Bregman 𝑓-Projection Operator with Applications to Variational Inequalities in Banach Spaces

Abstract

Using Bregman functions, we introduce the new concept of Bregman generalized 𝑓-projection operator Proj_𝐶^𝑓,𝑔∶𝐸^∗→𝐶, where 𝐸 is a reflexive Banach space with dual space 𝐸^∗;𝑓∶𝐸→ℝ∪{+∞} is a proper, convex, lower semicontinuous and bounded from below function; 𝑔∶𝐸→ℝ is a strictly convex and Gâteaux differentiable function; and 𝐶 is a nonempty, closed, and convex subset of 𝐸. The existence of a solution for a class of variational inequalities in Banach spaces is presented.

1. Introduction

Many nonlinear problems in functional analysis can be reduced to the search of fixed points of nonlinear operators. See, for example, [1–14] and the references therein. Let 𝐸 be a (real) Banach space with norm ‖⋅‖ and dual space 𝐸^∗. For any 𝑥 in 𝐸, we denote the value of 𝑥^∗ in 𝐸^∗ at 𝑥 by ⟨𝑥,𝑥^∗⟩. When {𝑥_𝑛}_𝑛∈ℕ is a sequence in 𝐸, we denote the strong convergence of {𝑥_𝑛}_𝑛∈ℕ to 𝑥∈𝐸 by 𝑥_𝑛→𝑥 and the weak convergence by 𝑥_𝑛⇀𝑥. Let 𝐶 be a nonempty subset of 𝐸 and 𝑇∶𝐶→𝐸 be a mapping. We denote by 𝐹(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥} the set of fixed points of 𝑇. Let 𝐶 be a nonempty, closed, and convex subset of a smooth Banach space 𝐸; let 𝑇 be a mapping from 𝐶 into itself. A point 𝑝∈𝐶 is said to be an asymptotic fixed point [15] of 𝑇 if there exists a sequence {𝑥_𝑛}_𝑛∈ℕ in 𝐶 which converges weakly to 𝑝 and lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0. We denote the set of all asymptotic fixed points of 𝑇 by 𝐹(𝑇). A point 𝑝∈𝐶 is called a strong asymptotic fixed point of 𝑇 if there exists a sequence {𝑥_𝑛}_𝑛∈ℕ in 𝐶 which converges strongly to 𝑝 and lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0. We denote the set of all strong asymptotic fixed points of 𝑇 by 𝐹(𝑇).

We recall the definition of Bregman distances. Let 𝑔∶𝐸→ℝ be a strictly convex and Gâteaux differentiable function on a Banach space 𝐸. The Bregman distance [16] (see also [17, 18]) 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. (1) It follows from the strict convexity of 𝑔 that 𝐷_𝑔(𝑥,𝑦)≥0 for all 𝑥,𝑦 in 𝐸. However, 𝐷_𝑔 might not be symmetric and 𝐷_𝑔 might not satisfy the triangular inequality.

When 𝐸 is a smooth Banach space, setting 𝑔(𝑥)=‖𝑥‖² for all 𝑥 in 𝐸, we have that ∇𝑔(𝑥)=2𝐽𝑥 for all 𝑥 in 𝐸. Here 𝐽 is the normalized duality mapping from 𝐸 into 𝐸^∗. Hence, 𝐷_𝑔(⋅,⋅) reduces to the usual map 𝜙(⋅,⋅) as 𝐷_𝑔𝑥,𝑦=𝜙𝑥,𝑦∶=‖𝑥‖²−2𝑥,𝐽𝑦+𝑦²,∀𝑥,𝑦∈𝐸. (2) If 𝐸 is a Hilbert space, then 𝐷_𝑔(𝑥,𝑦)=‖𝑥−𝑦‖².

Let 𝑔∶𝐸→ℝ be strictly convex and Gâteaux differentiable and 𝐶⊆𝐸 be nonempty. A mapping 𝑇∶𝐶→𝐸 is said to be(i)Bregman nonexpansive if 𝐷_𝑔𝑇𝑥,𝑇𝑦≤𝐷_𝑔𝑥,𝑦,∀𝑥,𝑦∈𝐶. (3)(ii)Bregman quasi-nonexpansive if 𝐹(𝑇)≠∅ and 𝐷_𝑔𝑝,𝑇𝑥≤𝐷_𝑔𝑝,𝑥,∀𝑥∈𝐶,∀𝑝∈𝐹(𝑇). (4)(iii)Bregman relatively nonexpansive if the following conditions are satisfied:(1)𝐹(𝑇) is nonempty;(2)𝐷_𝑔(𝑝,𝑇𝑣)≤𝐷_𝑔(𝑝,𝑣), ∀𝑝∈𝐹(𝑇),𝑣∈𝐶;(3)𝐹(𝑇)=𝐹(𝑇);(iv)Bregman weak relatively nonexpansive if the following conditions are satisfied:(1)𝐹(𝑇) is nonempty;(2)𝐷_𝑔(𝑝,𝑇𝑣)≤𝐷_𝑔(𝑝,𝑣), ∀𝑝∈𝐹(𝑇),𝑣∈𝐶;(3)𝐹(𝑇)=𝐹(𝑇).

It is clear that any Bregman relatively nonexpansive mapping is a Bregman quasi-nonexpansive mapping. It is also obvious that every Bregman relatively nonexpansive mapping is a Bregman weak relatively nonexpansive mapping, but the converse is not true in general; see, for example, [19]. Indeed, for any mapping 𝑇∶𝐶→𝐶 we have 𝐹(𝑇)⊂𝐹(𝑇)⊂𝐹(𝑇). If 𝑇 is Bregman relatively nonexpansive, then 𝐹(𝑇)=𝐹(𝑇)=𝐹(𝑇).

Let 𝐸 be a reflexive Banach space, let 𝑓∶𝐸→ℝ∪{+∞} be a proper, convex, lower semicontinuous function, let 𝑔∶𝐸→ℝ be strictly convex and Gâteaux differentiable, and let 𝐶⊆𝐸 be nonempty. We define a functional 𝐻∶𝐸×𝐸^∗→ℝ∪{+∞} by 𝐻𝑥,𝑥^∗=𝑔(𝑥)−𝑥,𝑥^∗+𝑔^∗𝑥^∗+𝑓(𝑥),𝑥∈𝐸,𝑥^∗∈𝐸^∗. (5) It could easily be seen that 𝐻 satisfies the following properties:(1)𝐻(𝑥,𝑥^∗) is convex and continuous with respect to 𝑥^∗ when 𝑥 is fixed;(2)𝐻(𝑥,𝑥^∗) is convex and lower semicontinuous with respect to 𝑥 when 𝑥^∗ is fixed.

Definition 1. Let 𝐸 be a Banach space with dual space 𝐸^∗, let 𝑓∶𝐸→ℝ∪{+∞} be a proper, convex, lower semicontinuous function, let 𝑔∶𝐸→ℝ be strictly convex and Gâteaux differentiable, and let 𝐶 be a nonempty, closed subset of 𝐸. We say that Proj_𝐶^𝑓,𝑔∶𝐸^∗→2^𝐶 is a Bregman generalized 𝑓-projection operator if Proj_𝐶^𝑓,𝑔=𝑧∈𝐶∶𝐻𝑧,𝑥^∗=inf_𝑦∈𝐶𝐻𝑦,𝑥^∗,∀𝑥^∗∈𝐸^∗. (6)

In this paper, using Bregman functions, we introduce the new concept of Bregman generalized 𝑓-projection operator Proj_𝐶^𝑓,𝑔∶𝐸^∗→𝐶, where 𝐸 is a reflexive Banach space with dual space 𝐸^∗, 𝑓∶𝐸→ℝ∪{+∞} is a proper, convex, lower semicontinuous, and bounded from below function, 𝑔∶𝐸→ℝ is a strictly convex and Gâteaux differentiable function, and 𝐶 is a nonempty, closed, and convex subset of 𝐸. The existence of a solution for a class of variational inequalities in Banach spaces is presented. Our results improve and generalize some known results in the current literature; see, for example, [20, 21].

