Strong Convergence of a Monotone Projection Algorithm in a Banach Space

Abstract

In this paper, a common solution problem is investigated based on a Bregman projection. Strong convergence of the monotone projection algorithm for monotone operators and bifunctions is obtained in a reflexive Banach space.

1. Introduction

Fixed point theory as an important branch of nonlinear analysis theory has been applied in the study of nonlinear phenomena. Lots of problems arising in economics, engineering, and physics can be studied based on fixed point techniques; see [1–21] and the references therein. Many well-known problems can be studied by using algorithms which are iterative in their nature. As an example, in computer tomography with limited data, each piece of information implies the existence of a convex set 𝐶_𝑚 in which the required solution lies. The problem of finding a point in the intersection ∩^𝑁_𝑚=1𝐶_𝑚, where 𝑁≥1 is some positive integer, is then of crucial interest, and it cannot be usually solved directly. Therefore, an iterative algorithm must be used to approximate such point. The well-known convex feasibility problem which captures applications in various disciplines such as image restoration and radiation therapy treatment planning is to find a point in the intersection of common fixed point sets of a family of nonlinear mappings see, for example, [22–28].

Interest in monotone operators stems mainly from their firm connection with equations of evolution which is an important class of nonlinear operators. It is known that many physically significant problems can be modelled by initial value problems of the form: 𝑥^′(𝑡)+𝐴𝑥(𝑡)=0,𝑥(0)=𝑥₀, (1) where 𝐴 is an monotone operator in an appropriate Banach space. Typical examples, where such evolution equations occur, can be found in the heat, wave, or Schrödinger’s equations. If 𝑥(𝑡) is dependent on 𝑡, then (1) is reduced to 𝐴𝑢=0, (2) whose solutions correspond to the equilibrium points of the system (1). Consequently, considerable research efforts have been devoted, especially within the past 40 years or so, to methods for finding approximate solutions (when they exist) of (2). An early fundamental result in the theory of monotone operators, due to Browder [29], states that the initial value problem (1) is solvable if 𝐴 is locally Lipschitz and accretive on 𝐸.

The Krasnoselskii-Mann iterative algorithm is efficient for treating zero points of monotone operators. However, it is weak convergence only. In many disciplines, including economics, image recovery, and control theory, problems arise in infinite dimension spaces. In such problems, strong convergence (norm convergence) is often much more desirable than weak convergence for it translates the physically tangible property that the energy ‖𝑥_𝑛−𝑥‖ of the error between the iterate 𝑥_𝑛 and the solution 𝑥 eventually becomes arbitrarily small. The importance of strong convergence is also underlined, where a convex function 𝑓 is minimized via the proximal-point algorithm; it is shown that the rate of convergence of the value sequence {𝑓(𝑥_𝑛)} is better when {𝑥_𝑛} converges strongly than when it converges weakly. Such properties have a direct impact when the process is executed directly in the underlying infinite dimensional space. Projection methods which were first introduced by Haugazeau [30] have been considered for treating zero points of monotone operators. The advantage of projection methods is that strong convergence of iterative sequences can be guaranteed without any compact assumptions.

In this paper, we study common zero points of a family of maximal monotone operators and common solutions of a system of equilibrium problems based on the Bregman projection. Strong convergence of the monotone projection algorithm is obtained in a reflexive Banach space.

2. Preliminaries

Let 𝐸 be a real Banach space, 𝐸^∗ the dual space of 𝐸, and 𝐶 a nonempty subset of 𝐸. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ, where ℝ denotes the set of real numbers. Recall the following equilibrium problem. Find 𝑥∈𝐶 such that 𝑓𝑥𝑦≥0,∀𝑦∈𝐶. (3) We use EP(𝑓) to denote the solution set of the equilibrium problem (3). That is, EP𝑓=𝑝∈𝐶∶𝑓𝑝,𝑦≥0,∀𝑦∈𝐶. (4) Given a mapping 𝐵∶𝐶→𝐸^∗, let 𝑓𝑥,𝑦=𝐵𝑥,𝑦−𝑥,∀𝑥,𝑦∈𝐶. (5) Then, 𝑥∈EP(𝑓) if and only if 𝑥 is a solution of the following variational inequality. Find 𝑥 such that 𝐵𝑥𝑦−𝑥≥0,∀𝑦∈𝐶. (6)

In order to study the solution of the equilibrium problem (3), we assume that 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,𝑦∈𝐶; (A3)limsup_𝑡↓0𝑓𝑡𝑧+(1−𝑡)𝑥,𝑦≤𝑓𝑥,𝑦,∀𝑥,𝑦,𝑧∈𝐶; (7)(A4) for each 𝑥∈𝐶, 𝑦↦𝑓(𝑥,𝑦) is convex and weakly lower semicontinuous.

