Strong Convergence Theorems for Quasi-Bregman Nonexpansive Mappings in Reflexive Banach Spaces

Abstract

We study a strong convergence for a common fixed point of a finite family of quasi-Bregman nonexpansive mappings in the framework of real reflexive Banach spaces. As a consequence, convergence for a common fixed point of a finite family of Bergman relatively nonexpansive mappings is discussed. Furthermore, we apply our method to prove strong convergence theorems of iterative algorithms for finding a common solution of a finite family equilibrium problem and a common zero of a finite family of maximal monotone mappings. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

1. Introduction

Throughout this paper, 𝐸 is a real reflexive Banach space with the dual space 𝐸^∗. The norm and the dual pair between 𝐸^∗ and 𝐸 are denoted by ‖⋅‖ and ⟨⋅,⋅⟩, respectively. We also assume that 𝑓∶𝐸→(−∞,+∞] is a proper, lower semicontinuous, and convex function. Denote the domain of 𝑓 by dom𝑓; that is, dom𝑓={𝑥∈𝐸∶𝑓(𝑥)<∞}. Let 𝑥∈ int(dom𝑓). The subdifferential of 𝑓 at 𝑥∈𝐸 is the convex set defined by 𝜕𝑓(𝑥)=𝑥^∗∈𝐸^∗∶𝑓(𝑥)+𝑥^∗,𝑦−𝑥≤𝑓𝑦,∀𝑦∈𝐸. (1) The Fenchel conjugate of 𝑓 is the function 𝑓^∗∶𝐸^∗→(−∞,+∞] defined by 𝑓^∗(𝑦)=sup{⟨𝑦,𝑥⟩−𝑓(𝑥)∶𝑥∈𝐸}.

Let 𝑥∈Dom(𝑓) and 𝑦∈𝐸. The right-hand derivative of 𝑓 at 𝑥 in the direction of 𝑦 is defined by 𝑓^∘(𝑥,𝑧)=lim_𝑡→0⁺(𝑓(𝑥+𝑡𝑦)−𝑓(𝑥))/𝑡. The function 𝑓 is called Gâteaux differentiable at 𝑥 if lim_𝑡→0⁺(𝑓(𝑥+𝑡𝑦)−𝑓(𝑥))/𝑡 exists for any 𝑦 and hence 𝑓^∘(𝑥,𝑦) coincides with ∇𝑓(𝑥), the value of the gradient ∇𝑓 of 𝑓 at 𝑥. The function 𝑓 is said to be Gâteaux differentiable if it is Gâteaux differentiable for any 𝑥∈ intdom𝑓. Furthermore, 𝑓 is said to be Fréchet differentiable at 𝑥 if this limit is attained uniformly in ‖𝑦‖=1 and it is called uniformly Fréchet differentiable on a subset 𝐶 of 𝐸 if the limit is attained uniformly for 𝑥∈𝐶 and ‖𝑦‖=1.

Let 𝑓∶𝐸→(−∞,+∞] be a Gâteaux differentiable function. The function 𝐷_𝑓∶dom𝑓× int(dom𝑓)→[0,+∞) defined by 𝐷_𝑓𝑥,𝑦∶=𝑓(𝑥)−𝑓𝑦−∇𝑓𝑦,𝑥−𝑦 (2) is called the Bregman distance with respect to 𝑓 [1].

When 𝐸 is a smooth Banach space, setting 𝑓(𝑥)=‖𝑥‖² for all 𝑥∈𝐸, we have ∇𝑓(𝑥)=2𝐽𝑥, for all 𝑥∈𝐸, where 𝐽 is the normalized duality mapping from 𝐸 into 2^𝐸∗, and hence 𝐷_𝑓(𝑥,𝑦) reduces to 𝐷_𝑓(𝑥,𝑦)=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+‖𝑦‖², for all 𝑥,𝑦∈𝐸, which is the Lyapunov function introduced by Alber [2]. If 𝐸=𝐻, a Hilbert space, 𝐽 is identity mapping and hence 𝐷_𝑓(𝑥,𝑦) becomes 𝐷_𝑓(𝑥,𝑦)=‖𝑥−𝑦‖², for all 𝑥, 𝑦∈𝐻.

Let 𝐶 be a nonempty and convex subset of int(dom𝑓) and let 𝑇∶𝐶→int(dom𝑓) be a mapping. 𝑇 is said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶, and 𝑇 is said to be quasinonexpansive if 𝐹(𝑇)≠∅ and ‖𝑇𝑥−𝑝‖≤‖𝑥−𝑝‖, for all 𝑥∈𝐾 and 𝑝∈𝐹(𝑇), where 𝐹(𝑇) stands for the fixed point set of 𝑇; that is, 𝐹(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥}. A point 𝑝∈𝐶 is said to be an asymptotic fixed point of 𝑇 (see [3]) 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 is denoted by 𝐹(𝑇).

A mapping 𝑇∶𝐶→ int(dom𝑓) with 𝐹(𝑇)≠∅ is called(i)quasi-Bregman nonexpansive [4] if 𝐷_𝑓𝑝,𝑇𝑥≤𝐷_𝑓𝑝,𝑥,∀𝑥∈𝐶,𝑝∈𝐹(𝑇); (3)(ii)Bregman relatively nonexpansive [4] if 𝐷_𝑓𝑝,𝑇𝑥≤𝐷_𝑓(𝑇),𝐹(𝑇)(𝑇);𝑝,𝑥,∀𝑥∈𝐶,𝑝∈𝐹=𝐹 (4)(iii)Bregman firmly nonexpansive [5] if, for all 𝑥,𝑦∈𝐶, ()≤(𝑥)𝑦∇𝑓𝑇𝑥−∇𝑓𝑇𝑦,𝑇𝑥−𝑇𝑦∇𝑓−∇𝑓,𝑇𝑥−𝑇𝑦 (5)or, equivalently, 𝐷_𝑓𝑇𝑥,𝑇𝑦+𝐷_𝑓𝑇𝑦,𝑇𝑥+𝐷_𝑓(𝑇𝑥,𝑥)+𝐷_𝑓𝑇𝑦,𝑦≤𝐷_𝑓𝑇𝑥,𝑦+𝐷_𝑓𝑇𝑦,𝑥. (6) Iterative methods for approximating fixed points of nonexpansive, quasinonexpansive mappings and their generalizations have been studied by various authors (see, e.g., [6–16] and the references therein) in Hilbert spaces. But extending this theory to Banach spaces encountered some difficulties because the useful examples of nonexpansive operators in Hilbert spaces are no longer nonexpansive in Banach spaces (e.g., the resolvent 𝑅_𝐴=(𝐼+𝑟𝐴)⁻¹, for 𝑟>0, of a monotone mapping 𝐴∶𝐶→2^𝐻 and the metric projection 𝑃_𝐶 onto a nonempty, closed, and convex subset 𝐶 of 𝐻). To overcome these difficulties, Bregman [1] discovered techniques with the use of Bregman distance function 𝐷_𝑓(⋅,⋅) instead of norm in the process of designing and analyzing feasibility and optimization problems. This opened a growing area of research for solving variational inequalities and approximating solutions or fixed points of nonlinear mappings (see, e.g., [1, 17–21] and the references therein).

