Approximation of Fixed Points of Weak Bregman Relatively Nonexpansive Mappings in Banach Spaces

Abstract

We introduce a concept of weak Bregman relatively nonexpansive mapping which is distinct from Bregman relatively nonexpansive mapping. By using projection techniques, we construct several modification of Mann type iterative algorithms with errors and Halpern-type iterative algorithms with errors to find fixed points of weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings in Banach spaces. The strong convergence theorems for weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings are derived under some suitable assumptions. The main results in this paper develop, extend, and improve the corresponding results of Matsushita and Takahashi (2005) and Qin and Su (2007).

1. Introduction

Throughout this paper, without other specifications, we denote by 𝑅 the set of real numbers. Let 𝐸 be 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. Let 𝑓∶𝐸→𝑅∪{+∞} be proper convex and lower semicontinuous. The Fenchel conjugate of 𝑓 is the function 𝑓^∗∶𝐸^∗→(−∞,+∞] defined by 𝑓^∗(𝜉)=sup𝜉,𝑥−𝑓(𝑥)∶𝑥∈𝐸. (1.1) We denote by dom𝑓 the domain of 𝑓, that is, dom𝑓={𝑥∈𝐸∶𝑓(𝑥)<+∞}. Let 𝐶 be a nonempty closed and convex subset of 𝐸and𝑇∶𝐶→𝐶 a nonlinear mapping. Denote by 𝐹(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥}, the set of fixed points of 𝑇. 𝑇 is said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶.

In 1967, Brègman [1] discovered an elegant and effective technique for the using of the so-called Bregman distance function 𝐷_𝑓 (see, Section 2, Definition 2.1) in the process of designing and analyzing feasibility and optimization algorithms. This opened a growing area of research in which Bregman's technique is applied in various ways in order to design and analyze iterative algorithms for solving not only feasibility and optimization problems, but also algorithms for solving variational inequalities, for approximating equilibria, for computing fixed points of nonlinear mappings, and so on (see, e.g., [1–25], and the references therein).

Nakajo and Takahashi [26] introduced the following modification of the Mann iteration method for a nonexpansive mapping 𝑇∶𝐶→𝐶 in a Hilbert space 𝐻 as follows:𝑥₀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑧−𝑦_𝑛≤𝑧−𝑥_𝑛,𝑄_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥_𝑛−𝑥₀,𝑥≤0_𝑛+1=P_{𝐶𝑛∩𝑄𝑛}𝑥₀,∀𝑛≥0, (1.2) where {𝛼_𝑛}⊂[0,1] and 𝑃_𝐶 is the metric projection from 𝐻 onto a closed and convex subset 𝐶 of 𝐻. They proved that {𝑥_𝑛} generated by (1.2) converges strongly to a fixed point of 𝑇 under some suitable assumptions. Motivated by Nakajo and Takahashi [26], Matsushita and Takahashi [27] introduced the following modification of the Mann iteration method for a relatively nonexpansive mapping 𝑇∶𝐶→𝐶 in a Banach space 𝐸 as follows:𝑥₀𝑦∈𝐶,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛,𝑄_𝑛=𝑧∈𝐶∶𝐽𝑥_𝑛−𝐽𝑥₀,𝑥_𝑛,𝑥−𝑧≤0_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀,∀𝑛≥0, (1.3) where {𝛼_𝑛}⊂[0,1], 𝜙(𝑦,𝑥)=‖𝑦‖²−2⟨𝑦,𝐽(𝑥)⟩+‖𝑥‖² for all 𝑥,𝑦∈𝐸, 𝐽 is the duality mapping of 𝐸 and Π_𝐶 is the generalized projection (see, e.g., [2, 3, 28]) from 𝐸 onto a closed and convex subset 𝐶 of 𝐸. They also proved that {𝑥_𝑛} generated by (1.3) converges strongly to a fixed point of 𝑇 under some suitable assumptions. Martinez-Yanes and Xu [29] gave a Halpern-type iterative algorithm for a nonexpansive mapping 𝑇∶𝐶→𝐶 as follows: 𝑥₀𝑦∈𝐶,_𝑛=𝛽_𝑛𝑥₀+1−𝛽_𝑛𝑇𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑧−𝑦_𝑛²≤𝑧−𝑥_𝑛²+𝛽_𝑛𝑥₀²+2𝑥_𝑛−𝑥₀,𝑧,𝑄_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥_𝑛−𝑥₀,𝑥≤0_𝑛+1=P_{𝐶𝑛∩𝑄𝑛}𝑥₀,∀𝑛≥0, (1.4) where {𝛽_𝑛}⊂[0,1]. They derived that {𝑥_𝑛} generated by (1.3) converges strongly to a fixed point of 𝑇 under some suitable assumptions. Qin and Su [30] generalized the results of Martinez-Yanes and Xu [29] to a uniformly convex and uniformly smooth Banach space for a relatively nonexpansive mapping and proposed the following iterative algorithm:𝑥₀𝑦∈𝐶,_𝑛=𝐽⁻¹𝛽_𝑛𝐽𝑥₀+1−𝛽_𝑛𝐽𝑇𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝜙𝑧,𝑦_𝑛≤𝛽_𝑛𝜙𝑧,𝑥₀+1−𝛽_𝑛𝜙𝑧,𝑥_𝑛,𝑄_𝑛=𝑧∈𝐶∶𝐽𝑥_𝑛−𝐽𝑥₀,𝑥_𝑛,𝑥−𝑧≤0_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀,∀𝑛≥0, (1.5) where {𝛽_𝑛}⊂[0,1], Π_𝐶 is the generalized projection (see, e.g., [2, 3, 28]) from 𝐸 onto a closed and convex subset 𝐶 of 𝐸. They also obtained that {𝑥_𝑛} generated by (1.5) converges strongly to a fixed point of 𝑇 under some suitable assumptions. In 2003, Butnariu et al. [13] studied several notions of convex analysis: uniformly convexity at a point, total convexity at a point, uniformly convexity on bounded sets, and sequential consistency, which are useful in establishing convergence properties for fixed point and optimization algorithms in infinite dimensional Banach spaces. They established connections between these concepts and used these relations in order to obtain improved convergence results concerning the outer Bregman projection algorithm for solving convex feasibility problems and the generalized proximal point algorithm for optimization in Banach spaces. In 2006, Butnariu and Resmerita [14] presented a Bregman-type iterative algorithms and studied the convergence of the Bregman-type iterative method of solving operator equations. Resmerita [19] investigated the existence of totally convex functions in Banach spaces and, further, established continuity and stability properties of Bregman projections. Very recently, by using Bregman projection, Reich and Sabach [21] presented the following algorithms for finding common zeroes of maximal monotone operators 𝐴_𝑖∶𝐸→2^𝐸∗(𝑖=1,2,…,𝑁) in reflexive Banach space 𝐸 as follows: 𝑥₀𝑦∈𝐸,_𝑖𝑛=Res^𝑓_𝜆𝑖𝑛𝑥_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=proj^𝑓_{𝐶𝑛∩𝑄𝑛}𝑥₀𝑥,∀𝑛≥0,₀𝜂∈𝐸,_𝑖𝑛=𝜉^𝑖_𝑛+1𝜆^𝑖_𝑛∇𝑓𝑦^𝑖_𝑛−∇𝑓𝑥_𝑛,𝜉^𝑖_𝑛∈𝐴_𝑖𝑦^𝑖_𝑛,𝜔^𝑖_𝑛=∇𝑓^∗𝜆^𝑖_𝑛𝜂^𝑖_𝑛+∇𝑓𝑥_𝑛,𝐶^𝑖_𝑛=𝑧∈𝐸∶𝐷_𝑓𝑧,𝑦^𝑖_𝑛≤𝐷_𝑓𝑧,𝜔^𝑖_𝑛,𝐶_𝑛=^𝑁_𝑖=1𝐶^𝑖_𝑛,𝑄_𝑛=𝑧∈𝐸∶∇𝑓𝑥₀−∇𝑓𝑥_𝑛,𝑧−𝑥_𝑛,𝑥≤0_𝑛+1=proj^𝑓_{𝐶𝑛∩𝑄𝑛}𝑥₀,∀𝑛≥0. (1.6)(1.7) Further, under some suitable conditions, they obtained two strong convergence theorems of maximal monotone operators in reflexive Banach spaces. Reich and Sabach [22] studied the convergence of two iterative algorithms for finitely many Bregman strongly nonexpansive operators in Banach spaces and obtained two strong convergence theorems for finitely many Bregman strongly nonexpansive operators under some assumptions. In [24], Reich and Sabach proposed the following algorithms for finding common fixed points of finitely many Bregman firmly nonexpansive operators 𝑇_𝑖∶𝐶→𝐶(𝑖=1,2,…,𝑁) in reflexive Banach space 𝐸 as follows: if ⋂^𝑁_𝑖=1𝐹(𝑇_𝑖)≠∅𝑥₀𝑄∈𝐸,_𝑖0𝑦=𝐸,𝑖=1,2,…,𝑁,_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=proj^𝑓_𝑄𝑛+1𝑥₀,∀𝑛≥0. (1.8)(1) Under some suitable conditions, they proved that the sequence {𝑥_𝑛} generated by (1.8) converges strongly to ⋂^𝑁_𝑖=1𝐹(𝑇_𝑖) and applied it to the solution of convex feasibility and equilibrium problems.