2. Properties of Bregman Functions and Bregman Distances

Let 𝐸 be a (real) Banach space, and let 𝑔∶𝐸→ℝ. For any 𝑥 in 𝐸, the gradient ∇𝑔(𝑥) is defined to be the linear functional in 𝐸^∗ such that 𝑦,∇𝑔(𝑥)=lim_𝑡→0𝑔𝑥+𝑡𝑦−𝑔(𝑥)𝑡,∀𝑦∈𝐸. (7) The function 𝑔 is said to be Gâteaux differentiable at 𝑥 if ∇𝑔(𝑥) is well defined, and 𝑔 is Gâteaux differentiable if it is Gâteaux differentiable everywhere on 𝐸. We call 𝑔 Fréchet differentiable at 𝑥 (see, for example, [22, page 13] or [23, page 508]) if, for all 𝜖>0, there exists 𝛿>0 such that 𝑔𝑦−𝑔(𝑥)−𝑦−𝑥,∇𝑔(𝑥)≤𝜖𝑦−𝑥whenever𝑦−𝑥≤𝛿. (8) The function 𝑔 is said to be Fréchet differentiable if it is Fréchet differentiable everywhere.

For any 𝑟>0, let 𝐵_𝑟∶={𝑧∈𝐸∶‖𝑧‖≤𝑟}. A function 𝑔∶𝐸→ℝ is said to be(i)strongly coercive if lim_{‖𝑥𝑛‖→+∞}𝑔𝑥_𝑛𝑥_𝑛=+∞; (9)(ii)locally bounded if 𝑔(𝐵_𝑟) is bounded for all 𝑟>0;(iii)locally uniformly smooth on 𝐸 ([24, pages 207, 221]) if the function 𝜎_𝑟∶[0,+∞)→[0,+∞], defined by 𝜎_𝑟(𝑡)=sup_{𝑥∈𝐵𝑟,𝑦∈𝑆𝐸,𝛼∈(0,1)}()+()𝑔(𝑥)×(𝛼(𝛼𝑔𝑥+1−𝛼𝑡𝑦1−𝛼𝑥−𝛼𝑡𝑦−𝑔1−𝛼))⁻¹, (10) satisfies lim_𝑡↓0𝜎_𝑟(𝑡)𝑡=0,∀𝑟>0; (11)(iv)locally uniformly convex on 𝐸 (or uniformly convex on bounded subsets of 𝐸 ([24, pages 203, 221])) if the gauge 𝜌_𝑟∶[0,+∞)→[0,+∞] of uniform convexity of 𝑔, defined by 𝜌_𝑟(𝑡)=inf_{𝑥,𝑦∈𝐵𝑟,‖𝑥−𝑦‖=𝑡,𝛼∈(0,1)}(𝑥)+()𝑔𝑦()𝑦×(𝛼(𝛼𝑔1−𝛼−𝑔𝛼𝑥+1−𝛼1−𝛼))⁻¹, (12) satisfies 𝜌_𝑟(𝑡)>0,∀𝑟,𝑡>0. (13)

For a locally uniformly convex map 𝑔∶𝐸→ℝ, we have 𝑔()𝑦(𝑥)+()𝑔𝑦()𝜌𝛼𝑥+1−𝛼≤𝛼𝑔1−𝛼−𝛼1−𝛼_𝑟𝑥−𝑦, (14) for all 𝑥,𝑦 in 𝐵_𝑟 and for all 𝛼 in (0,1).

Let 𝐸 be a Banach space and 𝑔∶𝐸→ℝ a strictly convex and Gâteaux differentiable function. By (1), the Bregman distance satisfies [16] 𝐷_𝑔(𝑥,𝑧)=𝐷_𝑔𝑥,𝑦+𝐷_𝑔+𝑦(𝑧𝑦,𝑧𝑥−𝑦,∇𝑔−∇𝑔),∀𝑥,𝑦,𝑧∈𝐸. (15) In particular, 𝐷_𝑔𝑥,𝑦=−𝐷_𝑔𝑦,𝑥+𝑦−𝑥,∇𝑔𝑦−∇𝑔(𝑥),∀𝑥,𝑦∈𝐸. (16)

We call a function 𝑔∶𝐸→(−∞,+∞] lower semicontinuous if {𝑥∈𝐸∶𝑔(𝑥)≤𝑟} is closed for all 𝑟 in ℝ. For a lower semicontinuous convex function 𝑔∶𝐸→ℝ, the subdifferential 𝜕𝑔 of 𝑔 is defined by 𝜕𝑔(𝑥)=𝑥^∗∈𝐸^∗∶𝑔(𝑥)+𝑦−𝑥,𝑥^∗≤𝑔𝑦,∀𝑦∈𝐸 (17) for all 𝑥 in 𝐸. It is well known that 𝜕𝑔⊂𝐸×𝐸^∗ is maximal monotone [25, 26]. For any lower semicontinuous convex function 𝑔∶𝐸→(−∞,+∞], the conjugate function 𝑔^∗ of 𝑔 is defined by 𝑔^∗𝑥^∗=sup_𝑥∈𝐸⟨𝑥,𝑥^∗⟩−𝑔(𝑥),∀𝑥^∗∈𝐸^∗. (18) It is well known that 𝑔(𝑥)+𝑔^∗𝑥^∗≥𝑥,𝑥^∗,∀𝑥,𝑥^∗∈𝐸×𝐸^∗,𝑥,𝑥^∗∈𝜕𝑔isequivalentto𝑔(𝑥)+𝑔^∗𝑥^∗=𝑥,𝑥^∗. (19)(20) We also know that if 𝑔∶𝐸→(−∞,+∞] is a proper lower semicontinuous convex function, then 𝑔^∗∶𝐸^∗→(−∞,+∞] is a proper weak* lower semicontinuous convex function. Here, saying 𝑔 is proper we mean that dom𝑔∶={𝑥∈𝐸∶𝑔(𝑥)<+∞}≠∅.

The following definition is slightly different from that in Butnariu and Iusem [22].

Definition 2 (see [23]). Let 𝐸 be a Banach space. A function 𝑔∶𝐸→ℝ is said to be a Bregman function if the following conditions are satisfied: (1)𝑔 is continuous, strictly convex, and Gâteaux differentiable;(2)the set {𝑦∈𝐸∶𝐷_𝑔(𝑥,𝑦)≤𝑟} is bounded for all 𝑥 in 𝐸 and 𝑟>0.

The following lemma follows from Butnariu and Iusem [22] and Ză‌linescu [24].

