Strong Convergence Theorem for Bregman Strongly Nonexpansive Mappings and Equilibrium Problems in Reflexive Banach Spaces

Abstract

By using a new hybrid method, a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of Bregman strongly nonexpansive mappings in a reflexive Banach space is proved.

1. Introduction

Throughout this paper, we denote by ℝ and ℝ⁺ the set of all real numbers and all nonnegative real numbers, respectively. We also assume that 𝐸 is a real reflexive Banach space, 𝐸^∗ is the dual space of 𝐸, 𝐶 is a nonempty closed convex subset of 𝐸, and ⟨⋅,⋅⟩ is the pairing between 𝐸 and 𝐸^∗. Let Θ be a bifunction from 𝐶×𝐶→ℝ. The equilibrium problem is to find 𝑥^∗∈𝐶suchthatΘ𝑥^∗,𝑦≥0,∀𝑦∈𝐶. (1) The set of such solutions 𝑥^∗ is denoted by EP(Θ).

Recall that a mapping 𝑇∶𝐶→𝐶 is said to be nonexpansive, if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶. (2) We denote by 𝐹(𝑇) the set of fixed points of 𝑇.

Numerous problems in physics, optimization, and economics reduce to find a solution of the equilibrium problem. Some methods have been proposed to solve the equilibrium problem in a Hilbert spaces; see, for instance, Blum and Oettli [1], Combettes and Hirstoaga [2], and Moudafi [3]. Recently, Tada and Takahashi [4, 5] and S. Takahashi and W. Takahashi [6] obtained weak and strong convergence theorems for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. In particular, Tada and Takahashi [4] established a strong convergence theorem for finding a common element of two sets by using the hybrid method introduced by Nakajo and Takahashi [7]. The authors also proved such a strong convergence theorem in a uniformly convex and uniformly smooth Banach space.

In this paper, motivated by Takahashi et al. [8], we prove a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a Bregman strongly nonexpansive mapping in a real reflexive Banach space by using the shrinking projection method. Using this theorem, we obtain two new strong convergence results for finding a solution of an equilibrium problem and a fixed point of Bregman strongly nonexpansive mappings in a real reflexive Banach space.

2. Preliminaries and Lemmas

In the sequel, we begin by recalling some preliminaries and lemmas which will be used in the proof.

Let 𝐸 be a real reflexive Banach space with the norm ‖⋅‖ and 𝐸^∗ the dual space of 𝐸. Throughout this paper, 𝑓∶𝐸→(−∞,+∞] is a proper, lower semicontinuous, and convex function. We denote by dom𝑓 the domain of 𝑓, that is, the set {𝑥∈𝐸∶𝑓(𝑥)<+∞}.

Let 𝑥∈  intdom𝑓. The subdifferential of 𝑓 at 𝑥 is the convex set defined by 𝜕𝑓(𝑥)=𝑥^∗∈𝐸^∗∶𝑓(𝑥)+𝑥^∗𝑦,,𝑦−𝑥≤𝑓,∀𝑦∈𝐸 (3) where the Fenchel conjugate of 𝑓 is the function 𝑓^∗∶𝐸^∗→(−∞,+∞] defined by 𝑓^∗𝑥^∗=sup𝑥^∗,𝑥−𝑓(𝑥)∶𝑥∈𝐸. (4) We know that the Young-Fenchel inequality holds: 𝑥^∗,𝑥≤𝑓(𝑥)+𝑓^∗𝑥^∗,∀𝑥∈𝐸,𝑥^∗∈𝐸^∗. (5)

A function 𝑓 on 𝐸 is coercive [9] if the sublevel set of 𝑓 is bounded; equivalently, lim_{‖𝑥‖→+∞}𝑓(𝑥)=+∞. (6)

A function 𝑓 on 𝐸 is said to be strongly coercive [10] if lim_{‖𝑥‖→+∞}𝑓(𝑥)‖𝑥‖=+∞. (7)

For any 𝑥∈intdom𝑓 and 𝑦∈𝐸, the right-hand derivative of 𝑓 at 𝑥 in the direction 𝑦 is defined by 𝑓^∘𝑥,𝑦∶=lim_𝑡→0+𝑓𝑥+𝑡𝑦−𝑓(𝑥)𝑡. (8)

The function 𝑓 is said to be Gâteaux differentiable at 𝑥 if lim_𝑡→0⁺((𝑓(𝑥+𝑡𝑦)−𝑓(𝑥))/𝑡) exists for any 𝑦. In this case, 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. The function 𝑓 is said to be Fréchet differentiable at 𝑥 if this limit is attained uniformly in ‖𝑦‖=1. Finally, 𝑓 is said to be uniformly Fréchet differentiable on a subset 𝐶 of 𝐸 if the limit is attained uniformly for 𝑥∈𝐶 and ‖𝑦‖=1. It is known that if 𝑓 is Gâteaux differentiable (resp., Fréchet differentiable) on intdom𝑓, then 𝑓 is continuous and its Gâteaux derivative ∇𝑓 is norm-to-weak∗ continuous (resp., continuous) on intdom𝑓 (see also [11, 12]). We will need the following result.

