐴‖²) and let {𝑒_𝑛} be a bounded sequence in ℋ₁. Suppose that the solution set Ω of SFFP(16) is nonempty. For any 𝑢∈ℋ₁, start with any 𝑥₁∈ℋ₁ and define a sequence {𝑥_𝑛} by 𝑥_𝑛+1=𝑎_𝑛𝑢+𝑏_𝑛𝑥_𝑛&+𝑐_𝑛𝑆_𝑛𝐼−𝛾_𝑛𝐴^∗𝐼−𝑇_𝑛𝐴𝑥_𝑛+𝑑_𝑛𝑒_𝑛. (47) Then the sequence {𝑥_𝑛} converges strongly to 𝑃_Ω𝑢 provided that the following conditions are satisfied: (i)lim_𝑛→∞𝑎_𝑛=lim_𝑛→∞(𝑑_𝑛/𝑎_𝑛)=0,  ∑^∞_𝑛=1𝑎_𝑛=∞,  ∑^∞_𝑛=1𝑑_𝑛<∞;(ii)lim_𝑛→∞inf𝑐_𝑛(2𝛾_𝑛−𝛾²_𝑛‖𝐴‖²)>0,  lim_𝑛→∞inf𝑏_𝑛𝑐_𝑛>0;(iii)there are two nonnegative real-valued functions 𝜅₁ and 𝜅₂ on ℕ with𝑆_𝑚𝑥−𝑥≤𝜅₁(𝑚)𝑆_𝑛𝑥−𝑥,∀𝑚∈ℕ,∀𝑛≥𝑚,∀𝑥∈ℋ₁,𝑇_𝑚𝑦−𝑦≤𝜅₂(𝑚)𝑇_𝑛𝑦−𝑦,∀𝑚∈ℕ,∀𝑛≥𝑚,∀𝑦∈ℋ₂. (48)

