Bregman Asymptotic Pointwise Nonexpansive Mappings in Banach Spaces

Abstract

We first introduce a new class of mappings called Bregman asymptotic pointwise nonexpansive mappings and investigate the existence and the approximation of fixed points of such mappings defined on a nonempty, bounded, closed, and convex subset C of a real Banach space E. Without using the original Opial property of a Banach space E, we prove weak convergence theorems for the sequences produced by generalized Mann and Ishikawa iteration processes for Bregman asymptotic pointwise nonexpansive mappings in a reflexive Banach space E. Our results are applicable in the function spaces 𝐿^𝑝, where 1<𝑝<∞ is a real number.

1. Introduction

Throughout this paper, we denote the set of real numbers and the set of positive integers by ℝ and ℕ, respectively. Let 𝐸 be a Banach space with the norm ‖⋅‖ and the dual space 𝐸^∗. For any 𝑥∈𝐸, we denote the value of 𝑥^∗∈𝐸^∗ at 𝑥 by ⟨𝑥,𝑥^∗⟩. Let {𝑥_𝑛}_𝑛∈ℕ be a sequence in 𝐸; we denote the strong convergence of {𝑥_𝑛}_𝑛∈ℕ to 𝑥∈𝐸 as 𝑛→∞ by 𝑥_𝑛→𝑥 and the weak convergence by 𝑥_𝑛⇀𝑥. The modulus 𝛿 of convexity of 𝐸 is denoted by 𝛿(𝜖)=inf1−𝑥+𝑦2∶‖𝑥‖≤1,𝑦≤1,𝑥−𝑦≥𝜖, (1) for every 𝜖 with 0≤𝜖≤2. A Banach space 𝐸 is said to be uniformly convex if 𝛿(𝜖)>0 for every 𝜖>0. Let 𝑆_𝐸={𝑥∈𝐸∶‖𝑥‖=1}. The norm of 𝐸 is said to be Gâteaux differentiable if for each 𝑥,𝑦∈𝑆_𝐸, the limit lim_𝑡→0𝑥+𝑡𝑦−‖𝑥‖𝑡 (2) exists. In this case, 𝐸 is called smooth. If the limit (2) is attained uniformly for all 𝑥,𝑦∈𝑆_𝐸, then 𝐸 is called uniformly smooth. The Banach space 𝐸 is said to be strictly convex if ‖(𝑥+𝑦)/2‖<1 whenever 𝑥,𝑦∈𝑆_𝐸 and 𝑥≠𝑦. It is well known that 𝐸 is uniformly convex if and only if 𝐸^∗ is uniformly smooth. It is also known that if 𝐸 is reflexive, then 𝐸 is strictly convex if and only if 𝐸^∗ is smooth; for more details, see [1, 2].

Let 𝐶 be a nonempty subset of 𝐸. Let 𝑇∶𝐶→𝐸 be a mapping. We denote the set of fixed points of 𝑇 by 𝐹(𝑇); that is, 𝐹(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥}. A mapping 𝑇∶𝐶→𝐸 is said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶. A mapping 𝑇∶𝐶→𝐸 is said to be quasi-nonexpansive if 𝐹(𝑇)≠∅ and ‖𝑇𝑥−𝑦‖≤‖𝑥−𝑦‖ for all 𝑥∈𝐶 and 𝑦∈𝐹(𝑇). The nonexpansivity plays an important role in the study of Mann iteration [3] for finding fixed points of a mapping 𝑇∶𝐶→𝐶. Recall that the Mann iteration is given by the following formula: 𝑥_𝑛+1=𝛾_𝑛𝑇𝑥_𝑛+1−𝛾_𝑛𝑥_𝑛,𝑥₁∈𝐶. (3) Here, {𝛾_𝑛}_𝑛∈ℕ is a sequence of real numbers in [0,1] satisfying some appropriate conditions. A more general iteration is the Ishikawa iteration [4], given by 𝑦_𝑛=𝛽_𝑛𝑇𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛,𝑥_𝑛+1=𝛾_𝑛𝑇𝑦_𝑛+1−𝛾_𝑛𝑥_𝑛, (4) where the sequences {𝛽_𝑛}_𝑛∈ℕ and {𝛾_𝑛}_𝑛∈ℕ satisfy some appropriate conditions. When all 𝛽_𝑛=0, the Ishikawa iteration reduces to the classical Mann iteration. Construction of fixed points of nonexpansive mappings via Mann’s and Ishikawa’s algorithms [3] has been extensively investigated in the literature (see, e.g., [5] and the references therein). A powerful tool in deriving weak or strong convergence of iterative sequences is due to Opial [6]. A Banach space 𝐸 is said to satisfy the Opial property [6] if for any weakly convergent sequence {𝑥_𝑛}_𝑛∈ℕ in 𝐸 with weak limit 𝑥, we have limsup_𝑛→∞𝑥_𝑛−𝑥<limsup_𝑛→∞𝑥_𝑛−𝑦, (5) for all 𝑦 in 𝐸 with 𝑦≠𝑥. It is well known that all Hilbert spaces, all finite dimensional Banach spaces, and the Banach spaces 𝑙^𝑝 (1≤𝑝<∞) satisfy the Opial property. However, not every Banach space satisfies the Opial property; see, for example, [7].

Let 𝐸 be a smooth, strictly convex, and reflexive Banach space and let 𝐽 be the normalized duality mapping of 𝐸. Let 𝐶 be a nonempty, closed, and convex subset of 𝐸. The generalized projection Π_𝐶 from 𝐸 onto 𝐶 [8] is defined and denoted by Π_𝐶(𝑥)=argmin_𝑦∈𝐶𝜙𝑦,𝑥, (6) where 𝜙(𝑥,𝑦)=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+‖𝑦‖². Let 𝐶 be a nonempty, closed, and convex subset of a smooth Banach space 𝐸, and let 𝑇 be a mapping from 𝐶 into itself.

1.1. Some Facts about Gradients

For any convex function 𝑔∶𝐸→(−∞,+∞] we denote the domain of 𝑔 by dom 𝑔={𝑥∈𝐸∶𝑔(𝑥)<∞}. For any 𝑥∈intdom𝑔 and any 𝑦∈𝐸, we denote by 𝑔^𝑜(𝑥,𝑦) the right-hand derivative of 𝑔 at 𝑥 in the direction 𝑦; that is, 𝑔^𝑜𝑥,𝑦=lim_𝑡↓0𝑔𝑥+𝑡𝑦−𝑔(𝑥)𝑡. (7) 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 𝑔^𝑜(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 this limit is attained uniformly in ‖𝑦‖=1. The function 𝑔 is Fréchet differentiable at 𝑥∈𝐸 (see, e.g., [9, page 13] or [10, page 508]) if for all 𝜖>0, there exists 𝛿>0 such that ‖𝑦−𝑥‖≤𝛿 implies that 𝑔𝑦−𝑔(𝑥)−𝑦−𝑥,∇𝑔(𝑥)≤𝜖𝑦−𝑥. (8) 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 (see, e.g., [9, Proposition 1.1.10]). Also, it is known that if 𝑔 is Fréchet differentiable, then ∇𝑔 is norm-to-norm continuous (see, [10, page 508]). The mapping ∇𝑔 is said to be weakly sequentially continuous if 𝑥_𝑛⇀𝑥 as 𝑛→∞ implies that ∇𝑔(𝑥_𝑛)⇀^∗∇𝑔(𝑥) as 𝑛→∞ (for more details, see [9, Theorem 3.2.4] or [10, page 508]). The function 𝑔 is said to be strongly coercive if lim_{𝑥𝑛→∞}𝑔𝑥_𝑛𝑥_𝑛=∞. (9) It is also said to be bounded on bounded subsets of 𝐸 if 𝑔(𝑈) is bounded for each bounded subset 𝑈 of 𝐸. Finally, 𝑔 is said to be uniformly Fréchet differentiable on a subset 𝑋 of 𝐸 if the limit (7) is attained uniformly for all 𝑥∈𝑋 and ‖𝑦‖=1.

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), (10) for all 𝑥^∗∈𝐸^∗. 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 is equivalent to 𝑔(𝑥)+𝑔^∗𝑥^∗=𝑥,𝑥^∗. (11) Here, 𝜕𝑔 is the subdifferential of 𝑔 [11, 12]. We also know that if 𝑔∶𝐸→(−∞,+∞] is a proper, lower semicontinuous, and convex function, then 𝑔^∗∶𝐸^∗→(−∞,+∞] is a proper, weak* lower semicontinuous, and convex function; see [2] for more details on convex analysis.

1.2. Some Facts about Bregman Distances

Let 𝐸 be a Banach space and let 𝐸^∗ be the dual space of 𝐸. Let 𝑔∶𝐸→ℝ be a convex and Gâteaux differentiable function. Then the Bregman distance [13, 14] 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. (12) It is clear that 𝐷_𝑔(𝑥,𝑦)≥0 for all 𝑥,𝑦∈𝐸. In that case when 𝐸 is a smooth Banach space, setting 𝑔(𝑥)=‖𝑥‖² for all 𝑥∈𝐸, we obtain that ∇𝑔(𝑥)=2𝐽𝑥 for all 𝑥∈𝐸 and hence 𝐷_𝑔(𝑥,𝑦)=𝜙(𝑥,𝑦) for all 𝑥,𝑦∈𝐸.

Let 𝐸 be a Banach space and let 𝐶 be a nonempty and convex subset of 𝐸. Let 𝑔∶𝐸→ℝ be a convex and Gâteaux differentiable function. Then, we know from [15] that for 𝑥∈𝐸 and 𝑥₀∈𝐶, 𝐷_𝑔(𝑥₀,𝑥)=min_𝑦∈𝐶𝐷_𝑔(𝑦,𝑥) if and only if 𝑦−𝑥₀,∇𝑔(𝑥)−∇𝑔𝑥₀≤0,∀𝑦∈𝐶. (13) Furthermore, 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∈𝐸, there exists a unique 𝑥₀∈𝐶 such that 𝐷_𝑔𝑥₀,𝑥=min_𝑦∈𝐶𝐷_𝑔𝑦,𝑥. (14) The Bregman projection proj^𝑔_𝐶 from 𝐸 onto 𝐶 is defined by proj^𝑔_𝐶(𝑥)=𝑥₀ for all 𝑥∈𝐸. It is also well known that proj^𝑔_𝐶 has the following property: 𝐷_𝑔𝑦,proj^𝑔_𝐶𝑥+𝐷_𝑔proj^𝑔_𝐶𝑥,𝑥≤𝐷_𝑔𝑦,𝑥, (15) for all 𝑦∈𝐶 and 𝑥∈𝐸 (see [9] for more details).