Lemma 1 (see [13]). If 𝑓∶𝐸→ℝ is uniformly Fréchet differentiable and bounded on bounded subsets of 𝐸, then ∇𝑓 is uniformly continuous on bounded subsets of 𝐸 from the strong topology of 𝐸 to the strong topology of 𝐸^∗.

Definition 2 (see [14]). The function 𝑓 is said to be(i)essentially smooth, if 𝜕𝑓 is both locally bounded and single valued on its domain,(ii)essentially strictly convex, if (𝜕𝑓)⁻¹ is locally bounded on its domain and 𝑓 is strictly convex on every convex subset of dom𝜕𝑓,(iii)Legendre if it is both essentially smooth and essentially strictly convex.

Remark 3. Let 𝐸 be a reflexive Banach space. Then we have the following.(i)𝑓 is essentially smooth if and only if 𝑓^∗ is essentially strictly convex (see [14, Theorem 5.4]).(ii)(𝜕𝑓)⁻¹=𝜕𝑓^∗ (see [12]).(iii)𝑓 is Legendre if and only if 𝑓^∗ is Legendre (see [14, 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 [14, Theorem 5.10]).

Examples of Legendre functions were given in [14, 15]. One important and interesting Legendre function is (1/𝑝)‖⋅‖^𝑝(1<𝑝<∞) when 𝐸 is a smooth and strictly convex Banach space. In this case, the gradient ∇𝑓 of 𝑓 is coincident with the generalized duality mapping of 𝐸; that is, ∇𝑓=𝐽_𝑝(1<𝑝<∞). In particular, ∇𝑓=𝐼 the identity mapping in Hilbert spaces. In the rest of this paper, we always assume that 𝑓∶𝐸→(−∞,+∞] is Legendre.

Let 𝑓∶𝐸→(−∞,+∞] be a convex and Gâteaux differentiable function. The function 𝐷_𝑓∶dom𝑓×intdom𝑓→[0,+∞) defined as 𝐷_𝑓𝑦,𝑥∶=𝑓𝑦−𝑓(𝑥)−∇𝑓(𝑥),𝑦−𝑥 (9) is called the Bregman distance with respect to 𝑓 [16].

Recall that the Bregman projection [17] of 𝑥∈intdom𝑓 onto the nonempty closed and convex set 𝐶⊂dom𝑓 is the necessarily unique vector 𝑃^𝑓_𝐶(𝑥)∈𝐶 satisfying 𝐷_𝑓𝑃^𝑓_𝐶(𝑥),𝑥=inf𝐷_𝑓𝑦,𝑥∶𝑦∈𝐶. (10) Concerning the Bregman projection, the following are well known.

Lemma 4 (see [18]). Let 𝐶 be a nonempty, closed, and convex subset of a reflexive Banach space 𝐸. Let 𝑓∶𝐸→ℝ be a Gâteaux differentiable and totally convex function and let 𝑥∈𝐸. Then(a)𝑧=𝑃^𝑓_𝐶(𝑥) if and only if ⟨∇𝑓(𝑥)−∇𝑓(𝑧),𝑦−𝑧⟩≤0, for all 𝑦∈𝐶.(b)𝐷_𝑓𝑦,𝑃^𝑓_𝐶(𝑥)+𝐷_𝑓𝑃^𝑓_𝐶(𝑥),𝑥≤𝐷_𝑓𝑦,𝑥,∀𝑥∈𝐸,𝑦∈𝐶. (11)

Let 𝑓∶𝐸→(−∞,+∞] be a convex and Gâteaux differentiable function. The modulus of total convexity of 𝑓 at 𝑥∈intdom𝑓 is the function 𝜈_𝑓(𝑥,⋅)∶[0,+∞)→[0,+∞] defined by 𝜈_𝑓(𝑥,𝑡)∶=inf𝐷_𝑓𝑦,𝑥∶𝑦∈dom𝑓,𝑦−𝑥=𝑡. (12)

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∈intdom𝑓 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. (13)

The next lemma will be useful in the proof of our main results.