Proof. Put 𝑝=𝑃_Ω𝑢. For simplicity, put 𝐺_𝑛=𝑆_𝑛(𝐼−𝛾_𝑛𝐴^∗(𝐼−𝑇_𝑛)𝐴). In view of Proposition 9, 𝐺_𝑛 is nonexpansive, so 𝑥_𝑛+1=𝑎−𝑝_𝑛𝑢−𝑝+𝑏_𝑛𝑥_𝑛−𝑝+𝑐_𝑛𝐺_𝑛𝑥_𝑛−𝑝+𝑑_𝑛𝑒_𝑛−𝑝≤𝑎_𝑛𝑢−𝑝+𝑏_𝑛+𝑐_𝑛×𝑥_𝑛−𝑝+𝑑_𝑛𝑒_𝑛−𝑝, (49) from which follows that {𝑥_𝑛} is a bounded sequence. Taking into account Lemma 3, we get 𝑥_𝑛+1−𝑝²=𝑎_𝑛𝑢−𝑝+𝑏_𝑛𝑥_𝑛−𝑝+𝑐_𝑛(𝐺_𝑛𝑥_𝑛−𝑝)+𝑑_𝑛(𝑒_𝑛−𝑝)²≤𝑏_𝑛(𝑥_𝑛−𝑝)+𝑐_𝑛(𝐺_𝑛𝑥_𝑛−𝑝)+𝑑_𝑛(𝑒_𝑛−𝑝)²+2𝑎_𝑛𝑢−𝑝,𝑥_𝑛+1=−𝑝1−𝑎_𝑛²𝑏_𝑛1−𝑎_𝑛𝑥_𝑛+𝑐−𝑝_𝑛1−𝑎_𝑛𝐺_𝑛𝑥_𝑛+𝑑−𝑝_𝑛1−𝑎_𝑛(𝑒_𝑛−𝑝)²+2𝑎_𝑛𝑢−𝑝,𝑥_𝑛+1≤−𝑝1−𝑎_𝑛²𝑏_𝑛1−𝑎_𝑛𝑥_𝑛−𝑝²+𝑐_𝑛1−𝑎_𝑛×𝐺_𝑛𝑥_𝑛−𝑝²+𝑑_𝑛1−𝑎_𝑛𝑒_𝑛−𝑝²−𝑏_𝑛𝑐_𝑛𝑥_𝑛−𝐺_𝑛𝑥_𝑛²+2𝑎_𝑛𝑢−𝑝,𝑥_𝑛+1−𝑝≤𝑏_𝑛𝑥_𝑛−𝑝²+𝑐_𝑛𝐺_𝑛𝑥_𝑛−𝑝²+𝑑_𝑛𝑒_𝑛−𝑝²−𝑏_𝑛𝑐_𝑛𝑥_𝑛−𝐺_𝑛𝑥_𝑛²+2𝑎_𝑛𝑢−𝑝,𝑥_𝑛+1−𝑝. (50) Meanwhile, we have by Lemma 14 that 𝐺_𝑛𝑥_𝑛−𝑝²=𝑆_𝑛𝐼−𝛾_𝑛𝐴^∗𝐼−𝑇_𝑛𝐴𝑥_𝑛−𝑆_𝑛𝐼−𝛾_𝑛𝐴^∗𝐼−𝑇_𝑛𝐴𝑝²≤𝐼−𝛾_𝑛𝐴^∗𝐼−𝑇_𝑛𝐴𝑥_𝑛−𝐼−𝛾_𝑛𝐴^∗𝐼−𝑇_𝑛𝐴𝑝²≤𝑥_𝑛−𝑝²−2𝛾_𝑛−𝛾²_𝑛‖𝐴‖²×(𝐼−𝑇_𝑛)𝐴𝑥_𝑛−(𝐼−𝑇_𝑛)𝐴𝑝². (51) Therefore, we deduce that 𝑥_𝑛+1−𝑝²≤𝑏_𝑛𝑥_𝑛−𝑝²+𝑐_𝑛𝑥_𝑛−𝑝²−2𝛾_𝑛−𝛾²_𝑛‖𝐴‖²×(𝐼−𝑇_𝑛)𝐴𝑥_𝑛−(𝐼−𝑇_𝑛)𝐴𝑝²+𝑑_𝑛𝑒_𝑛−𝑝²−𝑏_𝑛𝑐_𝑛𝑥_𝑛−𝐺_𝑛𝑥_𝑛²+2𝑎_𝑛𝑢−𝑝,𝑥_𝑛+1=𝑏−𝑝_𝑛+𝑐_𝑛𝑥_𝑛−𝑝²+𝑑_𝑛𝑒_𝑛−𝑝²+2𝑎_𝑛𝑢−𝑝,𝑥_𝑛+1−𝑝−𝑐_𝑛2𝛾_𝑛−𝛾²_𝑛‖𝐴‖²(𝐼−𝑇_𝑛)𝐴𝑥_𝑛²−𝑏_𝑛𝑐_𝑛𝑥_𝑛−𝐺_𝑛𝑥_𝑛². (52) We now carry on with the proof by considering the following two cases: (I) {‖𝑥_𝑛−𝑝‖} is eventually decreasing and (II) {‖𝑥_𝑛−𝑝‖} is not eventually decreasing.
Case I. Suppose that {‖𝑥_𝑛−𝑝‖} is eventually decreasing; that is, there is 𝑛₀∈ℕ such that {‖𝑥_𝑛−𝑝‖}_𝑛≥𝑛0 is decreasing. In this case, lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists in ℝ. From inequality (52) we have 𝑐_𝑛2𝛾_𝑛−𝛾²_𝑛‖𝐴‖²(𝐼−𝑇_𝑛)𝐴𝑥_𝑛²+𝑏_𝑛𝑐_𝑛𝑥_𝑛−𝐺_𝑛𝑥_𝑛²≤𝑏_𝑛+𝑐_𝑛𝑥_𝑛−𝑝²+𝑑_𝑛𝑒_𝑛−𝑝²+2𝑎_𝑛𝑢−𝑝,𝑥_𝑛+1−𝑝−𝑥_𝑛+1−𝑝², (53) which together with the boundedness of {𝑥_𝑛} and conditions (i) and (ii) implies lim_𝑛→∞𝑥_𝑛−𝐺_𝑛𝑥_𝑛=0,lim_𝑛→∞𝐴𝑥_𝑛−𝑇_𝑛𝐴𝑥_𝑛=0. (54) Then, an application of condition (iii) follows that for all 𝑚∈ℕ, lim_𝑛→∞𝑥_𝑛−𝐺_𝑚𝑥_𝑛=lim_𝑛→∞𝐴𝑥_𝑛−𝑇_𝑚𝐴𝑥_𝑛=0. (55) Since {𝑥_𝑛} is bounded, it has a subsequence {𝑥_𝑛𝑘} such that {𝑥_𝑛𝑘} converges weakly to some 𝑧∈ℋ and limsup_𝑛→∞𝑢−𝑝,𝑥_𝑛+1−𝑝=lim_𝑘→∞𝑢−𝑝,𝑥_𝑛𝑘=−𝑝𝑢−𝑝,𝑧−𝑝≤0, (56) where the last inequality follows from (24) since 𝑧∈Ω by Lemma 4. Choose 𝑀>0 so that sup{‖𝑒_𝑛−𝑝‖²+2‖𝑢−𝑝‖‖𝑥_𝑛+1−𝑝‖∶𝑛∈ℕ}≤𝑀. From (52) we have 𝑥_𝑛+1−𝑝²≤1−𝑎_𝑛+𝑑_𝑛𝑥_𝑛−𝑝²+𝑎_𝑛+𝑑_𝑛⋅2𝑢−𝑝,𝑥_𝑛+1−𝑝+𝑑_𝑛𝑒_𝑛−𝑝²+2𝑢−𝑝𝑥_𝑛+1≤𝑎−𝑝1−_𝑛+𝑑_𝑛𝑥_𝑛−𝑝²+𝑎_𝑛+𝑑_𝑛⋅2𝑢−𝑝,𝑥_𝑛+1−𝑝+𝑑_𝑛𝑀. (57) Accordingly, because of (56) and condition (i), we can apply Lemma 5 to inequality (57) with 𝑠_𝑛=‖𝑥_𝑛−𝑝‖², 𝛼_𝑛=𝑎_𝑛+𝑏_𝑛, 𝜇_𝑛=2⟨𝑢−𝑝,𝑥_𝑛+1−𝑝⟩, and 𝜈_𝑛=𝑑_𝑛𝑀 to conclude that lim_𝑛→∞𝑥_𝑛=𝑝. (58)
Case II. Suppose that {‖𝑥_𝑛−𝑝‖} is not eventually decreasing. In this case, by Lemma 6, there exists a nondecreasing sequence {𝑚_𝑘} in ℕ such that 𝑚_𝑘→∞ and max𝑥_𝑚𝑘−𝑝,𝑥_𝑘≤𝑥−𝑝_𝑚𝑘+1−𝑝,∀𝑘∈ℕ. (59) Then, it follows from (52) and (59) that 𝑥_𝑚𝑘−𝑝²≤𝑥_𝑚𝑘+1−𝑝²≤𝑏_𝑚𝑘+𝑐_𝑚𝑘𝑥_𝑚𝑘−𝑝²+𝑑_𝑚𝑘𝑒_𝑚𝑘−𝑝²+2𝑎_𝑚𝑘𝑢−𝑝,𝑥_𝑚𝑘+1−𝑝−𝑐_𝑚𝑘2𝛾_𝑚𝑘−𝛾²_𝑚𝑘‖𝐴‖²×(𝐼−𝑇_𝑚𝑘)𝐴𝑥_𝑚𝑘²−𝑏_𝑚𝑘𝑐_𝑚𝑘𝑥_𝑚𝑘−𝐺_𝑚𝑘𝑥_𝑚𝑘². (60) Therefore, 0≤𝑐_𝑚𝑘2𝛾_𝑚𝑘+𝛾²_𝑚𝑘‖𝐴‖²(𝐼−𝑇_𝑚𝑘)𝐴𝑥_𝑚𝑘²+𝑏_𝑚𝑘𝑐_𝑚𝑘𝑥_𝑚𝑘−𝐺_𝑚𝑘𝑥_𝑚𝑘²≤−1−𝑏_𝑚𝑘+𝑐_𝑚𝑘𝑥_𝑚𝑘−𝑝²+𝑑_𝑚𝑘𝑒_𝑚𝑘−𝑝²+2𝑎_𝑚𝑘𝑢−𝑝,𝑥_𝑚𝑘+1𝑎−𝑝=−_𝑚𝑘+𝑑_𝑚𝑘𝑥_𝑚𝑘−𝑝²+𝑑_𝑚𝑘𝑒_𝑚𝑘−𝑝²+2𝑎_𝑚𝑘𝑢−𝑝,𝑥_𝑚𝑘+1−𝑝, (61) and then proceeding just as in the proof in Case I, we obtain lim_𝑘→∞𝑥_𝑚𝑘−𝐺_𝑚𝑘𝑥_𝑚𝑘=lim_𝑘→∞𝐴𝑥_𝑚𝑘−𝑇_𝑚𝑘𝐴𝑥_𝑚𝑘=0, (62) which in conjunction with condition (iii) shows that for all 𝑚_𝑗lim_𝑘→∞𝑥_𝑚𝑘−𝐺_𝑚𝑗𝑥_𝑚𝑘=lim_𝑘→∞𝐴𝑥_𝑚𝑘−𝑇_𝑚𝑗𝐴𝑥_𝑚𝑘=0, (63) and then follows that limsup_𝑘→∞𝑢−𝑝,𝑥_𝑚𝑘+1−𝑝≤0. (64) From (60) we have 1−𝑏_𝑚𝑘+𝑐_𝑚𝑘𝑥_𝑚𝑘−𝑝²≤𝑑_𝑚𝑘𝑒_𝑚𝑘−𝑝²+2𝑎_𝑚𝑘𝑢−𝑝,𝑥_𝑚𝑘+1−𝑝, (65) and thus, 𝑥_𝑚𝑘−𝑝²≤𝑑_𝑚𝑘𝑎_𝑚𝑘+𝑑_𝑚𝑘𝑒_𝑚𝑘−𝑝²+2𝑎_𝑚𝑘𝑎_𝑚𝑘+𝑑_𝑚𝑘𝑢−𝑝,𝑥_𝑚𝑘+1≤𝑑−𝑝_𝑚𝑘𝑎_𝑚𝑘𝑒_𝑚𝑘−𝑝²+2𝑢−𝑝,𝑥_𝑚𝑘+1−𝑝. (66) Letting 𝑘→∞ and using (64) and condition (i), we obtain lim_𝑘→∞𝑥_𝑚𝑘−𝑝=0. (67) Also, since 𝑥_𝑚𝑘+1−𝑥_𝑚𝑘≤𝑎_𝑚𝑘𝑢−𝑥_𝑚𝑘+𝑐_𝑚𝑘𝐺_𝑚𝑘𝑥_𝑚𝑘−𝑥_𝑚𝑘+𝑑_𝑚𝑘𝑒_𝑚𝑘−𝑥_𝑚𝑘, (68) which together with (62) and condition (i) implies that lim_𝑘→∞‖𝑥_𝑚𝑘+1−𝑥_𝑚𝑘‖=0, and so lim_𝑘→∞𝑥_𝑚𝑘+1−𝑝=0 (69) by virtue of (67). Consequently, we conclude that lim_𝑘→∞‖𝑥_𝑘−𝑝‖=0 via (59) and (69). This completes the proof.