For any bounded subset 𝐴 of a reflexive Banach space 𝐸, we denote the Bregman diameter of 𝐴 by 𝐵diam(𝐴)∶=sup𝐷_𝑔𝑥,𝑦∶𝑥,𝑦∈𝐴<∞. (16)

1.3. Some Facts about Uniformly Convex Functions

Let 𝐸 be a Banach space and let 𝐵_𝑠∶={𝑧∈𝐸∶‖𝑧‖≤𝑠} for all 𝑠>0. Then a function 𝑔∶𝐸→ℝ is said to be uniformly convex on bounded subsets of 𝐸 ([16, Pages 203, 221]) if 𝜌_𝑠(𝑡)>0 for all 𝑠, 𝑡>0, where 𝜌_𝑠∶[0,+∞)→[0,∞] is defined by 𝜌_𝑠(𝑡)=inf_{𝑥,𝑦∈𝐵𝑠(),‖𝑥−𝑦‖=𝑡,𝛼∈0,1}(𝑥)+()𝑔𝑦()𝑦×(𝛼(𝛼𝑔1−𝛼−𝑔𝛼𝑥+1−𝛼1−𝛼))⁻¹, (17) for all 𝑡≥0. The function 𝜌_𝑠 is called the gauge of uniform convexity of 𝑔. The function 𝑔 is also said to be uniformly smooth on bounded subsets of 𝐸 ([16, Pages 207, 221]) if lim_𝑡↓0(𝜎_𝑠(𝑡)/𝑡)=0 for all 𝑠>0, where 𝜎_𝑠∶[0,+∞)→[0,∞] is defined by 𝜎_𝑠(𝑡)=sup_{𝑥∈𝐵𝑠,𝑦∈𝑆𝐸(),𝛼∈0,1}()+()𝑔(𝑥)×(𝛼(𝛼𝑔𝑥+1−𝛼𝑡𝑦1−𝛼𝑥−𝛼𝑡𝑦−𝑔1−𝛼))⁻¹, (18) for all 𝑡≥0. The function 𝑔 is said to be uniformly convex if the function 𝛿_𝑔∶[0,+∞)→[0,+∞], defined by 𝛿_𝑔(𝑡)1∶=sup2𝑔(𝑥)+12𝑔𝑦−𝑔𝑥+𝑦2∶𝑦−𝑥=𝑡, (19) satisfies that lim_𝑡↓0(𝜎_𝑠(𝑡)/𝑡)=0.

Remark 1. Let 𝐸 be a Banach space, let 𝑠>0 be a constant, and let 𝑔∶𝐸→ℝ be a convex function which is uniformly convex on bounded subsets. Then 𝑔()𝑦(𝑥)+()𝑔𝑦()𝜌𝛼𝑥+1−𝛼≤𝛼𝑔1−𝛼−𝛼1−𝛼_𝑠𝑥−𝑦, (20) for all 𝑥,𝑦∈𝐵_𝑠∶={𝑧∈𝐸∶‖𝑧‖≤𝑠} and 𝛼∈(0,1), where 𝜌_𝑠 is the gauge of uniform convexity of 𝑔.

Definition 2. 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, bounded, closed, and convex subset of 𝐸. A mapping 𝑇∶𝐶→𝐸 is said to be Bregman asymptotic pointwise nonexpansive if there exists a sequence of mappings 𝜃_𝑛∶𝐶→[0,∞) such that 𝐷_𝑔𝑇^𝑛𝑥,𝑇^𝑛𝑦≤𝜃_𝑛(𝑥)𝐷_𝑔𝑥,𝑦,∀𝑥,𝑦∈𝐶,limsup_𝑛→∞𝜃_𝑛(𝑥)≤1,∀𝑥∈𝐶. (21) Denoting 𝑎_𝑛(𝑥)=max{𝜃_𝑛(𝑥),1}, we note that without loss of generality we can assume that 𝑇 is Bregman asymptotic pointwise nonexpansive if 𝐷_𝑔𝑇^𝑛𝑥,𝑇^𝑛𝑦≤𝑎_𝑛(𝑥)𝐷_𝑔𝑥,𝑦,∀𝑥,𝑦∈𝐶,𝑛∈ℕ,lim_𝑛→∞𝑎_𝑛(𝑥)=1,𝑎_𝑛(𝑥)≥1,∀𝑥∈𝐶,𝑛∈ℕ. (22)(23) Define 𝑏_𝑛(𝑥)=𝑎_𝑛(𝑥)−1. In view of (23), we obtain lim_𝑛→∞𝑏_𝑛(𝑥)=0. (24) Next, we denote by ℬ𝒯(𝐶) the class of all Bregman asymptotic pointwise nonexpansive mappings 𝑇∶𝐶→𝐶.
Imposing some restrictions on the behavior of 𝑎_𝑛 and 𝑏_𝑛, we can define the following subclass of Bregman asymptotic pointwise nonexpansive mappings.

Definition 3. Let 𝐶 and ℬ𝒯(𝐶) be as in Definition 2. We define ℬ𝒯_𝑟(𝐶) as a class of all 𝑇∈ℬ𝒯(𝐶) such that ^∞_𝑛=1𝑏_𝑛(𝑥)<∞,∀𝑥∈𝐶,𝑎_𝑛isaboundedfunctionforevery𝑛inℕ. (25) Kirk and Xu [17] studied the existence of fixed points of asymptotic pointwise nonexpansive mappings with respect to the norm of a Banach space 𝐸. Recently, Kozlowski [18] proved weak and strong convergence theorems for asymptotic pointwise nonexpansive mappings in a Banach space. To see some other related works, we refer the reader to [19, 20].

In this paper, we first investigate the approximation of fixed points of a new class of Bregman asymptotic pointwise nonexpansive mappings defined on a nonempty, bounded, closed, and convex subset 𝐶 of a real Banach space 𝐸. Without using the Opial property of a Banach space 𝐸, we prove weak convergence theorems for the sequences produced by generalized Mann and Ishikawa iteration processes. Our results improve and generalize many known results in the current literature; see, for example, [18, 21].

2. Preliminaries

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

Definition 4 (see [10]). Let 𝐸 be a Banach space. The function 𝑔∶𝐸→ℝ is said to be a Bregman function if the following conditions are satisfied:(1)𝑔is continuous, strictly convex, and Gâteaux differentiable;(2)the set {𝑦∈𝐸∶𝐷_𝑔(𝑥,𝑦)≤𝑟} is bounded for all 𝑥∈𝐸 and 𝑟>0.

Lemma 5 (see [9, 16]). 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∈𝐸 and 𝑟>0;(4)dom𝑔^∗=𝐸^∗,𝑔^∗ is Gâteaux differentiable and ∇𝑔^∗=(∇𝑔)⁻¹.

We know the following two results; see [16, Proposition 3.6.4].

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

Theorem 7. Let 𝐸 be a reflexive Banach space and 𝑔∶𝐸→ℝ a continuous convex function which is strongly coercive. Then the following assertions are equivalent:(1)𝑔 is bounded on bounded subsets and uniformly smooth on bounded subsets of 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 uniformly convex on bounded subsets of 𝐸^∗.

Let 𝐸 be a Banach space and let 𝑔∶𝐸→ℝ be a convex and Gâteaux differentiable function. Then the Bregman distance [13, 14] does not satisfy the well known properties of a metric, but it does have the following important property, which is called the three point identity [22]: 𝐷_𝑔(𝑥,𝑧)=𝐷_𝑔𝑥,𝑦+𝐷_𝑔+𝑦(𝑧𝑦,𝑧𝑥−𝑦,∇𝑔−∇𝑔),∀𝑥,𝑦,𝑧∈𝐸. (26) In particular, it can easily be seen that 𝐷_𝑔𝑥,𝑦=−𝐷_𝑔𝑦,𝑥+𝑦−𝑥,∇𝑔𝑦−∇𝑔(𝑥),∀𝑥,𝑦∈𝐸. (27) Indeed, by letting 𝑧=𝑥 in (26) and taking into account that 𝐷_𝑔(𝑥,𝑥)=0, we get the desired result.

Lemma 8 (see [23]). Let 𝐸 be a Banach space and 𝑔∶𝐸→ℝ a Gâteaux differentiable function which is uniformly convex on bounded subsets of 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 9 (see [10, 24]). Let 𝐸 be a reflexive Banach space, 𝑔∶𝐸→ℝ a strongly coercive Bregman function, and 𝑉 the function defined by 𝑉𝑥,𝑥^∗=𝑔(𝑥)−𝑥,𝑥^∗+𝑔^∗𝑥^∗,𝑥∈𝐸,𝑥^∗∈𝐸^∗. (28) Then the following assertions hold:(1)𝐷_𝑔(𝑥,∇𝑔^∗(𝑥^∗))=𝑉(𝑥,𝑥^∗) for all 𝑥∈𝐸 and 𝑥^∗∈𝐸^∗;(2)𝑉(𝑥,𝑥^∗)+⟨∇𝑔^∗(𝑥^∗)−𝑥,𝑦^∗⟩≤𝑉(𝑥,𝑥^∗+𝑦^∗) for all 𝑥∈𝐸 and 𝑥^∗,𝑦^∗∈𝐸^∗.

Let 𝐶 and 𝐷 be nonempty subsets of a real Banach space 𝐸 with 𝐷⊂𝐶. A mapping 𝑅_𝐷∶𝐶→𝐷 is said to be sunny if 𝑅_𝐷𝑅_𝐷𝑥+𝑡𝑥−𝑅_𝐷𝑥=𝑅_𝐷𝑥, (29) for each 𝑥∈𝐸 and 𝑡≥0. A mapping 𝑅_𝐷∶𝐶→𝐷 is said to be a retraction if 𝑅_𝐷𝑥=𝑥 for each 𝑥∈𝐶.

Lemma 10 (see [25]). Suppose {𝑟_𝑘}_𝑘∈ℕ is a bounded sequence of real numbers and {𝑑_𝑘,𝑛}_{𝑘,𝑛∈ℕ} is a doubly index sequence of real numbers which satisfy limsup_𝑘→∞limsup_𝑛→∞𝑑_𝑘,𝑛≤0,𝑟_𝑘,𝑛≤𝑟_𝑘+𝑑_𝑘,𝑛, (30) for each 𝑘,𝑛≥1. Then {𝑟_𝑘}_𝑘∈ℕ converges to an 𝑟∈ℝ.

Let 𝐸 be a reflexive Banach space and let 𝑔∶𝐸→(−∞,+∞] be an admissible function, that is, a proper, lower-semicontinuous, convex, and Gâteaux differentiable function. Let 𝐶 be a nonempty, closed, and convex subset of 𝐸 and let {𝑥_𝑛}_𝑛∈ℕ be a bounded sequence in 𝐸. For any 𝑥 in 𝐸, we set Br𝑥_𝑛,𝑥=limsup_𝑛→∞𝐷_𝑔𝑥_𝑛,𝑥. (31) The Bregman asymptotic radius of {𝑥_𝑛}_𝑛∈ℕ relative to 𝐶 is defined by Br𝑥_𝑛,𝐶=infBr𝑥_𝑛,𝑥∶𝑥∈𝐶. (32) The Bregman asymptotic center of {𝑥_𝑛}_𝑛∈ℕ relative to 𝐶 is the set BA𝑥_𝑛,𝐶=𝑥∈𝐶∶Br𝑥_𝑛,𝑥=Br𝑥_𝑛,𝐶. (33)

The following Bregman Opial-like inequality has been proved in [26]. It is worth mentioning that the Bregman Opial-like inequality is different from the ordinary Opial inequality [6] and can be applied in uniformly convex Banach spaces.