Inspired and motivated by the works, we introduce the concept of weak Bregman relatively nonexpansive mappings in reflexive Banach space and give an example to illustrate the existence of weak Bregman relatively nonexpansive mapping and the difference between weak Bregman relatively nonexpansive mapping and Bregman relatively nonexpansive mapping. Secondly, by using the conception of the Bregman projection (see, e.g., [1, 13, 14]), we construct several modification of Mann type iterative algorithms with errors and Halpern-type iterative algorithms with errors to find fixed points of weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings in Banach spaces. The strong convergence theorems for weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings are derived under some suitable assumptions. Moreover, the convergence rate of our algorithms is faster than that of Matsushita and Takahashi [27] and Qin and Su [30]. The main results in this paper develop, extend, and improve the corresponding results in the literature.

2. Preliminaries

Let 𝐶 be a nonempty closed convex subset of a real reflexive Banach space 𝐸, and let 𝑇∶𝐶→𝐶 be a nonlinear mapping. A point 𝜔∈𝐶 is called an asymptotic fixed point of 𝑇 (see, e.g., [2, 3]) if 𝐶 contains a sequence {𝑥_𝑛} which converges weakly to 𝜔 such that lim_𝑛→∞‖𝑇𝑥_𝑛−𝑥_𝑛‖=0. A point 𝜔∈𝐶 is called an strong asymptotic fixed point of 𝑇 (see, e.g., [2, 3]) if 𝐶 contains a sequence {𝑥_𝑛} which converges strongly to 𝜔 such that lim_𝑛→∞‖𝑇𝑥_𝑛−𝑥_𝑛‖=0. We denote the sets of asymptotic fixed points and strong asymptotic fixed points of 𝑇 by 𝐹(𝑇) and 𝐹(𝑇), respectively. When {𝑥_𝑛} is a sequence in 𝐸, we denote strong convergence of {𝑥_𝑛} to 𝑥∈𝐸 by 𝑥_𝑛→𝑥. For any 𝑥∈int(dom𝑓) and 𝑦∈𝐸, the right-hand derivative of 𝑓 at 𝑥 in the direction 𝑦 defined by 𝑓⁰𝑥,𝑦∶=lim_𝑡↘0𝑓𝑥+𝑡𝑦−𝑓(𝑥)𝑡. (2.1)𝑓 is called Gâteaux differentiable at 𝑥 if, for all 𝑦∈𝐸,lim_𝑡↘0(𝑓(𝑥+𝑡𝑦)−𝑓(𝑥))/𝑡 exists. In this case, 𝑓⁰(𝑥,𝑦) coincides with ∇𝑓(𝑥), the value of the gradient of 𝑓 at 𝑥. 𝑓 is called Gâteaux differentiable if it is Gâteaux differentiable for any 𝑥∈int(dom𝑓). 𝑓 is called Fréchet differentiable at 𝑥 if this limit is attained uniformly for ‖𝑦‖=1. We say 𝑓 is uniformly Fréchet differentiable on a subset 𝐶 of 𝐸 if the limit is attained uniformly for 𝑥∈𝐶 and ‖𝑦‖=1.

Legendre function 𝑓∶𝐸→(−∞,+∞] is defined in [7]. From [7], if 𝐸 is a reflexive Banach space, then 𝑓 is Legendre if and only if it satisfies the following conditions (L1) and (L2):(L1) the interior of the domain of 𝑓, int(dom𝑓), is nonempty, 𝑓 is Gâteaux differentiable on int(dom𝑓), and dom𝑓=int(dom𝑓),(L2) the interior of the domain of 𝑓^∗, int(dom𝑓^∗), is nonempty, 𝑓^∗ is Gâteaux differentiable on int(dom𝑓^∗), and dom𝑓^∗=int(dom𝑓^∗).

Since 𝐸 is reflexive, we know that (𝜕𝑓)⁻¹=𝜕𝑓^∗ (see, e.g., [31]). This, by (L1) and (L2), implies ∇𝑓=∇𝑓^∗−1,ran∇𝑓=dom∇𝑓^∗=intdom𝑓^∗,ran∇𝑓^∗=dom∇𝑓=intdom𝑓. (2.2) By Theorem 5.4 [7], conditions (L1) and (L2) also yield that the functions 𝑓 and 𝑓^∗ are strictly convex on the interior of their respective domains. From now on, we assume that the convex function 𝑓∶𝐸→(−∞,+∞] is Legendre.

We first recall some definitions and lemmas which are needed in our main results.

Definition 2.1 (see [1, 13]). Let 𝑓∶𝐸→(−∞,+∞] be a Gâteaux differentiable and convex function. The function 𝐷_𝑓∶dom𝑓×int(dom𝑓)→[0,+∞), defined by 𝐷_𝑓𝑦,𝑥∶=𝑓𝑦−𝑓(𝑥)−∇𝑓(𝑥),𝑦−𝑥 (2.3) is called the Bregman distance with respect to 𝑓.