This theorem says that the sequence {𝑥_𝑛} converges strongly to a point of Ω which is the nearest to 𝑢. In particular, if 𝑢 is taken to be 0, then the limit point 𝑣 of the sequence {𝑥_𝑛} is the unique minimum solution of SFFP(16).

Corollary 16. Let {𝑎_𝑛}, {𝑏_𝑛}, and {𝑐_𝑛} be sequences in [0,1] with 𝑎_𝑛+𝑏_𝑛+𝑐_𝑛=1 and 𝑎_𝑛∈(0,1) for all 𝑛∈ℕ. Let {𝛾_𝑛} be a sequence in (0,2/‖𝐴‖²) and let {𝑒_𝑛} be a bounded sequence in ℋ₁. Suppose that the solution set Ω of SFFP(16) is nonempty. For any 𝑢∈ℋ₁, start with any 𝑥₁∈ℋ₁ and define a sequence {𝑥_𝑛} by 𝑥_𝑛+1=𝑎_𝑛𝑢+𝑏_𝑛𝑥_𝑛+𝑐_𝑛𝑆_𝑛𝐼−𝛾_𝑛𝐴^∗𝐼−𝑇_𝑛𝐴𝑥_𝑛+𝑒_𝑛. (70) Then, the sequence {𝑥_𝑛} converges strongly to 𝑃_Ω𝑢 provided that the following conditions are satisfied: (i)lim_𝑛→∞𝑎_𝑛=lim_𝑛→∞(𝑑_𝑛/𝑎_𝑛)=0,  ∑^∞_𝑛=1𝑎_𝑛=∞,  ∑^∞_𝑛=1𝑑_𝑛<∞;(ii)lim_𝑛→∞inf𝑐_𝑛(2𝛾_𝑛−𝛾²_𝑛‖𝐴‖²)>0,  lim_𝑛→∞inf𝑏_𝑛𝑐_𝑛>0;(iii)there are two nonnegative real-valued functions 𝜅₁ and 𝜅₂ on ℕ with𝑆_𝑚𝑥−𝑥≤𝜅₁(𝑚)𝑆_𝑛𝑥−𝑥,∀𝑚∈ℕ,∀𝑛≥𝑚,∀𝑥∈ℋ₁,𝑇_𝑚𝑦−𝑦≤𝜅₂(𝑚)𝑇_𝑛𝑦−𝑦,∀𝑚∈ℕ,∀𝑛≥𝑚,∀𝑦∈ℋ₂; (71)(72)(iv)either lim_𝑛→∞(‖𝑒_𝑛‖/𝑎_𝑛)=0 or ∑^∞_𝑛=1‖𝑒_𝑛‖<∞.