For any convex function 𝑔∶𝐸→(−∞,+∞], we denote the domain of 𝑔 by Dom(𝑔)={𝑥∈𝐸∶𝑔(𝑥)<∞}. For any 𝑥∈intDom(𝑔) and 𝑦∈𝐸, we denote by 𝑔⁰(𝑥,𝑦) the right-hand derivative of 𝑔 at 𝑥 in the direction 𝑦; that is, 𝑔⁰𝑥,𝑦=lim_𝑡→0+𝑔𝑥+𝑡𝑦−𝑔(𝑥)𝑡. (8) The function 𝑔 is said to be Gâteaux differentiable at 𝑥 if lim_𝑡→0⁺((𝑔(𝑥+𝑡𝑦)−𝑔(𝑥))/𝑡) exists for any 𝑦. In this case, 𝑔⁰(𝑥,𝑦) coincides with ∇𝑔(𝑥), the value of the gradient ∇𝑔 of 𝑔 at 𝑥. The function 𝑔 is said to be Gâteaux differentiable if it is Gâteaux differentiable everywhere. The function 𝑔 is said to be Fréchet differentiable at 𝑥 if the limit is attained uniformly in ‖𝑦‖=1. The function 𝑔 is said to be Fréchet differentiable if it is Fréchet differentiable everywhere. It is well known that if a continuous convex function 𝑔∶𝐸→ℝ is Gâteaux differentiable, then ∇𝑔 is norm-to-weak* continuous. If 𝑔∶𝐸→ℝ is Fréchet differentiable, then ∇𝑔 is norm-to-norm continuous; for more details, see [31] and the references therein; for more details, see [32] and the references therein. The function 𝑔 is said to be strongly coercive [32] if lim_{‖𝑥‖→∞}(𝑓(𝑥)/‖𝑥‖)=∞. It is said to be bounded on bounded subsets of 𝐸 if 𝑔(𝐵) is bounded for each bounded subset 𝐵 of 𝐸. Let 𝐸 be a reflexive Banach space. For any proper, lower semicontinuous, and convex function: 𝑔∶𝐸→(−∞,+∞], the conjugate function 𝑔^∗ of 𝑔 is defined by 𝑔^∗𝑥^∗=sup_𝑥∈𝐸𝑥,𝑥^∗−𝑔(𝑥),∀𝑥^∗∈𝐸^∗. (9) It is well known that 𝑔(𝑥)+𝑔^∗(𝑥^∗)≥⟨𝑥,𝑥^∗⟩ for all (𝑥,𝑥^∗)∈𝐸×𝐸^∗. It is also known that (𝑥,𝑥^∗)∈𝜕𝑔, where 𝜕𝑔 is the subdifferential of 𝑔, is equivalent to 𝑔(𝑥)+𝑔^∗𝑥^∗=𝑥,𝑥^∗. (10) If 𝑔 is a proper, lower semicontinuous, and convex function, then 𝑔^∗ is a proper, weak* lower semicontinuous, and convex function.

Next, we recall some facts about Bregman distance. Let 𝑔∶𝐸→ℝ be a convex and Gâteaux differentiable function. Then, Bregman distance corresponding to 𝑔 is the function 𝐷_𝑔∶𝐸×𝐸→ℝ defined by 𝐷_𝑔𝑥,𝑦=𝑔(𝑥)−𝑔𝑦−𝑥−𝑦,∇𝑔𝑦,∀𝑥,𝑦∈𝐸. (11) It is clear that 𝐷_𝑔(𝑥,𝑦)≥0 for all 𝑥,𝑦∈𝐸. If 𝐸 is smooth and 𝑔(𝑥)=‖𝑥‖² for all 𝑥∈𝐸, we obtain that ∇𝑔(𝑥)=2𝐽𝑥, where 𝐽 is the generalized duality mapping. If 𝐶 is a nonempty, closed and convex subset of a reflexive Banach space 𝐸 and 𝑔 is a strongly coercive Bregman function, then for each 𝑥∈𝐸, there exists a unique 𝑥₀∈𝐶 such that 𝐷_𝑔(𝑥₀,𝑥)=min_𝑦∈𝐶𝐷_𝑔(𝑦,𝑥). Bregman projection Proj^𝑔_𝐶 from 𝐸 onto 𝐶 is defined by Proj^𝑔_𝐶𝑥=𝑥₀ for all 𝑥∈𝐸. It is also well known [32] that Proj^𝑔_𝐶 has the following property: 𝐷_𝑔𝑦,Proj^𝑔_𝐶𝑥+𝐷_𝑔Proj^𝑔_𝐶𝑥,𝑥≤𝐷_𝑔𝑦,𝑥,∀𝑦∈𝐶,𝑥∈𝐸. (12) Let 𝐵_𝑟∶={𝑧∈𝐸∶‖𝑧‖<𝑟}. A function 𝑔∶𝐸→ℝ is said to be uniformly convex on bounded subsets of 𝐸 if 𝑓_𝑟(𝑡)>0 for all 𝑟,𝑡>0, where 𝑓_𝑟∶[0,∞]→∞[0,∞] is defined by 𝑓_𝑟(𝑡)=inf_{𝑥,𝑦∈𝐵𝑟,𝑥−𝑦=𝑡,𝛼∈(0,1)}(𝑥)+()𝑔𝑦()𝑦×(𝛼(𝛼𝑔1−𝛼−𝑔𝛼𝑥+1−𝛼1−𝛼))⁻¹,∀𝑡≥0. (13) The function 𝑓_𝑟 is called the gauge of the uniform convexity of 𝑔. If 𝑔∶𝐸→ℝ is a convex function which is uniformly convex on bounded subsets, then 𝑔()𝑦(𝑥)+()𝑔𝑦()𝑓𝛼𝑥+1−𝛼≤𝛼𝑔1−𝛼−𝛼1−𝛼_𝑟𝑥−𝑦 (14) for all 𝑥,𝑦∈𝐵_𝑟 and 𝛼∈(0,1), where 𝑓_𝑟 is the gauge of the uniform convexity of 𝑔.

Let 𝑇∶𝐶→𝐶 be a mapping. In this paper, we use 𝐹(𝑇) to denote the fixed point set of 𝑇. 𝑇 is said to be closed; if for any sequence {𝑥_𝑛}⊂𝐶 such that lim_𝑛→∞𝑥_𝑛=𝑥₀ and lim_𝑛→∞𝑇𝑥_𝑛=𝑦₀, then 𝑇𝑥₀=𝑦₀.