Remark 2.2 (see [24]). The Bregman distance has the following properties:(i)the three point identity, for any 𝑥∈dom𝑓 and 𝑦,𝑧∈int(dom𝑓),𝐷_𝑓𝑥,𝑦+𝐷_𝑓𝑦,𝑧−𝐷_𝑓(𝑥,𝑧)=∇𝑓(𝑧)−∇𝑓𝑦,𝑥−𝑦, (2.4)(ii) the four point identity, for any 𝑦,𝜔∈dom𝑓 and 𝑥,𝑧∈int(dom𝑓),𝐷_𝑓𝑦,𝑥−𝐷_𝑓𝑦,𝑧−𝐷_𝑓(𝜔,𝑥)+𝐷_𝑓(𝜔,𝑧)=∇𝑓(𝑧)−∇𝑓(𝑥),𝑦−𝜔. (2.5)

Definition 2.3 (see [1]). Let 𝑓∶𝐸→(−∞,+∞] be a Gâteaux differentiable and convex function. The Bregman projection of 𝑥∈int(dom𝑓) onto the nonempty closed and convex set 𝐶⊂dom𝑓 is the necessarily unique vector proj^𝑓_𝐶(𝑥)∈𝐶 satisfying 𝐷_𝑓proj^𝑓_𝐶(𝑥),𝑥=inf𝐷_𝑓𝑦,𝑥∶𝑦∈𝐶. (2.6)

Remark 2.4. (i) If 𝐸 is a Hilbert space and 𝑓(𝑦)=(1/2)‖𝑥‖² for all 𝑥∈𝐸, then the Bregman projection proj^𝑓_𝐶(𝑥) is reduced to the metric projection of 𝑥 onto 𝐶.
(ii) If 𝐸 is a smooth Banach space and 𝑓(𝑦)=(1/2)‖𝑥‖² for all 𝑥∈𝐸, then the Bregman projection proj^𝑓_𝐶(𝑥) is reduced to the generalized projection Π_𝐶(𝑥) (see, e.g. [3]) which defined by 𝜙Π_𝐶(𝑥),𝑥=min_𝑦∈𝐶𝜙𝑦,𝑥, (2.7) where 𝜙(𝑦,𝑥)=‖𝑦‖²−2⟨𝑦,𝐽(𝑥)⟩+‖𝑥‖²,𝐽 is the normalized duality mapping from E to 2^𝐸∗.

Definition 2.5 (see[12, 21]). Let 𝐶 be a nonempty closed and convex set of dom𝑓. The operator 𝑇∶𝐶→int(dom𝑓) with 𝐹(𝑇)≠∅ is called: (i)quasi-Bregman nonexpansive if 𝐷_𝑓(𝑢,𝑇𝑥)≤𝐷_𝑓(𝑢,𝑥),∀𝑥∈𝐶,𝑢∈𝐹(𝑇); (2.8)(ii) Bregman relatively nonexpansive if 𝐷_𝑓(𝑢,𝑇𝑥)≤𝐷_𝑓(𝑢,𝑥),∀𝑥∈𝐶,𝑢∈𝐹(𝑇), (2.9) and 𝐹(𝑇)=𝐹(𝑇),(iii) Bregman firmly nonexpansive if ∇𝑓(𝑇𝑥)−∇𝑓𝑇𝑦,𝑇𝑥−𝑇𝑦≤∇𝑓(𝑥)−∇𝑓𝑦,𝑇𝑥−𝑇𝑦,∀𝑥,𝑦∈𝐶, (2.10) or equivalently 𝐷_𝑓𝑇𝑥,𝑇𝑦+𝐷_𝑓𝑇𝑦,𝑇𝑥+𝐷_𝑓(𝑇𝑥,𝑥)+𝐷_𝑓𝑇𝑦,𝑦≤𝐷_𝑓𝑇𝑥,𝑦+𝐷_𝑓𝑇𝑦,𝑥,∀𝑥,𝑦∈𝐶. (2.11)

Definition 2.6. Let 𝐶 be a nonempty closed and convex set of dom𝑓. The operator 𝑇∶𝐶→int(dom𝑓) with 𝐹(𝑇)≠∅ is called weak Bregman relatively nonexpansive if 𝐹(𝑇)=𝐹(𝑇) and 𝐷_𝑓(𝑢,𝑇𝑥)≤𝐷_𝑓(𝑢,𝑥),∀𝑥∈𝐶,𝑢∈𝐹(𝑇). (2.12)

Remark 2.7. It is easy to see that each nonexpansive mapping 𝑇 is quasi-Bregman nonexpansive mapping with respect to 𝑓(𝑥)=(1/2)‖𝑥‖² for all 𝑥∈𝐸. Moreover, every relatively nonexpansive mapping 𝑇 also is Bregman relatively nonexpansive mapping, where 𝑇 is called relatively nonexpansive mapping (see, e.g., [32]) if the following conditions are satisfied: 𝐹(𝑇)=𝐹(𝑇)≠∅,𝜙(𝑢,𝑇𝑥)≤𝜙(𝑢,𝑥),∀𝑥∈𝐶,𝑢∈𝐹(𝑇). (2.13)
Now, we give an example which is weak Bregman relatively nonexpansive mapping but not Bregman relatively nonexpansive mapping.

Example 2.8. Let 𝐸=𝑙²,𝑓(𝑥)=(1/2)‖𝑥‖² for all 𝑥∈𝐸, where 𝑙²=𝜉=𝜉₁,𝜉₂,…,𝜉_𝑛,…∶^∞_𝑛=1𝜉_𝑛²<∞,𝜉=^∞_𝑛=1𝜉_𝑛^21/2,∀𝜉∈𝑙², (2.14) and for any 𝜉=(𝜉₁,𝜉₂,…,𝜉_𝑛,…),𝜇=(𝜇₁,𝜇₂,…,𝜇_𝑛,…)∈𝐸,⟨𝜉,𝜇⟩=∑^∞_𝑛=1𝜉_𝑛𝜇_𝑛. It is well known that 𝑙² is a Hilbert space. Let {𝑥_𝑛}⊂𝐸 be a sequence defined by 𝑥₀=(1,0,0,0,…), 𝑥₁=(1,1,0,0,…),𝑥₂=(1,0,1,0,…),…,𝑥_𝑛=(𝜉_𝑛,1,…,𝜉_𝑛,𝑘,…),…, where 𝜉_𝑛,𝑘=1,if𝑘=1,𝑛+1,0,otherwise, (2.15) for all 𝑛≥0.
Define a mapping 𝑇∶𝐸→𝐸 by 𝑇(𝑥)=𝑛𝑥_𝑛𝑛+1,if𝑥=𝑥_𝑛(∃𝑛≥1),−𝑥,if𝑥≠𝑥_𝑛(∀𝑛≥1), (2.16) for all 𝑛≥0. It is easy to see that 𝐹(𝑇)={0}, and so, {𝑥_𝑛} converges weakly to 𝑥₀. Indeed, for any 𝑔=(𝜁₁,𝜁₂…,𝜁_𝑘,…)∈𝐸, we have 𝑔𝑥_𝑛−𝑥₀=𝑔,𝑥_𝑛−𝑥₀=^∞_𝑘=2𝜁_𝑘𝜉_𝑛,𝑘=𝜁_𝑛+1. (2.17) From ∑^∞_𝑛=1|𝜁_𝑛|²<∞, it shows that lim_𝑛→∞𝜁_𝑛+1=0. Moreover, lim_𝑛→∞𝑔𝑥_𝑛−𝑥₀=lim_𝑛→∞𝜁_𝑛+1=0. (2.18) Next, for any 𝑚≠𝑛, one has ‖𝑥_𝑛−𝑥_𝑚‖=√2≠0; that is, {𝑥_𝑛} is not a Cauchy sequence. Owing to ‖𝑇𝑥_𝑛−𝑥_𝑛‖=‖𝑥_𝑛‖/(𝑛+1), we obtain lim_𝑛→∞𝑇𝑥_𝑛−𝑥_𝑛=0. (2.19) Then, 𝑥₀ is an asymptotic fixed point of 𝑇, but 𝑥₀∉𝐹(𝑇)={0}. So, 𝑇 is not Bregman relatively nonexpansive mapping.
For any strong convergent sequence {𝑦_𝑛}⊂𝑙² such that 𝑦_𝑛→𝑦₀ and ‖𝑇𝑦_𝑛−𝑦_𝑛‖→0 as 𝑛→∞. Then, there exists a sufficiently large nature number 𝑀 such that 𝑦_𝑛≠𝑥_𝑚 for any 𝑛,𝑚>𝑀. Thus, 𝑇𝑦_𝑛=−𝑦_𝑛 for 𝑛>𝑀, which implies that 2𝑦_𝑛→0 and 𝑦_𝑛→𝑦₀=0 as 𝑛→∞. That is, 𝑦₀=0 is a strong asymptotic fixed point of 𝑇, and so, 𝐹(𝑇)=𝐹(𝑇)={0}. Since 𝐷_𝑓()(0)()−()10,𝑇𝑥=𝑓−𝑓𝑇𝑥∇𝑓𝑇𝑥,0−𝑇𝑥=−2‖𝑇𝑥‖²+⟨⟩=1𝑇𝑥,𝑇𝑥2‖𝑇𝑥‖²≤12‖𝑥‖²=𝑓(0)−𝑓(𝑥)−∇𝑓(𝑥),0−𝑥=𝐷_𝑓(0,𝑥),𝑥∈𝐸. (2.20) Therefore, 𝑇 is a weak Bregman relatively nonexpansive mapping.