Lemma 11 (see [26]). Let 𝐸 be a Banach space and let 𝑔∶𝐸→(−∞,+∞] be a proper strictly convex function so that it is Gâteaux differentiable on intdom𝑔. Suppose {𝑥_𝑛}_𝑛∈ℕ is a sequence in dom𝑔 such that 𝑥_𝑛⇀𝑣 for some 𝑣∈intdom𝑔. Then limsup_𝑛→∞𝐷_𝑔𝑥_𝑛,𝑣<limsup_𝑛→∞𝐷_𝑔𝑥_𝑛,𝑦,∀𝑦∈intdom𝑔𝑤𝑖𝑡ℎ𝑦≠𝑣. (34)

Theorem 12 (see [16]). Let 𝑔∶𝐸→(−∞,+∞] be a function. Then the following assertions are equivalent:(1)𝑔 is convex and lower semicontinuous;(2)𝑔 is convex and weakly lower semicontinuous;(3)epi(𝑔) is convex and closed;(4)epi(𝑔) is convex and weakly closed,
where epi(𝑔)={(𝑥,𝑡)∈𝐸×ℝ∶𝑔(𝑥)≤𝑡} denotes the epigraph of 𝑔.

3. Fixed Point Theorems and Demiclosedness Principle

Proposition 13. Let 𝐸 be a reflexive Banach space and let 𝑔∶𝐸→ℝ be a strongly admissible function which is bounded on bounded subsets and uniformly convex on bounded subsets of 𝐸. Let 𝐶 be a nonempty, bounded, closed, and convex subset of 𝐸 and let 𝑇∈ℬ𝒯_𝑟(𝐶). Then 𝑇 has a fixed point. Moreover, 𝐹(𝑇) is closed and convex.

Proof. We first show that 𝐹(𝑇) is nonempty. Let 𝑥 in 𝐶 be fixed. We define a function 𝑓∶𝐶→[0,∞) by 𝑓𝑦=limsup_𝑛→∞𝐷_𝑔𝑦,𝑇^𝑛𝑥,𝑦∈𝐶. (35) In view of Remark 1, it is easy to see that 𝑓 is convex. Since 𝑔 is continuous, by Theorem 12 we conclude that the Bregman distance 𝐷_𝑔 is weakly lower-semicontinuous in the first argument. Since 𝐸 is a uniformly convex Banach space and 𝐶 is weakly compact, in view of [1] there exists a unique point 𝑧∈𝐶 such that 𝑓(𝑧)=min_𝑦∈𝐶𝑓𝑦. (36) We show that {𝑇^𝑛𝑧}_𝑛∈ℕ is convergent in norm. To this end, let 𝑠₁=sup{‖𝑇^𝑛𝑧‖,‖𝑧‖∶𝑛∈ℕ} and 𝜌_𝑠1∶𝐸→ℝ be the gauge of uniform convexity of 𝑔. For any 𝑘, 𝑚∈ℕ, put 𝑢_𝑘,𝑚=(1/2)𝑇^𝑘𝑧+(1/2)𝑇^𝑚𝑧. Then we have 𝑢_𝑘,𝑚∈𝐶. In view of Remark 1, we obtain 𝐷_𝑔𝑢_𝑘,𝑚,𝑇^{𝑘+𝑚+𝑛}𝑥=𝑔𝑢_𝑘,𝑚−𝑔𝑇^{𝑘+𝑚+𝑛}𝑥−𝑢_𝑘,𝑚−𝑇^{𝑘+𝑚+𝑛}𝑥,∇𝑔𝑇^{𝑘+𝑚+𝑛}𝑥1=𝑔2𝑇^𝑘1𝑧+2𝑇^𝑚𝑧−𝑔𝑇^{𝑘+𝑚+𝑛}𝑥−12𝑇^𝑘1𝑧+2𝑇^𝑚𝑧−𝑇^{𝑘+𝑚+𝑛}𝑥,∇𝑔𝑇^{𝑘+𝑚+𝑛}𝑥≤12𝑔𝑇^𝑘𝑧+12𝑔𝑇^𝑚𝑧−14𝜌_𝑠1𝑇^𝑘𝑧−𝑇^𝑚𝑧−𝑔𝑇^{𝑘+𝑚+𝑛}𝑥−12𝑇^𝑘𝑧−𝑇^{𝑘+𝑚+𝑛}𝑥,∇𝑔𝑇^{𝑘+𝑚+𝑛}𝑥−12𝑇^𝑚𝑧−𝑇^{𝑘+𝑚+𝑛}𝑥,∇𝑔𝑇^{𝑘+𝑚+𝑛}𝑥=12𝑔𝑇^𝑘𝑧−𝑔𝑇^{𝑘+𝑚+𝑛}𝑥−⟨𝑇^𝑘𝑧−𝑇^{𝑘+𝑚+𝑛}𝑥,∇𝑔𝑇^{𝑘+𝑚+𝑛}𝑥⟩+12𝑔𝑇^𝑚𝑧−𝑔𝑇^{𝑘+𝑚+𝑛}𝑥−𝑇^𝑚𝑧−𝑇^{𝑘+𝑚+𝑛}𝑥,∇𝑔𝑇^{𝑘+𝑚+𝑛}𝑥−14𝜌_𝑠1𝑇^𝑘𝑧−𝑇^𝑚𝑧=12𝐷_𝑔𝑇^𝑘𝑧,𝑇^{𝑘+𝑚+𝑛}𝑥+12𝐷_𝑔𝑇^𝑚𝑧,𝑇^{𝑘+𝑚+𝑛}𝑥−14𝜌_𝑠1𝑇^𝑘𝑧−𝑇^𝑚𝑧≤12𝑎_𝑘(𝑧)𝐷_𝑔𝑧,𝑇^𝑚+𝑛𝑥+12𝑎_𝑘(𝑧)𝐷_𝑔𝑧,𝑇^𝑘+𝑛𝑥−14𝜌_𝑠1𝑇^𝑘𝑧−𝑇^𝑚𝑧. (37) Applying to both sides of the above inequalities limsup_𝑛→∞ we obtain 𝑓𝑢_𝑘,𝑚≤12𝑎_𝑘(𝑧)+𝑎_𝑚(𝑧𝑓(𝑧)−1)4𝜌_𝑠1𝑇^𝑘𝑧−𝑇^𝑚𝑧. (38) This, together with 𝑓(𝑧)≤𝑓(𝑢_𝑘,𝑚), implies that 𝜌_𝑠1𝑇^𝑘𝑧−𝑇^𝑚𝑧1≤42𝑎_𝑘(𝑧)+𝑎_𝑚(𝑧)𝑓(𝑧)−𝑓𝑢_𝑘,𝑚1≤42𝑎_𝑘(𝑧)+𝑎_𝑚(𝑧)𝑓(𝑧)−𝑓(𝑧). (39) Letting 𝑘, 𝑚→∞ in (39) we conclude that lim_{𝑘,𝑚→∞}𝜌_𝑠1𝑇^𝑘𝑧−𝑇^𝑚𝑧=0. (40) From the properties of 𝜌_𝑠1, we deduce that lim_{𝑘,𝑚→∞}‖𝑇^𝑘𝑧−𝑇^𝑚𝑧‖=0. Thus, {𝑇^𝑘𝑧}_𝑘∈ℕ is a norm-Cauchy sequence and hence convergent. Let 𝑣=lim_𝑛→∞𝑇^𝑛𝑧. (41) Since 𝑇 is a Bregman asymptotic pointwise nonexpansive mapping, we have, for all 𝑛∈ℕ, 𝐷_𝑔𝑇^𝑛+1𝑧,𝑇𝑣≤𝑎₁𝑇^𝑛𝑧𝐷_𝑔𝑇^𝑛𝑧,𝑣. (42) Letting 𝑛→∞ in (42), we conclude that 𝑇𝑣=𝑣. This shows that 𝐹(𝑇)≠∅.
Now, we show that 𝐹(𝑇) is closed. Let {𝑝_𝑛}_𝑛∈ℕ be a sequence in 𝐹(𝑇) such that 𝑝_𝑛→𝑝 as 𝑛→∞. Then we have that {𝑝_𝑛}_𝑛∈ℕ is a bounded sequence in 𝐸. We claim that 𝑝∈𝐹(𝑇). Since 𝑔 is continuous, we conclude that 𝑔(𝑝_𝑛)→𝑔(𝑝) as 𝑛→∞. This implies that 𝐷_𝑔𝑝_𝑛,𝑝=𝑔𝑝_𝑛−𝑔𝑝−𝑝_𝑛𝑝≤𝑔𝑝−𝑝,∇𝑔_𝑛−𝑔𝑝+𝑝_𝑛𝑝().−𝑝∇𝑔⟶0𝑛⟶∞ (43) In view of the definition of 𝑇, we obtain 𝐷_𝑔𝑇𝑝,𝑝_𝑛≤𝑎_𝑛𝑝𝐷_𝑔𝑝,𝑝_𝑛⟶0(𝑛⟶∞). (44) This implies that 0=lim_𝑛→∞𝐷_𝑔𝑝_𝑛,𝑇𝑝=lim_𝑛→∞𝑔𝑝_𝑛−𝑔𝑇𝑝−𝑝_𝑛−𝑇𝑝,∇𝑔𝑇𝑝=𝐷_𝑔𝑝,𝑇𝑝. (45) It follows from Lemma 8 that 𝑇𝑝=𝑝. Thus we have 𝑝∈𝐹(𝑇).
Let us show that 𝐹(𝑇) is convex. For any 𝑝, 𝑞∈𝐹(𝑇), 𝑡∈(0,1), and 𝑛∈ℕ, we set 𝑥=𝑡𝑝+(1−𝑡)𝑞 and 𝑒_𝑛(𝑝,𝑞)=max{𝑎_𝑛(𝑝),𝑎_𝑛(𝑞)}. We prove that 𝑥∈𝐹(𝑇). By the definition of Bregman distance (see (12)), we get 𝐷_𝑔𝑥,𝑇^𝑛𝑥=𝑔(𝑥)−𝑔𝑇^𝑛𝑥−𝑥−𝑇^𝑛𝑥,∇𝑔𝑇^𝑛𝑥=𝑔(𝑥)−𝑔𝑇^𝑛𝑥−𝑡𝑝+(1−𝑡)𝑞−𝑇^𝑛𝑥,∇𝑔𝑇^𝑛𝑥=𝑔(𝑥)−𝑔𝑇^𝑛𝑥−𝑡𝑝−𝑇^𝑛𝑥,∇𝑔𝑇^𝑛𝑥−(1−𝑡)𝑞−𝑇^𝑛𝑥,∇𝑔𝑇^𝑛𝑥𝑝+()𝑔𝑞−𝑝+()𝑔𝑞(𝑥)𝑔𝑝𝑇+𝑡𝑔1−𝑡𝑡𝑔1−𝑡=𝑔+𝑡−𝑔^𝑛𝑥−𝑝−𝑇^𝑛𝑥,∇𝑔𝑇^𝑛𝑥+(1−𝑡)𝑔𝑞−𝑔𝑇^𝑛𝑥−𝑞−𝑇^𝑛𝑥,∇𝑔𝑇^𝑛𝑥=𝑔(𝑥)+𝑡𝐷_𝑔𝑝,𝑇^𝑛𝑥+(1−𝑡)𝐷_𝑔𝑞,𝑇^𝑛𝑥−𝑝+()𝑔𝑞(𝑥)𝑡𝑔1−𝑡≤𝑔+𝑡𝑎_𝑛𝑝𝐷_𝑔+()𝑎𝑝,𝑥1−𝑡_𝑛𝑞𝐷_𝑔𝑝−()𝑔𝑞(𝑥)𝑞,𝑥−𝑡𝑔1−𝑡≤𝑔+𝑡𝑒_𝑛𝑝,𝑞𝐷_𝑔+()𝑒𝑝,𝑥1−𝑡_𝑛𝑝,𝑞𝐷_𝑔𝑝−()𝑔𝑞(𝑥)𝑞,𝑥−𝑡𝑔1−𝑡=𝑔+𝑒_𝑛×𝑡𝑔𝑝(𝑥)−(𝑥+()𝑔𝑞(𝑥)−(𝑥𝑝−()𝑔𝑞(𝑥)𝑝,𝑞−𝑔𝑝−𝑥,∇𝑔)1−𝑡−𝑔𝑞−𝑥,∇𝑔)−𝑡𝑔1−𝑡=𝑔+𝑒_𝑛×(𝑥)−𝑡(𝑥−)(𝑥𝑝,𝑞−𝑔𝑝−𝑥,∇𝑔)(1−𝑡𝑞−𝑥,∇𝑔)+𝑒_𝑛𝑝+()𝑔𝑞𝑝−()𝑔𝑞(𝑥)𝑝,𝑞𝑡𝑔1−𝑡−𝑡𝑔1−𝑡=𝑔−𝑒_𝑛𝑝,𝑞𝑔(𝑥)−𝑒_𝑛×()(𝑥𝑝,𝑞𝑡𝑝+1−𝑡𝑞−𝑥,∇𝑔)+𝑒_𝑛𝑝+()𝑔𝑞𝑝−()𝑔𝑞=𝑒𝑝,𝑞𝑡𝑔1−𝑡−𝑡𝑔1−𝑡_𝑛(𝑥+𝑒𝑝,𝑞−1−𝑔)_𝑛𝑝+𝑒𝑝,𝑞−1𝑡𝑔_𝑛()𝑔𝑞=𝑒𝑝,𝑞−11−𝑡_𝑛×(𝑥)𝑝+()𝑔𝑞.𝑝,𝑞−1−𝑔+𝑡𝑔1−𝑡 (46) This implies that lim_𝑛→∞𝐷_𝑔(𝑥,𝑇^𝑛𝑥)=0. Thus for each 𝜖>0, there exists 𝑛₀∈ℕ such that 𝐷_𝑔𝑥,𝑇^𝑛𝑥≤𝜖,∀𝑛≥𝑛₀. (47) This means that the sequence {𝐷_𝑔(𝑥,𝑇^𝑛𝑥)}_𝑛∈ℕ is bounded. In view of Definition 4, we conclude that the sequence {𝑇^𝑛𝑥}_𝑛∈ℕ is bounded. Then, by Lemma 8, we obtain lim_𝑛→∞‖𝑥−𝑇^𝑛𝑥‖=0. Thus we have 𝑇^𝑛+1𝑥→𝑥; that is, 𝑇(𝑇^𝑛𝑥)→𝑥. On the other hand, in view of three-point identity (see (26)), we deduce that 𝐷_𝑔(𝑥,𝑇𝑥)=𝐷_𝑔𝑥,𝑇^𝑛+1𝑥+𝐷_𝑔𝑇^𝑛+1+𝑥,𝑇𝑥𝑥−𝑇^𝑛+1𝑥,∇𝑔𝑇^𝑛+1𝑥−∇𝑔(𝑇𝑥)≤𝐷_𝑔𝑥,𝑇^𝑛+1𝑥+𝑎₁𝑇^𝑛𝑥𝐷_𝑔𝑇^𝑛+𝑥,𝑥𝑥−𝑇^𝑛+1𝑥∇𝑔𝑇^𝑛+1𝑥−∇𝑔(𝑇𝑥). (48) Letting 𝑛→∞ in the above inequalities we deduce that 𝐷_𝑔(𝑥,𝑇𝑥)=0 and hence by Lemma 8 we conclude that 𝑇𝑥=𝑥, which completes the proof.