Let 𝑔∶𝐸→ℝ be a proper, lower semicontinuous and convex function. Recall that 𝑇∶𝐶→𝐶 is said to be Bregman quasi-nonexpansive, if and only if 𝐹(𝑇)≠∅,𝐷_𝑔𝑝,𝑇𝑥≤𝐷_𝑔,(𝑇).𝑝,𝑥∀𝑥∈𝐶,𝑝∈𝐹 (15)

A point 𝑝 in 𝐶 is said to be an asymptotic fixed point of 𝑇 if and only if 𝐶 contains a sequence {𝑥_𝑛}, which converges weakly to 𝑝 such that lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0. The set of asymptotic fixed points of 𝑇 will be denoted by 𝐹(𝑇). 𝑇 is said to be Bregman relatively nonexpansive if and only if 𝐹(𝑇)=𝐹(𝑇)≠∅ and 𝐷_𝑔(𝑝,𝑇𝑥)≤𝐷_𝑔(𝑝,𝑥) for all 𝑥∈𝐶 and 𝑝∈𝐹(𝑇).

Let 𝐴 be a multivalued operator from 𝐸 to 𝐸^∗ with domain Dom(𝐴)={𝑧∈𝐸∶𝐴𝑧≠∅} and range Ran(𝐴)=∪{𝐴𝑧∶𝑧∈Dom(𝐴)}. An operator 𝐴 is said to be monotone if and only if ⟨𝑥₁−𝑥₂,𝑦₁−𝑦₂⟩≥0 for each 𝑥_𝑖∈Dom(𝐴) and 𝑦_𝑖∈𝐴𝑥_𝑖, 𝑖=1,2. A monotone operator 𝐴 is said to be maximal if its graph Grap(𝐴)={(𝑥,𝑦)∶𝑦∈𝐴𝑥} is not properly contained in the graph of any other monotone operator. We know that if 𝐴 is a maximal monotone operator, then 𝐴⁻¹(0) is closed and convex.

Let 𝐸 be a reflexive, strictly convex, and smooth Banach space, and let 𝐴 be a maximal monotone operator from 𝐸 to 𝐸^∗. From Rockafellar [33], we find that 𝑠>0 and 𝑥∈𝐸; there exists a unique 𝑥_𝑠∈𝐷(𝐴) such that ∇𝑔𝑥∈∇𝑔𝑥_𝑠+𝑠𝐴𝑥_𝑠. If 𝐽_𝑠𝑥=𝑥_𝑠, then we can define a single-valued mapping 𝐽_𝑠∶𝐸→Dom(𝐴) by 𝐽_𝑠=(∇𝑔+𝑠𝐴)⁻¹∇𝑔 and such a 𝐽_𝑠 is called the resolvent of 𝐴. We know that 𝐽_𝑠 is closed and 𝐴⁻¹(0)=𝐹(𝐽_𝑠) for all 𝑠>0. From [34], we know that 𝐽_𝑠∶𝐸→Dom(𝐴) is a Bregman quasi-nonexpansive.

In this paper, we study a common solution problem based on Bregman projections. Strong convergence of the monotone projection algorithm for monotone operators and bifunctions is obtained in a reflexive Banach space.

In order to introduce our main results, we also need the following lemmas.

Lemma 1 (see [34]). Let 𝐸 be a Banach space and let 𝑔∶𝐸→ℝ be a Gâteaux differentiable function which is uniformly convex on bounded subsets of 𝐸. Let {𝑥_𝑛} and {𝑦_𝑛} be bounded sequences in 𝐸. Then, lim_𝑛→∞𝐷_𝑔𝑥_𝑛,𝑦_𝑛=0⟺lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0. (16)

Lemma 2 (see [35]). Let 𝐸 be a reflexive Banach space and let 𝑔∶𝐸→ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of 𝐸. Let 𝐶 be a nonempty, closed, and convex subset of 𝐸. Let 𝑇∶𝐶→𝐶 be a Bregman relatively nonexpansive mapping. Then, 𝐹(𝑇) is closed and convex.

Lemma 3 (see [36]). Let 𝐸 be a reflexive Banach space and let 𝑔∶𝐸→ℝ be a convex, continuous, and strongly coercive function which is bounded on bounded subsets and uniformly convex on bounded subsets of 𝐸. Let 𝐶 be a nonempty, closed, and convex subset of 𝐸 and let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4). Let 𝑟>0 and 𝑥∈𝐸. Then, there exists 𝑧∈𝐶 such that 𝑓(𝑧,𝑦)+1/𝑟⟨𝑦−𝑧,∇𝑔𝑧−∇𝑔𝑥⟩≥0, for all 𝑦∈𝐶. Define a mapping 𝑆_𝑟∶𝐸→𝐶 by 𝑆_𝑟𝑥={𝑧∈𝐶∶𝑓(𝑧,𝑦)+(1/𝑟)⟨𝑦−𝑧,∇𝑔𝑧−∇𝑥⟩, for all 𝑦∈𝐶}. Then, the following conclusions hold: (1)𝑆_𝑟 is single-valued; (2)𝑆_𝑟 is a Bregman firmly nonexpansive-type mapping; that is, for all 𝑥,𝑦∈𝐸, 𝑆_𝑟𝑥−𝑆_𝑟𝑦,∇𝑔𝑆_𝑟𝑥−∇𝑔𝑆_𝑟𝑦≤𝑆_𝑟𝑥−𝑆_𝑟𝑦,∇𝑥−∇𝑔𝑦; (17)(3)𝐹(𝑆_𝑟)=𝐸𝑃(𝑓); (4)𝑆_𝑟 is Bregman quasi-nonexpansive;(5)𝐷_𝑔𝑞,𝑆_𝑟𝑥+𝐷_𝑔𝑆_𝑟𝑥,𝑥≤𝜙𝑞,𝑥,∀𝑞∈𝐹𝑆_𝑟; (18)(6)𝐸𝑃(𝑓) is closed and convex.