Lemma 5 (see [19]). If 𝑥∈dom𝑓, then the following statements are equivalent.(i)The function 𝑓 is totally convex at 𝑥.(ii)For any sequence {𝑦_𝑛}⊂dom𝑓, lim_𝑛→+∞𝐷_𝑓𝑦_𝑛,𝑥=0⟹lim_𝑛→+∞𝑦_𝑛−𝑥=0. (14)

Recall that the function 𝑓 is called sequentially consistent [18] if, for any two sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝐸 such that the first one is bounded, lim_𝑛→+∞𝐷_𝑓𝑦_𝑛,𝑥_𝑛=0⟹lim_𝑛→+∞𝑦_𝑛−𝑥_𝑛=0. (15)

Lemma 6 (see [20]). The function 𝑓 is totally convex on bounded sets if and only if the function 𝑓 is sequentially consistent.

Lemma 7 (see [21]). Let 𝑓∶𝐸→ℝ be a Gâteaux differentiable and totally convex function. If 𝑥₀∈𝐸 and the sequence {𝐷_𝑓(𝑥_𝑛,𝑥₀)} is bounded, then the sequence {𝑥_𝑛} is bounded too.

Lemma 8 (see [21]). Let 𝑓∶𝐸→ℝ be a Gâteaux differentiable and totally convex function, 𝑥₀∈𝐸, and let 𝐶 be a nonempty, closed, and convex subset of 𝐸. Suppose that the sequence {𝑥_𝑛} is bounded and any weak subsequential limit of {𝑥_𝑛} belongs to 𝐶. If 𝐷_𝑓(𝑥_𝑛,𝑥₀)≤𝐷_𝑓(𝑃^𝑓_𝐶𝑥₀,𝑥₀) for any 𝑛∈ℕ, then {𝑥_𝑛} converges strongly to 𝑃^𝑓_𝐶𝑥₀.

Let 𝐶 be a convex subset of intdom𝑓 and let 𝑇 be a self-mapping of 𝐶. A point 𝑝∈𝐶 is called an asymptotic fixed point of 𝑇 (see [22, 23]) if 𝐶 contains a sequence {𝑥_𝑛} which converges weakly to 𝑝 such that lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0. We denote by 𝐹(𝑇) the set of asymptotic fixed points of 𝑇.

Definition 9. A mapping 𝑇 with a nonempty asymptotic fixed point set 𝐹(𝑇) is said to be(i)Bregman strongly nonexpansive (see [24, 25]) with respect to 𝐹(𝑇) if 𝐷_𝑓𝑝,𝑇𝑥≤𝐷_𝑓𝑝,𝑥,∀𝑥∈𝐶,𝑝∈𝐹(𝑇), (16) and if, whenever {𝑥_𝑛}⊂𝐶 is bounded, 𝑝∈𝐹(𝑇) and lim_𝑛→∞𝐷_𝑓𝑝,𝑥_𝑛−𝐷_𝑓𝑝,𝑇𝑥_𝑛=0, (17) it follows that lim_𝑛→∞𝐷_𝑓𝑥_𝑛,𝑇𝑥_𝑛=0. (18)(ii)Bregman firmly nonexpansive [26] if, for all 𝑥,𝑦∈𝐶, ()≤(𝑥)𝑦∇𝑓𝑇𝑥−∇𝑓𝑇𝑦,𝑇𝑥−𝑇𝑦∇𝑓−∇𝑓,𝑇𝑥−𝑇𝑦 (19) or, equivalently, 𝐷_𝑓𝑇𝑥,𝑇𝑦+𝐷_𝑓𝑇𝑦,𝑇𝑥+𝐷_𝑓(𝑇𝑥,𝑥)+𝐷_𝑓𝑇𝑦,𝑦≤𝐷_𝑓𝑇𝑥,𝑦+𝐷_𝑓𝑇𝑦,𝑥. (20)

The existence and approximation of Bregman firmly nonexpansive mappings were studied in [26]. It is also known that if 𝑇 is Bregman firmly nonexpansive and 𝑓 is Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of 𝐸, then 𝐹(𝑇)=𝐹(𝑇) and 𝐹(𝑇) is closed and convex (see [26]). It also follows that every Bregman firmly nonexpansive mapping is Bregman strongly nonexpansive with respect to 𝐹(𝑇)=𝐹(𝑇).