Lemma 3. Let 𝐸 be a reflexive Banach space and 𝑔∶𝐸→ℝ a strongly coercive Bregman function. Then (1)∇𝑔∶𝐸→𝐸^∗ is one-to-one, onto, and norm-to-weak* continuous;(2)⟨𝑥−𝑦,∇𝑔(𝑥)−∇𝑔(𝑦)⟩=0 if and only if 𝑥=𝑦;(3){𝑥∈𝐸∶𝐷_𝑔(𝑥,𝑦)≤𝑟} is bounded for all 𝑦 in 𝐸 and 𝑟>0;(4)𝑑𝑜𝑚𝑔^∗=𝐸^∗,𝑔^∗ is Gâteaux differentiable and ∇𝑔^∗=(∇𝑔)⁻¹.

The following two results follow from [24, Proposition 3.6.4].

Proposition 4. Let 𝐸 be a reflexive Banach space and let 𝑔∶𝐸→ℝ be a convex function which is locally bounded. The following assertions are equivalent:(1)𝑔 is strongly coercive and locally uniformly convex on 𝐸;(2)𝑑𝑜𝑚𝑔^∗=𝐸^∗,𝑔^∗ is locally bounded and locally uniformly smooth on 𝐸;(3)𝑑𝑜𝑚𝑔^∗=𝐸^∗,𝑔^∗ is Fréchet differentiable and ∇𝑔^∗ is uniformly norm-to-norm continuous on bounded subsets of 𝐸^∗.

Proposition 5. Let 𝐸 be a reflexive Banach space and 𝑔∶𝐸→ℝ a continuous convex function which is strongly coercive. The following assertions are equivalent: (1)𝑔 is locally bounded and locally uniformly smooth on 𝐸;(2)𝑔^∗ is Fréchet differentiable and ∇𝑔^∗ is uniformly norm-to-norm continuous on bounded subsets of 𝐸;(3)𝑑𝑜𝑚𝑔^∗=𝐸^∗,𝑔^∗ is strongly coercive and locally uniformly convex on 𝐸.

Let 𝐸 be a Banach space and let 𝐶 be a nonempty convex subset of 𝐸. Let 𝑔∶𝐸→ℝ be a strictly convex and Gâteaux differentiable function. Then, we know from [27] that for 𝑥 in 𝐸 and 𝑥₀ in 𝐶, we have 𝐷_𝑔𝑥₀,𝑥=min_𝑦∈𝐶𝐷_𝑔𝑦,𝑥iff𝑦−𝑥₀,∇𝑔(𝑥)−∇𝑔𝑥₀≤0,∀𝑦∈𝐶. (21) Further, if 𝐶 is a nonempty, closed, and convex subset of a reflexive Banach space 𝐸 and 𝑔∶𝐸→ℝ is a strongly coercive Bregman function, then, for each 𝑥 in 𝐸, there exists a unique 𝑥₀ in 𝐶 such that 𝐷_𝑔𝑥₀,𝑥=min_𝑦∈𝐶𝐷_𝑔𝑦,𝑥. (22) The Bregman projection proj^𝑔_𝐶 from 𝐸 onto 𝐶 defined by proj^𝑔_𝐶(𝑥)=𝑥₀ has the following property: 𝐷_𝑔𝑦,proj^𝑔_𝐶𝑥+𝐷_𝑔proj^𝑔_𝐶𝑥,𝑥≤𝐷_𝑔𝑦,𝑥,∀𝑦∈𝐶,∀𝑥∈𝐸. (23) See [22] for details.

Lemma 6 (see [9]). Let 𝐸 be a Banach space and 𝑔∶𝐸→ℝ a Gâteaux differentiable function which is locally uniformly convex on 𝐸. Let {𝑥_𝑛}_𝑛∈ℕ and {𝑦_𝑛}_𝑛∈ℕ be bounded sequences in 𝐸. Then the following assertions are equivalent:(1)lim_𝑛→∞𝐷_𝑔(𝑥_𝑛,𝑦_𝑛)=0;(2)lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0.

Lemma 7 (see [23, 28]). Let 𝐸 be a reflexive Banach space, let 𝑔∶𝐸→ℝ be a strongly coercive Bregman function, and let 𝑉 be the function defined by 𝑉𝑥,𝑥^∗=𝑔(𝑥)−𝑥,𝑥^∗+𝑔^∗𝑥^∗,∀𝑥∈𝐸,∀𝑥^∗∈𝐸^∗. (24) The following assertions hold: (1)𝐷_𝑔(𝑥,∇𝑔^∗(𝑥^∗))=𝑉(𝑥,𝑥^∗) for all 𝑥 in 𝐸 and 𝑥^∗ in 𝐸^∗;(2)𝑉(𝑥,𝑥^∗)+⟨∇𝑔^∗(𝑥^∗)−𝑥,𝑦^∗⟩≤𝑉(𝑥,𝑥^∗+𝑦^∗) for all 𝑥 in 𝐸 and 𝑥^∗,𝑦^∗ in 𝐸^∗.

It also follows from the definition that 𝑉 is convex in the second variable 𝑥^∗, and 𝑉𝑥,∇𝑔𝑦=𝐷_𝑔𝑥,𝑦. (25)

Lemma 8 (see [29, Proposition 23.1]). Let 𝐸 be a real Banach space and let 𝑓∶𝐸→ℝ∪{+∞} be a lower semicontinuous convex function. Then there exist 𝑥^∗∈𝐸^∗ and 𝑎∈ℝ such that 𝑓(𝑥)≥𝑥^∗(𝑥)+𝑎,∀𝑥∈𝐸. (26)

3. Properties of Bregman 𝑓-Projection Operator Proj_𝐶^𝑓,𝑔

Theorem 9. Let 𝐶 be a nonempty, closed, and convex subset of a reflexive Banach space 𝐸. Let 𝑓∶𝐸→ℝ∪{+∞} be a proper, convex, lower semicontinuous function and let 𝑔∶𝐸→ℝ be strictly convex, continuous, strongly coercive, Gâteaux differentiable, locally bounded, and locally uniformly convex on 𝐸. Then 𝑃𝑟𝑜𝑗_𝐶^𝑓,𝑔(𝑥^∗)≠∅ for all 𝑥^∗∈𝐸^∗.