Proof. Put 𝑝=𝑃_Ω𝑢 and 𝐺_𝑛=𝑆_𝑛[𝐼−𝛾_𝑛𝐴^∗(𝐼−𝑇_𝑛)𝐴]. Let 𝑧₁=𝑥₁ and define a sequence {𝑧_𝑛} iteratively by 𝑧_𝑛+1=𝑎_𝑛𝑢+𝑏_𝑛𝑧_𝑛+𝑐_𝑛𝐺_𝑛𝑧_𝑛. (73) We have lim_𝑛→∞𝑧_𝑛=𝑝 by Theorem 15. Since 𝑥_𝑛+1−𝑧_𝑛+1≤𝑏_𝑛𝑥_𝑛−𝑧_𝑛+𝑐_𝑛𝐺_𝑛𝑥_𝑛−𝐺_𝑛𝑧_𝑛+𝑒_𝑛≤𝑏_𝑛+𝑐_𝑛𝑥_𝑛−𝑧_𝑛+𝑒_𝑛=1−𝑎_𝑛𝑥_𝑛−𝑧_𝑛+𝑒_𝑛, (74) the limit lim_𝑛→∞‖𝑥_𝑛−𝑧_𝑛‖=0 follows by applying Lemma 5 to (74), and thus, lim_𝑛→∞𝑥_𝑛=𝑝. (75)

If the sequence {𝑆_𝑛} (resp., {𝑇_𝑛}) of firmly nonexpansive mappings consists of a single mapping 𝑆 (resp., 𝑇), then {𝑆_𝑛} and {𝑇_𝑛} obviously verify condition (iii), and hence, we have the following corollary.

Corollary 17. Let {𝑎_𝑛}, {𝑏_𝑛}, {𝑐_𝑛}, and {𝑑_𝑛} be sequences in [0,1] with 𝑎_𝑛+𝑏_𝑛+𝑐_𝑛+𝑑_𝑛=1 and 𝑎_𝑛∈(0,1) for all 𝑛∈ℕ. Let {𝛾_𝑛} be a sequence in (0,2/‖𝐴‖²) and let {𝑒_𝑛} be a bounded sequences in ℋ₁. Assume that the solution set Ω of SFFP(5) is nonempty. For any 𝑢∈ℋ₁, start with an arbitrary 𝑥₁∈ℋ₁ and define the sequence {𝑥_𝑛} by 𝑥_𝑛+1=𝑎_𝑛𝑢+𝑏_𝑛𝑥_𝑛+𝑐_𝑛𝑆𝐼−𝛾_𝑛𝐴^∗(𝐼−𝑇)𝐴𝑥_𝑛+𝑑_𝑛𝑒_𝑛. (76) Then, {𝑥_𝑛} converges strongly to 𝑃_Ω𝑢 provided that the following conditions are satisfied: (i)lim_𝑛→∞𝑎_𝑛=lim_𝑛→∞(𝑑_𝑛/𝑎_𝑛)=0,  ∑^∞_𝑛=1𝑎_𝑛=∞,  ∑^∞_𝑛=1𝑑_𝑛<∞;(ii)lim_𝑛→∞inf𝑐_𝑛(2𝛾_𝑛−𝛾²_𝑛‖𝐴‖²)>0,  lim_𝑛→∞inf𝑏_𝑛𝑐_𝑛>0.

When the sequence {𝛾_𝑛} is taken to be a constant 𝛾∈(0,2/‖𝐴‖²), then because 𝑆[𝐼−𝛾_𝑛𝐴^∗(𝐼−𝑇)𝐴] is an averaged mapping, we can apply Corollary 3.4 of Huang and Hong [26] to obtain the following result.

Theorem 18. Let {𝑎_𝑛}, {𝑏_𝑛}, {𝑐_𝑛}, and {𝑑_𝑛} be sequences in [0,1] with 𝑎_𝑛+𝑏_𝑛+𝑐_𝑛+𝑑_𝑛=1 and 𝑎_𝑛∈(0,1) for all 𝑛∈ℕ. Suppose that 𝛾∈(0,2/‖𝐴‖²) and suppose that {𝑒_𝑛} is a bounded sequence in ℋ₁. Assume that the solution set Ω of SFFP(5) is nonempty. For any 𝑢∈ℋ₁, start with an arbitrary 𝑥₁∈ℋ₁ and define the sequence {𝑥_𝑛} by 𝑥_𝑛+1=𝑎_𝑛𝑢+𝑏_𝑛𝑥_𝑛+𝑐_𝑛𝑆𝐼−𝛾𝐴^∗(𝐼−𝑇)𝐴𝑥_𝑛+𝑑_𝑛𝑒_𝑛. (77) Then, {𝑥_𝑛} converges strongly to 𝑃_Ω𝑢 provided that the following conditions are satisfied: (i)lim_𝑛→∞𝑎_𝑛=lim_𝑛→∞(𝑑_𝑛/𝑎_𝑛)=0,  ∑^∞_𝑛=1𝑎_𝑛=∞,  ∑^∞_𝑛=1𝑑_𝑛<∞;(ii)lim_𝑛→∞inf𝑏_𝑛>0,  lim_𝑛→∞inf𝑐_𝑛>0.

Since both condition (ii) of Corollary 17 and Theorem 18 are equivalent provided that 𝛾_𝑛=𝛾 for every 𝑛∈ℕ, Theorem 18 also follows from Corollary 17.