The existence and approximation of fixed points of Bregman firmly nonexpansive mappings were studied in [5]. It is also known that if 𝑇 is Bregman firmly nonexpansive and 𝑓 is Legendre function which is bounded, uniformly Frêchet differentiable, and totally convex on bounded subsets of 𝐸, then 𝐹(𝑇)=𝐹(𝑇) and 𝐹(𝑇) is closed and convex (see [5]). It also follows that every Bregman firmly nonexpansive mapping is Bregman relatively nonexpansive and hence quasi-Bregman nonexpansive mapping.

A Bregman projection [1] of 𝑥∈int(dom𝑓) onto the nonempty closed and convex set 𝐶⊂ dom𝑓 is the unique vector 𝑃^𝑓_𝐶(𝑥)∈𝐶 satisfying 𝐷_𝑓𝑃^𝑓_𝐶(𝑥),𝑥=inf𝐷_𝑓𝑦,𝑥∶𝑦∈𝐶. (7)

Remark 1. If 𝐸 is a smooth Banach space and 𝑓(𝑥)=‖𝑥‖² for all 𝑥∈𝐸, then the Bregman projection 𝑃^𝑓_𝐶(𝑥) reduces to the generalized projection Π_𝐶(𝑥) (see, e.g., [2]) which is defined by 𝜙Π_𝐶(𝑥),𝑥=min_𝑦∈𝐶𝜙𝑦,𝑥, (8) where 𝜙(𝑦,𝑥)=‖𝑦‖²−2⟨𝑦,𝐽𝑥⟩+‖𝑥‖².
Very recently, by using Bregman projection, Reich and Sabach [4] introduced an algorithm for finding a common zero of many finitely maximal monotone mappings 𝐴_𝑖∶𝐸→𝐸(𝑖=1,2,…,𝑁) satisfying ℱ∶=⋂^𝑁_𝑖=1𝐴_𝑖⁻¹(0)≠∅ in a reflexive Banach space 𝐸 as follows: 𝑥₀𝑦∈𝐸,chosenarbitrarily,_𝑖𝑛=Res^𝑓_{𝜆𝑖𝑛𝐴𝑖}𝑥_𝑛+𝑒^𝑖_𝑛,𝐶^𝑖_𝑛=𝑧∈𝐸∶𝐷_𝑓𝑧,𝑦^𝑖_𝑛≤𝐷_𝑓𝑧,𝑥_𝑛+𝑒^𝑖_𝑛,𝐶_𝑛∶=^𝑁_𝑖=1𝐶^𝑖_𝑛,𝑄_𝑛=𝑧∈𝐸∶∇𝑓𝑥₀−∇𝑓𝑥_𝑛,𝑧−𝑥_𝑛,𝑥≤0_𝑛+1=𝑃^𝑓_{𝐶𝑛∩𝑄𝑛}𝑥₀,∀𝑛≥0, (9) where Res^𝑓_𝑟𝐴∶=(∇𝑓+𝑟𝐴)⁻¹(∇𝑓). Under suitable conditions, they proved that if, for each 𝑖=1,2,…,𝑁, we have that liminf_𝑛→∞𝜆^𝑖_𝑛>0 and the sequences of errors {𝑒^𝑖_𝑛}⊂𝐸 satisfy liminf_𝑛→∞𝑒^𝑖_𝑛=0, then the sequence {𝑥_𝑛} converges strongly to 𝑃^𝑓_ℱ(𝑥₀).
In [22], Reich and Sabach proposed an algorithm for finding a common fixed point of many finitely Bregman firmly nonexpansive mappings 𝑇_𝑖∶𝐶→𝐶(𝑖=1,2,…,𝑁) satisfying ⋂^𝑁_𝑖=1𝐹(𝑇_𝑖)≠∅ in a reflexive Banach space 𝐸 as follows: 𝑥₀𝑄∈𝐸,chosenarbitrarily,_𝑖0𝑦=𝐸,_𝑖𝑛=𝑇_𝑖𝑥_𝑛+𝑒^𝑖_𝑛,𝑄^𝑖_𝑛+1=𝑧∈𝑄^𝑖_𝑛∶∇𝑓𝑥_𝑛+𝑒^𝑖_𝑛−∇𝑓𝑦^𝑖_𝑛,𝑧−𝑦^𝑖_𝑛,𝑄≤0_𝑛=^𝑁_𝑖=1𝑄^𝑖_𝑛,𝑥_𝑛+1=𝑃^𝑓_𝑄𝑛+1𝑥₀,∀𝑛≥0. (10) They proved that, under suitable conditions, the sequence {𝑥_𝑛} generated by (10), converges strongly to ⋂^𝑁_𝑖=1𝐹(𝑇_𝑖) and applied it to the solution of convex feasibility and equilibrium problems.

Remark 2. But it is worth mentioning that the iteration processes (9) and (10) seem difficult in the sense that, at each stage of iteration, the set(s) 𝐶_𝑛 and/or 𝑄_𝑛 are/is computed and the next iterate is taken as the Bregman projection of 𝑥₀ onto the intersection of 𝐶_𝑛 and/or 𝑄_𝑛. This seems difficult to do in applications.

In this paper, we investigate an iterative scheme for finding a common fixed point of a finite family of quasi-Bregman nonexpansive mappings in reflexive Banach spaces. We prove strong convergence theorems for the sequences produced by the method. Furthermore, we apply our method to prove strong convergence theorems for finding a solution of a finite family of equilibrium problems and for finding a common zero of a finite family of maximal monotone mappings. Our results improve and generalize many known results in the current literature (see, e.g., [4, 23])

2. Preliminaries

Let 𝑓∶𝐸→(−∞,+∞] be a Gâteaux differentiable function. The function 𝑓 is said to be essentially smooth if 𝜕𝑓 is both locally bounded and single-valued on its domain. It is called essentially strictly convex, if (𝜕𝑓)⁻¹ is locally bounded on its domain and is strictly convex on every convex subset of dom𝜕𝑓. 𝑓 is said to be a Legendre, if it is both essentially smooth and essentially strictly convex. When the subdifferential of 𝑓 is single-valued, it coincides with the gradient 𝜕𝑓=∇𝑓 (see [24]).

We note that if 𝐸 is a reflexive Banach space, then we have the following.(i)𝑓 is essentially smooth if and only if 𝑓^∗ is essentially strictly convex (see [25, Theorem 5.4]).(ii)(𝜕𝑓)⁻¹=𝜕𝑓^∗ (see [26]).(iii)𝑓 is Legendre if and only if 𝑓^∗ is Legendre (see [25, Corollary 5.5]).(iv)If 𝑓 is Legendre, then ∇𝑓 is a bijection satisfying ∇𝑓=(∇𝑓^∗)⁻¹, ran∇𝑓=dom∇𝑓^∗=intdom𝑓^∗, and ran∇𝑓^∗=dom∇𝑓=intdom𝑓 (see [25, Theorem 5.10]).When 𝐸 is a smooth and strictly convex Banach space, one important and interesting example of Legendre function is 𝑓(𝑥)∶=(1/𝑝)‖𝑥‖^𝑝(1<𝑝<∞). In this case the gradient ∇𝑓 of 𝑓 coincides with the generalized duality mapping of 𝐸; that is, ∇𝑓=𝐽_𝑝(1<𝑝<∞). In particular, ∇𝑓=𝐼, the identity mapping in Hilbert spaces.