Definition 2.9 (see [12]). Let 𝑓∶𝐸→(−∞,+∞] be a convex and Gâteaux differentiable function. 𝑓 is called: (i)totally convex at 𝑥∈int(dom𝑓) if its modulus of total convexity at 𝑥; that is, the function 𝜈_𝑓∶int(dom𝑓)×[0,+∞)→[0,+∞) defined by 𝜈_𝑓(𝑥,𝑡)∶=inf𝐷_𝑓𝑦,𝑥∶𝑦∈dom𝑓,𝑦−𝑥=𝑡 (2.21) is positive whenever 𝑡>0,(ii)totally convex if, it is totally convex at every point 𝑥∈int(dom𝑓),(iii)totally convex on bounded sets if 𝜈_𝑓(𝐵,𝑡) is positive 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∶int(dom𝑓)×[0,+∞)→[0,+∞) defined by 𝜈_𝑓(𝐵,𝑡)∶=inf𝜈_𝑓(𝑥,𝑡)∶𝑥∈𝐵∩dom𝑓. (2.22)

Definition 2.10 (see [12, 21]). The function 𝑓∶𝐸→(−∞,+∞] is called:(i)cofinite if dom𝑓^∗=𝐸^∗,(ii)sequentially consistent if, for any two sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝐸 such that the first is bounded, and lim_𝑛→∞𝐷_𝑓𝑦_𝑛,𝑥_𝑛=0⟹lim_𝑛→∞𝑦_𝑛−𝑥_𝑛=0. (2.23)

Lemma 2.11 (see [21, Proposition 2.3]). If 𝑓∶𝐸→(−∞,+∞] is Fréchet differentiable and totally convex, then 𝑓 is cofinite.

Lemma 2.12 (see [14, Theorem 2.10]). Let 𝑓∶𝐸→(−∞,+∞] be a convex function whose domain contains at least two points. Then, the following statements hold: (i)𝑓 is sequentially consistent if and only if it is totally convex on bounded sets, (ii)if 𝑓 is lower semicontinuous, then 𝑓 is sequentially consistent if and only if it is uniformly convex on bounded sets, (iii)if 𝑓 is uniformly strictly convex on bounded sets, then it is sequentially consistent and the converse implication holds when 𝑓 is lower semicontinuous, Fréchet differentiable on its domain, and the Fréchet derivative 𝑓′ is uniformly continuous on bounded sets.

Lemma 2.13 (see [20, Proposition 2.1]). Let 𝑓∶𝐸→𝑅 be a 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^∗.

Lemma 2.14 (see [21, Lemma 3.1]). Let 𝑓∶𝐸→𝑅 be a Gâteaux differentiable and totally convex function. If 𝑥₀∈𝐸 and the sequence {𝐷_𝑓(𝑥_𝑛,𝑥₀)}^∞_𝑛=1 is bounded, then the sequence {𝑥_𝑛}^∞_𝑛=1 is also bounded.

Lemma 2.15 (see [21, Proposition 2.2]). Let f∶𝐸→𝑅 be a Gâteaux differentiable and totally convex function, 𝑥₀∈𝐸, and let 𝐶 be a nonempty closed convex subset of 𝐸. Suppose that the sequence {𝑥_𝑛}^∞_𝑛=1 is bounded and any weak subsequential limit of {𝑥_𝑛}^∞_𝑛=1 belongs to 𝐶. If 𝐷_𝑓(𝑥_𝑛,𝑥₀)≤𝐷_𝑓(proj^𝑓_𝐶(𝑥₀),𝑥₀) for any 𝑛∈𝑁, then {𝑥_𝑛}^∞_𝑛=1 converges strongly to proj^𝑓_𝐶(𝑥₀).

In [23], Reich and Sabach proved the following result.

Lemma 2.16 (see[23, Lemma 15.5]). Let 𝑓∶𝐸→(−∞,+∞] be a Legendre function. Let 𝐶 be a nonempty closed convex subset of int(dom𝑓) and 𝑇∶𝐶→𝐶 a Bregman firmly nonexpansive mapping with respect to 𝑓. Then, 𝐹(𝑇) is closed and convex.

Motivated by Lemma 2.16, we get the similar result for quasi-Bregman nonexpansive mapping.

Proposition 2.17. Let 𝑓∶𝐸→(−∞,+∞] be a Legendre function. Let 𝐶 be a nonempty closed convex subset of int(dom𝑓) and 𝑇∶𝐶→𝐶 a quasi-Bregman nonexpansive mapping with respect to 𝑓. Then, 𝐹(𝑇) is closed and convex.