We now turn to SFFP(5) for another algorithm, which essentially follows the argument of Wang and Xu [27]. For the sake of completeness, we still give a detailed proof.

Theorem 19. Let {𝑎_𝑛} be a sequence in (0,1). Suppose that 𝛾∈(0,2/‖𝐴‖²) and assume that the solution set Ω of SFFP(5) is nonempty. Start with any 𝑥₁∈ℋ₁ and define a sequence {𝑥_𝑛} by 𝑥_𝑛+1=𝑆1−𝑎_𝑛𝐼−𝛾𝐴^∗(𝐼−𝑇)𝐴𝑥_𝑛. (78) Then, the sequence {𝑥_𝑛} converges strongly to the minimum norm solution of SFFP(5) provided that the following conditions are satisfied: (i)lim_𝑛→∞𝑎_𝑛=0;(ii)∑^∞_𝑛=1𝑎_𝑛=∞;(iii)either  ∑^∞_𝑛=1|𝑎_𝑛+1−𝑎_𝑛|<∞ or  lim_𝑛→∞(|𝑎_𝑛+1−𝑎_𝑛|/𝑎_𝑛)=0.

Proof. Put  𝑈=𝐼−𝛾𝐴^∗(𝐼−𝑇)𝐴, 𝐺=𝑆𝑈, and 𝐺_𝑛=𝑆[(1−𝑎_𝑛)]𝑈 for all 𝑛∈ℕ. Then, 𝑥_𝑛+1=𝐺_𝑛𝑥_𝑛=𝑆1−𝑎_𝑛𝑈𝑥_𝑛,𝐺_𝑛𝑥−𝐺_𝑛𝑦≤1−𝑎_𝑛𝑥−𝑦,∀𝑥,𝑦∈ℋ₁. (79) Take 𝑝∈Ω. From Proposition 9, we have 𝐺_𝑛𝑝−𝑝=𝑆1−𝑎_𝑛𝑈≤𝑝−𝑆𝑈𝑝1−𝑎_𝑛𝑈𝑝−𝑈𝑝=𝑎_𝑛𝑈𝑝=𝑎_𝑛𝑝. (80) Hence, 𝑥_𝑛+1−𝑝=𝐺_𝑛𝑥_𝑛≤𝐺−𝑝_𝑛𝑥_𝑛−𝐺_𝑛𝑝+𝐺_𝑛≤𝑝−𝑝1−𝑎_𝑛𝑥_𝑛−𝑝+𝑎_𝑛𝑝, (81) from which follows that {𝑥_𝑛} is bounded and so is {𝑈𝑥_𝑛}. Choose 𝑀>0 so that ‖𝑈𝑥_𝑛‖≤𝑀 for all 𝑛∈ℕ. We have 𝑥_𝑛+1−𝑥_𝑛=𝐺_𝑛𝑥_𝑛−𝐺_𝑛−1𝑥_𝑛−1≤𝐺_𝑛𝑥_𝑛−𝐺_𝑛𝑥_𝑛−1+𝐺_𝑛𝑥_𝑛−1−𝐺_𝑛−1𝑥_𝑛−1=1−𝑎_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑆1−𝑎_𝑛𝑈𝑥_𝑛−1−𝑆1−𝑎_𝑛−1𝑈𝑥_𝑛−1≤1−𝑎_𝑛𝑥_𝑛−𝑥_𝑛−1+1−𝑎_𝑛𝑈𝑥_𝑛−1−1−𝑎_𝑛−1𝑈𝑥_𝑛−1=1−𝑎_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑎_𝑛−𝑎_𝑛−1𝑈𝑥_𝑛−1≤1−𝑎_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑀𝑎_𝑛−𝑎_𝑛−1. (82) In view of conditions (i), (ii), and (iii), we can apply Lemma 5 to (82) to get lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0, (83) and then from 𝑥_𝑛−𝐺𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝐺_𝑛𝑥_𝑛−𝐺𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛+1+1−𝑎_𝑛𝑈𝑥_𝑛−𝑈𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝑀𝑈𝑥_𝑛, (84) we see that lim_𝑛→∞𝑥_𝑛−𝑇𝑥_𝑛=0. (85) Consequently, the demiclosedness principle ensures that each weak limit point of {𝑥_𝑛} is a fixed point of the averaged mapping 𝑆𝑈. And then we conclude from Proposition 9 that each weak limit point of {𝑥_𝑛} lies in Ω.
Let 𝑥 be the minimum norm element of Ω; that is, 𝑥=𝑃_Ω(0). We shall show that {𝑥_𝑛} converges strongly to 𝑥. To see this, we compute ‖𝑥_𝑛+1−𝑥‖² as follows: 𝑥_𝑛+1−𝑥²=𝑆1−𝑎_𝑛𝑈𝑥_𝑛−𝑆𝑈𝑥²≤(1−𝑎_𝑛)𝑈𝑥_𝑛−𝑈𝑥²=1−𝑎_𝑛𝑈𝑥_𝑛−𝑥²=1−𝑎_𝑛𝑈𝑥_𝑛−𝑥−𝑎_𝑛𝑥²=1−𝑎_𝑛²𝑈𝑥_𝑛−𝑥²+2𝑎_𝑛1−𝑎_𝑛𝑈𝑥_𝑛−𝑥,−𝑥+𝑎²_𝑛‖𝑥‖²≤1−𝑎_𝑛𝑥_𝑛−𝑥²+𝑎_𝑛21−𝑎_𝑛𝑈𝑥_𝑛−𝑥,−𝑥+𝑎_𝑛‖𝑥‖². (86) If limsup_𝑛→∞2(1−𝑎_𝑛)⟨𝑈𝑥_𝑛−𝑥,−𝑥⟩+𝑎_𝑛‖𝑥‖²≤0, then an application of Lemma 5 to (86) yields that lim_𝑛→∞𝑥_𝑛=𝑥. Hence, to complete the proof, it suffices to show that limsup_𝑛→∞2(1−𝑎_𝑛)⟨𝑈𝑥_𝑛−𝑥,−𝑥⟩+𝑎_𝑛‖𝑥‖²≤0. For this, taking into account Proposition 8, we can write 𝑈=(1−𝛽)𝐼+𝛽𝑉 for some 𝛽∈(0,1) and some nonexpansive mapping 𝑉. Then, from 𝑥_𝑛+1−𝑝²=𝑆(1−𝑎_𝑛)𝑈𝑥_𝑛−𝑝²≤(1−𝑎_𝑛)𝑈𝑥_𝑛−𝑝²=1−𝑎_𝑛𝑈𝑥_𝑛−𝑝−𝑎_𝑛𝑝²≤1−𝑎_𝑛𝑈𝑥_𝑛−𝑝²+𝑎_𝑛𝑝²≤1−𝛽𝑥_𝑛−𝑝+𝛽𝑉𝑥_𝑛−𝑝²+𝑎_𝑛𝑝²=1−𝛽𝑥_𝑛−𝑝²+𝛽𝑉𝑥_𝑛−𝑝²−𝛽1−𝛽𝑥_𝑛−𝑉𝑥_𝑛²+𝑎_𝑛𝑝²≤𝑥_𝑛−𝑝²−𝛽1−𝛽𝑥_𝑛−𝑉𝑥_𝑛²+𝑎_𝑛𝑝², (87) we obtain 𝛽1−𝛽𝑥_𝑛−𝑉𝑥_𝑛²≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+𝑎_𝑛𝑝², (88) which ensures that lim_𝑛→∞‖𝑥_𝑛−𝑉𝑥_𝑛‖=0, and hence, lim_𝑛→∞𝑥_𝑛−𝑈𝑥_𝑛=0 (89) once we note that 𝐼−𝑈=𝛽(𝐼−𝑉). Since {𝑥_𝑛} is bounded, we can take a subsequence {𝑥_𝑛𝑖} so that it is weakly convergent to 𝑥^∗∈Ω and limsup_𝑛→∞𝑥_𝑛−𝑥,−𝑥=lim_𝑖→∞𝑥_𝑛𝑖𝑥=𝑥,−_∗−𝑥,−𝑥≤0, (90) where the last inequality comes from the characterization inequality of a metric projection. Now, applying (89) and (90) to the equality 𝑈𝑥_𝑛−𝑥,−𝑥=𝑈𝑥_𝑛−𝑥_𝑛,−𝑥+𝑥_𝑛−𝑥,−𝑥, (91) we obtain limsup_𝑛→∞𝑈𝑥_𝑛−𝑥,−𝑥≤0, (92) and thus, limsup_𝑛→∞21−𝑎_𝑛𝑈𝑥_𝑛−𝑥,−𝑥+𝑎_𝑛‖𝑥‖²≤0. (93) This completes the proof.