Let 𝐸 be a Banach space and let 𝐵_𝑟∶={𝑧∈𝐸∶‖𝑧‖≤𝑟}, for all 𝑟>0 and 𝑆_𝐸={𝑥∈𝐸∶‖𝑥‖=1}. Then a function 𝑓∶𝐸→ℝ is said to be uniformly convex on bounded subsets of 𝐸 [27, page 203] if 𝜌_𝑟(𝑡)>0, for all 𝑟,𝑡>0, where 𝜌_𝑟∶[0,∞)→[0,∞] is defined by 𝜌_𝑟(𝑡)∶=inf_{𝑥,𝑦∈𝐵𝑟,‖𝑥−𝑦‖=𝑡,𝛼∈(0,1)}(𝑥)+()𝑓𝑦()𝑦×(𝛼(𝛼𝑓1−𝛼−𝑓𝛼𝑥+1−𝛼1−𝛼))⁻¹, (11) for all 𝑡≥0. The function 𝜌_𝑟 is called the gauge of uniform convexity of 𝑓. The function 𝑓 is also said to be uniformly smooth on bounded subsets of 𝐸 [27, page 207] if lim_𝑡→0⁺(𝜎_𝑟(𝑡)/𝑡)=0, for all 𝑟>0, where 𝜎_𝑟∶[0,∞)→[0,∞] is defined by 𝜎_𝑟(𝑡)∶=sup_{𝑥∈𝐵𝑟,𝑦∈𝑆𝐸,𝛼∈(0,1)}()+()𝑓(𝑥×(𝛼(𝛼𝑓𝑥+1−𝛼𝑡𝑦1−𝛼𝑥−𝛼𝑡𝑦−𝑓)1−𝛼))⁻¹, (12) for all 𝑡≥0.

In the sequel, we will need the following lemmas.

Lemma 3 (see [28]). Let 𝐸 be a Banach space, let 𝑟>0 be a constant, and let 𝑓∶𝐸→ℝ be a uniformly convex on bounded subsets of 𝐸. Then 𝑓^𝑛_𝑘=0𝛼_𝑘𝑥_𝑘≤^𝑛_𝑘=0𝛼_𝑘𝑓𝑥_𝑘−𝛼_𝑖𝛼_𝑗𝜌_𝑟𝑥_𝑖−𝑦_𝑗, (13) for all 𝑖,𝑗∈{0,1,2,…,𝑛},  𝑥_𝑘∈𝐵𝑟,𝛼_𝑘∈(0,1), and 𝑘=0,1,2,…,𝑛 with ∑^𝑛_𝑘=0𝛼_𝑘=1, where 𝜌_𝑟 is the gauge of uniform convexity of 𝑓.

Lemma 4 (see [24]). Let 𝑓∶𝐸→(−∞,+∞] be a proper, lower semicontinuous, and convex function; then 𝑓^∗∶𝐸^∗→(−∞,+∞] is a proper, weak^∗ lower semicontinuous, and convex function. Thus, for all 𝑧∈𝐸, we have 𝐷_𝑓𝑧,∇𝑓^∗𝑁_𝑖=1𝑡_𝑖∇𝑓𝑥_𝑖≤^𝑁_𝑖=1𝑡_𝑖𝐷_𝑓𝑧,𝑥_𝑖. (14)

Lemma 5 (see [29]). Let 𝑓∶𝐸→ℝ be a Gâteaux differentiable on int(dom𝑓) such that ∇𝑓^∗ is bounded on bounded subsets of dom𝑓^∗. Let 𝑥^∗∈𝑋 and {𝑥_𝑛}⊂𝐸. If {𝐷_𝑓(𝑥,𝑥_𝑛)} is bounded, so is the sequence {𝑥_𝑛}.

A function 𝑓 on 𝐸 is coercive [30] if the sublevel set of 𝑓 is bounded; equivalently, lim_{‖𝑥‖→∞}𝑓(𝑥)=∞. A function 𝑓 on 𝐸 is said to be strongly coercive [27] if lim_{‖𝑥‖→∞}𝑓(𝑥)/‖𝑥‖=∞.

Lemma 6 (see [27]). Let 𝐸 be a reflexive Banach space and let 𝑓∶𝐸→ℝ be a continuous convex function which is strongly coercive. Then the following assertions are equivalent.(1)𝑓 is bounded on bounded subsets and uniformly smooth on bounded subsets of 𝐸;(2)𝑓^∗ is Fréchet differentiable and ∇𝑓^∗ is uniformly norm-to-norm continuous on bounded subsets of 𝐸^∗;(3)dom𝑓=𝐸^∗, 𝑓^∗ is strongly coercive and uniformly convex on bounded subsets of 𝐸^∗.

Lemma 7 (see [5]). Let 𝑓∶𝐸→(−∞,+∞] be a Legendre function. Let 𝐶 be a nonempty closed convex subset of int(dom𝑓) and let 𝑇∶𝐶→𝐶 be a quasi-Bregman nonexpansive mapping. Then 𝐹(𝑇) is closed and convex.

Let 𝑓∶𝐸→(−∞,+∞] be a Gâteaux differentiable function. The modulus of total convexity of 𝑓 at 𝑥∈ dom𝑓 is the function 𝜈_𝑓(𝑥,⋅)∶[0,+∞)→[0,+∞] defined by 𝜈_𝑓(𝑥,𝑡)∶=inf𝐷_𝑓𝑦,𝑥∶𝑦∈dom𝑓,𝑦−𝑥=𝑡. (15) The function 𝑓 is called totally convex at 𝑥 if 𝜈_𝑓(𝑥,𝑡)>0, whenever 𝑡>0. The function 𝑓 is called totally convex if it is totally convex at any point 𝑥∈int(dom𝑓) and is said to be totally convex on bounded sets if 𝜈_𝑓(𝐵,𝑡)>0 for any nonempty bounded subset 𝐵 of 𝐸 and 𝑡>0, where the modulus of total convexity of the function 𝑓 on the set 𝐵 is the function 𝜈_𝑓∶intdom𝑓×[0,+∞)→[0,+∞] defined by 𝜈_𝑓(𝐵,𝑡)∶=inf𝜈_𝑓(𝑥,𝑡)∶𝑥∈𝐵∩dom𝑓. (16) We know that 𝑓 is totally convex on bounded sets if and only if 𝑓 is uniformly convex on bounded sets (see [18, Theorem 2.10]). The following lemmas will be useful in the proof of our main results.