Proof. Let 𝑥^∗∈𝐸^∗ and 𝜆=inf_𝑦∈𝐶𝐻(𝑦,𝑥^∗). Then there exists a sequence {𝑥_𝑛}_𝑛∈ℕ⊂𝐶 such that 𝜆=lim_𝑛→∞𝐻(𝑥_𝑛,𝑥^∗). We consider the following two possible cases.
Case 1. If 𝐶 is bounded, then there exists a subsequence {𝑥_𝑛𝑗}_𝑗∈ℕ of {𝑥_𝑛}_𝑛∈ℕ and 𝑥∈𝐶 such that 𝑥_𝑛𝑗⇀𝑥 as 𝑗→∞. Since 𝐻(𝑧,𝑥^∗) is convex and lower semicontinuous with respect to 𝑧, we deduce that 𝐻(𝑧,𝑥^∗) is convex and weakly lower semicontinuous with respect to 𝑧. This implies that 𝐻𝑥,𝑥^∗≤liminf_𝑛→∞𝐻𝑥_𝑛,𝑥^∗=lim_𝑛→∞𝐻𝑥_𝑛,𝑥^∗=inf_𝑦∈𝐶𝐻𝑥_𝑛,𝑥^∗ (27) and hence 𝑥∈Proj_𝐶^𝑓,𝑔(𝑥^∗). This shows that Proj_𝐶^𝑓,𝑔≠∅.
Case 2. Assume that 𝐶 is unbounded. Since 𝑓∶𝐶→ℝ∪{+∞} is proper, convex, and lower semicontinuous, we know that the function 𝑓_𝐶∶𝐸→ℝ∪{+∞}, defined by 𝑓_𝐶(𝑥)=𝑓(𝑥),if𝑥∈𝐶,+∞,if𝑥∉𝐶, (28) is proper, convex, and lower semicontinuous. In view of Lemma 8, there exist 𝑥^∗∈𝐸^∗ and 𝑎∈ℝ such that 𝑓_𝐶(𝑥)≥𝑥,𝑥^∗+𝑎,∀𝑥∈𝐸. (29) This implies that for any 𝑥^∗∈𝐸^∗ and 𝑥∈𝐶𝐻𝑥,𝑥^∗=𝑔(𝑥)−𝑥,𝑥^∗+𝑔^∗𝑥^∗(𝑥)(𝑥)+𝑓≥𝑔+𝑔^∗𝑥^∗+𝑎. (30) Next, we show that {𝑥_𝑛}_𝑛∈ℕ is bounded. If not, then there exists a subsequence {𝑥_𝑛𝑗}_𝑗∈ℕ of {𝑥_𝑛}_𝑛∈ℕ such that ‖𝑥_𝑛𝑘‖→+∞ as 𝑘→∞. Since 𝑔 is strongly coercive, we conclude that lim_{‖𝑥𝑛𝑘‖→+∞}𝐻𝑥_𝑛𝑘,𝑥^∗𝑥_𝑛𝑘≥lim_{‖𝑥𝑛𝑘‖→+∞}𝑔𝑥_𝑛𝑘𝑥_𝑛𝑘=+∞. (31) This implies that lim_{‖𝑥𝑛𝑘‖→+∞}𝐻𝑥_𝑛𝑘,𝑥^∗=+∞. (32) Since 𝑓 is proper in 𝐶, we obtain that 𝜆=inf_𝑦∈𝐶𝐻(𝑦,𝑥^∗)=lim_𝑛→∞𝐻(𝑥_𝑛,𝑥^∗)<+∞ which contradicts (31). By a similar argument, as in Case 1, we can prove that Proj_𝐶^𝑓,𝑔(𝑥^∗)≠∅ which completes the proof.

Theorem 10. Let 𝐶 be a nonempty, closed, and convex subset of a reflexive Banach space 𝐸. Let 𝑔∶𝐸→ℝ be strictly convex, continuous, strongly coercive, Gâteaux differentiable, locally bounded, and locally uniformly convex on 𝐸. Then the following assertions hold:(i)for any given 𝑥^∗∈𝐸^∗, 𝑃𝑟𝑜𝑗_𝐶^𝑓,𝑔(𝑥^∗) is a nonempty, closed, and convex subset of 𝐶;(ii)𝑃𝑟𝑜𝑗_𝐶^𝑓,𝑔 is monotone; that is, for any 𝑥^∗,𝑦^∗∈𝐸^∗, 𝑥∈𝑃𝑟𝑜𝑗_𝐶^𝑓,𝑔(𝑥^∗) and 𝑦∈𝑃𝑟𝑜𝑗_𝐶^𝑓,𝑔(𝑦^∗), 𝑥−𝑦,𝑥^∗−𝑦^∗≥0; (33)(iii)For any given 𝑥^∗∈𝐸^∗, 𝑥∈𝑃𝑟𝑜𝑗_𝐶^𝑓,𝑔(𝑥^∗) if and only if 𝑥−𝑦,𝑥^∗−∇𝑔(𝑥)+𝑓𝑦−𝑓(𝑥)≥0; (34)

Proof. (i) Let 𝑥^∗∈𝐸^∗ be fixed. In view of Theorem 9, we conclude that Proj_𝐶^𝑓,𝑔(𝑥^∗)≠∅. According to (20) we have 𝑔(𝑥)+𝑔^∗(𝑥^∗)−⟨𝑥,𝑥^∗⟩≥0,∀(𝑥,𝑥^∗)∈𝐸×𝐸^∗. Let us prove that Proj_𝐶^𝑓,𝑔(𝑥^∗) is closed. Let {𝑥_𝑛}_𝑛∈ℕ⊂Proj_𝐶^𝑓,𝑔(𝑥^∗) and 𝑥_𝑛→𝑥 as 𝑛→∞. In view of (6), we deduce that 𝐺𝑥,𝑥^∗≤liminf_𝑛→∞𝐻𝑥_𝑛,𝑥^∗=liminf_𝑛→∞𝐻𝑥_𝑛,𝑥^∗=inf_𝑦∈𝐶𝐻𝑦,𝑥^∗. (35) This implies that 𝑥∈Proj_𝐶^𝑓,𝑔(𝑥^∗) and hence Proj_𝐶^𝑓,𝑔(𝑥^∗) is closed. Next, we show that Proj_𝐶^𝑓,𝑔(𝑥^∗) is convex. Let 𝑥₁,𝑥₂∈Proj_𝐶^𝑓,𝑔(𝑥^∗) and 0≤𝑡≤1. By the property (2) of the functional 𝐻, we obtain 𝐻𝑡𝑥₁+(1−𝑡)𝑥₂,𝑥^∗≤𝑡𝐻𝑥₁,𝑥^∗+(1−𝑡)𝐻𝑥₂,𝑥^∗=𝑡inf_𝑦∈𝐶𝐻𝑦,𝑥^∗+(1−𝑡)inf_𝑦∈𝐶𝐻𝑦,𝑥^∗=inf_𝑦∈𝐶𝐻𝑦,𝑥^∗. (36) Thus, we have 𝑡𝑥₁+(1−𝑡)𝑥₂∈Proj_𝐶^𝑓,𝑔(𝑥^∗) and hence Proj_𝐶^𝑓,𝑔(𝑥^∗) is convex.
(ii) Let 𝑥^∗₁,𝑥^∗₂∈𝐸^∗, 𝑥₁∈Proj_𝐶^𝑓,𝑔(𝑥^∗₁), and 𝑥₂∈Proj_𝐶^𝑓,𝑔(𝑥^∗₂). Then we have 𝑔𝑥₁−𝑥₁,𝑥^∗₁+𝑔^∗𝑥^∗₁+𝑓𝑥₁≤𝑔𝑥₂−𝑥₂,𝑥^∗₂+𝑔^∗𝑥^∗₂+𝑓𝑥₂,𝑔𝑥₂−𝑥₂,𝑥^∗₂+𝑔^∗𝑥^∗₂+𝑓𝑥₂≤𝑔𝑥₁−𝑥₁,𝑥^∗₁+𝑔^∗𝑥^∗₁+𝑓𝑥₁. (37) In view of (37), we conclude that Proj_𝐶^𝑓,𝑔(𝑥^∗) is monotone.
(iii) It is a simple matter to see that 𝑥∈Proj_𝐶^𝑓,𝑔(𝑥^∗) implies that 𝑥^∗−∇𝑔(𝑥),𝑥−𝑦+𝑓𝑦−𝑓(𝑥)≥0,∀𝑦∈𝐶. (38) To this end, let 𝑦∈𝐶 and 𝑡∈(0,1] be arbitrarily chosen. By the definition of Proj_𝐶^𝑓,𝑔(𝑥^∗) we see that 𝐻𝑥,𝑥^∗≤𝐻𝑥+𝑡𝑦−𝑥,𝑥^∗. (39) Therefore, 𝑔(𝑥)+𝑔^∗𝑥^∗−𝑥,𝑥^∗+𝑓(𝑥)≤𝑔𝑥+𝑡𝑦−𝑥+𝑔^∗𝑥^∗−𝑥+𝑡𝑦−𝑥,𝑥^∗+𝑓𝑥+𝑡𝑦−𝑥≤𝑔𝑥+𝑡𝑦−𝑥+𝑔^∗𝑥^∗−𝑥+𝑡𝑦−𝑥,𝑥^∗+𝑡𝑓𝑦+(1−𝑡)𝑓(𝑥) (40) and hence 𝑡𝑦−𝑥,𝑥^∗≤𝑔𝑥+𝑡𝑦−𝑥+𝑡𝑓𝑦−𝑓(𝑥). (41) On the other hand, by the definition of Bregman distance, we obtain that 𝑔(𝑥)+𝑔𝑥+𝑡𝑦−𝑥≥𝑡𝑥−𝑦,∇𝑔𝑥+𝑡𝑦−𝑥. (42) This, together with (41), implies that 𝑥−𝑦,∇𝑔𝑥+𝑡𝑦−𝑥≥𝑓(𝑥)−𝑓𝑦+𝑥−𝑦,𝑥^∗. (43) Since ∇𝑔 is demi-continuous, letting 𝑡→0 in (43), we conclude that 𝑥−𝑦,∇𝑔(𝑥)−𝑥^∗+𝑓𝑦−𝑓(𝑥)≥0. (44) Conversely, assume that 𝑥−𝑦,∇𝑔(𝑥)−𝑥^∗+𝑓𝑦−𝑓(𝑥)≥0,∀𝑦∈𝐾. (45) This implies that 𝑔𝑦(𝑥)≥(𝑥≥−𝑔𝑥−𝑦,∇𝑔)𝑥−𝑦,𝑥^∗+𝑓𝑦−𝑓(𝑥)≥0∀𝑦∈𝐾. (46)