5. Applications

In this section, we shall apply the results in Section 4 to approximate zeros of maximal monotone operators and solutions of equilibrium problems.

Theorem 20. Suppose that 𝑀 and 𝑁 are two maximal monotone operators on ℋ₁ and ℋ₂, respectively, and suppose that 𝐴∶ℋ₁→ℋ₂ is a bounded linear operator with adjoint 𝐴^∗. Suppose further that {𝑎_𝑛}, {𝑏_𝑛}, {𝑐_𝑛}, and {𝑑_𝑛} are sequences in [0,1] with 𝑎_𝑛+𝑏_𝑛+𝑐_𝑛+𝑑_𝑛=1 and 𝑎_𝑛∈(0,1) for all 𝑛∈ℕ. Let {𝛼_𝑛} and {𝛽_𝑛} be sequences in (0,∞), {𝛾_𝑛} a sequence in (0,2/‖𝐴‖²), and {𝑒_𝑛} a bounded sequence in ℋ₁. Assume that the solution set Ω of the problem 𝑓𝑖𝑛𝑑𝑥^∗∈ℋ₁𝑠𝑜𝑡ℎ𝑎𝑡𝑥^∗∈𝑀⁻¹0,𝐴𝑥^∗∈𝑁⁻¹0 (94) is nonempty. For any 𝑢∈ℋ₁, start with an arbitrary 𝑥₁∈ℋ and define a sequence {𝑥_𝑛} by 𝑥_𝑛+1=𝑎_𝑛𝑢+𝑏_𝑛𝑥_𝑛+𝑐_𝑛𝐽^𝑀_𝛼𝑛𝐼−𝛾_𝑛𝐴^∗𝐼−𝐽^𝑁_𝛽𝑛𝑥_𝑛+𝑑_𝑛𝑒_𝑛. (95) Then, the sequence {𝑥_𝑛} converges strongly to 𝑃_Ω𝑢 provided that the following conditions are satisfied: (i)lim_𝑛→∞𝑎_𝑛=lim_𝑛→∞(𝑑_𝑛/𝑎_𝑛)=0, ∑^∞_𝑛=1𝑎_𝑛=∞, ∑^∞_𝑛=1𝑑_𝑛<∞;(ii)lim_𝑛→∞inf𝑐_𝑛(2𝛾_𝑛−𝛾²_𝑛‖𝐴‖²)>0, lim_𝑛→∞inf𝑏_𝑛𝑐_𝑛>0;(iii)lim_𝑛→∞inf𝛼_𝑛>0, lim_𝑛→∞inf𝛽_𝑛>0.