3. Main Results

Theorem 4. Let 𝐸 be a reflexive Banach space and let 𝑔∶𝐸→𝑅 be a strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and smooth on bounded subsets of 𝐸. Let 𝐶 be a nonempty, closed, and convex subset of 𝐸 and let Λ be an index set. Let {𝑠_𝑖} be a positive real number sequence. Let 𝑓_𝑖 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4) and let 𝐴_𝑖∶𝐸→𝐸^∗ be a maximal monotone operator such that Dom(𝐴_𝑖)⊂𝐶 for every 𝑖∈Λ. Assume that the common solution set 𝐶𝑆𝑆∶=∩_𝑖∈Λ𝐴_𝑖⁻¹(0)⋂∩_𝑖∈Λ𝐸𝐹(𝑓_𝑖) is nonempty. Let {𝑥_𝑛} be a sequence generated in the following manner: 𝑥₀𝐶∈𝐸𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_(1,𝑖)=𝐶,𝐶₁=∩_𝑖∈Λ𝐶_(1,𝑖),𝑥₁=𝑃𝑟𝑜𝑗^𝑔_𝐶1𝑥₀,𝑦_(𝑛,𝑖)=∇𝑔^∗𝛼_(𝑛,𝑖)∇𝑔𝑥_𝑛+1−𝛼_(𝑛,𝑖)∇𝑔𝐽^𝐴𝑖_𝑠𝑖𝑥_𝑛,𝑢_(𝑛,𝑖)𝑓∈𝐶𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡_𝑖𝑢_(𝑛,𝑖)+1,𝑦𝑟_(𝑛,𝑖)𝑦−𝑢_(𝑛,𝑖),∇𝑔𝑢_(𝑛,𝑖)−∇𝑔𝑦_(𝑛,𝑖)𝐶≥0,∀𝑦∈𝐶,_{(𝑛+1,𝑖)}=𝑧∈𝐶_(𝑛,𝑖)∶𝐷_𝑔𝑧,𝑢_(𝑛,𝑖)≤𝐷_𝑔𝑧,𝑥_𝑛,𝐶_𝑛+1=∩_𝑖∈Λ𝐶_{(𝑛+1,𝑖)},𝑥_𝑛+1=𝑃𝑟𝑜𝑗^𝑔_𝐶𝑛+1𝑥₁, (19) where ∇𝑔 is the right-hand derivative of 𝑔, 𝐽^𝐴𝑖_𝑠𝑖=(∇𝑔+𝑠_𝑖𝐴_𝑖)⁻¹∇𝑔, {𝛼_(𝑛,𝑖)} is a real sequence in [0,1] such that liminf_𝑛→∞𝛼_(𝑛,𝑖)(1−𝛼_(𝑛,𝑖))>0, and {𝑟_(𝑛,𝑖)} is a real sequence in [𝑎,∞), where 𝑎 is some positive real number, for every 𝑖∈Λ. Then, the sequence {𝑥_𝑛} converges strongly to 𝑃𝑟𝑜𝑗^𝑔_𝐶𝑆𝑆𝑥₁, where 𝑃𝑟𝑜𝑗^𝑔_𝐶𝑆𝑆 is the Bregman projection from 𝐸 onto 𝐶𝑆𝑆.