4. Applications to Variational Inequalities

In this section, we investigate the existence of solution to the following variational inequality problem: find the point 𝑥∈𝐶 such that 𝑦−𝑥,𝐴𝑥+𝑓𝑦−𝑓(𝑥)≥0,∀𝑦∈𝐶, (47) where 𝐶 is a nonempty, closed, and convex subset of the Banach space 𝐸, and 𝐴∶𝐶→𝐸^∗ and 𝑓∶𝐶→ℝ∪{+∞} are two mappings.

Definition 11 (KKM mapping [30]). Let 𝐶 be a nonempty subset of a linear space 𝑋. A set-valued mapping 𝐺∶𝐶→2^𝑋 is called a KKM mapping if, for any finite subset {𝑦₁,𝑦₂,…,𝑦_𝑛} of 𝐶, we have co𝑦₁,𝑦₂,…,𝑦_𝑛⊂^𝑛_𝑖=1𝐺𝑦_𝑖, (48) where co{𝑦₁,𝑦₂,…,𝑦_𝑛} denotes the convex hull of {𝑦₁,𝑦₂,…,𝑦_𝑛}.

Lemma 12 (Fan KKM Theorem [30]). Let 𝐶 be a nonempty convex subset of a Hausdorff topological vector 𝑋 and let 𝐺∶𝐶→2^𝑋 be a KKM mapping with closed values. If there exists a point 𝑦₀∈𝐶 such that 𝐺(𝑦₀) is a compact subset of 𝐶, then ⋂_𝑦∈𝐶𝐺(𝑦)≠∅.

Theorem 13. Let 𝐶 be a nonempty, closed, and convex subset of a reflexive Banach space 𝐸 with dual space 𝐸^∗. Let 𝑔∶𝐸→ℝ be strictly convex, continuous, strongly coercive, Gâteaux differentiable, locally bounded and locally uniformly convex on 𝐸. Let 𝐴∶𝐶→𝐸^∗ be a continuous mapping and 𝑓∶𝐸→ℝ∪{+∞} be a proper, convex, lower semicontinuous function. If there exists an element 𝑦₀∈𝐶 such that 𝑥∈𝐶∶𝑦₀(𝑥)(𝑥)(𝑥)𝑦−𝑥,∇𝑔−𝐴𝑥+𝑔+𝑓≤𝑔₀+𝑓𝑦₀ (49) is a compact subset of 𝐶, then the variational inequality (47) has a solution.

Proof. In view of Theorem 10, we need to prove that the following inclusion has a solution: 𝑥∈Proj_𝐶^𝑓,𝑔∇𝑔(𝑥)−𝐴𝑥. (50) We define a set-valued mapping 𝑉∶𝐶→2^𝐶 by 𝑉𝑦=𝑥∈𝐶∶𝐻𝑥,∇𝑔(𝑥)−𝐴𝑥≤𝐻𝑦,∇𝑔(𝑥)−𝐴𝑥. (51) It is obvious that, for any 𝑦∈𝐶, 𝑉(𝑦)≠∅. Let us prove that 𝑉(𝑦) is closed for any 𝑦∈𝐶. Let {𝑥_𝑛}_𝑛∈ℕ⊂𝑉(𝑦) and 𝑥_𝑛→𝑥 as 𝑛→∞. Then, 𝐻𝑥_𝑛,∇𝑔𝑥_𝑛−𝐴𝑥_𝑛≤𝐻𝑦,∇𝑔𝑥_𝑛−𝐴𝑥_𝑛. (52) This implies that −𝑥_𝑛,∇𝑔𝑥_𝑛−𝐴𝑥_𝑛+𝑔𝑥_𝑛+𝑓𝑥_𝑛≤−𝑦,∇𝑔𝑥_𝑛−𝐴𝑥_𝑛+𝑔𝑦+𝑓𝑦. (53) Since ∇𝑔 and 𝐴 are continuous and 𝑓 is lower semicontinuous, we conclude that −(𝑥)(𝑥)(𝑥)(𝑥)𝑦𝑦.𝑥,∇𝑔−𝐴𝑥+𝑔+𝑓≤−𝑦,∇𝑔−𝐴𝑥+𝑔+𝑓 (54) Therefore, 𝐻𝑥,∇𝑔(𝑥)−𝐴𝑥≤𝐻𝑦,∇𝑔(𝑥)−𝐴𝑥, (55) which implies that 𝑥∈𝑉(𝑦). Now, we prove that 𝑉∶𝐶→2^𝐶 is a KKM mapping. Indeed, suppose 𝑦₁,𝑦₂,…,𝑦_𝑛∈𝐶 and 0<𝑎₁,𝑎₂,…,𝑎_𝑛≤1 with ∑^𝑛_𝑖=1𝑎_𝑖=1. Let 𝑧=∑^𝑛_𝑖=1𝑎_𝑖𝑦_𝑖. In view of the property (2) of 𝐻, we obtain 𝐻𝑧,∇𝑔(𝑧)−𝐴𝑧=𝐻^𝑛_𝑖=1𝑎_𝑖𝑦_𝑖,∇𝑔(𝑧)−𝐴𝑧≤^𝑛_𝑖=1𝑎_𝑖𝐻𝑦_𝑖,∇𝑔(𝑧)−𝐴𝑧 (56) and hence 𝐻𝑧,∇𝑔(𝑧)−𝐴𝑧≤max_{1≤𝑖≤𝑛}𝐻𝑦_𝑖,∇𝑔(𝑧)−𝐴𝑧. (57) Hence there exists at least one number 𝑗=1,2,…,𝑛, such that 𝐻𝑧,∇𝑔(𝑧)−𝐴𝑧≤𝐻𝑦_𝑗,∇𝑔(𝑧)−𝐴𝑧. (58) that is, 𝑧∈𝑉(𝑦). Thus, 𝑉 is a KKM mapping.
If 𝑥∈𝑉(𝑦₀), then 𝐻(𝑧,∇𝑔(𝑧)−𝐴𝑧)≤𝐻(𝑦₀,∇𝑔(𝑧)−𝐴𝑧). By the definition of 𝐻, we obtain −(𝑥)(𝑥)(𝑥)𝑦𝑥,∇𝑔−𝐴𝑥+𝑔+𝑓≤−₀,∇𝑔(𝑥)−𝐴𝑥+𝑔𝑦₀+𝑓𝑦₀ (59) which is equivalent to 𝑦₀−𝑥,∇𝑔(𝑥)−𝐴𝑥+𝑔(𝑥)+𝑓(𝑥)≤𝑔𝑦₀+𝑓𝑦₀. (60) Therefore, 𝑉𝑦₀=𝑥∈𝐶∶∇𝑔(𝑥)−𝐴𝑥,𝑦₀(𝑥)(𝑥)𝑦−𝑥+𝑔+𝑓≤𝑔₀+𝑓𝑦₀. (61) In view of (49), we deduce that 𝑉(𝑦₀) is compact. It follows from Lemma 12 that ⋂_𝑦∈𝐶𝑉(𝑦)≠∅. Hence there exists at least one 𝑥₀∈⋂_𝑦∈𝐶𝑉(𝑦)); that is, 𝐻𝑥₀,∇𝑔𝑥₀−𝐴𝑥₀≤𝐻𝑦,∇𝑔𝑥₀−𝐴𝑥₀,∀𝑦∈𝐶. (62) In view of the definition of Bregman 𝑓-projection operator Proj_𝐶^𝑓,𝑔, we conclude that 𝑥₀∈Proj_𝐶^𝑓,𝑔∇𝑔𝑥₀−𝐴𝑥₀. (63) This completes the proof.