Lemma 14. 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, bounded, closed, and convex subset of 𝐸 and let 𝑇∈ℬ𝒯_𝑟(𝐶). If {𝑥_𝑛}_𝑛∈ℕ is a sequence of 𝐶 such that lim_𝑛→∞‖𝑇𝑥_𝑛−𝑥_𝑛‖=0, then, for any 𝑚∈ℕ, lim_𝑛→∞‖𝑇^𝑚𝑥_𝑛−𝑥_𝑛‖=0.

Proof. In view of (25), there exists a finite constant 𝑀₁>0 such that sup𝑎^𝑚−1_𝑗=1𝑗(𝑥)∶𝑥∈𝐶≤𝑀₁. (49) It follows from three-point identity (see (26)) that 𝐷_𝑔𝑇^𝑚𝑥_𝑛,𝑥_𝑛=𝐷_𝑔𝑇^𝑚𝑥_𝑛,𝑇^𝑚−1𝑥_𝑛+𝐷_𝑔𝑇^𝑚−1𝑥_𝑛,𝑥_𝑛+𝑇^𝑚𝑥_𝑛−𝑇^𝑚−1𝑥_𝑛,∇𝑔𝑇^𝑚−1𝑥_𝑛−∇𝑔𝑥_𝑛≤𝑎_𝑚−1𝑥_𝑛𝐷_𝑔𝑇𝑥_𝑛,𝑥_𝑛+𝐷_𝑔𝑇^𝑚−1𝑥_𝑛,𝑇^𝑚−2𝑥_𝑛+𝐷_𝑔𝑇^𝑚−2𝑥_𝑛,𝑥_𝑛+𝑇^𝑚−1𝑥_𝑛−𝑇^𝑚−2𝑥_𝑛,∇𝑔𝑇^𝑚−2𝑥_𝑛−∇𝑔𝑥_𝑛+𝑀₂𝑇^𝑚𝑥_𝑛−𝑇^𝑚−1𝑥_𝑛⋮≤^𝑚−1_𝑗=1𝑎_𝑗𝑥_𝑛+1𝐷_𝑔𝑇𝑥_𝑛,𝑥_𝑛+2𝑀₂^𝑚−1_𝑗=1𝑇^𝑚𝑥_𝑛−𝑇^𝑚−1𝑥_𝑛, (50) where 𝑀₂∶=sup{‖∇𝑔(𝑇^𝑗𝑥_𝑛)‖∶𝑗=1,2,…,𝑚and𝑛∈ℕ}. This, together with Lemma 8, implies that lim_𝑛→∞‖𝑇^𝑚𝑥_𝑛−𝑥_𝑛‖=0. This completes the proof.

Theorem 15. 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 and uniformly smooth on bounded subsets of 𝐸. Let 𝐶 be a nonempty, bounded, closed, and convex subset of 𝐸 and let 𝑇∈ℬ𝒯_𝑟(𝐶). If {𝑥_𝑛}_𝑛∈ℕ converges weakly to 𝑧 and lim_𝑛→∞‖𝑇𝑥_𝑛−𝑥_𝑛‖=0, then 𝑇𝑧=𝑧. That is, 𝐼−𝑇 is demiclosed at zero, where 𝐼 is the identity mapping on 𝐸.