Lemma 10 (see [27]). Let 𝐸 be a real reflexive Banach space and 𝑓∶𝐸→(−∞,+∞] a proper lower semicontinuous 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. (21)

In order to solve the equilibrium problem, let us assume that a bifunction Θ∶𝐶×𝐶→ℝ satisfies the following conditions [28]:(A₁)Θ(𝑥,𝑥)=0, for all 𝑥∈𝐶.(A₂)Θ is monotone; that is, Θ(𝑥,𝑦)+Θ(𝑦,𝑥)≤0, for all 𝑥,𝑦∈𝐶.(A₃) lim sup_𝑡↓0Θ(𝑥+𝑡(𝑧−𝑥),𝑦)≤Θ(𝑥,𝑦) for all 𝑥,𝑧,𝑦∈𝐶.(A₄) The function 𝑦↦Θ(𝑥,𝑦) is convex and lower semicontinuous.

The resolvent of a bifunction Θ [29] is the operator Res^𝑓_Θ∶𝐸→2^𝐶 defined by Res^𝑓_Θ(𝑥)=+(𝑧)(𝑥).𝑧∈𝐶∶Θ𝑧,𝑦∇𝑓−∇𝑓,𝑦−𝑧≥0,∀𝑦∈𝐶 (22)

From Lemma 1 in [24], if 𝑓∶𝐸→(−∞,+∞] is a strongly coercive and Gâteaux differentiable function and Θ satisfies conditions (A₁–A₄), then dom (Res^𝑓_Θ)=𝐸. We also know the following lemma which gives us some characterizations of the resolvent Res^𝑓_Θ.

Lemma 11 (see [24]). Let 𝐸 be a real reflexive Banach space and 𝐶 a nonempty closed convex subset of 𝐸. Let 𝑓∶𝐸→(−∞,+∞] be a Legendre function. If the bifunction Θ∶𝐶×𝐶→ℝ satisfies the conditions (A₁)–(A₄), then the followings hold:(i)𝑅𝑒𝑠^𝑓_Θ is single-valued;(ii)𝑅𝑒𝑠^𝑓_Θ is a Bregman firmly nonexpansive operator;(iii)𝐹(𝑅𝑒𝑠^𝑓_Θ)=𝐸𝑃(Θ);(iv)𝐸𝑃(Θ) is a closed and convex subset of 𝐶;(v)for all 𝑥∈𝐸 and for all 𝑞∈𝐹(𝑅𝑒𝑠^𝑓_Θ), we have 𝐷_𝑓𝑞,𝑅𝑒𝑠^𝑓_Θ(𝑥)+𝐷_𝑓𝑅𝑒𝑠^𝑓_Θ(𝑥),𝑥≤𝐷_𝑓𝑞,𝑥. (23)

3. Strong Convergence Theorem

In this section, we proved a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and a fixed point of Bregman strongly nonexpansive mapping in a real reflexive Banach space by using the shrinking projection method.

Theorem 12. Let 𝐶 be a nonempty, closed, and convex subset of a real reflexive Banach space 𝐸 and 𝑓∶𝐸→ℝ a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of 𝐸. Let 𝑔 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A₁)–(A₄) and let 𝑇 be a Bregman strongly nonexpansive mapping from 𝐶 into itself such that 𝐹(𝑇)=𝐹(𝑇) and 𝐺=𝐹(𝑇)∩𝐸𝑃(𝑔)≠∅. Let {𝑥_𝑛} be a sequence generated by 𝑥₀=𝑥∈𝐶,𝐶₀=𝐶 and 𝑦_𝑛=∇𝑓^∗𝛼_𝑛∇𝑓𝑥_𝑛+1−𝛼_𝑛∇𝑓𝑇𝑥_𝑛,𝑢_𝑛𝑔𝑢∈𝐶𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡_𝑛,𝑦+∇𝑓𝑢_𝑛−∇𝑓𝑦_𝑛,𝑦−𝑢_𝑛𝐶≥0,∀𝑦∈𝐶,_𝑛+1=𝑧∈𝐶_𝑛∶𝐷_𝑓𝑧,𝑢_𝑛≤𝐷_𝑓𝑧,𝑥_𝑛,𝑥_𝑛+1=𝑃^𝑓_𝐶𝑛+1𝑥 (24) for every 𝑛∈ℕ∪{0}, where {𝛼_𝑛}⊂[0,1] satisfies liminf_𝑛→∞(1−𝛼_𝑛)>0. Then, {𝑥_𝑛} converges strongly to 𝑃^𝑓_{𝐹(𝑇)∩𝐸𝑃(𝑔)}𝑥, where 𝑃^𝑓_{𝐹(𝑇)∩𝐸𝑃(𝑔)} is the Bregman projection of 𝐸 onto 𝐹(𝑇)∩𝐸𝑃(𝑔).