Proof. In view of Lemma 3, we find that the common solution set CSS is closed and convex. Next, we prove that 𝐶_𝑛 is closed and convex. It suffices to show that, for each fixed but arbitrary 𝑖∈Λ, 𝐶_(𝑛,𝑖) is closed and convex. This can be proved by induction into 𝑛. It is obvious that 𝐶_(1,𝑖)=𝐶 is closed and convex. Assume that 𝐶_(𝑘,𝑖) is closed and convex for some 𝑘≥1. For 𝑧₁,𝑧₂∈𝐶_{(𝑘+1,𝑖)}, we see that 𝑧₁,𝑧₂∈𝐶_(𝑘,𝑖). It follows that 𝑧=𝑡𝑧₁+(1−𝑡)𝑧₂∈𝐶_(𝑘,𝑖), where 𝑡∈(0,1). Notice that 𝐷_𝑔𝑧₁,𝑢_(𝑘,𝑖)≤𝐷_𝑔𝑧₁,𝑥_𝑘,𝐷_𝑔𝑧₂,𝑢_(𝑘,𝑖)≤𝐷_𝑔𝑧₂,𝑥_𝑘. (20) The above inequalities are equivalent to 𝑧₁,∇𝑔(𝑥)−∇𝑔𝑢_(𝑘,𝑖)≤𝑔𝑢_(𝑘,𝑖)−𝑔𝑥_𝑘−𝑢_(𝑘,𝑖),∇𝑔𝑢_(𝑘,𝑖)+(𝑥,𝑧𝑥,∇𝑔)₂,∇𝑔(𝑥)−∇𝑔𝑢_(𝑘,𝑖)≤𝑔𝑢_(𝑘,𝑖)−𝑔𝑥_𝑘−𝑢_(𝑘,𝑖),∇𝑔𝑢_(𝑘,𝑖)+𝑥,∇𝑔(𝑥). (21) Multiplying 𝑡 and (1−𝑡) on both sides of (21), respectively, yields that 𝑧,∇𝑔(𝑥)−∇𝑔𝑢_(𝑘,𝑖)≤𝑔𝑢_(𝑘,𝑖)−𝑔𝑥_𝑘−𝑢_(𝑘,𝑖),∇𝑔𝑢_(𝑘,𝑖)+⟨𝑥,∇𝑔(𝑥)⟩. (22) That is, 𝐷_𝑔𝑧,𝑢_(𝑘,𝑖)≤𝐷_𝑔𝑧,𝑥_𝑘, (23) where 𝑧∈𝐶_(𝑘,𝑖). This implies that 𝐶_{(𝑘+1,𝑖)} is closed and convex. We conclude that 𝐶_(𝑛,𝑖) is closed and convex. This in turn implies that 𝐶_𝑛=∩_𝑖∈Λ𝐶_(𝑛,𝑖) is closed and convex. This implies that Proj^𝑔_𝐶𝑛+1𝑥₁ is well defined.
Next, we show that CSS⊂𝐶_𝑛. CSS⊂𝐶₁=𝐶 is clear. Suppose that CSS⊂𝐶_(𝑘,𝑖) for some positive integer 𝑘. For any 𝑤∈CSS⊂𝐶_(𝑘,𝑖), we see that 𝐷_𝑔𝑤,𝑢_(𝑘,𝑖)=𝐷_𝑔𝑤,𝑆_{𝑟(𝑘,𝑖)}𝑦_(𝑘,𝑖)≤𝐷_𝑔𝑤,𝑦_(𝑘,𝑖)=𝐷_𝑔𝑤,∇𝑔^∗𝛼_(𝑘,𝑖)∇𝑔𝑥_𝑘+1−𝛼_(𝑘,𝑖)∇𝑔𝐽^𝐴𝑖_𝑠𝑖𝑥_𝑘=𝑉𝑤,𝛼_(𝑘,𝑖)∇𝑔𝑥_𝑘+1−𝛼_(𝑘,𝑖)∇𝑔𝐽^𝐴𝑖_𝑠𝑖𝑥_𝑘=𝑔(𝑤)−𝛼_(𝑘,𝑖)𝑤,∇𝑔𝑥_𝑘−1−𝛼_(𝑘,𝑖)×𝑤,∇𝑔𝐽^𝐴𝑖_𝑠𝑖𝑥_𝑘+𝑔^∗𝛼_(𝑘,𝑖)∇𝑔𝑥_𝑘+1−𝛼_(𝑘,𝑖)∇𝑔𝐽^𝐴𝑖_𝑠𝑖𝑥_𝑘≤𝛼_(𝑘,𝑖)𝑉𝑤,∇𝑔𝑥_𝑘+1−𝛼_(𝑘,𝑖)𝑉𝑤,∇𝑔𝐽^𝐴𝑖_𝑠𝑖𝑥_𝑘≤𝛼_(𝑘,𝑖)𝐷_𝑔𝑤,𝑥_𝑘+1−𝛼_(𝑘,𝑖)𝐷_𝑔𝑤,𝐽^𝐴𝑖_𝑠𝑖𝑥_𝑘≤𝐷_𝑔𝑤,𝑥_𝑘, (24) which shows that 𝑤∈𝐶_{(𝑘+1,𝑖)}. This implies that CSS⊂𝐶_(𝑛,𝑖). This yields that CSS⊂∩_𝑖∈Λ𝐶_(𝑛,𝑖). This completes the proof that CSS⊂𝐶_𝑛.
Next, we show that the sequence {𝑥_𝑛} is bounded. It follows from (12) that 𝐷_𝑔𝑥_𝑛,𝑥₁≤𝐷_𝑔Proj^𝑔_CSS𝑥₁,𝑥₁−𝐷_𝑔Proj^𝑔_CSS𝑥₁,𝑥_𝑛≤𝐷_𝑔Proj^𝑔_CSS𝑥₁,𝑥₁. (25) This implies that the sequence {𝐷_𝑔(𝑥_𝑛,𝑥₁)} is bounded. It follows that the sequence {𝑥_𝑛} is also bounded. In view of the construction of the set 𝐶_𝑛, we find that 𝐶_𝑚⊂𝐶_𝑛 and 𝑥_𝑚=proj^𝑔_𝐶𝑚𝑥₁∈𝐶_𝑚⊂𝐶_𝑛 for any positive integer 𝑚≥𝑛. It follows from (12) that 𝐷_𝑔𝑥_𝑚,𝑥_𝑛=𝐷_𝑔𝑥_𝑚,Proj^𝑔_𝐶𝑛𝑥₁≤𝐷_𝑔𝑥_𝑚,𝑥₁−𝐷_𝑔Proj^𝑔_𝐶𝑛𝑥₁,𝑥₁=𝐷_𝑔𝑥_𝑚,𝑥₁−𝐷_𝑔𝑥_𝑛,𝑥₁. (26) It follows that 𝐷_𝑔𝑥_𝑛,𝑥₁≤𝐷_𝑔𝑥_𝑛,𝑥₁+𝐷_𝑔𝑥_𝑚,𝑥_𝑛≤𝐷_𝑔𝑥_𝑚,𝑥₁. (27) This shows that the sequence {𝐷_𝑔(𝑥_𝑛,𝑥₁)} is nondecreasing. Hence, the limit lim_𝑛→∞𝐷_𝑔(𝑥_𝑛,𝑥₁) exists. In view of 𝑥_𝑛=Proj^𝑔_𝐶𝑛𝑥₁, we find that 𝐷_𝑔𝑥_𝑛,𝑥₁≤𝐷_𝑔𝑥_𝑛+1,𝑥₁. (28) This concludes that lim_𝑛→∞𝐷_𝑔(𝑥_𝑛,𝑥₁) exists. Let 𝑚,𝑛→∞ in (26); we find that 𝐷_𝑔(𝑥_𝑚,𝑥_𝑛)→0. In view of Lemma 1, we obtain that ‖𝑥_𝑚−𝑥_𝑛‖→0 as 𝑚,𝑛→∞. This yields that {𝑥_𝑛} is a Cauchy sequence. Since 𝐶 is closed and convex, we see that there exists an 𝑥∈𝐶; that is, lim_𝑛→∞𝑥_𝑛−𝑥=0. (29) Next, we show that 𝑥∈CSS. It follows from (26) that lim_𝑛→∞𝐷_𝑔𝑥_𝑛+1,𝑥_𝑛=0. (30) Therefore, we obtain that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (31) Since 𝑥_𝑛+1∈𝐶_𝑛+1, we find that 𝐷_𝑔𝑥_𝑛+1,𝑢_(𝑛,𝑖)≤𝐷_𝑔𝑥_𝑛+1,𝑥_𝑛. (32) In