Proof. Let the function 𝜙∶𝐸→[0,∞) be defined by 𝜙(𝑥)=limsup_𝑛→∞𝐷_𝑔𝑥_𝑛,𝑥(𝑥∈𝐸). (51) For any 𝑚,𝑛∈ℕ with 𝑚>2, in view of three-point identity (see (26)), we obtain 𝐷_𝑔𝑇^𝑚𝑥_𝑛,𝑥=𝐷𝑇^𝑚𝑥_𝑛,𝑇^𝑚−1𝑥_𝑛+𝐷_𝑔𝑇^𝑚−1𝑥_𝑛+𝑇,𝑥_𝑚𝑥_𝑛−𝑇^𝑚−1𝑥_𝑛,∇𝑔𝑇^𝑚−1𝑥_𝑛−∇𝑔(𝑥)≤𝑎_𝑚−1𝑥_𝑛𝐷_𝑔𝑇𝑥_𝑛,𝑥_𝑛+𝐷𝑇^𝑚−1𝑥_𝑛,𝑇^𝑚−2𝑥_𝑛+𝐷_𝑔𝑇^𝑚−2𝑥_𝑛+𝑇,𝑥_𝑚−1𝑥_𝑛−𝑇^𝑚−2𝑥_𝑛,∇𝑔𝑇^𝑚−2𝑥_𝑛−∇𝑔𝑥_𝑛+𝑀^′₂𝑇^𝑚𝑥_𝑛−𝑇^𝑚−1𝑥_𝑛⋮≤^𝑚−1_𝑗=1𝑎_𝑗(𝑥)+1𝐷_𝑔𝑇𝑥_𝑛,𝑥_𝑛+𝐷_𝑔𝑇𝑥_𝑛,𝑥+𝑀^′₂^𝑚−1_𝑗=1𝑇^𝑚𝑥_𝑛−𝑇^𝑚−1𝑥_𝑛, (52) where 𝑀^′₂∶=sup{‖∇𝑔(𝑇^𝑗𝑥_𝑛)‖∶𝑗=1,2,…,𝑚and𝑛∈ℕ}. This, together with Lemma 8, implies that limsup_𝑛→∞𝐷_𝑔𝑇^𝑚𝑥_𝑛,𝑥≤limsup_𝑛→∞𝐷_𝑔𝑥_𝑛,𝑥=𝜙(𝑥). (53) In view of (25), there exists a finite constant 𝑀₃>0 such that sup𝑎^𝑚−1_𝑗=1𝑗(𝑥)∶𝑥∈𝐶≤𝑀₃, (54) where 𝑀₄∶=sup{‖∇𝑔(𝑇^𝑚−1𝑥_𝑛)‖∶𝑚=1,2,…,𝑚and𝑛∈ℕ}. This, together with Lemma 8, implies that lim_𝑛→∞‖𝑇^𝑚𝑥_𝑛−𝑥_𝑛‖=0. Employing Lemma 8, we conclude that 𝜙(𝑥)=limsup_𝑛→∞𝐷_𝑔𝑥_𝑛,𝑥≤limsup_𝑛→∞𝐷_𝑔𝑥_𝑛,𝑇^𝑚𝑥_𝑛+limsup_𝑛→∞𝐷_𝑔𝑇^𝑚𝑥_𝑛,𝑥+limsup_𝑛→∞𝑀^′₂𝑥_𝑛−𝑇^𝑚𝑥_𝑛≤limsup_𝑛→∞𝐷_𝑔𝑇^𝑚𝑥_𝑛,𝑥≤𝜙(𝑥). (55) This means that 𝜙(𝑥)=limsup_𝑛→∞𝐷_𝑔𝑇^𝑚𝑥_𝑛,𝑥. (56) Since 𝑇 is a Bregman asymptotic pointwise nonexpansive mapping, it follows that 𝜙𝑇^𝑚𝑥≤𝑎_𝑚𝑥_𝑛𝜙(𝑥),∀𝑥∈𝐶. (57) In view of (57), we deduce that limsup_𝑚→∞𝜙𝑇^𝑚𝑧≤𝜙(𝑧). (58) By the Bregman Opial-like inequality ((34)) we obtain that for any 𝑥≠𝑧𝜙(𝑧)=limsup_𝑛→∞𝐷_𝑔𝑥_𝑛,𝑧<limsup_𝑛→∞𝐷_𝑔𝑥_𝑛,𝑥=𝜙(𝑥). (59) This shows that 𝜙(𝑧)=inf{𝜙(𝑥)∶𝑥∈𝐶}. Thus we have 𝜙𝑇^𝑚𝑧=𝜙(𝑧). (60) Put 𝑠₂=sup{‖∇𝑔(𝑧)‖,‖∇𝑔(𝑇^𝑚𝑧)‖∶𝑚∈ℕ}, 𝑧_𝑚=∇𝑔^∗((1/2)∇𝑔(𝑧)+(1/2)∇𝑔(𝑇^𝑚𝑧)), and 𝑢_𝑚=proj^𝑔_𝐶(𝑧_𝑚) for all 𝑚∈ℕ. Then we have 𝑢_𝑚∈𝐶. In view of Remark 1, we obtain a continuous strictly increasing convex function 𝜌^∗_𝑠2∶[0,+∞)→[0,+∞) with 𝜌^∗_𝑠2(0)=0 such that 𝐷_𝑔𝑥_𝑛,𝑢_𝑚=𝐷_𝑔𝑥_𝑛,proj^𝑔_𝐶𝑧_𝑚≤𝐷_𝑔𝑥_𝑛,𝑧_𝑚=𝑔𝑥_𝑛−𝑔𝑧_𝑚−𝑥_𝑛−𝑧_𝑚,∇𝑔𝑧_𝑚=𝑔𝑥_𝑛+𝑔^∗𝑧_𝑚−𝑧_𝑚,∇𝑔𝑧_𝑚−𝑥_𝑛,∇𝑔𝑧_𝑚+𝑧_𝑚,∇𝑔𝑧_𝑚=𝑔𝑥_𝑛+𝑔^∗12(𝑧)+1∇𝑔2∇𝑔𝑇^𝑚𝑧−𝑥_𝑛,12(𝑧)+1∇𝑔2∇𝑔𝑇^𝑚𝑧≤12𝑔𝑥_𝑛+12𝑔𝑥_𝑛+12𝑔^∗∇𝑔𝑥_𝑛+12𝑔^∗∇𝑔𝑇^𝑚𝑧−14𝜌^∗_𝑠2∇𝑔(𝑧)−∇𝑔𝑇^𝑚𝑧−12𝑥_𝑛,∇𝑔𝑥_𝑛−12𝑥_𝑛,∇𝑔𝑇^𝑚𝑧=12𝑔𝑥_𝑛+𝑔^∗∇𝑔(𝑧)−𝑥_𝑛,∇𝑔𝑥_𝑛+12𝑔𝑥_𝑛+𝑔^∗∇𝑔𝑇^𝑚𝑧−𝑥_𝑛,∇𝑔𝑇^𝑚𝑧−14𝜌^∗_𝑠2∇𝑔(𝑧)−∇𝑔𝑇^𝑚𝑧=12𝑔𝑥_𝑛−𝑔(𝑧)+𝑥_𝑛,∇𝑔(𝑧)−𝑥_𝑛(𝑧+1,∇𝑔)2𝑔𝑥_𝑛−𝑔𝑇^𝑚𝑧+𝑇^𝑚𝑧,∇𝑔𝑇^𝑚𝑧−𝑥_𝑛,∇𝑔𝑇^𝑚𝑧−14𝜌^∗_𝑠2∇𝑔(𝑧)−∇𝑔𝑇^𝑚𝑧=12𝐷_𝑔𝑥_𝑛+1,𝑧2𝐷_𝑔𝑥_𝑛,𝑇^𝑚𝑧−14𝜌^∗_𝑠2∇𝑔(𝑧)−∇𝑔𝑇^𝑚𝑧. (61) Applying to both sides limsup_𝑛→∞ and remembering that 𝜙(𝑧)=inf{𝜙(𝑥)∶𝑥∈𝐶} we obtain 𝜙(𝑧)≤12𝜙(𝑧)+12𝜙𝑇^𝑚𝑧−14𝜌^∗_𝑠2∇𝑔(𝑧)−∇𝑔𝑇^𝑚𝑧. (62) This implies that 𝜌^∗_𝑠2∇𝑔(𝑧)−∇𝑔𝑇^𝑚𝑧≤2𝜙𝑇^𝑚𝑧−2𝜙(𝑧). (63) Letting 𝑚→∞ in (63) we conclude that lim_𝑚→∞𝜌^∗_𝑠2∇𝑔(𝑧)−∇𝑔𝑇^𝑚𝑧=0. (64) From the properties of 𝜌^∗_𝑠2, we deduce that lim_𝑚→∞‖∇𝑔(𝑧)−∇𝑔(𝑇^𝑚𝑧)‖. Since ∇𝑔^∗ is uniformly norm-to-norm continuous on bounded subsets of 𝐸^∗, we arrive at lim_𝑚→∞‖𝑧−𝑇^𝑚𝑧‖=0. Thus we have 𝑇^𝑚+1𝑧→𝑧; that is, 𝑇(𝑇^𝑚𝑧)→𝑧. On the other hand, in view of three-point identity (see (26)), we deduce that 𝐷_𝑔(𝑧,𝑇𝑧)=𝐷_𝑔𝑧,𝑇^𝑚+1𝑧+𝐷_𝑔𝑇^𝑚+1+𝑧,𝑇𝑧𝑧−𝑇^𝑚+1𝑧,∇𝑔𝑇^𝑚+1𝑧−∇𝑔(𝑇𝑧)≤𝐷_𝑔𝑧,𝑇^𝑚+1𝑧+𝑎₁𝑇^𝑚𝑧𝐷_𝑔𝑇^𝑚+𝑧,𝑧𝑧−𝑇^𝑚+1𝑧∇𝑔𝑇^𝑚+1𝑧−∇𝑔(𝑇𝑧). (65) Letting 𝑚→∞ in the above inequalities we deduce that 𝐷_𝑔(𝑧,𝑇𝑧)=0 and hence by Lemma 8 we conclude that 𝑇𝑧=𝑧, which completes the proof.

4. Weak Convergence Theorems of Generalized Mann Iteration Process

In this section, we prove weak convergence theorems of generalized Mann iteration process in a reflexive Banach space.

Definition 16. 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, bounded, closed, and convex subset of 𝐸 and let 𝑇∈ℬ𝒯_𝑟(𝐶). Let {𝑛_𝑘}_𝑘∈ℕ be an increasing sequence in ℕ and let {𝛾_𝑘}_𝑘∈ℕ⊂(0,1) be such that liminf_𝑘→𝛾_𝑘(1−𝛾_𝑘)>0. The generalized Mann iteration process generated by the mapping 𝑇, the sequence {𝛾_𝑘}_𝑘∈ℕ, and the sequence {𝑛_𝑘}_𝑘∈ℕ, denoted by 𝑔𝑀(𝑇,{𝛾_𝑘}_𝑘∈ℕ,{𝑛_𝑘}_𝑘∈ℕ), is defined by the following iterative formula: 𝑥_𝑘+1=𝛾_𝑘𝑇^𝑛𝑘𝑥_𝑘+1−𝛾_𝑘𝑥_𝑘,where𝑥₁∈𝐶ischosenarbitrarily. (66)

Definition 17. We say that a generalized Mann iteration process 𝑔𝑀(𝑇,{𝛾_𝑘}_𝑘∈ℕ,{𝑛_𝑘}_𝑘∈ℕ) is well defined if limsup_𝑘→∞𝑎_𝑛𝑘𝑥_𝑘=1. (67)

Remark 18. Observe that by the definition of Bregman asymptotic pointwise nonexpansive, lim_𝑘→∞𝑎_𝑛𝑘(𝑥)=1 for every 𝑥∈𝐶. Hence we can always select a subsequence {𝑎_𝑛𝑘}_𝑘∈ℕ of {𝑎_𝑛}_𝑛∈ℕ such that (67) holds. In other words, by a suitable choice of {𝑛_𝑘}_𝑘∈ℕ we can always make 𝑔𝑀(𝑇,{𝛾_𝑘}_𝑘∈ℕ,{𝑛_𝑘}_𝑘∈ℕ) well defined.

We will prove a series of lemmas necessary for the proof of the generalized Mann process convergence theorem.

Lemma 19. 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, bounded, closed, and convex subset of 𝐸. Let 𝑇∈ℬ𝒯_𝑟(𝐶) and let {𝑛_𝑘}_𝑘∈ℕ⊂ℕ. Let {𝛾_𝑘}_𝑘∈ℕ⊂(0,1) such that liminf_𝑘→∞𝛾_𝑘(1−𝛾_𝑘)>0. Let 𝑤∈𝐹(𝑇) and let 𝑔𝑀(𝑇,{𝛾_𝑘}_𝑘∈ℕ,{𝑛_𝑘}_𝑘∈ℕ) be a generalized Mann process. Then there exists 𝜆∈ℝ such that lim_𝑘→∞𝐷(𝑥_𝑘,𝑤)=𝜆.

Proof. Let 𝑤∈𝐹(𝑇) be arbitrary chosen. In view of (66), we obtain 𝐷_𝑔𝑥_𝑘+1,𝑤≤𝛾_𝑘𝐷_𝑔𝑇^𝑛𝑘𝑥_𝑘+,𝑤1−𝛾_𝑘𝐷_𝑔𝑥_𝑘,𝑤≤𝛾_𝑘𝐷_𝑔𝑇^𝑛𝑘𝑥_𝑘,𝑇^𝑛𝑘(𝑤)+1−𝛾_𝑘𝐷_𝑔𝑥_𝑘,𝑤≤𝛾_𝑘1+𝑏_𝑛𝑘𝑥_𝑘𝐷_𝑔𝑥_𝑘+,𝑤1−𝛾_𝑘𝐷_𝑔𝑥_𝑘,𝑤≤𝛾_𝑘𝑏_𝑛𝑘𝑥_𝑘𝐷_𝑔𝑥_𝑘,𝑤+𝐷_𝑔𝑥_𝑘,𝑤≤𝑏_𝑛𝑘𝑥_𝑘𝐵diam(𝐶)+𝐷_𝑔𝑥_𝑘,𝑤. (68) This implies that for every 𝑛∈ℕ, 𝐷_𝑔𝑥_𝑘+𝑛,𝑤≤𝐷_𝑔𝑥_𝑘,𝑤+𝐵diam(𝐶)^{𝑘+𝑛−1}_𝑖=𝑘𝑏_𝑛𝑖𝑥_𝑖. (69) Put 𝑟_𝑝=𝐷_𝑔(𝑥_𝑝,𝑤) for every 𝑝∈ℕ and 𝑑_𝑘,𝑛=𝐵diam(𝐶)∑^{𝑘+𝑛−1}_𝑖=𝑘𝑏_𝑛𝑖(𝑥_𝑖). Since 𝑇∈ℬ𝒯_𝑟(𝐶), we obtain that limsup_𝑘→∞limsup_𝑛→∞𝑑_𝑘,𝑛=0. In view of Lemma 11, there exists 𝜆∈ℝ such that lim_𝑘→∞𝐷_𝑔(𝑥_𝑘,𝑤)=𝜆. This completes the proof.