Proof. We divide the proof of Theorem 12 into five steps.
(I) We first prove that 𝐺 and 𝐶_𝑛 both are closed and convex subset of 𝐶 for all 𝑛≥0. In fact, it follows from Lemma 11 and by Reich and Sabach [26] that EP(𝑔) and 𝐹(𝑇) both are closed and convex. Therefore, 𝐺 is a closed and convex subset in 𝐶. Furthermore, it is obvious that 𝐶₀=𝐶 is closed and convex. Suppose that 𝐶_𝑛 is closed and convex for some 𝑛≥1. Since the inequality 𝐷_𝑓(𝑧,𝑢_𝑛)≤𝐷_𝑓(𝑧,𝑥_𝑛) is equivalent to ∇𝑓𝑥_𝑛,𝑧−𝑥_𝑛−∇𝑓𝑢_𝑛,𝑧−𝑢_𝑛≤𝑓𝑢_𝑛−𝑓𝑥_𝑛. (25)
Therefore, we have 𝐶_𝑛+1=𝑧∈𝐶_𝑛∶⟨∇𝑓𝑥_𝑛,𝑧−𝑥_𝑛⟩−⟨∇𝑓𝑢_𝑛,𝑧−𝑢_𝑛⟩≤𝑓𝑢_𝑛−𝑓𝑥_𝑛. (26)
This implies that 𝐶_𝑛+1 is closed and convex. The desired conclusions are proved. These in turn show that 𝑃^𝑓_{𝐹(𝑇)∩EP(𝑔)}𝑥 and 𝑃^𝑓_𝐶𝑛𝑥 are well defined.
(II) we prove that 𝐺∶=𝐹(𝑇)∩EP(𝑔)⊂𝐶_𝑛 for all 𝑛≥0.
Indeed, it is obvious that 𝐺=𝐹(𝑇)∩EP(𝑔)⊂𝐶₀=𝐶. Suppose that 𝐺⊂𝐶_𝑛 for some 𝑛∈ℕ. Let 𝑢∈𝐺⊂𝐶_𝑛; since 𝑢_𝑛=Res^𝑓_𝑔(𝑦_𝑛), by Lemma 11 and (21), we have 𝐷_𝑓𝑢,𝑢_𝑛=𝐷_𝑓𝑢,Res^𝑓_𝑔𝑦_𝑛≤𝐷_𝑓𝑢,𝑦_𝑛=𝐷_𝑓𝑢,∇𝑓^∗𝛼_𝑛∇𝑓𝑥_𝑛+1−𝛼_𝑛∇𝑓𝑇𝑥_𝑛≤𝛼_𝑛𝐷_𝑓𝑢,𝑥_𝑛+1−𝛼_𝑛𝐷_𝑓𝑢,𝑇𝑥_𝑛≤𝛼_𝑛𝐷_𝑓𝑢,𝑥_𝑛+1−𝛼_𝑛𝐷_𝑓𝑢,𝑥_𝑛=𝐷_𝑓𝑢,𝑥_𝑛. (27) Hence, we have 𝑢∈𝐶_𝑛+1. This implies that 𝐹(𝑇)∩EP𝑔⊂𝐶_𝑛,∀𝑛∈ℕ∪{0}. (28) So, {𝑥_𝑛} is well defined.
(III) We prove that {𝑥_𝑛} is a bounded sequence in 𝐶.
By the definition of 𝐶_𝑛, we have 𝑥_𝑛=𝑃^𝑓_𝐶𝑛𝑥 for all 𝑛≥0. It follows from Lemma 4(b) that 𝐷_𝑓𝑥_𝑛,𝑥=𝐷_𝑓𝑃^𝑓_𝐶𝑛𝑥,𝑥≤𝐷_𝑓(𝑢,𝑥)−𝐷_𝑓𝑢,𝑃^𝑓_𝐶𝑛𝑥≤𝐷_𝑓(𝑢,𝑥),∀𝑛≥0,𝑢∈𝐺. (29) This implies that {𝐷_𝑓(𝑥_𝑛,𝑥)} is bounded. By Lemma 7, {𝑥_𝑛} is bounded. Since 𝑓∶𝐸→ℝ is uniformly Fréchet differentiable and bounded on bounded subsets of 𝐸, by Lemma 1∇𝑓 is uniformly continuous and bounded on bounded subsets of 𝐸. This implies that {∇𝑓(𝑥_𝑛)} is bounded.
(IV) Now we proved that lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0.
From 𝑥_𝑛+1∈𝐶_𝑛+1⊂𝐶_𝑛 and 𝑥_𝑛=𝑃^𝑓_𝐶𝑛𝑥, we have 𝐷_𝑓𝑥_𝑛,𝑥≤𝐷_𝑓𝑥_𝑛+1,𝑥,∀𝑛∈ℕ∪{0}. (30) Thus, {𝐷_𝑓(𝑥_𝑛,𝑥)} is nondecreasing. So, the limit of {𝐷_𝑓(𝑥_𝑛,𝑥)} exists. Since 𝐷_𝑓(𝑥_𝑛+1,𝑥_𝑛)=𝐷_𝑓(𝑥_𝑛+1,𝑃^𝑓_𝐶𝑛𝑥)≤𝐷_𝑓(𝑥_𝑛+1,𝑥)−𝐷_𝑓(𝑃^𝑓_𝐶𝑛𝑥,𝑥)=𝐷_𝑓(𝑥_𝑛+1,𝑥)−𝐷_𝑓(𝑥_𝑛,𝑥) for all 𝑛≥0, we have lim_𝑛→∞𝐷_𝑓(𝑥_𝑛+1,𝑥_𝑛)=0. From 𝑥_𝑛+1=𝑃^𝑓_𝐶𝑛+1𝑥∈𝐶_𝑛+1, we have 𝐷_𝑓𝑥_𝑛+1,𝑢_𝑛≤𝐷_𝑓𝑥_𝑛+1,𝑥_𝑛,∀𝑛∈ℕ∪{0}. (31) Therefore, we have lim_𝑛→∞𝐷_𝑓𝑥_𝑛+1,𝑢_𝑛=0. (32) From Lemma 5, we have lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=lim_𝑛→∞𝑥_𝑛+1−𝑢_𝑛=0. (33) So, we have lim_𝑛→∞𝑥_𝑛−𝑢_𝑛=0. (34) This means that the sequence {𝑢_𝑛} is bounded. Since 𝑓 is uniformly Fréchet differentiable, it follows from Lemma 1 that ∇𝑓 is uniformly continuous. Therefore, we have lim_𝑛→∞∇𝑓𝑥_𝑛−∇𝑓𝑢_𝑛=0. (35)
Since 𝑓 is uniformly Fréchet differentiable on bounded subsets of 𝐸, then 𝑓 is uniformly continuous on bounded subsets of 𝐸 (see [30, Theorem 1.8]). It follows that lim_𝑛→∞𝑓𝑥_𝑛−𝑓𝑢_𝑛=0. (36)