Proof. Without loss of generality, set 𝐹(𝑇) is nonempty. Firstly, we show that 𝐹(𝑇) is closed. Let {𝑥_𝑛}^∞_𝑛=0 be a sequence in 𝐹(𝑇) such that 𝑥_𝑛→𝑥. By the definition of quasi-Bregman nonexpansive mapping, we have 𝐷_𝑓𝑥_𝑛,𝑇𝑥≤𝐷_𝑓𝑥_𝑛,𝑥,𝑛≥0. (2.24) Since 𝑓∶𝐸→(−∞,+∞] is a Legendre function, 𝑓 is continuous at 𝑥∈𝐶⊂int(dom𝑓). Then, from the definition of Bregman distance, lim_𝑛→∞𝐷_𝑓𝑥_𝑛,𝑇𝑥=𝐷_𝑓(𝑥,𝑇𝑥),lim_𝑛→∞𝐷_𝑓𝑥_𝑛,𝑥=𝐷_𝑓(𝑥,𝑥)=0. (2.25) From (2.24) and (2.25), it follows that 𝐷_𝑓(𝑥,𝑇𝑥)=0, and so, from [7, Lemma 7.3(vi), page 642], 𝑇𝑥=𝑥. Therefore, 𝑥∈𝐹(𝑇), and so, 𝐹(𝑇) is closed.
We now show that 𝐹(𝑇) is convex. For any 𝑥,𝑦∈𝐹(𝑇) and 𝑡∈(0,1), it yields that 𝑧=𝑡𝑥+(1−𝑡)𝑦∈𝐶. From the definition of quasi-Bregman nonexpansive mapping, it follows that 𝐷_𝑓()(𝑧)()−()()()𝑦(𝑧)𝑧,𝑇𝑧=𝑓−𝑓𝑇𝑧∇𝑓𝑇𝑧,𝑡𝑥+1−𝑡𝑦−𝑇𝑡𝑥+1−𝑡=𝑓+𝑡𝐷_𝑓(𝑥,𝑇𝑧)+(1−𝑡)𝐷_𝑓(𝑥)−()𝑓𝑦(𝑧)𝑦,𝑇𝑧−𝑡𝑓1−𝑡≤𝑓+𝑡𝐷_𝑓(𝑥,𝑧)+(1−𝑡)𝐷_𝑓(𝑥)−()𝑓𝑦=(𝑧)()𝑦𝑦,𝑧−𝑡𝑓1−𝑡∇𝑓,𝑧−𝑡𝑥−1−𝑡=0. (2.26) Again, from [7, Lemma 7.3(vi), page 642], we get 𝑇𝑧=𝑧. Therefore, 𝐹(𝑇) is convex. This completes the proof.

From the definitions of Bregman distance and the Fenchel conjugate of 𝑓, we have the following result.

Lemma 2.18. Let 𝑓∶𝐸→(−∞,+∞] be a Gâteaux differentiable and proper convex lower semicontinuous. Then, for all 𝑧∈𝐸, 𝐷_𝑓𝑧,∇𝑓^∗𝑁_𝑖=1𝑡_𝑖∇𝑓𝑥_𝑖≤^𝑁_𝑖=1𝑡_𝑖𝐷_𝑓𝑧,𝑥_𝑖, (2.27) where {𝑥_𝑖}^𝑁_𝑖=1⊂𝐸 and {𝑡_𝑖}^𝑁_𝑖=1⊂(0,1) with ∑^𝑁_𝑖=1𝑡_𝑖=1.

Lemma 2.19 (see [14, Corollary 4.4]). Let 𝑓∶𝐸→(−∞,+∞] be a Gâteaux differentiable and totally convex on int(dom𝑓). Let 𝑥∈int(dom𝑓) and 𝐶⊂int(dom𝑓) a nonempty closed convex set. If 𝑥∈𝐶, then the following statements are equivalent:(i)the vector 𝑥 is the Bregman projection of 𝑥 onto 𝐶 with respect to 𝑓,(ii) the vector 𝑥 is the unique solution of the variational inequality ∇𝑓(𝑥)−∇𝑓(𝑧),𝑧−𝑦≥0,∀𝑦∈𝐶, (2.28)(iii) the vector 𝑥 is the unique solution of the inequality 𝐷_𝑓𝑦,𝑧+𝐷_𝑓(𝑧,𝑥)≤𝐷_𝑓𝑦,𝑥,∀𝑦∈𝐶. (2.29)

3. Main Results

In this section, we introduce several modification of Mann-type iterative algorithms with errors and Halpern-type iterative algorithms with errors to find fixed points of weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings in Banach spaces. The strong convergence theorems for weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings are proved under some suitable conditions.

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a real reflexive Banach space 𝐸 and 𝑓∶𝐸→𝑅 a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subset of 𝐸, and let 𝑇∶𝐶→𝐶 be a weak Bregman relatively nonexpansive mapping such that 𝐹(𝑇)≠∅. Define a sequence {𝑥_𝑛} in 𝐶 by the following algorithm: 𝑥₀∈𝐶,𝑄₀𝑧=𝐶,_𝑛=∇𝑓^∗𝛽_𝑛∇𝑓𝑇𝑥_𝑛+𝑒_𝑛+1−𝛽_𝑛∇𝑓𝑥_𝑛+𝑒_𝑛,𝑦_𝑛=∇𝑓^∗𝛼_𝑛∇𝑓𝑥_𝑛+𝑒_𝑛+1−𝛼_𝑛∇𝑓𝑧_𝑛,𝐶_𝑛=𝑧∈𝐶_𝑛−1∩𝑄_𝑛−1∶𝐷_𝑓𝑧,𝑦_𝑛≤𝐷_𝑓𝑧,𝑥_𝑛+𝑒_𝑛,𝐶₀=𝑧∈𝐶∶𝐷_𝑓𝑧,𝑦₀≤𝐷_𝑓𝑧,𝑥₀,𝑄_𝑛=𝑧∈𝐶_𝑛−1∩𝑄_𝑛−1∶∇𝑓𝑥₀−∇𝑓𝑥_𝑛,𝑧−𝑥_𝑛,𝑥≤0_𝑛+1=proj^𝑓_{𝐶𝑛∩𝑄𝑛}𝑥₀,∀𝑛≥0, (3.1) where {𝛼_𝑛},{𝛽_𝑛}⊂[0,1] such that liminf_𝑛→∞(1−𝛼_𝑛)𝛽_𝑛>0, and {𝑒_𝑛} is an error sequence in 𝐸 with 𝑒_𝑛→0 as 𝑛→∞. Then, the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to the point proj^𝑓_𝐹(𝑇)(𝑥₀), where proj^𝑓_𝐹(𝑇)(𝑥₀) is the Bregman projection of 𝐶 onto 𝐹(𝑇).