Lemma 20. 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, bounded, closed, and convex subset of 𝐸. Let 𝑇∈ℬ𝒯_𝑟(𝐶) and let {𝑛_𝑘}_𝑘∈ℕ⊂ℕ. Let {𝛾_𝑘}_𝑘∈ℕ⊂(0,1) such that liminf_𝑘→∞𝛾_𝑘(1−𝛾_𝑘)>0 and let 𝑔𝑀(𝑇,{𝛾_𝑘}_𝑘∈ℕ,{𝑛_𝑘}_𝑘∈ℕ) be a generalized Mann process. Then lim_𝑘→∞𝐷_𝑔𝑇^𝑛𝑘𝑥_𝑘,𝑥_𝑘=0,lim_𝑘→∞𝐷_𝑔𝑥_𝑘+1,𝑥_𝑘=0. (70)

Proof. In view of Proposition 13, we conclude that 𝐹(𝑇)≠∅. Let 𝑤∈𝐹(𝑇) be fixed. It follows from Lemma 19 that there exists 𝜆∈ℝ such that lim_𝑘→∞𝐷_𝑔(𝑥_𝑘,𝑤)=𝜆. Let 𝑠₃=sup{‖𝑇^𝑛𝑘(𝑥_𝑘)‖,‖𝑥_𝑘‖∶𝑘∈ℕ} and let 𝜌_𝑠3∶𝐸→ℝ be the gauge of uniform convexity of 𝑔. By the definition of 𝑇, we obtain 𝐷_𝑔𝑥_𝑘+1,𝑤=𝐷_𝑔𝛾_𝑘𝑇^𝑛𝑘𝑥_𝑘+1−𝛾_𝑘𝑥_𝑘,𝑤≤𝛾_𝑘𝐷_𝑔𝑇^𝑛𝑘𝑥_𝑘,𝑤+1−𝛾_𝑘𝐷_𝑔𝑥_𝑘,𝑤=𝛾_𝑘𝐷_𝑔𝑇^𝑛𝑘𝑥_𝑘,𝑇^𝑛𝑘(𝑤)+1−𝛾_𝑘𝐷_𝑔𝑥_𝑘,𝑤≤𝛾_𝑘𝑎_𝑛𝑘𝑥_𝑘𝐷_𝑔𝑥_𝑘,𝑤+1−𝛾_𝑘𝐷_𝑔𝑥_𝑘,𝑤−𝛾_𝑘1−𝛾_𝑘𝜌_𝑠3𝑇^𝑛𝑘𝑥_𝑘−𝑥_𝑘=𝛾_𝑘𝑎_𝑛𝑘𝑥_𝑘−1𝐷_𝑔𝑥_𝑘,𝑤+𝐷_𝑔𝑥_𝑘,𝑤−𝛾_𝑘1−𝛾_𝑘𝜌_𝑠3𝑇^𝑛𝑘𝑥_𝑘−𝑥_𝑘. (71) This implies that 𝛾_𝑘1−𝛾_𝑘𝜌_𝑠3𝑇^𝑛𝑘𝑥_𝑘−𝑥_𝑘≤𝑎_𝑛𝑘𝑥_𝑘−1𝐷_𝑔𝑥_𝑘,𝑤+𝐷_𝑔𝑥_𝑘,𝑤−𝐷_𝑔𝑥_𝑘+1,𝑤. (72) Letting 𝑘→∞ in (72) we conclude that lim_𝑘→∞𝜌_𝑠3𝑇^𝑛𝑘𝑥_𝑘−𝑥_𝑘=0. (73) From the properties of 𝜌_𝑠3, we deduce that lim_𝑘→∞‖𝑇^𝑛𝑘(𝑥_𝑘)−𝑥_𝑘‖=0. Employing Lemma 8, we conclude that lim_𝑘→∞𝐷_𝑔𝑇^𝑛𝑘𝑥_𝑘,𝑥_𝑘=0,lim_𝑘→∞𝐷_𝑔𝑥_𝑘+1,𝑥_𝑘=0. (74) This completes the proof.

In the next lemma, we prove that under suitable assumption the sequence {𝑥_𝑘}_𝑘∈ℕ becomes an approximate fixed point sequence, which will provide an important step in the proof of the generalized Mann iteration process convergence. First, we need to recall the following notions.

Definition 21. A strictly increasing sequence {𝑛_𝑖}_𝑖∈ℕ⊂ℕ is called quasiperiodic if the sequence {𝑛_𝑖+1−𝑛_𝑖}_𝑖∈ℕ is bounded or equivalently if there exists a number 𝑝∈ℕ such that any block of consecutive natural numbers must contain a term of the sequence {𝑛_𝑖}_𝑖∈ℕ. The smallest of such numbers will be called a quasi period of {𝑛_𝑖}_𝑖∈ℕ.

Lemma 22. 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, bounded, closed, and convex subset of 𝐸. Let 𝑇∈ℬ𝒯_𝑟(𝐶) and let {𝑛_𝑘}_𝑘∈ℕ⊂ℕ. Let {𝛾_𝑘}_𝑘∈ℕ⊂(0,1) such that liminf_𝑘→∞𝛾_𝑘(1−𝛾_𝑘)>0. Let {𝑛_𝑘}_𝑘∈ℕ⊂ℕ be such that the generalized Mann process 𝑔𝑀(𝑇,{𝛾_𝑘}_𝑘∈ℕ,{𝑛_𝑘}_𝑘∈ℕ) is well defined. If, in addition, the set of indices 𝒥={𝑗∈ℕ∶𝑛_𝑗+1=1+𝑛_𝑗} is quasi-periodic, then {𝑥_𝑘}_𝑘∈ℕ is an approximate fixed point sequence; that is, lim_𝑘→∞𝑇𝑥_𝑘−𝑥_𝑘=0. (75)

Proof. In view of (66), we have 𝑥_𝑘+1−𝑥_𝑘=𝛾_𝑘𝑇^𝑛𝑘𝑥_𝑘−𝑥_𝑘. (76) This, together with Lemmas 8 and 20, implies that lim_𝑘→∞𝑥_𝑘+1−𝑥_𝑘=0. (77) In view of Lemma 8, we conclude that lim_𝑚→∞𝐷_𝑔𝑥_𝑘+1,𝑥_𝑘=0. (78) Let 𝑝∈ℕ be a quasi-period of 𝒥. We first prove that ‖𝑥_𝑘−𝑇(𝑥_𝑘)‖→0 as 𝑘→∞ through 𝒥. Since 𝑛_𝑘+1=𝑛_𝑘+1 for such 𝑘, we obtain 𝐷_𝑔𝑇^𝑛𝑘+1𝑥_𝑘+1,𝑇^𝑛𝑘+1𝑥_𝑘=𝐷_𝑔𝑇^𝑛𝑘+1𝑥_𝑘+1,𝑇^𝑛𝑘+1𝑥_𝑘≤𝑎_𝑛𝑘+1𝑥_𝑘+1𝐷_𝑔𝑥_𝑘+1,𝑥_𝑘,𝐷_𝑔𝑇𝑇^𝑛𝑘𝑥_𝑘,𝑇𝑥_𝑘≤𝑀₅𝐷_𝑔𝑇^𝑛𝑘𝑥_𝑘,𝑥_𝑘, (79) where 𝑀₅=sup{𝑎₁(𝑥)∶𝑥∈𝐶}<∞. This implies that lim_𝑘→∞𝐷_𝑔𝑇^𝑛𝑘+1𝑥_𝑘+1,𝑇^𝑛𝑘+1𝑥_𝑘=0,lim_𝑘→∞𝐷_𝑔𝑇𝑇^𝑛𝑘𝑥_𝑘,𝑇𝑥_𝑘=0. (80) It follows from Lemma 8 that lim_𝑘→∞𝑇^𝑛𝑘+1𝑥_𝑘+1−𝑇^𝑛𝑘+1𝑥_𝑘=0,lim_𝑘→∞𝑇𝑇^𝑛𝑘𝑥_𝑘−𝑇𝑥_𝑘=0. (81) On the other hand, we have 𝑥_𝑘−𝑇𝑥_𝑘≤𝑥_𝑘−𝑥_𝑘+1+𝑥_𝑘+1−𝑇^𝑛𝑘+1𝑥_𝑘+1+𝑇^𝑛𝑘+1𝑥_𝑘+1−𝑇^𝑛𝑘+1𝑥_𝑘+𝑇𝑇^𝑛𝑘𝑥_𝑘−𝑇𝑥_𝑘. (82) Thus, we obtain ‖𝑥_𝑘−𝑇(𝑥_𝑘)‖→0 as 𝑘→∞ through 𝒥. In view of Lemma 8, we conclude that 𝐷_𝑔(𝑥_𝑘,𝑇(𝑥_𝑘))→0 as 𝑘→∞ through 𝒥. Now, let 𝜖>0 be fixed. It follows from 𝐷_𝑔(𝑥_𝑘,𝑇(𝑥_𝑘))→0 as 𝑘→∞ through 𝒥 that there exists 𝑁₀∈ℕ such that 𝐷_𝑔𝑥_𝑘,𝑇𝑥_𝑘<𝜖3,∀𝑘≥𝑁₀. (83) Since 𝒥 is quasi-periodic, for any 𝑘∈ℕ there exists 𝑗_𝑘∈𝒥 such that |𝑘−𝑗_𝑘|≤𝑝. Assume that 𝑘−𝑝≤𝑗_𝑘≤𝑘 (the proof of the other case is identical). Since 𝑇 is a Bregman 𝑀₆-Lipschitzian mapping where 𝑀₆=sup{𝑎₁(𝑥)∶𝑥∈𝐶}, there exists 0<𝛿<𝜖/3 such that 𝐷_𝑔<𝜖𝑇𝑥,𝑇𝑦3if𝐷_𝑔𝑥,𝑦<𝛿. (84) In view of (66) and (83), there exists 𝑁₁∈ℕ such that 𝑥_𝑘+1−𝑥_𝑘<𝛿𝑝,∀𝑘≥𝑁₁. (85) We also obtain 𝑥_𝑘−𝑥_𝑗𝑘≤𝑥_𝑘−𝑥_𝑘−1+𝑥_𝑘−1−𝑥_𝑘−2+⋯+𝑥_𝑗𝑘−1−𝑥_𝑗𝑘𝛿≤𝑝𝑝=𝛿. (86) This implies that lim_𝑘→∞𝐷_𝑔𝑥_𝑘,𝑥_𝑗𝑘=0,lim_𝑘→∞𝐷_𝑔𝑇𝑥_𝑘,𝑇𝑥_𝑗𝑘=0. (87) If follows from (77)-(78) that 𝑥_𝑘−𝑇𝑥_𝑘≤𝑥_𝑘−𝑥_𝑗𝑘+𝑥_𝑗𝑘−𝑇𝑥_𝑗𝑘+𝑇𝑥_𝑗𝑘−𝑇𝑥_𝑘𝜖≤𝛿+3+𝜖3<𝜖. (88) This completes the proof.