Proof. For any 𝑛∈ℕ, letting 𝑆_𝑛=𝐽^𝑀_𝛼𝑛 and 𝑇_𝑛=𝐽^𝑁_𝛽𝑛, then we have by Lemma 2(b) that 𝑀⁻¹0=Fix(𝑆_𝑛) and 𝑁⁻¹0=Fix(𝑇_𝑛) for all 𝑛∈ℕ, and hence, as shown in Section 1, the problem (94) becomes SFFP(16). Since all the requirements of Theorem 15 are satisfied except condition (iii), we have to check this condition. Because liminf_𝑛→∞𝛼_𝑛=0, liminf_𝑛→∞𝛽_𝑛=0 by assumption, we may assume that there is 𝜏∈(0,1) so that 𝜏<𝛼_𝑛 and 𝜏<𝛽_𝑛 for all 𝑛∈ℕ. Let 𝜅₁(𝑛)=2+(𝛼_𝑛/𝜏) and 𝜅₂(𝑛)=2+(𝛽_𝑛/𝜏). Then, by virtue of the resolvent identity and the nonexpansiveness of 𝐽^𝑀_𝛼𝑚, one has for all 𝑚∈ℕ that 𝐽^𝑀_𝛼𝑛𝑥−𝐽^𝑀_𝛼𝑚𝑥=𝐽^𝑀_𝛼𝑚𝛼_𝑚𝛼_𝑛𝛼𝑥+1−^𝑚𝛼_𝑛𝐽^𝑀_𝛼𝑛𝑥−𝐽^𝑀_𝛼𝑚𝑥≤𝛼1−^𝑚𝛼_𝑛𝐽^𝑀_𝛼𝑛≤𝛼𝑥−𝑥1+^𝑚𝛼_𝑛𝐽^𝑀_𝛼𝑛𝑥−𝑥, (96) and thus, 𝐽^𝑀_𝛼𝑚𝑥−𝑥≤𝐽^𝑀_𝛼𝑚𝑥−𝐽^𝑀_𝛼𝑛𝑥+𝐽^𝑀_𝛼𝑛≤𝛼𝑥−𝑥1+^𝑚𝛼_𝑛+1𝐽^𝑀_𝛼𝑛≤𝛼𝑥−𝑥2+^𝑚𝜏𝐽^𝑀_𝛼𝑛𝑥−𝑥=𝜅₁(𝑚)𝐽^𝑀_𝛼𝑛𝑥−𝑥,∀𝑛≥𝑚,∀𝑥∈ℋ. (97) The same argument shows for all 𝑚∈ℕ that 𝐽^𝑁_𝛽𝑚𝑥−𝑥≤𝜅₂(𝑚)𝐽^𝑁_𝛽𝑛𝑥−𝑥,∀𝑛≥𝑚,∀𝑥∈ℋ. (98) Therefore, condition (iii) of Theorem 15 is true for {𝐽^𝑀_𝛼𝑛} and {𝐽^𝑁_𝛽𝑛}, and then the desired conclusion follows from Theorem 15.

Replacing 𝑆 and 𝑇 with 𝐽^𝑀_𝛼 and 𝐽^𝑁_𝛽, respectively, in Theorem 19, we obtain the following result.

Theorem 21. Suppose that 𝑀 and 𝑁 are two maximal monotone operators on ℋ₁ and ℋ₂, respectively, and suppose that 𝐴∶ℋ₁→ℋ₂ is a bounded linear operator with adjoint 𝐴^∗. Let {𝑎_𝑛} be a sequence in (0,1), 𝛾∈(0,2/‖𝐴‖²), and 𝛼,𝛽∈(0,∞) and assume that the solution set Ω of problem (94) is nonempty. Start with any 𝑥₁∈ℋ₁ and define a sequence {𝑥_𝑛} by 𝑥_𝑛+1=𝐽^𝑀_𝛼1−𝑎_𝑛𝐼−𝛾𝐴^∗𝐼−𝐽^𝑁_𝛽𝐴𝑥_𝑛. (99) Then, the sequence {𝑥_𝑛} converges strongly to the minimum norm solution of problem (94) provided that the following conditions are satisfied: (i)lim_𝑛→∞𝑎_𝑛=0;(ii)∑^∞_𝑛=1𝑎_𝑛=∞;(iii)either ∑^∞_𝑛=1|𝑎_𝑛+1−𝑎_𝑛|<∞ or lim_𝑛→∞(|𝑎_𝑛+1−𝑎_𝑛|/𝑎_𝑛)=0.

Recall some facts in Section 1. For a nonempty closed convex subset 𝐶 of ℋ, let 𝑓∶𝐶×𝐶→ℝ be a function satisfying conditions (A1)–(A4), which are described in Section 1. The solution set of the equilibrium problem find𝑥^∗∈𝐶suchthat𝑓𝑥^∗,𝑦≥0,∀𝑦∈𝐶 (100) is denoted by EP(𝑓), which is equal to Fix(𝐽^𝑓_𝛼) for any 𝛼>0, where 𝐽^𝑓_𝛼 is a function on ℋ into 𝐶 defined by 𝐽^𝑓_𝛼+1𝑥=𝑧∈𝐶∶𝑓𝑧,𝑦𝛼𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶 (101) for all 𝑥∈ℋ. 𝐽^𝑓_𝛼 is a single-valued firmly nonexpansive mapping.