Lemma 8 (see [31]). The function 𝑓∶𝐸→(−∞,+∞) is totally convex on bounded subsets of 𝐸 if and only if, for any two sequences {𝑥_𝑛} and {𝑦_𝑛} in int(dom𝑓) and dom𝑓, respectively, such that the first one is bounded, lim_𝑛→∞𝐷_𝑓𝑦_𝑛,𝑥_𝑛=0⟹lim_𝑛→∞𝑦_𝑛−𝑥_𝑛=0. (17)

Lemma 9 (see [27, 32]). Let 𝑓∶𝐸→ℝ be a strongly coercive and uniformly convex on bounded subsets of 𝐸; then 𝑓^∗ is bounded on bounded sets and uniformly Fréchet differentiable on bounded subsets of 𝐸^∗.

Lemma 10 (see [33]). Let 𝑓∶𝐸→(−∞,+∞] be uniformly Fréchet differentiable and bounded on bounded sets of 𝐸; then ∇𝑓 is uniformly continuous on bounded subsets of 𝐸 from the strong topology of 𝐸 to the strong topology of 𝐸^∗.

Lemma 11 (see [18]). Let 𝐶 be a nonempty, closed, and convex subset of 𝐸. Let 𝑓∶𝐸→ℝ be a Gâteaux differentiable and totally convex function and let 𝑥∈𝐸; then (i)𝑧=𝑃^𝑓_𝐶(𝑥) if and only if ⟨∇𝑓(𝑥)−∇𝑓(𝑧),𝑦−𝑧⟩≤0,∀𝑦∈𝐶.(ii)𝐷_𝑓(𝑦,𝑃^𝑓_𝐶(𝑥))+𝐷_𝑓(𝑃^𝑓_𝐶(𝑥),𝑥)≤𝐷_𝑓(𝑦,𝑥),∀𝑦∈𝐶.

Let 𝑓∶𝐸→ℝ be a Legendre and Gâteaux differentiable function. Following [2, 34], we make use of the function 𝑉_𝑓∶𝐸×𝐸^∗→[0,+∞) associated with 𝑓, which is defined by 𝑉_𝑓𝑥,𝑥^∗=𝑓(𝑥)−𝑥,𝑥^∗+𝑓^∗𝑥^∗,∀𝑥∈𝐸,𝑥^∗∈𝐸^∗. (18) Then 𝑉_𝑓 is nonnegative and 𝑉_𝑓𝑥,𝑥^∗=𝐷_𝑓𝑥,∇𝑓^∗𝑥^∗∀𝑥∈𝐸,𝑥^∗∈𝐸^∗. (19) Moreover, by the subdifferential inequality, 𝑉_𝑓𝑥,𝑥^∗+𝑦^∗,∇𝑓^∗𝑥^∗−𝑥≤𝑉_𝑓𝑥,𝑥^∗+𝑦^∗, (20) where ∀𝑥∈𝐸 and 𝑥^∗,𝑦^∗∈𝐸^∗ (see [35]).

Lemma 12 (see [36]). Let {𝑎_𝑛} be a sequence of nonnegative real numbers satisfying the following relation: 𝑎_𝑛+1≤1−𝛼_𝑛𝑎_𝑛+𝛼_𝑛𝛿_𝑛,𝑛≥𝑛₀, (21) where {𝛼_𝑛}⊂(0,1) and {𝛿_𝑛}⊂ℝ satisfying the following conditions: lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=1𝛼_𝑛=∞ and limsup_𝑛→∞𝛿_𝑛≤0. Then, lim_𝑛→∞𝑎_𝑛=0.

Lemma 13 (see [37]). Let {𝑎_𝑛} be sequences of real numbers such that there exists a subsequence {𝑛_𝑖} of {𝑛} such that 𝑎_𝑛𝑖<𝑎_𝑛𝑖+1 for all 𝑖∈ℕ. Then there exists an increasing sequence {𝑚_𝑘}⊂ℕ such that 𝑚_𝑘→∞ and the following properties are satisfied by all (sufficiently large) numbers 𝑘∈ℕ: 𝑎_𝑚𝑘≤𝑎_𝑚𝑘+1,𝑎_𝑘≤𝑎_𝑚𝑘+1. (22) In fact, 𝑚_𝑘 is the largest number 𝑛 in the set {1,2,…,𝑘} such that the condition 𝑎_𝑛≤𝑎_𝑛+1 holds.

3. Main Results

We now prove the following theorem.

Theorem 14. Let 𝑓∶𝐸→ℝ be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of 𝐸. Let 𝐶 be a nonempty, closed, and convex subset of int(dom𝑓) and let 𝑇_𝑖∶𝐶→𝐶, for 𝑖=1,2,…,𝑁, be a finite family of quasi-Bregman nonexpansive mappings such that 𝐹(𝑇_𝑖)=𝐹(𝑇_𝑖), for 𝑖=1,2,…,𝑁. Assume that ℱ∶=⋂^𝑁_𝑖=1𝐹(𝑇_𝑖) is nonempty. For 𝑢,𝑥₀∈𝐶 let {𝑥_𝑛} be a sequence generated by 𝑦_𝑛=𝑃^𝑓_𝐶∇𝑓^∗𝛼_𝑛∇𝑓(𝑢)+1−𝛼_𝑛∇𝑓𝑥_𝑛,𝑥_𝑛+1=∇𝑓^∗𝛽₀∇𝑓𝑦_𝑛+^𝑁_𝑖=1𝛽_𝑖∇𝑓𝑇_𝑖𝑦_𝑛,∀𝑛≥0, (23) where {𝛼_𝑛}⊂(0,1) and {𝛽_𝑖}^𝑁_𝑖=0⊂[𝑐,𝑑]⊂(0,1) satisfying lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=1𝛼_𝑛=∞, and ∑^𝑁_𝑖=0𝛽_𝑖=1. Then, {𝑥_𝑛} converges strongly to 𝑝=𝑃^𝑓_ℱ(𝑢).