Proof. By Proposition 2.17, it follows that 𝐹(𝑇) is a nonempty closed and convex subset of 𝐸. It is easy to verify that 𝐶₀,𝐶₁,𝑄₀, and 𝑄₁ are closed and convex. Suppose that 𝐶_𝑘 and 𝑄_𝑘(𝑘≥1) are closed and convex. Then, 𝐶_𝑘∩𝑄_𝑘 is closed and convex. For any 𝑧∈𝐶_𝑘+1,𝑦∈𝑄_𝑘+1, 𝐷_𝑓𝑧,𝑦_𝑘+1≤𝐷_𝑓𝑧,𝑥_𝑘+1+𝑒_𝑘+1⟺𝑓(𝑧)−𝑓𝑦_𝑘+1−∇𝑓𝑦_𝑘+1,𝑧−𝑦_𝑘+1≤𝑓(𝑧)−𝑓𝑥_𝑘+1+𝑒_𝑘+1−∇𝑓𝑥_𝑘+1+𝑒_𝑘+1,𝑧−𝑥_𝑘+1+𝑒_𝑘+1⟺∇𝑓𝑥_𝑘+1+𝑒_𝑘+1,𝑧−𝑥_𝑘+1+𝑒_𝑘+1−∇𝑓𝑦_𝑘+1,𝑧−𝑦_𝑘+1≤𝑓𝑦_𝑘+1−𝑓𝑥_𝑘+1+𝑒_𝑘+1⟺∇𝑓𝑥_𝑘+1+𝑒_𝑘+1−∇𝑓𝑦_𝑘+1,𝑧−𝑦_𝑘+1≤𝑓𝑦_𝑘+1−𝑓𝑥_𝑘+1+𝑒_𝑘+1−∇𝑓𝑥_𝑘+1+𝑒_𝑘+1,𝑦_𝑘+1−𝑥_𝑘+1+𝑒_𝑘+1⟺∇𝑓𝑥_𝑘+1+𝑒_𝑘+1−∇𝑓𝑦_𝑘+1,𝑧−𝑦_𝑘+1≤𝐷_𝑓𝑦_𝑘+1,𝑥_𝑘+1+𝑒_𝑘+1,∇𝑓𝑥₀−∇𝑓𝑥_𝑘,𝑦−𝑥_𝑘≤0, (3.2) which implies that 𝐶_𝑘+1 and 𝑄_𝑘+1 are closed and convex. As a consequence, 𝐶_𝑛 and 𝑄_𝑛 are closed and convex for all 𝑛≥0. Taking 𝑝∈𝐹(𝑇) arbitrarily, 𝐷_𝑓𝑝,𝑦_𝑛=𝐷_𝑓𝑝,∇𝑓^∗𝛼_𝑛∇𝑓𝑥_𝑛+𝑒_𝑛+1−𝛼_𝑛∇𝑓𝑧_𝑛≤𝛼_𝑛𝐷_𝑓𝑝,𝑥_𝑛+𝑒_𝑛+1−𝛼_𝑛𝐷_𝑓𝑝,𝑧_𝑛=𝛼_𝑛𝐷_𝑓𝑝,𝑥_𝑛+𝑒_𝑛+1−𝛼_𝑛𝐷_𝑓𝑝,∇𝑓^∗𝛽_𝑛∇𝑓𝑇𝑥_𝑛+𝑒_𝑛+1−𝛽_𝑛∇𝑓𝑥_𝑛+𝑒_𝑛≤𝛼_𝑛𝐷_𝑓𝑝,𝑥_𝑛+𝑒_𝑛+1−𝛼_𝑛1−𝛽_𝑛𝐷_𝑓𝑝,𝑥_𝑛+𝑒_𝑛+𝛽_𝑛𝐷_𝑓𝑝,𝑇𝑥_𝑛+𝑒_𝑛≤𝛼_𝑛𝐷_𝑓𝑝,𝑥_𝑛+𝑒_𝑛+1−𝛼_𝑛1−𝛽_𝑛𝐷_𝑓𝑝,𝑥_𝑛+𝑒_𝑛+𝛽_𝑛𝐷_𝑓𝑝,𝑥_𝑛+𝑒_𝑛=𝐷_𝑓𝑝,𝑥_𝑛+𝑒_𝑛, (3.3) that is, 𝑝∈𝐶_𝑛, and so, 𝐹(𝑇)⊂𝐶_𝑛 for all 𝑛≥0. We now show that 𝐹(𝑇)⊂𝑄_𝑛 for all 𝑛≥0. Clearly, 𝐹(𝑇)⊂𝑄₀=𝐶. Assume that 𝐹(𝑇)⊂𝑄_𝑘 for all 𝑘≥0. Note that 𝑥_𝑘+1=proj^𝑓_{𝐶𝑘∩𝑄𝑘}(𝑥₀), and we have ∇𝑓𝑥₀−∇𝑓𝑥_𝑘+1,𝑥_𝑘+1−𝑧≥0,𝑧∈𝐶_𝑘∩𝑄_𝑘. (3.4) Therefore, ∇𝑓𝑥₀−∇𝑓𝑥_𝑘+1,𝑥_𝑘+1−𝑝≥0,𝑝∈𝐹⊂𝐶_𝑘∩𝑄_𝑘, (3.5) which yields that 𝑝∈𝑄_𝑘+1. Then, 𝐹(𝑇)⊂𝑄_𝑛 for all 𝑛≥0. Consequently, 𝐹(𝑇)⊂𝐶_𝑛∩𝑄_𝑛 and 𝐶_𝑛∩𝑄_𝑛 is nonempty closed and convex for all 𝑛≥0. Moreover, {𝑥_𝑛} is well defined.
Secondly, we show that {𝑥_𝑛} is a Cauchy sequence and bounded. Since ∇𝑓𝑥₀−∇𝑓𝑥_𝑛,𝑧−𝑥_𝑛≤0,∀𝑧∈𝑄_𝑛, (3.6) it follows that 𝑥_𝑛=proj^𝑓_𝑄𝑛(𝑥₀). Therefore, by 𝑥_𝑛+1=proj^𝑓_{𝐶𝑛∩𝑄𝑛}(𝑥₀)∈𝑄_𝑛, 𝐷_𝑓𝑥_𝑛,𝑥₀≤𝐷_𝑓𝑥_𝑛+1,𝑥₀. (3.7) Taking 𝑝∈𝐹(𝑇) arbitrarily. From Lemma 2.19, it yields that 𝐷_𝑓𝑝,proj^𝑓_𝑄𝑛𝑥₀+𝐷_𝑓proj^𝑓_𝑄𝑛𝑥₀,𝑥₀≤𝐷_𝑓𝑝,𝑥₀. (3.8) Moreover, one has 𝐷_𝑓𝑥_𝑛,𝑥₀≤𝐷_𝑓𝑝,𝑥₀−𝐷_𝑓𝑝,𝑥_𝑛≤𝐷_𝑓𝑝,𝑥₀. (3.9) Hence, {𝐷_𝑓(𝑥_𝑛,𝑥₀)} is bounded and so {𝑥_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛} are also bounded. From (3.7), it shows that lim_𝑛→∞𝐷_𝑓(𝑥_𝑛,𝑥₀) exists. In the light of 𝑥_𝑚∈𝑄_𝑚−1⊂𝑄_𝑛 for any 𝑚>𝑛, by Lemma 2.19, 𝐷_𝑓𝑥_𝑚,proj^𝑓_𝑄𝑛𝑥₀+𝐷_𝑓proj^𝑓_𝑄𝑛𝑥₀,𝑥₀≤𝐷_𝑓𝑥_𝑚,𝑥₀, (3.10) that is, 𝐷_𝑓𝑥_𝑚,𝑥_𝑛≤𝐷_𝑓𝑥_𝑚,𝑥₀−𝐷_𝑓𝑥_𝑛,𝑥₀. (3.11) Consequently, one has lim_𝑛→∞𝐷_𝑓𝑥_𝑚,𝑥_𝑛=0. (3.12) Since 𝑓 is totally convex on bounded subsets of 𝐸, by Lemma 2.12 and (3.12), we have lim_𝑛→∞𝑥_𝑚−𝑥_𝑛=0. (3.13) Thus, {𝑥_𝑛} is a Cauchy sequence, and so, lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (3.14) Since 𝑒_𝑛→0 as 𝑛→∞, one has lim_𝑛→∞𝑥_𝑛+1+𝑒_𝑛+1−𝑥_𝑛+𝑒_𝑛=0,lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛+𝑒_𝑛=0. (3.15) Let 𝑥_𝑛→𝜔∈𝐶. Then, 𝑥_𝑛+𝑒_𝑛→𝜔.
Thirdly, we show that {𝑥_𝑛} converges strongly to a point of 𝐹(𝑇). Since 𝑓 is uniformly Fréchet differentiable on bounded subsets of 𝐸, from Lemma 2.12, ∇𝑓 is norm-to-norm uniformly continuous on bounded subsets of 𝐸. So, by (3.15), lim_𝑛→∞∇𝑓𝑥_𝑛+1−∇𝑓𝑥_𝑛+𝑒_𝑛=0. (3.16) It follows from 𝑥_𝑛+1∈𝐶_𝑛 that 𝐷_𝑓𝑥_𝑛+1,𝑦_𝑛≤𝐷_𝑓𝑥_𝑛+1,𝑥_𝑛+𝑒_𝑛. (3.17) By the uniformly Fréchet differentiable of 𝑓 on bounded subsets of 𝐸, 𝑓 is also uniformly continuous on bounded subsets of 𝐸. Hence, from (3.12) and lim_𝑛→∞𝑒_𝑛=0, lim_𝑛→∞𝐷_𝑓𝑥_𝑛+1,𝑥_𝑛+𝑒_𝑛=lim_𝑛→∞𝑓𝑥_𝑛+1−𝑓𝑥_𝑛+𝑒_𝑛−∇𝑓𝑥_𝑛+𝑒_𝑛,𝑥_𝑛+1−𝑥_𝑛+𝑒_𝑛=0. (3.18) As a consequence, lim_𝑛→∞𝐷_𝑓(𝑥_𝑛+1,𝑦_𝑛)=0 and so, lim_𝑛→∞‖𝑥_𝑛+1−𝑦_𝑛‖=0. Moreover, one has lim_𝑛→∞∇𝑓𝑥_𝑛+1−∇𝑓𝑦_𝑛=0. (3.19) Since ‖𝑥_𝑛−𝑦_𝑛‖≤‖𝑥_𝑛−𝑥_𝑛+1‖+‖𝑥_𝑛+1−𝑦_𝑛‖,‖𝑥_𝑛−𝑦_𝑛‖→0 and 𝑦_𝑛→𝜔 as 𝑛→∞. Noticing that ∇𝑓𝑥_𝑛+1−∇𝑓𝑦_𝑛=∇𝑓𝑥_𝑛+1−𝛼_𝑛∇𝑓𝑥_𝑛+𝑒_𝑛+1−𝛼_𝑛∇𝑓𝑧_𝑛≥1−𝛼_𝑛∇𝑓𝑥_𝑛+1−∇𝑓𝑧_𝑛−𝛼_𝑛∇𝑓𝑥_𝑛+1−∇𝑓𝑥_𝑛+𝑒_𝑛=1−𝛼_𝑛∇𝑓𝑥_𝑛+1−𝛽_𝑛∇𝑓𝑇𝑥_𝑛+𝑒_𝑛+1−𝛽_𝑛∇𝑓𝑥_𝑛+𝑒_𝑛−𝛼_𝑛∇𝑓𝑥_𝑛+1−∇𝑓𝑥_𝑛+𝑒_𝑛≥−𝛼_𝑛∇𝑓𝑥_𝑛+1−∇𝑓𝑥_𝑛+𝑒_𝑛+1−𝛼_𝑛𝛽_𝑛∇𝑓𝑥_𝑛+1−∇𝑓𝑇𝑥_𝑛+𝑒_𝑛−1−𝛼_𝑛1−𝛽_𝑛∇𝑓𝑥_𝑛+1−∇𝑓𝑥_𝑛+𝑒_𝑛. (3.20) Therefore, 1−𝛼_𝑛𝛽_𝑛‖∇𝑓𝑥_𝑛+1−∇𝑓𝑇𝑥_𝑛+𝑒_𝑛‖≤‖∇𝑓𝑥_𝑛+1−∇𝑓𝑦_𝑛‖+𝛼_𝑛‖∇𝑓𝑥_𝑛+1−∇𝑓𝑥_𝑛+𝑒_𝑛‖+1−𝛼_𝑛1−𝛽_𝑛‖∇𝑓𝑥_𝑛+1−∇𝑓𝑥_𝑛+𝑒_𝑛‖. (3.21) In view of liminf_𝑛→∞(1−𝛼_𝑛)𝛽_𝑛>0 and from both (3.16) and (3.19), one has lim_𝑛→∞∇𝑓𝑥_𝑛+1−∇𝑓𝑇𝑥_𝑛+𝑒_𝑛=0. (3.22) Furthermore, we have lim_𝑛→∞𝑥_𝑛+1−𝑇𝑥_𝑛+𝑒_𝑛=0, (3.23) and so, by (3.14), lim_𝑛→∞𝑥_𝑛+𝑒_𝑛−𝑇𝑥_𝑛+𝑒_𝑛=0. (3.24) Since 𝑥_𝑛→𝜔 and 𝑒_𝑛→0, we get 𝜔∈𝐹(𝑇)=𝐹(𝑇).
Finally, we show 𝜔=proj^𝑓_𝐹(𝑇)(𝑥₀). Since proj^𝑓_𝐹(𝑇)(𝑥₀)∈𝐹(𝑇)⊂𝐶_𝑛∩𝑄_𝑛, it follows from 𝑥_𝑛+1=proj^𝑓_{(𝐶𝑛∩𝑄𝑛)}(𝑥₀) that 𝐷_𝑓(𝑥_𝑛+1,𝑥₀)≤𝐷_𝑓(proj^𝑓_𝐹(𝑇)(𝑥₀),𝑥₀). By Lemma 2.15, 𝑥_𝑛→proj^𝑓_𝐹(𝑇)(𝑥₀) as 𝑛→∞. Therefore, {𝑥_𝑛} and {𝑦_𝑛} converge strongly to proj^𝑓_𝐹(𝑇)(𝑥₀). This completes the proof.