view of (30), we find that lim_𝑛→∞𝐷_𝑔𝑥_𝑛+1,𝑢_(𝑛,𝑖)=0. (33) It follows that lim_𝑛→∞𝑥_𝑛+1−𝑢_(𝑛,𝑖)=0. (34) Notice that 𝑥_𝑛−𝑢_(𝑛,𝑖)≤𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑢_(𝑛,𝑖). (35) It follows from (29) and (34) that lim_𝑛→∞‖𝑥_𝑛−𝑢_(𝑛,𝑖)‖=0. In view of Lemma 1, we find that lim_𝑛→∞𝐷_𝑔𝑢_(𝑛,𝑖),𝑥_𝑛=0. (36) Since ∇𝑔 is uniformly norm-to-norm continuous on any bounded subset of 𝐸, we find that lim_𝑛→∞∇𝑔𝑢_(𝑛,𝑖)−∇𝑔𝑥_𝑛=0. (37) Notice that 𝐷_𝑔𝑤,𝑥_𝑛−𝐷_𝑔𝑤,𝑢_(𝑛,𝑖)=𝐷_𝑔𝑢_(𝑛,𝑖),𝑥_𝑛+𝑤−𝑢_(𝑛,𝑖),∇𝑔𝑢_(𝑛,𝑖)−∇𝑔𝑥_𝑛≤𝐷_𝑔𝑢_(𝑛,𝑖),𝑥_𝑛+𝑤−𝑢_(𝑛,𝑖)∇𝑔𝑢_(𝑛,𝑖)−∇𝑔𝑥_𝑛. (38) It follows from (36) and (37) that lim_𝑛→∞𝐷_𝑔𝑤,𝑥_𝑛−𝐷_𝑔𝑤,𝑢_(𝑛,𝑖)=0. (39) In view of (14), we find that 𝐷_𝑔𝑤,𝑢_(𝑛,𝑖)=𝐷_𝑔𝑤,𝑆_{𝑟(𝑛,𝑖)}𝑦_(𝑛,𝑖)≤𝐷_𝑔𝑤,𝑦_(𝑛,𝑖)=𝐷_𝑔𝑤,∇𝑔^∗𝛼_(𝑛,𝑖)∇𝑔𝑥_𝑛+1−𝛼_(𝑛,𝑖)∇𝑔𝐽^𝐴𝑖_𝑠𝑖𝑥_𝑛=𝑉𝑤,𝛼_(𝑛,𝑖)∇𝑔𝑥_𝑛+1−𝛼_(𝑛,𝑖)∇𝑔𝐽^𝐴𝑖_𝑠𝑖𝑥_𝑛=𝑔(𝑤)−𝛼_(𝑛,𝑖)𝑤,∇𝑔𝑥_𝑛−1−𝛼_(𝑛,𝑖)×𝑤,∇𝑔𝐽^𝐴𝑖_𝑠𝑖𝑥_𝑛+𝑔^∗𝛼_(𝑛,𝑖)∇𝑔𝑥_𝑛+1−𝛼_(𝑛,𝑖)∇𝑔𝐽^𝐴𝑖_𝑠𝑖𝑥_𝑛≤𝛼_(𝑛,𝑖)𝑉𝑤,∇𝑔𝑥_𝑛+1−𝛼_(𝑛,𝑖)𝑉𝑤,∇𝑔𝐽^𝐴𝑖_𝑠𝑖𝑥_𝑛−𝜌∇𝑔𝐽^𝐴𝑖_𝑠𝑖𝑥_𝑛−∇𝑔𝑥_𝑛≤𝛼_(𝑛,𝑖)𝐷_𝑔𝑤,𝑥_𝑛+1−𝛼_(𝑛,𝑖)𝐷_𝑔𝑤,𝐽^𝐴𝑖_𝑠𝑖𝑥_𝑛−𝛼_(𝑛,𝑖)1−𝛼_(𝑛,𝑖)×𝜌∇𝑔𝐽^𝐴𝑖_𝑠𝑖𝑥_𝑛−∇𝑔𝑥_𝑛≤𝐷_𝑔𝑤,𝑥_𝑛−𝛼_(𝑛,𝑖)1−𝛼_(𝑛,𝑖)×𝜌∇𝑔𝐽^𝐴𝑖_𝑠𝑖𝑥_𝑛−∇𝑔𝑥_𝑛, (40) where 𝜌 is the gauge of the uniform convexity of 𝑔^∗. It follows that 𝛼_(𝑛,𝑖)1−𝛼_(𝑛,𝑖)𝜌∇𝑔𝐽^𝐴𝑖_𝑠𝑖𝑥_𝑛−∇𝑔𝑥_𝑛≤𝐷_𝑔𝑤,𝑥_𝑛−𝐷_𝑔𝑤,𝑢_(𝑛,𝑖). (41) In view of the restrictions on the sequence {𝛼_(𝑛,𝑖)}, we find from (39) that lim_𝑛→∞‖∇𝑔(𝑥_𝑛)−∇𝑔(𝐽^𝐴𝑖_𝑠𝑖𝑥_𝑛)‖=0. Since ∇𝑔^∗ is uniformly norm-to-norm continuous on bounded subsets of 𝐸^∗, we obtain that lim_𝑛→∞𝑥_𝑛−𝐽^𝐴𝑖_𝑠𝑖𝑥_𝑛=0. (42) Since 𝐽^𝐴𝑖_𝑠𝑖 is closed Bregman quasi-nonexpansive, we find that 𝐽^𝐴𝑖_𝑠𝑖𝑥=𝑥. This proves that 𝑥∈∩_𝑖∈Λ𝐴_𝑖⁻¹(0).
Next, we show that 𝑥∈∩_𝑖∈ΛEF(𝑓_𝑖). Notice that 𝐷_𝑔𝑢_(𝑛,𝑖),𝑦_(𝑛,𝑖)≤𝐷_𝑔𝑤,𝑦_(𝑛,𝑖)−𝐷_𝑔𝑤,𝑢_(𝑛,𝑖)≤𝐷_𝑔𝑤,𝑥_𝑛−𝐷_𝑔𝑤,𝑢_(𝑛,𝑖). (43) It follows from (39) that lim_𝑛→∞𝐷_𝑔𝑢_(𝑛,𝑖),𝑦_(𝑛,𝑖)=0. (44) Therefore, we find that lim_𝑛→∞‖𝑢_(𝑛,𝑖)−𝑦_(𝑛,𝑖)‖=0. Since ∇𝑔 is uniformly norm-to-norm continuous on any bounded sets, we have lim_𝑛→∞‖∇𝑔(𝑢_(𝑛,𝑖))−∇(𝑦_(𝑛,𝑖))‖=0. From the assumption 𝑟_(𝑛,𝑖)≥𝑎, we see that lim_𝑛→∞∇𝑔𝑢_(𝑛,𝑖)−∇𝑔𝑦_(𝑛,𝑖)𝑟_(𝑛,𝑖)=0. (45) In view of 𝑢_(𝑛,𝑖)=𝑆_{𝑟(𝑛,𝑖)}𝑦_(𝑛,𝑖), we see that 𝑓_𝑖𝑢_(𝑛,𝑖)+1,𝑦𝑟_(𝑛,𝑖)𝑦−𝑢_(𝑛,𝑖),∇𝑔𝑢_(𝑛,𝑖)−∇𝑔𝑦_(𝑛,𝑖)≥0,∀𝑦∈𝐶. (46) It follows from (A2) that 𝑦−𝑢_(𝑛,𝑖)∇𝑔𝑢_(𝑛,𝑖)−∇𝑔𝑦_(𝑛,𝑖)𝑟_𝑛,𝑖≥1𝑟_(𝑛,𝑖)𝑦−𝑢_(𝑛,𝑖),∇𝑔𝑢_(𝑛,𝑖)−∇𝑔𝑦_(𝑛,𝑖)≥𝑓_𝑖𝑦,𝑢_(𝑛,𝑖),∀𝑦∈𝐶. (47) In view of (A4), we find from (45) that 𝑓_𝑖𝑦,𝑥≤0,∀𝑦∈𝐶. (48) For 0<𝑡_𝑖<1 and 𝑦∈𝐶, define 𝑦_(𝑡,𝑖)=𝑡_𝑖𝑦+(1−𝑡_𝑖)𝑥. It follows that 𝑦_(𝑡,𝑖)∈𝐶, which yields that 𝑓(𝑦_(𝑡,𝑖),𝑥)≤0. It follows from (A1) and (A4) that 0=𝑓𝑦_(𝑡,𝑖),𝑦_(𝑡,𝑖)≤𝑡_𝑖𝑓𝑦_(𝑡,𝑖),𝑦+1−𝑡_𝑖𝑓𝑦_(𝑡,𝑖),𝑥≤𝑡_𝑖𝑓𝑦_(𝑡,𝑖),𝑦. (49) That is, 𝑓𝑦_(𝑡,𝑖),𝑦≥0. (50) Letting 𝑡_𝑖↓0, we obtain from (A3) that 𝑓_𝑖(𝑥,𝑦)≥0, for all 𝑦∈𝐶. This implies that 𝑥∈EP(𝑓_𝑖) for every 𝑖∈Λ. This shows that 𝑥∈CSS.
Finally, we prove that 𝑥=Proj^𝑔_CSS𝑥₁. In view of 𝑥_𝑛=Proj^𝑔_𝐶𝑛𝑥₁, we conclude that 𝑧−𝑥_𝑛,∇𝑔𝑥_𝑛−∇𝑔𝑥₁,∀𝑧∈𝐶_𝑛. (51) Since CSC⊂𝐶_𝑛, we arrive at 𝑤−𝑥_𝑛,∇𝑔𝑥_𝑛−∇𝑔𝑥₁,∀𝑤∈CSC. (52) Letting 𝑛→∞ in the above inequality, we see that 𝑤−𝑥,∇𝑔(𝑥)−∇𝑔𝑥₁≥0,∀𝑤∈CSS. (53) This yields that 𝑥=Proj^𝑔_CSS𝑥₁. This completes the proof.