Proof. Lemma 7 ensures that each 𝐹(𝑇_𝑖), for 𝑖∈{1,2,…,𝑁}, and ℱ are closed and convex. Thus, 𝑃^𝑓_ℱ is well defined. Let 𝑝=𝑃^𝑓_ℱ(𝑢). Then, from (23), Lemmas 11 and 4, and property of 𝐷_𝑓 we get that 𝐷_𝑓𝑝,𝑦_𝑛=𝐷_𝑓𝑝,𝑃^𝑓_𝐶∇𝑓^∗𝛼_𝑛∇𝑓(𝑢)+1−𝛼_𝑛∇𝑓𝑥_𝑛≤𝐷_𝑓𝑝,∇𝑓^∗𝛼_𝑛∇𝑓(𝑢)+1−𝛼_𝑛∇𝑓𝑥_𝑛=𝛼_𝑛𝐷_𝑓𝑝,𝑢+1−𝛼_𝑛𝐷_𝑓𝑝,𝑥_𝑛. (24) Moreover, from (23), (18), and (19) we get that 𝐷_𝑓𝑝,𝑥_𝑛+1=𝐷_𝑓𝑝,∇𝑓^∗𝛽₀∇𝑓𝑦_𝑛+^𝑁_𝑖=1𝛽_𝑖∇𝑓𝑇_𝑖𝑦_𝑛=𝑉𝑝,𝛽₀∇𝑓𝑦_𝑛+^𝑁_𝑖=1𝛽_𝑖∇𝑓𝑇_𝑖𝑦_𝑛=𝑓𝑝−𝑝,𝛽₀∇𝑓𝑦_𝑛+^𝑁_𝑖=1𝛽_𝑖∇𝑓𝑇_𝑖𝑦_𝑛+𝑓^∗𝛽₀∇𝑓𝑦_𝑛+^𝑁_𝑖=1𝛽_𝑖∇𝑓𝑇_𝑖𝑦_𝑛. (25) Since 𝑓 is uniformly Fréchet differentiable function we have that 𝑓 is uniformly smooth and hence by Theorem 3.5.5 of [27] we get that 𝑓^∗ is uniformly convex. This, with Lemmas 3 and (24), gives that 𝐷_𝑓𝑝,𝑥_𝑛+1≤𝑓𝑝−𝛽₀𝑝,∇𝑓𝑦_𝑛−^𝑁_𝑖=1𝛽_𝑖𝑝,∇𝑓𝑇_𝑖𝑦_𝑛+𝛽₀𝑓^∗∇𝑓𝑦_𝑛+^𝑁_𝑖=1𝛽_𝑖𝑓^∗∇𝑓𝑇_𝑖𝑦_𝑛−𝛽₀𝛽_𝑖𝜌^∗_𝑟∇𝑓𝑦_𝑛−∇𝑓𝑇_𝑖𝑥_𝑛=𝛽₀𝑉𝑝,∇𝑓𝑦_𝑛+^𝑁_𝑖=1𝛽_𝑖𝑉𝑝,∇𝑓𝑇𝑦_𝑛−𝛽₀𝛽_𝑖𝜌^∗_𝑟∇𝑓𝑦_𝑛−∇𝑓𝑇_𝑖𝑥_𝑛=𝛽₀𝐷_𝑓𝑝,𝑦_𝑛+^𝑁_𝑖=1𝛽_𝑖𝐷_𝑓𝑝,𝑇𝑦_𝑛−𝛽₀𝛽_𝑖𝜌^∗_𝑟∇𝑓𝑦_𝑛−∇𝑓𝑇_𝑖𝑥_𝑛≤𝛽₀𝐷_𝑓𝑝,𝑦_𝑛+^𝑁_𝑖=1𝛽_𝑖𝐷_𝑓𝑝,𝑦_𝑛−𝛽₀𝛽_𝑖𝜌^∗_𝑟∇𝑓𝑦_𝑛−∇𝑓𝑇_𝑖𝑥_𝑛≤𝐷_𝑓𝑝,𝑦_𝑛−𝛽₀𝛽_𝑖𝜌^∗_𝑟∇𝑓𝑦_𝑛−∇𝑓𝑇_𝑖𝑦_𝑛≤𝐷_𝑓𝑝,𝑦_𝑛,≤𝛼_𝑛𝐷_𝑓𝑝,𝑢+1−𝛼_𝑛𝐷_𝑓𝑝,𝑥_𝑛, (26)(27) for each 𝑖∈{1,2,…,𝑁}. Thus, by induction, 𝐷_𝑓𝑝,𝑥_𝑛+1≤max𝐷_𝑓𝑝,𝑢,𝐷_𝑓𝑝,𝑥₀,∀𝑛≥0, (28) which implies that {𝑥_𝑛} is bounded. Now, let 𝑧_𝑛=∇𝑓^∗(𝛼_𝑛∇𝑓(𝑢)+(1−𝛼_𝑛)∇𝑓(𝑥_𝑛)). Then we note that 𝑦_𝑛=𝑃^𝑓_𝐶𝑧_𝑛. Since 𝑓 is strongly coercive, uniformly convex, uniformly Fréchet differentiable, and bounded on bounded subsets of 𝐸 by Lemmas 10, 9, and 6 we have that ∇𝑓 and ∇𝑓^∗ are bounded on bounded sets and hence {𝑧_𝑛} and {𝑦_𝑛} are bounded. In addition, using (19), (20), and property of 𝐷_𝑓 we obtain that 𝐷_𝑓𝑝,𝑦_𝑛≤𝐷_𝑓𝑝,𝑧_𝑛=𝑉𝑝,∇𝑓𝑧_𝑛≤𝑉𝑝,∇𝑓𝑧_𝑛−𝛼_𝑛∇𝑓(𝑢)−∇𝑓𝑝+𝛼_𝑛∇𝑓(𝑢)−∇𝑓𝑝,𝑧_𝑛−𝑝=𝐷_𝑓𝑝,∇𝑓^∗𝛼_𝑛∇𝑓𝑝+1−𝛼_𝑛∇𝑓𝑥_𝑛+𝛼_𝑛∇𝑓(𝑢)−∇𝑓𝑝,𝑧_𝑛−𝑝≤𝛼_𝑛𝐷_𝑓𝑝,𝑝+1−𝛼_𝑛𝐷_𝑓𝑝,𝑥_𝑛+𝛼_𝑛∇𝑓(𝑢)−∇𝑓𝑝,𝑧_𝑛=−𝑝1−𝛼_𝑛𝐷_𝑓𝑝,𝑥_𝑛+𝛼_𝑛∇𝑓(𝑢)−∇𝑓𝑝,𝑧_𝑛−𝑝. (29) Furthermore, from (26) and (29) we have that 𝐷_𝑓𝑝,𝑥_𝑛+1≤1−𝛼_𝑛𝐷_𝑓𝑝,𝑥_𝑛+𝛼_𝑛∇𝑓(𝑢)−∇𝑓𝑝,𝑧_𝑛−𝑝−𝛽₀𝛽_𝑖𝜌^∗_𝑟∇𝑓𝑦_𝑛−∇𝑓𝑇_𝑖𝑦_𝑛≤1−𝛼_𝑛𝐷_𝑓𝑝,𝑥_𝑛+𝛼_𝑛∇𝑓(𝑢)−∇𝑓𝑝,𝑧_𝑛−𝑝. (30)(31) Now, we consider two cases.
Case 1. Suppose that there exists 𝑛₀∈ℕ such that {𝐷_𝑓(𝑝,𝑥_𝑛)} is nonincreasing for all 𝑛≥𝑛₀. In this situation, {𝐷_𝑓(𝑝,𝑥_𝑛)} is convergent. Then, from (30) we have that 𝛽₀𝛽_𝑖𝜌^∗_𝑟∇𝑓𝑦_𝑛−∇𝑓𝑇_𝑖𝑦_𝑛⟶0, (32) which implies, by the property of 𝜌^∗_𝑟, that ∇𝑓𝑦_𝑛−∇𝑓𝑇_𝑖𝑦_𝑛⟶0,as𝑛⟶∞. (33) Now, since 𝑓 is a strongly coercive and uniformly convex on bounded subsets of 𝐸, 𝑓^∗ is uniformly Fréchet differentiable on bounded subsets of 𝐸^∗ (see [27, Proposition 3.6.2]) and 𝑓 is Legendre; we have by Lemma 10 that 𝑦_𝑛−𝑇_𝑖𝑦_𝑛⟶0,as𝑛⟶∞, (34) for each 𝑖∈{1,2,…,𝑁}.