From the definition of the Bregman distance, we obtain that 𝐷_𝑓𝑢,𝑥_𝑛−𝐷_𝑓𝑢,𝑢_𝑛=𝑓(𝑢)−𝑓𝑥_𝑛−⟨∇𝑓𝑥_𝑛,𝑢−𝑥_𝑛⟩−𝑓(𝑢)−𝑓𝑢_𝑛−⟨∇𝑓𝑢_𝑛,𝑢−𝑢_𝑛⟩=𝑓𝑢_𝑛−𝑓𝑥_𝑛+⟨∇𝑓𝑢_𝑛−∇𝑓𝑥_𝑛,𝑢−𝑢_𝑛⟩+⟨∇𝑓𝑥_𝑛,𝑥_𝑛−𝑢_𝑛⟩ (37) for any 𝑢∈𝐺.
It follows from (34)–(37) that lim_𝑛→∞𝐷_𝑓𝑢,𝑥_𝑛−𝐷_𝑓𝑢,𝑢_𝑛=0. (38) On the other hand, from 𝑢_𝑛=Res^𝑓_𝑔𝑦_𝑛 and Lemma 11(v), for any 𝑢∈𝐺 we have that 𝐷_𝑓𝑢_𝑛,𝑦_𝑛=𝐷_𝑓Res^𝑓_𝑔𝑦_𝑛,𝑦_𝑛≤𝐷_𝑓𝑢,𝑦_𝑛−𝐷_𝑓𝑢,Res^𝑓_𝑔𝑦_𝑛≤𝐷_𝑓𝑢,𝑥_𝑛−𝐷_𝑓𝑢,Res^𝑓_𝑔𝑦_𝑛=𝐷_𝑓𝑢,𝑥_𝑛−𝐷_𝑓𝑢,𝑢_𝑛. (39) So, we have from (38) that lim_𝑛→∞𝐷_𝑓𝑢_𝑛,𝑦_𝑛=0. (40) From Lemma 5, we have lim_𝑛→∞𝑢_𝑛−𝑦_𝑛=0. (41) So, from (34) and (41),we have lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0. (42) This means that the sequence {𝑦_𝑛} is bounded. Since 𝑓 is uniformly Fréchet differentiable, it follows from Lemma 1 that lim_𝑛→∞∇𝑓𝑥_𝑛−∇𝑓𝑦_𝑛=0. (43) Since 𝑓 is uniformly Fréchet differentiable on bounded subsets of 𝐸, then 𝑓 is uniformly continuous on bounded subsets of 𝐸 (see [30]). It follows that lim_𝑛→∞𝑓𝑥_𝑛−𝑓𝑦_𝑛=0. (44)
From the definition of the Bregman distance, we obtain that 𝐷_𝑓𝑢,𝑦_𝑛−𝐷_𝑓𝑢,𝑥_𝑛=𝑓(𝑢)−𝑓𝑦_𝑛−∇𝑓𝑦_𝑛,𝑢−𝑦_𝑛−𝑓(𝑢)−𝑓𝑥_𝑛−∇𝑓𝑥_𝑛,𝑢−𝑥_𝑛=𝑓𝑥_𝑛−𝑓𝑦_𝑛−∇𝑓𝑦_𝑛−∇𝑓𝑥_𝑛,𝑢−𝑦_𝑛+∇𝑓𝑥_𝑛,𝑦_𝑛−𝑥_𝑛 (45) for any 𝑢∈𝐺.
It follows from (42) to (45) that lim_𝑛→∞𝐷_𝑓𝑢,𝑦_𝑛−𝐷_𝑓𝑢,𝑥_𝑛=0. (46)
On the other hand, for any 𝑢∈𝐺 we have 𝐷_𝑓𝑢,𝑦_𝑛−𝐷_𝑓𝑢,𝑥_𝑛=𝐷_𝑓𝑢,∇𝑓^∗𝛼_𝑛∇𝑓𝑥_𝑛+1−𝛼_𝑛∇𝑓𝑇𝑥_𝑛−𝐷_𝑓𝑢,𝑥_𝑛≤𝛼_𝑛𝐷_𝑓𝑢,𝑥_𝑛+1−𝛼_𝑛𝐷_𝑓𝑢,𝑇𝑥_𝑛−𝐷_𝑓𝑢,𝑥_𝑛=1−𝛼_𝑛𝐷_𝑓𝑢,𝑇𝑥_𝑛−𝐷_𝑓𝑢,𝑥_𝑛. (47) This together with (46), (16), and lim_𝑛→∞𝛼_𝑛<1 shows that lim_𝑘→∞𝐷_𝑓𝑢,𝑇𝑥_𝑛−𝐷_𝑓𝑢,𝑥_𝑛=0. (48) Since 𝑇 is Bregman strongly nonexpansive, it follows from (48) that lim_𝑛→∞𝑥_𝑛−𝑇𝑥_𝑛=0. (49)
(V) Next, we prove that every weak subsequential limit of {𝑥_𝑛} belongs to 𝐺=𝐹(𝑇)∩EP(𝑔).
Since {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that 𝑥_𝑛𝑘⇀𝑥^∗. Since 𝑇 is a Bregman strongly nonexpansive mapping with 𝐹(𝑇)=𝐹(𝑇), we have 𝑥^∗∈𝐹(𝑇).
From 𝑥_𝑛𝑘⇀𝑥^∗ and (34), we have 𝑢_𝑛𝑘⇀𝑥^∗.
By 𝑢_𝑛=Res^𝑓_𝑔𝑦_𝑛, we have 𝑔𝑢_𝑛,𝑦+∇𝑓𝑢_𝑛−∇𝑓𝑦_𝑛,𝑦−𝑢_𝑛≥0,∀𝑦∈𝐶. (50) Replacing 𝑛 by 𝑛_𝑘, we have from (A₂) that ∇𝑓𝑢_𝑛𝑘−∇𝑓𝑦_𝑛𝑘,𝑦−𝑢_𝑛𝑘≥−𝑔𝑢_𝑛𝑘,𝑦≥𝑔𝑦,𝑢_𝑛𝑘,∀𝑦∈𝐶. (51) Since 𝑔(𝑥,⋅) is convex and lower semicontinuous, it is also weakly lower semicontinuous. So, letting 𝑘→∞, we have from (35), (43), and (A₄) that 𝑔𝑦,𝑥^∗≤0,∀𝑦∈𝐶. (52) For 𝑡∈(0,1] and 𝑦∈𝐶, letting 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑥^∗, there are 𝑦_𝑡∈𝐶 and 𝑔(𝑦_𝑡,𝑥^∗)≤0. By condition (A₁) and (A₄), we have 0=𝑔𝑦_𝑡,𝑦_𝑡≤𝑡𝑔𝑦_𝑡,𝑦+(1−𝑡)𝑔𝑦_𝑡,𝑥^∗≤𝑡𝑔𝑦_𝑡,𝑦. (53) Dividing both sides of the above equation by 𝑡, we have 𝑔(𝑦_𝑡,𝑦)≥0, for all 𝑦∈𝐶. Letting 𝑡↓0, from condition (A₃), we have 𝑔𝑥^∗,𝑦≥0,∀𝑦∈𝐶. (54) Therefore, 𝑥^∗∈EP(𝑔).
(VI) Now, we prove 𝑥_𝑛→𝑃^𝑓_{𝐹(𝑇)∩EP(𝑔)}𝑥.
Let 𝑤=𝑃^𝑓_{𝐹(𝑇)∩EP(𝑔)}𝑥. From 𝑤∈𝐹(𝑇)∩EP(𝑔)⊂𝐶_𝑛+1, we have 𝐷_𝑓(𝑥_𝑛+1,𝑥)≤𝐷_𝑓(𝑤,𝑥). Therefore, Lemma 8 implies that {𝑥_𝑛} converges strongly to 𝑤=𝑃^𝑓_{𝐹(𝑇)∩EP(𝑔)}𝑥, as claimed. This completes the proof of Theorem 12.