If the index set is singleton, then we have the following result.

Corollary 5. Let 𝐸 be a reflexive Banach space and let 𝑔∶𝐸→𝑅 be a strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and smooth on bounded subsets of 𝐸. Let 𝐶 be a nonempty, closed, and convex subset of 𝐸 and let 𝑠 be a positive real number. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4) and let 𝐴∶𝐸→𝐸^∗ be a maximal monotone operator such that Dom(𝐴)⊂𝐶. Assume that the common solution set 𝐶𝑆𝑆∶=𝐴⁻¹(0)⋂𝐸𝐹(𝑓) is nonempty. Let {𝑥_𝑛} be a sequence generated in the following manner: 𝑥₀𝐶∈𝐸𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,₁𝑥=𝐶,₁=𝑃𝑟𝑜𝑗^𝑔_𝐶1𝑥₀,𝑦_𝑛=∇𝑔^∗𝛼_𝑛∇𝑔𝑥_𝑛+1−𝛼_𝑛∇𝑔𝐽^𝐴_𝑠𝑥_𝑛,𝑢_𝑛𝑓𝑢∈𝐶𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,∇𝑔𝑢_𝑛−∇𝑔𝑦_𝑛𝐶≥0,∀𝑦∈𝐶,_𝑛+1=𝑧∈𝐶_𝑛∶𝐷_𝑔𝑧,𝑢_𝑛≤𝐷_𝑔𝑧,𝑥_𝑛,𝑥_𝑛+1=𝑃𝑟𝑜𝑗^𝑔_𝐶𝑛+1𝑥₁, (54) where ∇𝑔 is the right-hand derivative of 𝑔. 𝐽^𝐴_𝑠=(∇𝑔+𝑠𝐴)⁻¹∇𝑔, {𝛼_𝑛} is a real sequence in [0,1] such that liminf_𝑛→∞𝛼_𝑛(1−𝛼_𝑛)>0, and {𝑟_𝑛} is a real sequence in [𝑎,∞), where 𝑎 is some positive real number. Then, the sequence {𝑥_𝑛} converges strongly to 𝑃𝑟𝑜𝑗^𝑔_𝐶𝑆𝑆𝑥₁, where 𝑃𝑟𝑜𝑗^𝑔_𝐶𝑆𝑆 is the Bregman projection from 𝐸 onto 𝐶𝑆𝑆.