Furthermore, Lemma 11, property of 𝐷_𝑓, and the fact that 𝛼_𝑛→0, as 𝑛→∞, imply that 𝐷_𝑓𝑥_𝑛,𝑦_𝑛=𝐷_𝑓𝑥_𝑛,𝑃^𝑓_𝐶𝑧_𝑛≤𝐷_𝑓𝑥_𝑛,𝑧_𝑛=𝐷_𝑓𝑥_𝑛,∇𝑓^∗𝛼_𝑛∇𝑓(𝑢)+1−𝛼_𝑛∇𝑓𝑥_𝑛≤𝛼_𝑛𝐷_𝑓𝑥_𝑛,𝑢+1−𝛼_𝑛𝐷_𝑓𝑥_𝑛,𝑥_𝑛≤𝛼_𝑛𝐷_𝑓𝑥_𝑛+,𝑢1−𝛼_𝑛𝐷_𝑓𝑥_𝑛,𝑥_𝑛⟶0as𝑛⟶∞, (35) and hence by Lemma 8 we get that 𝑥_𝑛−𝑦_𝑛⟶0,𝑥_𝑛−𝑧_𝑛⟶0,𝑛⟶∞. (36) Since {𝑧_𝑛} is bounded and 𝐸 is reflexive, we choose a subsequence {𝑧_𝑛𝑗} of {𝑧_𝑛} such that 𝑧_𝑛𝑗⇀𝑧 and limsup_𝑛→∞⟨∇𝑓(𝑢)−∇𝑓(𝑝),𝑧_𝑛−𝑝⟩=lim_𝑗→∞⟨∇𝑓(𝑢)−∇𝑓(𝑝),𝑧_𝑛𝑗−𝑝⟩. Then, from (36) we get that 𝑦_𝑛𝑗⇀𝑧,as𝑗⟶∞. (37) Thus, from (34) and the fact that 𝐹(𝑇_𝑖)=𝐹(𝑇_𝑖) we obtain that 𝑧∈𝐹(𝑇_𝑖), for each 𝑖∈{1,2,…,𝑁} and hence 𝑧∈⋂^𝑁_𝑖=1𝐹(𝑇_𝑖).
Therefore, by Lemma 11, we immediately obtain that limsup_𝑛→∞⟨∇𝑓(𝑢)−∇𝑓(𝑝),𝑧_𝑛−𝑝⟩=lim_𝑗→∞⟨∇𝑓(𝑢)−∇𝑓(𝑝),𝑧_𝑛𝑗−𝑝⟩=⟨∇𝑓(𝑢)−∇𝑓(𝑝),𝑧−𝑝⟩≤0. It follows from Lemma 12 and (31) that 𝐷_𝑓(𝑝,𝑥_𝑛)→0, as 𝑛→∞. Consequently, by Lemma 8 we obtain that 𝑥_𝑛→𝑝.
Case 2. Suppose that there exists a subsequence {𝑛_𝑗} of {𝑛} such that 𝐷_𝑓𝑝,𝑥_𝑛𝑗<𝐷_𝑓𝑝,𝑥_𝑛𝑗+1, (38) for all 𝑗∈ℕ. Then, by Lemma 13, there exists a nondecreasing sequence {𝑚_𝑘}⊂ℕ such that 𝑚_𝑘→∞, 𝐷_𝑓(𝑝,𝑥_𝑚𝑘)≤𝐷_𝑓(𝑝,𝑥_𝑚𝑘+1) and 𝐷_𝑓(𝑝,𝑥_𝑘)≤𝐷_𝑓(𝑝,𝑥_𝑚𝑘+1), for all 𝑘∈ℕ. Then from (30) and the fact that 𝛼_𝑛→0 we obtain that 𝜌^∗_𝑟∇𝑓𝑦_𝑚𝑘−∇𝑓𝑇_𝑖𝑦_𝑚𝑘⟶0,as𝑘⟶∞, (39) for each 𝑖∈{1,2,…,𝑁}. Thus, following the method of proof of Case 1, we obtain that 𝑦_𝑚𝑘−𝑇_𝑖𝑦_𝑚𝑘→0, 𝑥_𝑚𝑘−𝑦_𝑚𝑘→0, 𝑥_𝑚𝑘−𝑧_𝑚𝑘→0, as 𝑘→∞, and hence we obtain that limsup_𝑘→∞∇𝑓(𝑢)−∇𝑓𝑝,𝑧_𝑚𝑘−𝑝≤0. (40) Then from (31) we have that 𝐷_𝑓𝑝,𝑥_𝑚𝑘+1≤1−𝛼_𝑚𝑘𝐷_𝑓𝑝,𝑥_𝑚𝑘+𝛼_𝑚𝑘∇𝑓(𝑢)−∇𝑓𝑝,𝑧_𝑚𝑘−𝑝. (41) Now, since 𝐷_𝑓(𝑝,𝑥_𝑚𝑘)≤𝐷_𝑓(𝑝,𝑥_𝑚𝑘+1), inequality (41) implies that 𝛼_𝑚𝑘𝐷_𝑓𝑝,𝑥_𝑚𝑘≤𝐷_𝑓𝑝,𝑥_𝑚𝑘−𝐷_𝑓𝑝,𝑥_𝑚𝑘+1+𝛼_𝑚𝑘∇𝑓(𝑢)−∇𝑓𝑝,𝑧_𝑚𝑘−𝑝≤𝛼_𝑚𝑘∇𝑓(𝑢)−∇𝑓𝑝,𝑧_𝑚𝑘−𝑝. (42) In particular, since 𝛼_𝑚𝑘>0, we get 𝐷_𝑓𝑝,𝑥_𝑚𝑘≤∇𝑓(𝑢)−∇𝑓𝑝,𝑧_𝑚𝑘−𝑝. (43) Then, from (40) we obtain 𝐷_𝑓(𝑝,𝑥_𝑚𝑘)→0, as 𝑘→∞. This together with (41) gives 𝐷_𝑓(𝑝,𝑥_𝑚𝑘+1)→0, as 𝑘→∞. But 𝐷_𝑓(𝑝,𝑥_𝑘)≤𝐷_𝑓(𝑝,𝑥_𝑚𝑘+1), for all 𝑘∈𝑁; thus we obtain that 𝑥_𝑘→𝑝. Therefore, from the above two cases, we can conclude that {𝑥_𝑛} converges strongly to 𝑝=𝑃^𝑓_ℱ(𝑢) and the proof is complete.