Theorem 3.2. Let 𝐶 be a nonempty closed convex subset of a real reflexive Banach space 𝐸 and 𝑓∶𝐸→𝑅 a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subset of 𝐸, and let 𝑇∶𝐶→𝐶 be a Bregman relatively nonexpansive mapping such that 𝐹(𝑇)≠∅. Assume that {𝛼_𝑛},{𝛽_𝑛}⊂[0,1] such that liminf_𝑛→∞(1−𝛼_𝑛)𝛽_𝑛>0, and {𝑒_𝑛} is an error sequence in 𝐸 with 𝑒_𝑛→0 as 𝑛→∞. Then, the sequences {𝑥_𝑛} and {𝑦_𝑛} generated by (3.1) converge strongly to the point proj^𝑓_𝐹(𝑇)(𝑥₀), where proj^𝑓_𝐹(𝑇)(𝑥₀) is the Bregman projection of 𝐶 onto 𝐹(𝑇).

Proof. As in the proof of Theorem 3.1, we know that the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝜔∈𝐶, and so, lim_𝑛→∞𝑥_𝑛+𝑒_𝑛−𝑇𝑥_𝑛+𝑒_𝑛=0. (3.25) Then, for any subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} converges weakly to 𝜔, lim_𝑘→∞𝑥_𝑛𝑘+𝑒_𝑛𝑘−𝑇𝑥_𝑛𝑘+𝑒_𝑛𝑘=0. (3.26) Therefore, 𝜔∈𝐹(𝑇)=𝐹(𝑇). By the similar proof of Theorem 3.2, the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to proj^𝑓_𝐹(𝑇)(𝑥₀). This completes the proof.

If 𝛼_𝑛≡0, 𝑒_𝑛≡0, and 𝑓(𝑥)=(1/2)‖𝑥‖² for all 𝑥∈𝐸, 𝑛≥0, then from Remark 2.4 and Theorem 3.1, we have the following result.