Remark 6. Corollary 5 improves Theorem 4 of Qin et al. [24] in the following aspects: (1) the framework of spaces is relaxed; (2) a more general notion of Bregman function is considered instead of the generalized duality mapping; (3) a more general notion of Bregman projection is considered instead of the generalized projection operator; (4) families of operators extend from a finite family to an infinite family.

Finally, we give some results in the framework of Hilbert spaces.

Theorem 7. Let 𝐸 be a Hilbert space and let 𝐶 be a nonempty, closed, and convex subset of 𝐸. Let Λ be an index set. Let {𝑠_𝑖} be a positive real number sequence. Let 𝑓_𝑖 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4) and let 𝐴_𝑖∶𝐸→𝐸 be a maximal monotone operator such that Dom(𝐴_𝑖)⊂𝐶 for every 𝑖∈Λ. Assume that the common solution set 𝐶𝑆𝑆∶=∩_𝑖∈Λ𝐴_𝑖⁻¹(0)⋂∩_𝑖∈Λ𝐸𝐹(𝑓_𝑖) is nonempty. Let {𝑥_𝑛} be a sequence generated in the following manner: 𝑥₀𝐶∈𝐸𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_(1,𝑖)𝐶=𝐶,₁=∩_𝑖∈Λ𝐶_(1,𝑖),𝑥₁=𝑃𝑟𝑜𝑗_𝐶1𝑥₀,𝑦_(𝑛,𝑖)=𝛼_(𝑛,𝑖)𝑥_𝑛+1−𝛼_(𝑛,𝑖)𝐽^𝐴𝑖_𝑠𝑖𝑥_𝑛,𝑢_(𝑛,𝑖)𝑓∈𝐶𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡_𝑖𝑢_(𝑛,𝑖)+1,𝑦𝑟_(𝑛,𝑖)𝑦−𝑢_(𝑛,𝑖),𝑢_(𝑛,𝑖)−𝑦_(𝑛,𝑖)𝐶≥0,∀𝑦∈𝐶,_{(𝑛+1,𝑖)}=𝑧∈𝐶_(𝑛,𝑖)∶𝑧−𝑢_(𝑛,𝑖)²≤𝑧−𝑥_𝑛²,𝐶_𝑛+1=∩_𝑖∈Λ𝐶_{(𝑛+1,𝑖)},𝑥_𝑛+1=𝑃𝑟𝑜𝑗_𝐶𝑛+1𝑥₁, (55) where 𝐽^𝐴𝑖_𝑠𝑖=(𝐼+𝑠_𝑖𝐴_𝑖)⁻¹, {𝛼_(𝑛,𝑖)} is a real sequence in [0,1] such that liminf_𝑛→∞𝛼_(𝑛,𝑖)(1−𝛼_(𝑛,𝑖))>0, and {𝑟_(𝑛,𝑖)} is a real sequence in [𝑎,∞), where 𝑎 is some positive real number, for every 𝑖∈Λ. Then, the sequence {𝑥_𝑛} converges strongly to 𝑃𝑟𝑜𝑗_𝐶𝑆𝑆𝑥₁, where 𝑃𝑟𝑜𝑗_𝐶𝑆𝑆 is the metric projection from 𝐸 onto 𝐶𝑆𝑆.

Proof. Putting 𝑔(𝑥)=(1/2)‖𝑥‖², we find that ∇𝑔(𝑥)=𝑥 for all 𝑥∈𝐸. It follows that 𝐷_𝑔(𝑥,𝑦)=‖𝑥−𝑦‖² for all 𝑥,𝑦∈𝐸. From Theorem 4, we can immediately obtain the desired conclusion.

If the index set is singleton, then we have the following result.

Corollary 8. Let 𝐸 be a Hilbert space and let 𝐶 be a nonempty, closed, and convex subset of 𝐸. Let 𝑠 be a positive real number. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4) and let 𝐴∶𝐸→𝐸 be a maximal monotone operator such that Dom(𝐴)⊂𝐶. Assume that the common solution set 𝐶𝑆𝑆∶=𝐴⁻¹(0)⋂𝐸𝐹(𝑓) is nonempty. Let {𝑥_𝑛} be a sequence generated in the following manner: 𝑥₀𝐶∈𝐸𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,₁𝑥=𝐶,₁=𝑃𝑟𝑜𝑗_𝐶1𝑥₀,𝑦_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝐽^𝐴_𝑠𝑥_𝑛,𝑢_𝑛𝑓𝑢∈𝐶𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑦_𝑛𝐶≥0,∀𝑦∈𝐶,_𝑛+1=𝑧∈𝐶_𝑛∶𝑧−𝑢_𝑛²≤𝑧−𝑥_𝑛²,𝑥_𝑛+1=𝑃𝑟𝑜𝑗_𝐶𝑛+1𝑥₁, (56) where 𝐽^𝐴_𝑠=(𝐼+𝑠𝐴)⁻¹, {𝛼_𝑛} is a real sequence in [0,1] such that liminf_𝑛→∞𝛼_𝑛(1−𝛼_𝑛)>0 and {𝑟_𝑛} is a real sequence in [𝑎,∞), where 𝑎 is some positive real number. Then, the sequence {𝑥_𝑛} converges strongly to 𝑃𝑟𝑜𝑗_𝐶𝑆𝑆𝑥₁, where 𝑃𝑟𝑜𝑗_𝐶𝑆𝑆 is the metric projection from 𝐸 onto 𝐶𝑆𝑆.