If, in Theorem 14, we consider a single quasi-Bregman nonexpansive mapping, we get the following corollary.

Corollary 15. Let 𝑓∶𝐸→ℝ be a coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of 𝐸. Let 𝐶 be a nonempty, closed, and convex subset of int(dom𝑓) and let 𝑇∶𝐶→𝐶 be a quasi-Bregman nonexpansive mapping such that 𝐹(𝑇)=𝐹(𝑇)≠∅. Let {𝑥_𝑛} be a sequence generated by 𝑦_𝑛=𝑃^𝑓_𝐶∇𝑓^∗𝛼_𝑛∇𝑓(𝑢)+1−𝛼_𝑛∇𝑓𝑥_𝑛,𝑥_𝑛+1=∇𝑓^∗𝛽∇𝑓𝑦_𝑛+1−𝛽∇𝑓𝑇𝑦_𝑛,∀𝑛≥0, (44) where {𝛼_𝑛}⊂(0,1) and 𝛽∈(0,1) satisfying lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1=∞. Then {𝑥_𝑛} converges strongly to 𝑝=𝑃^𝑓_𝐹(𝑇)(𝑢).

If, in Theorem 14, we assume that each 𝑇_𝑖, for 𝑖∈{1,2,…,𝑁}, is Bregman relatively nonexpansive mapping, then we have that 𝐹(𝑇_𝑖)=𝐹(𝑇_𝑖) for each 𝑖∈{1,2,…,𝑁}. Thus, we get the following corollary.

Corollary 16. Let 𝑓∶𝐸→ℝ be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of 𝐸. Let 𝐶 be a nonempty, closed, and convex subset of int(dom𝑓) and let 𝑇_𝑖∶𝐶→𝐶, for 𝑖=1,2,…,𝑁, be a finite family of Bregman relatively nonexpansive mappings. Assume that ℱ∶=⋂^𝑁_𝑖=1𝐹(𝑇_𝑖) is nonempty. For 𝑢,𝑥₀∈𝐶 let {𝑥_𝑛} be a sequence generated by 𝑦_𝑛=𝑃^𝑓_𝐶∇𝑓^∗𝛼_𝑛∇𝑓(𝑢)+1−𝛼_𝑛∇𝑓𝑥_𝑛,𝑥_𝑛+1=∇𝑓^∗𝛽₀∇𝑓𝑦_𝑛+^𝑁_𝑖=1𝛽_𝑖∇𝑓𝑇_𝑖𝑦_𝑛,∀𝑛≥0, (45) where {𝛼_𝑛}⊂(0,1) and {𝛽_𝑖}^𝑁_𝑖=0⊂[𝑐,𝑑]⊂(0,1) satisfying lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=1𝛼_𝑛=∞, and ∑^𝑁_𝑖=0𝛽_𝑖=1. Then, {𝑥_𝑛} converges strongly to 𝑝=𝑃^𝑓_ℱ(𝑢).

If, in Theorem 14, we assume that each 𝑇_𝑖(𝑖=1,2,…,𝑁) is Bregman firmly nonexpansive, then we get the following corollary.

Corollary 17. Let 𝑓∶𝐸→ℝ be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of 𝐸. Let 𝐶 be a nonempty, closed, and convex subset of int(dom𝑓) and let 𝑇_𝑖∶𝐶→𝐶, for 𝑖=1,2,…,𝑁, be a finite family of Bregman firmly nonexpansive mappings with ⋂^𝑁_𝑖=1𝐹(𝑇_𝑖)≠∅. Let {𝑥_𝑛} be a sequence generated by 𝑦_𝑛=𝑃^𝑓_𝐶∇𝑓^∗𝛼_𝑛∇𝑓(𝑢)+1−𝛼_𝑛∇𝑓𝑥_𝑛,𝑥_𝑛+1=∇𝑓^∗𝛽₀∇𝑓𝑦_𝑛+^𝑁_𝑖=1𝛽_𝑖∇𝑓𝑇_𝑖𝑦_𝑛,∀𝑛≥0, (46) where {𝛼_𝑛}⊂(0,1) and {𝛽_𝑖}^𝑁_𝑖=0⊂[𝑐,𝑑]⊂(0,1) satisfying lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=1𝛼_𝑛=∞, and ∑^𝑁_𝑖=0𝛽_𝑖=1. Then {𝑥_𝑛} converges strongly to 𝑝=𝑃^𝑓_ℱ(𝑢).

4. Applications

4.1. Equilibrium Problems

Let 𝑓∶𝐸→(−∞,+∞] be a coercive Legendre function. Let 𝐶 be a nonempty, closed, and convex subset of int(dom𝑓). Let 𝐻∶𝐶×𝐶→ℝ be a bifunction with the following conditions (see example [38]).(B1)𝐻(𝑥,𝑥)=0, for all 𝑥∈𝐶;(B2)𝐻 is monotone; that is, 𝐻(𝑥,𝑦)+𝐻(𝑦,𝑥)≤0, for all 𝑥,𝑦∈𝐶;(B3)for all 𝑥,𝑦,𝑧∈𝐶, limsup_𝑡→0+𝐻(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝐻(𝑥,𝑦);(B4)for each 𝑥∈𝐶, 𝐻(𝑥,⋅) is convex and lower semicontinuous.The equilibrium problem corresponding to 𝐻 is to find 𝑥^∗∈𝐶 such that 𝐻𝑥^∗,𝑦≥0,∀𝑦∈𝐶. (47) The set of solutions of (47) is denoted by EP(𝐻). Equilibrium problem is a unified model of several problems, namely, variational inequality problem, complementary problem, saddle point problem, optimization problem, fixed point problem, and so forth; see [20, 30, 38–43].

The resolvent of a bifunction 𝐻∶𝐶×𝐶→ℝ [39] is the operator Res^𝑓_𝐻∶𝐸→2^𝐶, defined by Res^𝑓_𝐻(𝑥)=+(𝑧)(𝑥).𝑧∈𝐶∶𝐻𝑧,𝑦∇𝑓−∇𝑓,𝑦−𝑧≥0,∀𝑦∈𝐶 (48) We know the following lemma in [23].

Lemma 18. Let 𝑓∶𝐸→(−∞,+∞] be a coercive Legendre function. Let 𝐶 be a nonempty, closed, and convex subset of int(dom𝑓). If the bifunction 𝐻∶𝐶×𝐶→ℝ satisfies conditions (B1)–(B4), then(1)Res^𝑓_𝐻 is single-valued and dom(Res^𝑓_𝑔))=𝐸;(2)Res^𝑓_𝐻 is a Bregman firmly nonexpansive mapping;(3)𝐸𝑃(𝐻) is a closed and convex subset of 𝐶 and 𝐸𝑃(𝐻)=𝐹(Res^𝑓_𝐻);(4)For all 𝑥∈𝐸 and for all 𝑢∈𝐹(Res^𝑓_𝐻), we have 𝐷_𝑓𝑢,Res^𝑓_𝐻(𝑥)+𝐷_𝑓Res^𝑓_𝐻(𝑥),𝑥≤𝐷_𝑓(𝑢,𝑥). (49)

In addition, if 𝑓 is uniformly Fréchet differentiable and bounded on bounded subsets of 𝐸, then by Reich and Sabach [5] we have that 𝐹(Res^𝑓_𝐻)=𝐹(Res^𝑓_𝐻). Thus, considering 𝑇_𝑖=Res^𝑓_𝐻𝑖, for 𝑖=1,2,…,𝑁, in Corollary 17, we get the following result.