Theorem 14. Let 𝐸 be a reflexive Banach space and 𝑔∶𝐸→ℝ a strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets of 𝐸. Let 𝑓∶𝐸→ℝ∪{+∞} be a proper, convex, lower semicontinuous function. Let 𝐶 be a nonempty, closed, and convex subset of 𝐸 and let 𝑇∶𝐶→𝐶 be a Bregman weak relatively nonexpansive mapping. Let {𝛼_𝑛}_{𝑛∈ℕ∪{0}} be a sequence in (0,1) such that liminf_𝑛→∞𝛼_𝑛(1−𝛼_𝑛)>0. Let {𝑥_𝑛}_{𝑛∈ℕ∪{0}} be a sequence generated by 𝑥₀𝐶=𝑥∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,₀𝑦=𝐶,_𝑛=∇𝑔^∗𝛼_𝑛∇𝑔𝑥_𝑛+1−𝛼_𝑛∇𝑔𝑇𝑥_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝐻𝑧,∇𝑔𝑦_𝑛≤𝐻𝑧,∇𝑔𝑥_𝑛,𝑥_𝑛+1=𝑃𝑟𝑜𝑗^𝑔_𝐶𝑛+1𝑥,𝑛∈ℕ∪{0}, (64) where ∇𝑔 is the gradient of 𝑔. Then {𝑥_𝑛}_𝑛∈ℕ, {𝑇𝑥_𝑛}_𝑛∈ℕ, and {𝑦_𝑛}_𝑛∈ℕ converge strongly to 𝑃𝑟𝑜𝑗^𝑔_𝐹𝑥₀.