Theorem 22. Suppose that 𝐶 and 𝑄 are two nonempty closed convex subsets of ℋ₁ and ℋ₂, respectively, and suppose that 𝐴∶ℋ₁→ℋ₂ is a bounded linear operator with adjoint 𝐴^∗. Let 𝑓∶𝐶×𝐶→ℝ and 𝑔∶𝑄×𝑄→ℝ be functions satisfying conditions (A1)–(A4). Suppose further that {𝑎_𝑛}, {𝑏_𝑛}, {𝑐_𝑛}, and {𝑑_𝑛} are sequences in [0,1] with 𝑎_𝑛+𝑏_𝑛+𝑐_𝑛+𝑑_𝑛=1 and 𝑎_𝑛∈(0,1) for all 𝑛∈ℕ. Let {𝛼_𝑛} and {𝛽_𝑛} be sequences in (0,∞), {𝛾_𝑛} a sequence in (0,2/‖𝐴‖²), and {𝑒_𝑛} a bounded sequence in ℋ₁. Assume that the solution set Ω of the problem 𝑓𝑖𝑛𝑑𝑥^∗∈ℋ₁𝑠𝑜𝑡ℎ𝑎𝑡𝑥^∗∈𝐸𝑃𝑓,𝐴𝑥^∗∈𝐸𝑃𝑔 (102) is nonempty. For any 𝑢∈ℋ₁, start with an arbitrary 𝑥₁∈ℋ and define a sequence {𝑥_𝑛} by 𝑥_𝑛+1=𝑎_𝑛𝑢+𝑏_𝑛𝑥_𝑛+𝑐_𝑛𝐽^𝑓_𝛼𝑛𝐼−𝛾_𝑛𝐴^∗𝐼−𝐽^𝑔_𝛽𝑛𝐴𝑥_𝑛+𝑑_𝑛𝑒_𝑛. (103) Then, the sequence {𝑥_𝑛} converges strongly to 𝑃_Ω𝑢 provided that the following conditions are satisfied: (i)lim_𝑛→∞𝑎_𝑛=lim_𝑛→∞(𝑑_𝑛/𝑎_𝑛)=0, ∑^∞_𝑛=1𝑎_𝑛=∞, ∑^∞_𝑛=1𝑑_𝑛<∞;(ii)lim_𝑛→∞inf𝑐_𝑛(2𝛾_𝑛−𝛾²_𝑛‖𝐴‖²)>0, lim_𝑛→∞inf𝑏_𝑛𝑐_𝑛>0;(iii)lim_𝑛→∞inf𝛼_𝑛>0, lim_𝑛→∞inf𝛽_𝑛>0.

Proof. Define two set-valued mappings 𝑀_𝑓 and 𝑁_𝑔 on ℋ₁ and ℋ₂, respectively, by 𝑀_𝑓(𝑥)=𝑧∈ℋ₁≥𝑁∶𝑓𝑥,𝑦𝑦−𝑥,𝑧,∀𝑦∈𝐶,∀𝑥∈𝐶,∅,∀𝑥∉𝐶;_𝑔(𝑢)=𝑤∈ℋ₂()≥⟨⟩}∶𝑔𝑢,𝑣𝑣−𝑢,𝑤,∀𝑣∈𝑄,∀𝑢∈𝑄,∅,∀𝑢∉𝑄. (104) As shown in Takahashi et al. [21], the set-valued mapping 𝑀_𝑓 (resp., 𝑁_𝑔) is a maximal monotone operator with 𝒟(𝑀_𝑓)⊆𝐶 (resp., 𝒟(𝑁_𝑔)⊆𝑄), and 𝐽^𝑀𝑓_𝛼=𝐽^𝑓_𝛼=EP(𝑓) for any 𝛼>0 (resp., 𝐽^𝑁𝑔_𝛽=𝐽^𝑔_𝛽=EP(𝑔) for any 𝛽>0). With 𝑀=𝑀_𝑓 and 𝑁=𝑁_𝑔 in Theorem 20, the desired conclusion follows.

If ℋ₁=ℋ₂=ℋ, 𝐶=𝑄, 𝑓=𝑔, and 𝐴 is the identity transformation on ℋ, then (102) is reduced to the usual equilibrium problem, and we have the following corollary.

Corollary 23. Suppose that 𝐶 is a nonempty closed convex subsets of ℋ and 𝑓∶𝐶×𝐶→ℝ is a function satisfying conditions (A1)–(A4). Let {𝑎_𝑛}, {𝑏_𝑛}, {𝑐_𝑛}, and {𝑑_𝑛} be sequences in [0,1] with 𝑎_𝑛+𝑏_𝑛+𝑐_𝑛+𝑑_𝑛=1 and 𝑎_𝑛∈(0,1) for all 𝑛∈ℕ. Let {𝛼_𝑛} and {𝛽_𝑛} be sequences in (0,∞), and let {𝛾_𝑛} be a sequence in (0,2/‖𝐴‖²) and suppose that {𝑒_𝑛} is a bounded sequence in ℋ₁. Assume that the solution set Ω of the problem 𝑓𝑖𝑛𝑑𝑥^∗∈ℋ𝑠𝑜𝑡ℎ𝑎𝑡𝑥^∗∈𝐸𝑃𝑓 (105) is nonempty. For any 𝑢∈ℋ₁, start with an arbitrary 𝑥₁∈ℋ and define a sequence {𝑥_𝑛} by 𝑥_𝑛+1=𝑎_𝑛𝑢+𝑏_𝑛𝑥_𝑛+𝑐_𝑛𝐽^𝑓_𝛼𝑛𝐼−𝛾_𝑛𝐼−𝐽^𝑓_𝛽𝑛𝑥_𝑛+𝑑_𝑛𝑒_𝑛. (106) Then, the sequence {𝑥_𝑛} converges strongly to 𝑃_Ω𝑢 provided that the following conditions are satisfied: (i)lim_𝑛→∞𝑎_𝑛=lim_𝑛→∞(𝑑_𝑛/𝑎_𝑛)=0, ∑^∞_𝑛=1𝑎_𝑛=∞, ∑^∞_𝑛=1𝑑_𝑛<∞;(ii)lim_𝑛→∞inf𝑐_𝑛(2𝛾_𝑛−𝛾²_𝑛‖𝐴‖²)>0, lim_𝑛→∞inf𝑏_𝑛𝑐_𝑛>0;(iii)lim_𝑛→∞inf𝛼_𝑛>0, lim_𝑛→∞inf𝛽_𝑛>0.

As shown in Blum and Oettli [7] that the variational inequalities, convex differentiable optimization, Nash equilibria in noncooperative games and so on can be formulated as equilibrium problems, we see that the pervious corollary may be applied to approximate solutions of all aforementioned problems. The readers may readily write down the details.

Just as Theorem 21 to Theorem 20, there are similar results corresponding to Theorem 22 and Corollary 23. We leave the work to readers.