Corollary 13. Let 𝐶 be a nonempty, closed, and convex subset of a real reflexive Banach space 𝐸 and 𝑓∶𝐸→ℝ a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of 𝐸. Let 𝑔 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A₁)–(A₄). Let {𝑥_𝑛} be a sequence generated by 𝑥₀=𝑥∈𝐶,𝐶₀=𝐶, and 𝑢_𝑛𝑔𝑢∈𝐶𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡_𝑛,𝑦+∇𝑓𝑢_𝑛−∇𝑓𝑥_𝑛,𝑦−𝑢_𝑛𝐶≥0,∀𝑦∈𝐶,_𝑛+1=𝑧∈𝐶_𝑛∶𝐷_𝑓𝑧,𝑢_𝑛≤𝐷_𝑓𝑧,𝑥_𝑛,𝑥_𝑛+1=𝑃^𝑓_𝐶𝑛+1𝑥 (55) for every 𝑛∈ℕ∪{0}. Then, {𝑥_𝑛} converges strongly to 𝑃^𝑓_{𝐸𝑃(𝑔)}𝑥, where 𝑃^𝑓_{𝐸𝑃(𝑔)} is the Bregman projection of 𝐸 onto 𝐸𝑃(𝑔).

Proof. Putting 𝑇=𝐼 in Theorem 12, we obtain Corollary 13.