Proof. We divide the proof into several steps.
Step 1. We prove that 𝐶_𝑛 is closed and convex for each 𝑛∈ℕ∪{0}.
It is clear that 𝐶₀=𝐶 is closed and convex. Let 𝐶_𝑚 be closed and convex for some 𝑚∈ℕ. For 𝑧∈𝐶_𝑚, we see that 𝐻𝑧,∇𝑔𝑦_𝑚≤𝐻𝑧,∇𝑔𝑥_𝑚 (65) is equivalent to 𝑧,∇𝑔𝑥_𝑚−∇𝑔𝑦_𝑚≤𝑔𝑦_𝑚−𝑔𝑥_𝑚+𝑥_𝑚,∇𝑔𝑥_𝑚−𝑦_𝑚,∇𝑔𝑦_𝑚. (66) It could easily be seen that 𝐶_𝑚+1 is closed and convex. Therefore, 𝐶_𝑛 is closed and convex for each 𝑛∈ℕ∪{0}.
Step 2. We claim that 𝐹⊂𝐶_𝑛 for all 𝑛∈ℕ∪{0}.
It is obvious that 𝐹⊂𝐶₀=𝐶. Assume now that 𝐹⊂𝐶_𝑚 for some 𝑚∈ℕ. Employing Lemma 7, for any 𝑤∈𝐹⊂𝐶_𝑚, we obtain 𝐻𝑤,∇𝑔𝑦_𝑚=𝐻𝑤,∇𝑔𝑦_𝑚=𝑔(𝑤)−𝑤,∇𝑔𝑦_𝑚+𝑔^∗∇𝑔𝑦_𝑚+𝑓(𝑤)=𝑉𝑤,𝛼_𝑚∇𝑔𝑥_𝑚+1−𝛼_𝑚∇𝑔𝑇𝑥_𝑚(𝑤)(𝑤)−+𝑓=𝑔𝑤,𝛼_𝑚∇𝑔𝑥_𝑚+1−𝛼_𝑚∇𝑔𝑇𝑥_𝑚+𝑔^∗𝛼_𝑚∇𝑔𝑥_𝑚+1−𝛼_𝑚∇𝑔𝑇𝑥_𝑚+𝑓(𝑤)≤𝛼_𝑚𝑔(𝑤)+1−𝛼_𝑚𝑔(𝑤)+𝛼_𝑚𝑔^∗∇𝑔𝑥_𝑚+1−𝛼_𝑚𝑔^∗∇𝑔𝑇𝑥_𝑚+𝑓(𝑤)=𝛼_𝑚𝑉𝑤,∇𝑔𝑥_𝑚+1−𝛼_𝑚𝑉𝑤,∇𝑔𝑇𝑥_𝑚+𝑓(𝑤)=𝛼_𝑚𝐷_𝑔𝑤,𝑥_𝑚+1−𝛼_𝑚𝐷_𝑔𝑤,𝑇𝑥_𝑚+𝑓(𝑤)≤𝛼_𝑚𝐷_𝑔𝑤,𝑥_𝑚+1−𝛼_𝑚𝐷_𝑔𝑤,𝑥_𝑚+𝑓(𝑤)=𝐷_𝑔𝑤,𝑥_𝑚(𝑤)𝑥+𝑓=𝑉𝑤,∇𝑔_𝑚(𝑤)𝑥+𝑓=𝐻𝑤,∇𝑔_𝑚. (67) This proves that 𝑤∈𝐶_𝑚+1 and hence 𝐹⊂𝐶_𝑛 for all 𝑛∈ℕ∪{0}.
Step 3. We prove that {𝑥_𝑛}_𝑛∈ℕ,{𝑦_𝑛}_𝑛∈ℕ, and {𝑇𝑥_𝑛}_𝑛∈ℕ are bounded sequences in 𝐶.
Since 𝑥_𝑛=proj^𝑔_𝐶𝑛𝑥, we get that 𝐻𝑥_𝑛,∇𝑔(𝑥)≤𝐻𝑤,∇𝑔(𝑥) (68) for each 𝑤∈𝐹(𝑇). This implies that the sequence {𝐻(𝑤,∇𝑔(𝑥_𝑛))}_𝑛∈ℕ is bounded and hence there exists 𝑀₁>0 such that 𝐻𝑥_𝑛,∇𝑔(𝑥)≤𝑀₁,∀𝑛∈ℕ. (69) We claim that the sequence {𝑥_𝑛}_𝑛∈ℕ is bounded. Assume on the contrary that ‖𝑥_𝑛‖→∞ as 𝑛→∞. In view of Lemma 8, there exist 𝑥^∗∈𝐸^∗ and 𝑎∈ℝ such that 𝑓(𝑥)≥𝑥_𝑛,𝑥^∗+𝑎,∀𝑛∈ℕ. (70) From the definition of Bregman distance, it follows that 𝑀₁≥𝐻𝑥_𝑛(𝑥)𝑥,∇𝑔=𝑔_𝑛−𝑔(𝑥)−𝑥_𝑛−𝑥,∇𝑔(𝑥)+𝑓𝑥_𝑛≥𝑔𝑥_𝑛−𝑔(𝑥)−𝑥_𝑛,∇𝑔(𝑥)−𝑥^∗+(𝑥𝑥𝑥,∇𝑔)+𝑎≥𝑔_𝑛−𝑔(𝑥)−𝑥_𝑛∇𝑔(𝑥)−𝑥^∗+𝑥,∇𝑔(𝑥)+𝑎,∀𝑛∈ℕ. (71) Without loss of generality, we may assume that ‖𝑥_𝑛‖≠0 for each 𝑛∈ℕ. This implies that 𝑀₁𝑥_𝑛≥𝑔𝑥_𝑛𝑥_𝑛−𝑔(𝑥)𝑥_𝑛−∇𝑔(𝑥)−𝑥^∗+𝑥,∇𝑔(𝑥)𝑥_𝑛+𝑎𝑥_𝑛,∀𝑛∈ℕ. (72) Since 𝑔 is strongly coercive, by letting 𝑛→∞ in (72), we conclude that 0≥∞, which is a contradiction. Therefore, {𝑥_𝑛}_𝑛∈ℕ is bounded. Since {𝑇_𝑛}_𝑛∈ℕ is an infinite family of Bregman weak relatively nonexpansive mappings from 𝐶 into itself, we have for any 𝑞∈𝐹 that 𝐷_𝑔𝑞,𝑇𝑥_𝑛≤𝐷_𝑔𝑞,𝑥_𝑛,∀𝑛∈ℕ. (73) This, together with Definition 2 and the boundedness of {𝑥_𝑛}_𝑛∈ℕ, implies that the sequence {𝑇_𝑛𝑥_𝑛}_𝑛∈ℕ is bounded.
Step 4. We show that 𝑥_𝑛→𝑣 for some 𝑣∈𝐹, where 𝑣=proj^𝑔_𝐹𝑥.
From Step 3 we know that {𝑥_𝑛}_𝑛∈ℕ is bounded. By the construction of 𝐶_𝑛, we conclude that 𝐶_𝑚⊂𝐶_𝑛 and 𝑥_𝑚=proj^𝑔_𝐶𝑚𝑥∈𝐶_𝑚⊂𝐶_𝑛 for any positive integer 𝑚≥𝑛. This, together with (23), implies that 𝐷_𝑔𝑥_𝑚,𝑥_𝑛=𝐷_𝑔𝑥_𝑚,proj^𝑔_𝐶𝑛𝑥≤𝐷_𝑔𝑥_𝑚,𝑥−𝐷_𝑔proj^𝑔_𝐶𝑛𝑥,𝑥=𝐷_𝑔𝑥_𝑚,𝑥−𝐷_𝑔𝑥_𝑛,𝑥. (74) In view of (21), we conclude that 𝐷_𝑔𝑥_𝑛,𝑥=𝐷_𝑔proj^𝑔_𝐶𝑛𝑥,𝑥≤𝐷_𝑔(𝑤,𝑥)−𝐷_𝑔𝑤,𝑥_𝑛≤𝐷_𝑔(𝑤,𝑥),∀𝑤∈𝐹⊂𝐶_𝑛,𝑛∈ℕ∪{0}. (75) It follows from (75) that the sequence {𝐷_𝑔(𝑥_𝑛,𝑥)}_𝑛∈ℕ is bounded and hence there exists 𝑀₂>0 such that 𝐷_𝑔𝑥_𝑛,𝑥≤𝑀₂,∀𝑛∈ℕ. (76) In view of (64), we conclude that 𝐷_𝑔𝑥_𝑛,𝑥≤𝐷_𝑔𝑥_𝑛,𝑥+𝐷_𝑔𝑥_𝑚,𝑥_𝑛≤𝐷_𝑔𝑥_𝑚,𝑥,∀𝑚≥𝑛. (77) This proves that {𝐷_𝑔(𝑥_𝑛,𝑥)}_𝑛∈ℕ is an increasing sequence in ℝ and hence the limit lim_𝑛→∞𝐷_𝑔(𝑥_𝑛,𝑥) exists. Letting 𝑚,𝑛→∞ in (74), we deduce that 𝐷_𝑔(𝑥_𝑚,𝑥_𝑛)→0. In view of Lemma 6, we obtain that ‖𝑥_𝑚−𝑥_𝑛‖→0 as 𝑚,𝑛→∞. This means that {𝑥_𝑛}_𝑛∈ℕ is a Cauchy sequence. Since 𝐸 is a Banach space and 𝐶 is closed and convex, we conclude that there exists 𝑣∈𝐶 such that lim_𝑛→∞𝑥_𝑛−𝑣=0. (78) Now, we show that 𝑣∈𝐹. In view of Lemma 6 and (78), we obtain lim_𝑛→∞𝐷_𝑔𝑥_𝑛+1,𝑥_𝑛=0. (79) Since 𝑥_𝑛+1∈𝐶_𝑛+1, we conclude that 𝐷_𝑔𝑥_𝑛+1,𝑦_𝑛≤𝐷_𝑔𝑥_𝑛+1,𝑥_𝑛. (80) This, together with (79), implies that