Theorem 23. 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. Let 𝐶 be a nonempty, bounded, closed and convex subset of 𝐸. Let 𝑇∈ℬ𝒯_𝑟(𝐶) and let {𝑛_𝑘}_𝑘∈ℕ⊂ℕ. Let {𝛾_𝑘}_𝑘∈ℕ⊂(0,1) such that liminf_𝑘→∞𝛾_𝑘(1−𝛾_𝑘)>0. Let {𝑛_𝑘}_𝑘∈ℕ⊂ℕ be such that the generalized Mann process 𝑔𝑀(𝑇,{𝛾_𝑘}_𝑘∈ℕ,{𝑛_𝑘}_𝑘∈ℕ) is well defined. If, in addition, the set of indices 𝒥={𝑗∈ℕ∶𝑘_𝑗+1=1+𝑘_𝑗} is quasi-periodic, then the sequence {𝑥_𝑘}_𝑘∈ℕ generated by 𝑔𝑀(𝑇,{𝛾_𝑘}_𝑘∈ℕ,{𝑛_𝑘}_𝑘∈ℕ) converges weakly to a fixed point of 𝑇.

Proof. In view of Lemma 22, we obtain lim_𝑘→∞𝑇𝑥_𝑘−𝑥_𝑘=0. (89) Let 𝑦,𝑧∈𝐶 be weak cluster points of the sequence {𝑥_𝑘}_𝑘∈ℕ. Then there exist subsequences {𝑦_𝑘}_𝑘∈ℕ and {𝑧_𝑘}_𝑘∈ℕ of {𝑥_𝑘}_𝑘∈ℕ such that 𝑦_𝑘⇀𝑦 and 𝑧_𝑘⇀𝑧 as 𝑘→∞. In view of (89) and Theorem 15, we conclude that 𝑇𝑦=𝑦 and 𝑇𝑧=𝑧. It follows from Lemma 19 that there exist real numbers 𝜆₁ and 𝜆₂ such that lim_𝑘→∞𝐷_𝑔𝑦_𝑘,𝑦=𝜆₁,lim_𝑘→∞𝐷_𝑔𝑧_𝑘,𝑧=𝜆₂. (90) We claim that 𝑦=𝑧. Assume on the contrary that 𝑦≠𝑧. By the Bregman Opial-like property we obtain 𝜆₁=limsup_𝑘→∞𝐷_𝑔𝑦_𝑘,𝑦<limsup_𝑘→∞𝐷_𝑔𝑦_𝑘,𝑧=limsup_𝑘→∞𝐷_𝑔𝑧_𝑘,𝑧<limsup_𝑘→∞𝐷_𝑔𝑧_𝑘,𝑦=limsup_𝑘→∞𝐷_𝑔𝑦_𝑘,𝑦=𝜆₁. (91) This is a contradiction and hence there exists 𝑤∈𝐶 such that 𝑥_𝑘⇀𝑤 as 𝑘→∞. Since 𝐶 is weakly sequentially compact, such a weak cluster point 𝑤 is unique. In view of Theorem 15, we conclude that 𝑇(𝑤)=𝑤, which completes the proof.

5. Weak Convergence of Generalized Ishikawa Iteration Process

The two-step Ishikawa iteration process is a generalization of the one-step Mann iteration process. The Ishikawa iteration process provides more flexibility in defining the algorithm parameters which is important from numerical implementation perspective.

Definition 24. 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. Let 𝐶 be a nonempty, bounded, closed, and convex subset of 𝐸. Let 𝑇∈ℬ𝒯_𝑟(𝐶) and let {𝑛_𝑘}_𝑘∈ℕ⊂ℕ be an increasing sequence. Let {𝛽_𝑘}_𝑘∈ℕ⊂(0,1) and {𝛾_𝑘}_𝑘∈ℕ⊂(0,1) be sequences of real numbers such that liminf_𝑘→∞𝛽_𝑘(1−𝛽_𝑘)>0 and liminf_𝑘→∞𝛾_𝑘(1−𝛾_𝑘)>0. The generalized Ishikawa iteration process generated by the mapping 𝑇, the sequences {𝛽_𝑘}_𝑘∈ℕ⊂(0,1), and {𝛾_𝑘}_𝑘∈ℕ⊂(0,1), and the sequence {𝑛_𝑘}_𝑘∈ℕ denoted by 𝑔𝐼(𝑇,{𝛽_𝑘}_𝑘∈ℕ,{𝛾_𝑘}_𝑘∈ℕ,{𝑛_𝑘}_𝑘∈ℕ) is defined by the following iterative scheme: 𝑥₁𝑦∈𝐶chosenarbitrarily,_𝑘=𝛽_𝑘𝑇^𝑛𝑘𝑥_𝑘+1−𝛽_𝑘𝑥_𝑘,𝑥_𝑘+1=𝛾_𝑘𝑇^𝑛𝑘𝑦_𝑘+1−𝛾_𝑘𝑥_𝑘. (92)

Definition 25. We say that a generalized Ishikawa iteration process 𝑔𝐼(𝑇,{𝛽_𝑘}_𝑘∈ℕ,{𝛾_𝑘}_𝑘∈ℕ,{𝑛_𝑘}_𝑘∈ℕ) is well defined if limsup_𝑘→∞𝑎_𝑛𝑘𝑥_𝑘=limsup_𝑘→∞𝑎_𝑛𝑘𝑦_𝑘=1. (93)

Lemma 26. 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, bounded, closed, and convex subset of 𝐸. Let 𝑇∈ℬ𝒯_𝑟(𝐶) and let {𝑛_𝑘}_𝑘∈ℕ⊂ℕ. Let {𝛽_𝑘}_𝑘∈ℕ⊂(0,1) and {𝛾_𝑘}_𝑘∈ℕ⊂(0,1) be sequences of real numbers such that liminf_𝑘→∞𝛽_𝑘(1−𝛽_𝑘)>0 and liminf_𝑘→∞𝛾_𝑘(1−𝛾_𝑘)>0. Let 𝑤∈𝐹(𝑇) and 𝑔𝐼(𝑇,{𝛽_𝑘}_𝑘∈ℕ,{𝛾_𝑘}_𝑘∈ℕ,{𝑛_𝑘}_𝑘∈ℕ) be a generalized Ishikawa iteration process. Then there exists 𝜃∈ℝ such that lim_𝑘→∞𝐷_𝑔(𝑥_𝑘,𝑤)=𝜃.