Theorem 19. Let 𝑓∶𝐸→ℝ be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of 𝐸. Let 𝐶 be a nonempty, closed, and convex subset of int(dom𝑓) and let 𝐻_𝑖∶𝐶→𝐶, for 𝑖=1,2,…,𝑁, be bifunctions which satisfy the conditions (B1)–(B4) and ℱ∶=⋂^𝑁_𝑖=1𝐸𝑃(𝐻_𝑖) is nonempty. For 𝑥₀∈𝐶 let {𝑥_𝑛} be a sequence generated by 𝑦_𝑛=∇𝑓^∗𝛼_𝑛∇𝑓(𝑢)+1−𝛼_𝑛∇𝑓𝑥_𝑛,𝑥_𝑛+1=∇𝑓^∗𝛽₀∇𝑓𝑦_𝑛+^𝑁_𝑖=1𝛽_𝑖∇𝑓𝑇_𝑖𝑦_𝑛,∀𝑛≥0, (50) where 𝑇_𝑖=Res^𝑓_𝐻𝑖, {𝛼_𝑛}⊂(0,1), and {𝛽_𝑖}^𝑁_𝑖=0⊂[𝑐,𝑑]⊂(0,1) satisfy lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=1𝛼_𝑛=∞, and ∑^𝑁_𝑖=0𝛽_𝑖=1. Then, {𝑥_𝑛} converges strongly to 𝑝=𝑃^𝑓_ℱ(𝑢).

4.2. Zeroes of Maximal Monotone Operators

In this section we present an algorithm for finding a common zero of a finite family of maximal monotone mappings.

Let 𝐴∶𝐸→2^𝐸∗ be a mapping with range 𝑅(𝐴)={𝑥^∗∈𝐸^∗∶𝑥^∗∈𝐴𝑥} and domain 𝐷(𝐴)={𝑥∈𝐸∶𝐴𝑥≠∅}. Then, 𝐴 is said to be monotone if, for any 𝑥,𝑦∈dom(𝐴), we have 𝑥^∗∈𝐴𝑥,𝑦^∗∈𝐴𝑦⟹𝑥−𝑦,𝑥^∗−𝑦^∗≥0. (51) It is said to be maximal monotone if 𝐴 is monotone and the graph of 𝐴 is not a proper subset of the graph of any other monotone mappings, where graph of 𝐴 is given by 𝐺(𝐴)∶={(𝑥,𝑥^∗)∈𝐸×𝐸^∗∶𝑥^∗∈𝐴𝑥}. It is known that if 𝐴 is maximal monotone, then the set 𝐴⁻¹(0)={𝑧∈𝐸∶0∈𝐴𝑧} is closed and convex.

Let 𝐴∶𝐶⊆𝐸→2^𝐸∗ be maximal monotone mapping and 𝑓 is Legendre function which is bounded, uniformly Fréchet differentiable on bounded subsets of 𝐸. The resolvent of 𝐴 with respect to 𝑓 is the mapping Res^𝑓_𝐴∶𝐸→2^𝐸 defined by Res^𝑓_𝐴=∇𝑓+𝐴⁻¹∘∇𝑓. (52) It is known that Res^𝑓_𝐴 is single-valued, Bregman firmly nonexpansive and 𝐹(Res^𝑓_𝐴)=𝐴⁻¹(0) (see [17]). Furthermore, the result by Reich and Sabach [5] implies that 𝐹(Res^𝑓_𝐴)=𝐹(Res^𝑓_𝐴). If, in Theorem 14, we assume that 𝑇_𝑖=Res^𝑓_𝐴𝑖, then we obtain an algorithm for finding a common zero of a finite family of maximal monotone mappings.

Theorem 20. Let 𝑓∶𝐸→ℝ be a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of 𝐸. Let 𝐶 be a nonempty, closed, and convex subset of int(dom𝑓) and let 𝐴_𝑖∶𝐶→2^𝐸∗,𝑖=1,2,…,𝑁, be 𝑁 maximal monotone mappings with ℱ∶=⋂^𝑁_𝑖=1𝐴_𝑖⁻¹(0)≠∅. Then, for each 𝑥₁,𝑢∈𝐸, the sequence {𝑥_𝑛} is defined by 𝑦_𝑛=∇𝑓^∗𝛼_𝑛∇𝑓(𝑢)+1−𝛼_𝑛∇𝑓𝑥_𝑛,𝑥_𝑛+1=∇𝑓^∗𝛽₀∇𝑓𝑦_𝑛+^𝑁_𝑖=1𝛽_𝑖∇𝑓𝑇_𝑖𝑦_𝑛,∀𝑛≥0, (53) where 𝑇_𝑖=Res^𝑓_𝐻𝑖, {𝛼_𝑛}⊂(0,1), and {𝛽_𝑖}^𝑁_𝑖=0⊂[𝑐,𝑑]⊂(0,1) satisfy lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=1𝛼_𝑛=∞, and ∑^𝑁_𝑖=0𝛽_𝑖=1. Then, {𝑥_𝑛} converges strongly to 𝑝=𝑃^𝑓_ℱ(𝑢).

For a common point of a solution of equilibrium problem and a zero of maximal monotone mapping we have the following corollary.

Corollary 21. Let 𝑓∶𝐸→ℝ be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of 𝐸. Let 𝐶 be a nonempty, closed, and convex subset of int(dom𝑓) and let 𝐻∶𝐶→𝐶 be bifunction which satisfy the conditions (B1)–(B4) and let 𝐴∶𝐶→2^𝐸∗ be a maximal monotone mapping. Assume that ℱ∶=𝐸𝑃(𝐻_𝑖)∩𝐴⁻¹(0) is nonempty. For 𝑥₀∈𝐶 let {𝑥_𝑛} be a sequence generated by 𝑦_𝑛=∇𝑓^∗𝛼_𝑛∇𝑓(𝑢)+1−𝛼_𝑛∇𝑓𝑥_𝑛,𝑥_𝑛+1=∇𝑓^∗𝛽₀∇𝑓𝑦_𝑛+𝛽₁∇𝑓Res^𝑓_𝐻𝑦_𝑛+𝛽₂∇𝑓Res^𝑓_𝐴𝑦_𝑛,∀𝑛≥0, (54) where {𝛼_𝑛}⊂(0,1) and {𝛽_𝑖}²_𝑖=1⊂[𝑐,𝑑]⊂(0,1) satisfying lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=1𝛼_𝑛=∞, and ∑²_𝑖=0𝛽_𝑖=1. Then, {𝑥_𝑛} converges strongly to 𝑝=𝑃^𝑓_ℱ(𝑢).

Remark 22. Corollary 17 and Corollary 4.3 improve Theorem 1 of Reich and Sabach [23] and Theorem 4.2 of Reich and Sabach [4], respectively, in the sense that at each stage the computation of 𝐶_𝑛 or 𝑄_𝑛 is not required.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. 170/130/1434. The authors, therefore, acknowledge with thanks DSR technical and financial support. Also, they would like to thank the anonymous reviewers for their valuable comments and suggestions.