lim_𝑛→∞𝐷_𝑔𝑥_𝑛+1,𝑦_𝑛=0. (81) It follows from Lemma 6, (79), and (81) that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0,lim_𝑛→∞𝑥_𝑛+1−𝑦_𝑛=0. (82) In view of (78), we get lim_𝑛→∞𝑦_𝑛−𝑢=0. (83) From (78) and (83), it follows that lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0. (84) Since ∇𝑔 is uniformly norm-to-norm continuous on any bounded subset of 𝐸, we obtain lim_𝑛→∞‖∇𝑔𝑥_𝑛−∇𝑔𝑦_𝑛‖=0. (85) Applying Lemma 6 we derive that lim_𝑛→∞𝐷_𝑔𝑦_𝑛,𝑥_𝑛=0. (86) It follows from the three-point identity (see (14)) that for any 𝑤∈𝐹𝐷_𝑔𝑤,𝑥_𝑛−𝐷_𝑔𝑤,𝑦_𝑛=𝐷_𝑔𝑤,𝑦_𝑛+𝐷_𝑔𝑦_𝑛,𝑥_𝑛+𝑤−𝑦_𝑛,∇𝑔𝑦_𝑛−∇𝑔𝑥_𝑛−𝐷_𝑔𝑤,𝑦_𝑛=𝐷_𝑔𝑦_𝑛,𝑥_𝑛−𝑤−𝑦_𝑛,∇𝑔𝑦_𝑛−∇𝑔𝑥_𝑛≤𝐷_𝑔𝑦_𝑛,𝑥_𝑛+𝑤−𝑦_𝑛∇𝑔𝑦_𝑛−∇𝑔𝑥_𝑛⟶0 (87) as 𝑛→∞.
The function 𝑔 is bounded on bounded subsets of 𝐸 and, thus, ∇𝑔 is also bounded on bounded subsets of 𝐸^∗ (see, e.g., [22, Proposition 1.1.11], for more details). This implies that the sequences {∇𝑔(𝑥_𝑛)}_𝑛∈ℕ, {∇𝑔(𝑦_𝑛)}_𝑛∈ℕ, and {∇𝑔(𝑇𝑥_𝑛)∶𝑛∈ℕ∪{0}} are bounded in 𝐸^∗.
In view of Proposition 4(3), we know that dom 𝑔^∗=𝐸^∗ and 𝑔^∗ is strongly coercive and uniformly convex on bounded subsets of 𝐸^∗. Let 𝑠₁=sup{‖∇𝑔(𝑥_𝑛)‖,‖∇𝑔(𝑇𝑥_𝑛)‖∶𝑛∈ℕ∪{0}} and 𝜌^∗_𝑠1∶𝐸^∗→ℝ be the gauge of uniform convexity of the conjugate function 𝑔^∗. We prove that for any 𝑤∈𝐹𝐷_𝑔𝑤,𝑦_𝑛≤𝐷_𝑔𝑤,𝑥_𝑛−𝛼_𝑛1−𝛼_𝑛𝜌^∗_𝑠1×∇𝑔𝑥_𝑛−∇𝑔𝑇𝑥_𝑛. (88)
Let us show (88). For any given 𝑤∈𝐹(𝑇), in view of the definition of the Bregman distance (see (2)) and Lemma 6, we obtain 𝐷_𝑔𝑤,𝑦_𝑛=𝐷_𝑔𝑤,∇𝑔^∗𝛼_𝑛∇𝑔𝑥_𝑛+1−𝛼_𝑛∇𝑔𝑇𝑥_𝑛=𝑉𝑤,𝛼_𝑛∇𝑔𝑥_𝑛+1−𝛼_𝑛∇𝑔𝑇𝑥_𝑛=𝑔(𝑤)−𝑤,𝛼_𝑛∇𝑔𝑥_𝑛+1−𝛼_𝑛∇𝑔𝑇𝑥_𝑛+𝑔^∗𝛼_𝑛∇𝑔𝑥_𝑛+1−𝛼_𝑛∇𝑔𝑇𝑥_𝑛≤𝛼_𝑛𝑔(𝑤)+1−𝛼_𝑛𝑔(𝑤)−𝛼_𝑛𝑤,∇𝑔𝑥_𝑛−1−𝛼_𝑛𝑤,∇𝑔𝑇𝑥_𝑛+𝛼_𝑛𝑔^∗∇𝑔𝑥_𝑛+1−𝛼_𝑛𝑔^∗∇𝑔𝑇𝑥_𝑛−𝛼_𝑛1−𝛼_𝑛𝜌^∗_𝑠1∇𝑔𝑥_𝑛−∇𝑔𝑇𝑥_𝑛=𝛼_𝑛𝑉𝑤,∇𝑔𝑥_𝑛+1−𝛼_𝑛𝑉𝑤,∇𝑔𝑇𝑥_𝑛−𝛼_𝑛1−𝛼_𝑛𝜌^∗_𝑠1∇𝑔𝑥_𝑛−∇𝑔𝑇_𝑛𝑥_𝑛=𝛼_𝑛𝐷_𝑔𝑤,𝑥_𝑛+1−𝛼_𝑛𝐷_𝑔𝑤,𝑇𝑥_𝑛−𝛼_𝑛1−𝛼_𝑛𝜌^∗_𝑠1∇𝑔𝑥_𝑛−∇𝑔𝑇𝑥_𝑛≤𝛼_𝑛𝐷_𝑔𝑤,𝑥_𝑛+1−𝛼_𝑛𝐷_𝑔𝑤,𝑥_𝑛−𝛼_𝑛1−𝛼_𝑛𝜌^∗_𝑠1∇𝑔𝑥_𝑛−∇𝑔𝑇𝑥_𝑛=𝐷_𝑔𝑤,𝑥_𝑛−𝛼_𝑛1−𝛼_𝑛𝜌^∗_𝑠1∇𝑔𝑥_𝑛−∇𝑔𝑇𝑥_𝑛. (89) In view of (87), we get that 𝐷_𝑔𝑤,𝑥_𝑛−𝐷_𝑔𝑤,𝑦_𝑛⟶0as𝑛⟶∞. (90) In view of (87) and (88), we conclude that 𝛼_𝑛1−𝛼_𝑛𝜌^∗_𝑠1∇𝑔𝑥_𝑛−∇𝑔𝑇𝑥_𝑛≤𝐷_𝑔𝑤,𝑥_𝑛−𝐷_𝑔𝑤,𝑦_𝑛⟶0 (91) as 𝑛→∞. From the assumption liminf_𝑛→∞𝛼_𝑛(1−𝛼_𝑛)>0, we get lim_𝑛→∞𝜌^∗_𝑠1∇𝑔𝑥_𝑛−∇𝑔𝑇𝑥_𝑛=0. (92) Therefore, from the property of 𝜌^∗_𝑠1 we deduce that lim_𝑛→∞∇𝑔𝑥_𝑛−∇𝑔𝑇𝑥_𝑛=0. (93) Since ∇𝑔^∗ is uniformly norm-to-norm continuous on bounded subsets of 𝐸^∗, we arrive at lim_𝑛→∞𝑥_𝑛−𝑇𝑥_𝑛=0. (94) This implies that 𝑣∈𝐹(𝑇).
Finally, we show that 𝑣=proj^𝑔_𝐹𝑥. From 𝑥_𝑛=proj^𝑔_𝐶𝑛𝑥, we conclude that 𝑧−𝑥_𝑛,∇𝑔𝑥_𝑛−∇𝑔(𝑥)≥0,∀𝑧∈𝐶_𝑛. (95) Since 𝐹⊂𝐶_𝑛 for each 𝑛∈ℕ, we obtain 𝑧−𝑥_𝑛,∇𝑔𝑥_𝑛−∇𝑔(𝑥)≥0,∀𝑧∈𝐹. (96) Letting 𝑛→∞ in (96), we deduce that 𝑧−𝑣,∇𝑔(𝑢)−∇𝑔(𝑥)≥0,∀𝑧∈𝐹. (97) In view of (21), we have 𝑣=proj^𝑔_𝐹𝑥, which completes the proof.

Remark 15. Theorem 14 improves Theorem 4.1 of [20] in the following aspects.(1)For the structure of Banach spaces, we extend the duality mapping to more general case, that is, a convex, continuous, and strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets.(2)For the mappings, we extend the mapping from a relatively nonexpansive mapping to a Bregman weak relatively nonexpansive mapping. We remove the assumption 𝐹(𝑇)=𝐹(𝑇) on the mapping 𝑇 and extend the result to a Bregman weak relatively nonexpansive mapping, where 𝐹(𝑇) is the set of asymptotic fixed points of the mapping 𝑇.(3)Theorems 9 and 10 extend and improve corresponding results of [20].

Conflict of Interests

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

Acknowledgment

This research was partially supported by a grant from NSC.