Proof. Let 𝑀₇>1 be fixed. Since lim_𝑘→∞𝑎_𝑛𝑘(𝑦_𝑘)=1, there exists 𝑘₀∈ℕ such that for any 𝑘>𝑘₀, 𝑎_𝑛𝑘(𝑦_𝑘)≤𝑀₇. Let 𝑠₄=sup{‖𝑇^𝑛𝑘(𝑥_𝑘)‖,‖𝑥_𝑘‖∶𝑘∈ℕ} and let 𝜌_𝑠4∶𝐸→ℝ be the gauge of uniform convexity of 𝑔. By the definition of 𝑇 and in view of (93), we obtain 𝐷_𝑔𝑥_𝑘+1,𝑤=𝐷_𝑔𝛾_𝑘𝑇^𝑛𝑘𝑦_𝑘+1−𝛾_𝑘𝑥_𝑘,𝑤≤𝛾_𝑘𝐷_𝑔𝑇^𝑛𝑘𝑦_𝑘,𝑤+1−𝛾_𝑘𝐷_𝑔𝑥_𝑘,𝑤=𝛾_𝑘𝐷_𝑔𝑇^𝑛𝑘𝑦_𝑘,𝑇^𝑛𝑘(𝑤)+1−𝛾_𝑘𝐷_𝑔𝑥_𝑘,𝑤≤𝛾_𝑘𝑎_𝑛𝑘𝑦_𝑘𝐷_𝑔𝑦_𝑘,𝑤+1−𝛾_𝑘𝐷_𝑔𝑥_𝑘,𝑤≤𝛾_𝑘𝑎_𝑛𝑘𝑦_𝑘𝛽_𝑘𝐷_𝑔𝑇^𝑛𝑘𝑥_𝑘+,𝑤1−𝛽_𝑘𝐷_𝑔𝑥_𝑘,𝑤−𝜌_𝑠4𝑇^𝑛𝑘𝑥_𝑘−𝑥_𝑘+1−𝛾_𝑘𝐷_𝑔𝑥_𝑘,𝑤=𝛾_𝑘𝑎_𝑛𝑘(𝑤)𝛽_𝑘𝐷_𝑔𝑇^𝑛𝑘𝑥_𝑘,𝑇^𝑛𝑘(𝑤+)1−𝛽_𝑘𝐷_𝑔𝑥_𝑘,𝑤−𝜌_𝑠4𝑇^𝑛𝑘𝑥_𝑘−𝑥_𝑘+1−𝛾_𝑘𝐷_𝑔𝑥_𝑘,𝑤≤𝛾_𝑘𝑎_𝑛𝑘𝑦_𝑘𝛽_𝑘𝑎_𝑛𝑘𝑥_𝑘𝐷_𝑔𝑥_𝑘+,𝑤1−𝛽_𝑘𝐷_𝑔𝑥_𝑘,𝑤−𝜌_𝑠4𝑇^𝑛𝑘𝑥_𝑘−𝑥_𝑘+1−𝛾_𝑘𝐷_𝑔𝑥_𝑘,𝑤=𝛾_𝑘𝑎_𝑛𝑘𝑦_𝑘𝛽_𝑘𝑎_𝑛𝑘𝑥_𝑘−1𝐷_𝑔𝑥_𝑘,𝑤+𝛽_𝑘𝐷_𝑔𝑥_𝑘+,𝑤1−𝛽_𝑘𝐷_𝑔𝑥_𝑘,𝑤−𝜌_𝑠4𝑇^𝑛𝑘𝑥_𝑘−𝑥_𝑘+1−𝛾_𝑘𝐷_𝑔𝑥_𝑘,𝑤=𝛾_𝑘𝛽_𝑘𝑎_𝑛𝑘𝑦_𝑘𝑎_𝑛𝑘𝑥_𝑘−1𝐷_𝑔𝑥_𝑘,𝑤+𝛾_𝑘𝑎_𝑛𝑘𝑦_𝑘𝐷_𝑔𝑥_𝑘,𝑤−𝛾_𝑘𝜌_𝑠4𝑇^𝑛𝑘𝑥_𝑘−𝑥_𝑘+1−𝛾_𝑘𝐷_𝑔𝑥_𝑘,𝑤≤𝛾_𝑘𝑎_𝑛𝑘𝑦_𝑘𝑎_𝑛𝑘𝑥_𝑘−1𝐷_𝑔𝑥_𝑘,𝑤+𝛾_𝑘𝑎_𝑛𝑘𝑦_𝑘𝐷_𝑔𝑥_𝑘,𝑤+1−𝛾_𝑘𝐷_𝑔𝑥_𝑘,𝑤−𝛾_𝑘𝜌_𝑠4𝑇^𝑛𝑘𝑥_𝑘−𝑥_𝑘=𝛾_𝑘𝑎_𝑛𝑘𝑦_𝑘𝑎_𝑛𝑘𝑥_𝑘𝐷_𝑔𝑥_𝑘,𝑤+1−𝛾_𝑘𝐷_𝑔𝑥_𝑘,𝑤−𝛾_𝑘𝜌_𝑠4𝑇^𝑛𝑘𝑥_𝑘−𝑥_𝑘=𝛾_𝑘𝑎_𝑛𝑘𝑦_𝑘𝑎_𝑛𝑘𝑥_𝑘𝐷_𝑔𝑥_𝑘,𝑤−𝐷_𝑔𝑥_𝑘,𝑤+𝛾_𝑘𝐷_𝑔𝑥_𝑘+,𝑤1−𝛾_𝑘𝐷_𝑔𝑥_𝑘,𝑤−𝛾_𝑘𝛽_𝑘𝜌_𝑠4𝑇^𝑛𝑘𝑥_𝑘−𝑥_𝑘≤𝑎_𝑛𝑘𝑦_𝑘𝑎_𝑛𝑘𝑥_𝑘−1𝐷_𝑔𝑥_𝑘,𝑤+𝐷_𝑔𝑥_𝑘,𝑤−𝛾_𝑘𝜌_𝑠4𝑇^𝑛𝑘𝑥_𝑘−𝑥_𝑘=𝑎_𝑛𝑘𝑦_𝑘𝑎_𝑛𝑘𝑥_𝑘𝐷_𝑔𝑥_𝑘,𝑤−𝛾_𝑘𝜌_𝑠4𝑇^𝑛𝑘𝑥_𝑘−𝑥_𝑘=𝑎_𝑛𝑘𝑦_𝑘𝑎_𝑛𝑘𝑥_𝑘−1𝐷_𝑔𝑥_𝑘+𝑎,𝑤_𝑛𝑘𝑥_𝑘−1𝐷_𝑔𝑥_𝑘,𝑤+𝐷_𝑔𝑥_𝑘,𝑤−𝛾_𝑘𝜌_𝑠4‖𝑇^𝑛𝑘𝑥_𝑘−𝑥_𝑘‖≤𝑀₇𝑏_𝑛𝑘𝑥_𝑘𝐵diam(𝐶)+𝐷_𝑔𝑥_𝑘,𝑤−𝛾_𝑘𝜌_𝑠4𝑇^𝑛𝑘𝑥_𝑘−𝑥_𝑘. (94) This implies that for every 𝑛∈ℕ, 𝐷_𝑔𝑥_𝑘+𝑛,𝑤≤𝐷_𝑔𝑥_𝑘,𝑤+𝑀₇𝐵diam(𝐶)^{𝑘+𝑛−1}_𝑖=𝑘𝑏_𝑛𝑖𝑥_𝑖. (95) Put 𝑟_𝑝=𝐷_𝑔(𝑥_𝑝,𝑤) for every 𝑝∈ℕ and 𝑑_𝑘,𝑛=𝑀₇𝐵diam(𝐶)∑^{𝑘+𝑛−1}_𝑖=𝑘𝑏_𝑛𝑖(𝑥_𝑖). Since 𝑇∈ℬ𝒯_𝑟(𝐶), we obtain that limsup_𝑘→∞limsup_𝑛→∞𝑑_𝑘,𝑛=0. In view of Lemma 11, there exists 𝜃∈ℝ such that lim_𝑘→∞𝐷_𝑔(𝑥_𝑘,𝑤)=𝜃. Let 𝑀₈>1 be fixed. Since lim_𝑘→∞𝑎_𝑛𝑘(𝑥_𝑘)=1, there exists 𝑘₀∈ℕ such that for any 𝑘>𝑘₀, 𝑎_𝑛𝑘(𝑥_𝑘)≤𝑀₈. Therefore, by the same argument, as in the proof of Lemma 19, we conclude that for 𝑘>𝑘₀ and 𝑛>1𝐷_𝑔𝑥_𝑘+𝑛,𝑤≤𝐷_𝑔𝑥_𝑘,𝑤+𝑀₇𝐵diam(𝐶)^{𝑘+𝑛−1}_𝑖=𝑘𝑏_𝑛𝑖𝑥_𝑖≤𝐷_𝑔𝑥_𝑘,𝑤+𝑀₇𝐵diam(𝐶)^{𝑘+𝑛−1}_𝑖=𝑘𝑏_𝑛𝑖𝑥_𝑖. (96) By the same manner as in the proof of Lemma 19, we deduce that there exists 𝜃∈ℝ such that lim_𝑘→∞𝐷_𝑔(𝑥_𝑘,𝑤)=𝜃, which completes the proof.

Lemma 27. 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. Let 𝐶 be a nonempty, bounded, closed, and convex subset of 𝐸. Let 𝑇∈ℬ𝒯_𝑟(𝐶) and let {𝑛_𝑘}_𝑘∈ℕ⊂ℕ. Let {𝛽_𝑘}_𝑘∈ℕ⊂(0,1) and {𝛾_𝑘}_𝑘∈ℕ⊂(0,1) be sequences of real numbers such that liminf_𝑘→∞𝛾_𝑘(1−𝛾_𝑘)>0 and liminf_𝑘→∞𝛽_𝑘(1−𝛽_𝑘)>0. Let 𝑤∈𝐹(𝑇) and 𝑔𝐼(𝑇,{𝛾_𝑘}_𝑘∈ℕ,{𝛽_𝑘}_𝑘∈ℕ,{𝑛_𝑘}_𝑘∈ℕ) be a generalized Ishikawa iteration process. Then lim_𝑘→∞𝐷𝑇^𝑛𝑘𝑦_𝑘,𝑥_𝑘=0,lim_𝑘→∞𝐷_𝑔𝑥_𝑘+1,𝑥_𝑘=0. (97)

Proof. In view of Proposition 13, 𝐹(𝑇)≠∅. Take any 𝑤∈𝐹(𝑇) arbitrarily chosen. Then, by Lemma 26, there exists 𝜃∈ℝ such that lim_𝑘→∞𝐷_𝑔(𝑥_𝑘,𝑤)=𝜃. By the same arguments, as in the proof of Lemma 26, we conclude that 𝐷_𝑔𝑥_𝑘+1,𝑤≤𝑀₇𝑏_𝑛𝑘𝑥_𝑘𝐵diam(𝐶)+𝐷_𝑔𝑥_𝑘,𝑤−𝛾_𝑘𝜌_𝑠4𝑇^𝑛𝑘𝑥_𝑘−𝑥_𝑘. (98) This implies that 𝛾_𝑘𝜌_𝑠4𝑇^𝑛𝑘𝑥_𝑘−𝑥_𝑘≤𝑀₇𝑏_𝑛𝑘𝑥_𝑘𝐵diam(𝐶)+𝐷_𝑔𝑥_𝑘,𝑤−𝐷_𝑔𝑥_𝑘+1,𝑤. (99) This implies that lim_𝑘→∞𝜌_𝑠4𝑇^𝑛𝑘𝑥_𝑘−𝑥_𝑘=0. (100) Therefore, from the property of 𝜌_𝑠4 we deduce that lim_𝑘→∞𝑇^𝑛𝑘𝑥_𝑘−𝑥_𝑘=0. (101) In a similar way, as in the proof of Lemma 20, we can prove that lim_𝑘→∞𝑇𝑥_𝑘−𝑥_𝑘=0. (102) This completes the proof.

Lemma 28. 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. Let 𝐶 be a nonempty, bounded, closed, and convex subset of 𝐸. Let 𝑇∈ℬ𝒯_𝑟(𝐶) and let {𝑛_𝑘}_𝑘∈ℕ⊂ℕ. Let {𝛾_𝑘}_𝑘∈ℕ⊂(0,1) such that liminf_𝑘→∞𝛾_𝑘(1−𝛾_𝑘)>0. Let {𝑛_𝑘}_𝑘∈ℕ⊂ℕ be such that the generalized Ishikawa process 𝑔𝐼(𝑇,{𝛾_𝑘}_𝑘∈ℕ,{𝛽_𝑘}_𝑘∈ℕ,{𝑛_𝑘}_𝑘∈ℕ) is well defined. If, in addition, the set of indices 𝒥={𝑗∈ℕ∶𝑛_𝑗+1=1+𝑛_𝑗} is quasi-periodic, then {𝑥_𝑘}_𝑘∈ℕ is an approximate fixed point sequence; that is, lim_𝑘→∞𝑇𝑥_𝑘−𝑥_𝑘=0. (103)

Theorem 29. 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. Let 𝐶 be a nonempty, bounded, closed, and convex subset of 𝐸. Let 𝑇∈ℬ𝒯_𝑟(𝐶) and let {𝑛_𝑘}_𝑘∈ℕ⊂ℕ. Let {𝛾_𝑘}_𝑘∈ℕ⊂(0,1) such that liminf_𝑘→∞𝛾_𝑘(1−𝛾_𝑘)>0. Let {𝑛_𝑘}_𝑘∈ℕ⊂ℕ be such that the generalized Ishikawa process 𝑔𝐼(𝑇,{𝛾_𝑘}_𝑘∈ℕ,{𝛽_𝑘}_𝑘∈ℕ,{𝑛_𝑘}_𝑘∈ℕ) is well defined. If, in addition, the set of indices 𝒥={𝑗∈ℕ∶𝑘_𝑗+1=1+𝑘_𝑗} is quasi-periodic, then the sequence {𝑛_𝑘}_𝑘∈ℕ generated by 𝑔𝐼(𝑇,{𝛾_𝑘}_𝑘∈ℕ,{𝛽_𝑘}_𝑘∈ℕ,{𝑛_𝑘}_𝑘∈ℕ) converges weakly to a fixed point of 𝑇.

Remark 30. Theorem 29 improves Theorems 3.1 and 4.1 of [18] 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, strongly coercive Bregman function which is bounded on bounded sets and uniformly convex and uniformly smooth on bounded sets.(2)For the mappings, we extend the mapping from an asymptotic pointwise nonexpansive mapping to a Bregman asymptotic pointwise nonexpansive mapping.(3)Since we do not need the weak sequential continuity of the duality mapping in Theorems 23 and 29 as was the case in [18], we can apply Theorem 29 in the Lebesgue space 𝐿^𝑝 where 1<𝑝<∞ and 𝑝≠2 while this space is not applicable for Theorems 3.1, 4.1, and 5.1 of [18].

Authors’ Contribution

All authors read and approved the final paper.

Conflict of Interests

The authors declare that they have no competing interests.

Acknowledgments

The authors would like to thank the editor and the referees for sincere evaluation and constructive comments which improved the paper considerably.