Corollary 14. Let 𝐶 be a nonempty, closed, and convex subset of a real reflexive Banach space 𝐸 and 𝑓∶𝐸→ℝ a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of 𝐸. let 𝑇 be a Bregman strongly nonexpansive mapping from 𝐶 into itself such that 𝐹(𝑇)=𝐹(𝑇) and 𝐺=𝐹(𝑇)∩𝐸𝑃(𝑔)≠∅. Let {𝑥_𝑛} be a sequence generated by 𝑥₀=𝑥∈𝐶,𝐶₀=𝐶, and 𝑢_𝑛=𝑃^𝑓_𝐶∇𝑓^∗𝛼_𝑛∇𝑓𝑥_𝑛+1−𝛼_𝑛∇𝑓𝑇𝑥_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝐷_𝑓𝑧,𝑢_𝑛≤𝐷_𝑓𝑧,𝑥_𝑛,𝑥_𝑛+1=𝑃^𝑓_𝐶𝑛+1𝑥 (56) for every 𝑛∈ℕ∪{0}, where {𝛼_𝑛}⊂[0,1] satisfies liminf_𝑛→∞(1−𝛼_𝑛)>0. Then, {𝑥_𝑛} converges strongly to 𝑃^𝑓_𝐹(𝑇)𝑥, where 𝑃^𝑓_𝐹(𝑇) is the Bregman projection of 𝐸 onto 𝐹(𝑇).

Proof. Putting 𝑔(𝑥,𝑦)=0 for all 𝑥,𝑦∈𝐶 in Theorem 12, we obtain Corollary 14.

Acknowledgments

This work was supported by Scientific Research Fund of Sichuan Provincial Education Department (11ZB146) and Yunnan University of Finance and Economics.