Corollary 3.3. Let 𝐶 be a nonempty closed convex subset of a real reflexive, smooth, and strictly convex Banach space 𝐸, and let 𝑇∶𝐶→𝐶 be a relatively nonexpansive mapping such that 𝐹(𝑇)≠∅. Define a sequence {𝑥_𝑛} in 𝐶 by the following algorithm: 𝑥₀∈𝐶,𝑄₀𝑦=𝐶,_𝑛=𝐽⁻¹𝛽_𝑛𝐽𝑇𝑥_𝑛+1−𝛽_𝑛𝐽𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶_𝑛−1∩𝑄_𝑛−1∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛,𝐶₀=𝑧∈𝐶∶𝜙𝑧,𝑦₀≤𝜙𝑧,𝑥₀,𝑄_𝑛=𝑧∈𝐶_𝑛−1∩𝑄_𝑛−1∶𝐽𝑥₀−𝐽𝑥_𝑛,𝑧−𝑥_𝑛,𝑥≤0_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀,∀𝑛≥0, (3.27) where 𝐽 is the duality mapping on 𝐸, {𝛽_𝑛}⊂[0,1] such that liminf_𝑛→∞𝛽_𝑛>0. Then, the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to the point Π_𝐹(𝑇)(𝑥₀), where Π_𝐹(𝑇)(𝑥₀) is the generalized projection (see, e.g., [2, 3, 28]) of 𝐶 onto 𝐹(𝑇).
In [27], Matsushita and Takahashi proved the following result.

Theorem MT (see [27, Theorem 3.1]). Let 𝐶 be a nonempty closed convex subset of a real uniformly convex and uniformly smooth Banach space 𝐸, and let 𝑇∶𝐶→𝐶 be a relatively nonexpansive mapping such that 𝐹(𝑇)≠∅. Assume that {𝛼_𝑛} is a sequence of real numbers such that 0≤𝛼_𝑛<1 and limsup_𝑛→∞𝛼_𝑛<1. Then, the sequence {𝑥_𝑛} generated by (1.3) converges strongly to the point Π_𝐹(𝑇)(𝑥₀), where Π_𝐹(𝑇)(𝑥₀) is the generalized projection (see, e.g., [2, 3, 28]) of 𝐶 onto 𝐹(𝑇).

Remark 3.4. Corollary 3.3 extends Theorem MT [27] from uniformly convex and uniformly smooth Banach spaces to reflexive, smooth, and strictly convex Banach space.
Now, we investigate convergence theorems for Halpern-type iterative algorithms with errors.

Theorem 3.5. Let 𝐶 be a nonempty closed convex subset of a real reflexive Banach space 𝐸 and 𝑓∶𝐸→𝑅 a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subset of 𝐸, and let 𝑇∶𝐶→𝐶 be a weak Bregman relatively nonexpansive mapping such that 𝐹(𝑇)≠∅. Define a sequence {𝑥_𝑛} in 𝐶 by the following algorithm: 𝑥₀∈𝐶,𝑄₀𝑧=𝐶,_𝑛=∇𝑓^∗𝛽_𝑛∇𝑓𝑥₀+1−𝛽_𝑛∇𝑓𝑇𝑥_𝑛+𝑒_𝑛,𝑦_𝑛=∇𝑓^∗𝛼_𝑛∇𝑓𝑧_𝑛+1−𝛼_𝑛∇𝑓𝑥_𝑛+𝑒_𝑛,𝐶_𝑛=𝑧∈𝐶_𝑛−1∩𝑄_𝑛−1∶𝐷_𝑓𝑧,𝑦_𝑛≤1−𝛼_𝑛𝛽_𝑛𝐷_𝑓𝑧,𝑥_𝑛+𝑒_𝑛+𝛼_𝑛𝛽_𝑛𝐷_𝑓𝑧,𝑥₀,𝐶₀=𝑧∈𝐶∶𝐷_f𝑧,𝑦₀≤𝐷_𝑓𝑧,𝑥₀,𝑄_𝑛=𝑧∈𝐶_𝑛−1∩𝑄_𝑛−1∶∇𝑓𝑥₀−∇𝑓𝑥_𝑛,𝑧−𝑥_𝑛,𝑥≤0_𝑛+1=proj^𝑓_{𝐶𝑛∩𝑄𝑛}𝑥₀,∀𝑛≥0, (3.28) where {𝛼_𝑛},{𝛽_𝑛}⊂[0,1] such that liminf_𝑛→∞𝛼_𝑛>0 and lim_𝑛→∞𝛽_𝑛=0, and {𝑒_𝑛} is an error sequence in 𝐸 with 𝑒_𝑛→0 as 𝑛→∞. Then, the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to the point proj^𝑓_𝐹(𝑇)(𝑥₀), where proj^𝑓_𝐹(𝑇)(𝑥₀) is the Bregman projection of 𝐶 onto 𝐹(𝑇).

Proof. By Proposition 2.17, it follows that 𝐹(𝑇) is a nonempty closed and convex subset of 𝐸. It is easy to see that 𝐶_𝑛 is closed and 𝑄_𝑛 is closed and convex for all 𝑛≥0. For any 𝑧∈𝐶_𝑛, 𝑛≥1, 𝐷_𝑓𝑧,𝑦_𝑛≤1−𝛼_𝑛𝛽_𝑛𝐷_𝑓𝑧,𝑥_𝑛+𝑒_𝑛+𝛼_𝑛𝛽_𝑛𝐷_𝑓𝑧,𝑥₀⟺𝑓(𝑧)−𝑓𝑦_𝑛−∇𝑓𝑦_𝑛,𝑧−𝑦_𝑛≤1−𝛼_𝑛𝛽_𝑛𝑓(𝑧)−𝑓𝑥_𝑛+𝑒_𝑛−∇𝑓𝑥_𝑛+𝑒_𝑛,𝑧−𝑥_𝑛−𝑒_𝑛+𝛼_𝑛𝛽_𝑛𝑓(𝑧)−𝑓𝑥₀−∇𝑓𝑥₀,𝑧−𝑥₀⟺1−𝛼_𝑛𝛽_𝑛𝑓𝑥_𝑛+𝑒_𝑛+𝛼_𝑛𝛽_𝑛𝑓𝑥₀−𝑓𝑦_𝑛≤∇𝑓𝑦_𝑛,𝑧−𝑦_𝑛−1−𝛼_𝑛𝛽_𝑛∇𝑓𝑥_𝑛+𝑒_𝑛,𝑧−𝑥_𝑛−𝑒_𝑛−𝛼_𝑛𝛽_𝑛∇𝑓𝑥₀,𝑧−𝑥₀, (3.29) which implies that 𝐶_𝑛 is closed and convex for all 𝑛≥1. Since, for any 𝑧∈𝐶₀, 𝐷_𝑓𝑧,𝑦₀≤𝐷_𝑓𝑧,𝑥₀⟺𝑓(𝑧)−𝑓𝑦₀−∇𝑓𝑦₀,𝑧−𝑦₀≤𝑓(𝑧)−𝑓𝑥₀−∇𝑓𝑥₀,𝑧−𝑥₀⟺∇𝑓𝑥₀,𝑧−𝑥₀−∇𝑓𝑦₀,𝑧−𝑦₀≤𝑓𝑦₀−𝑓𝑥₀⟺∇𝑓𝑥₀,𝑧−𝑥₀−∇𝑓𝑦₀,𝑧−𝑥₀+𝑥₀−𝑦₀≤𝑓𝑦₀−𝑓𝑥₀⟺∇𝑓𝑥₀−∇𝑓𝑦₀,𝑧−𝑥₀+𝐷_𝑓𝑥₀,𝑦₀≤0, (3.30) which shows that 𝐶₀ is closed and convex. As a consequence, 𝐶_𝑛 is closed and convex for all 𝑛≥0. Taking 𝑝∈𝐹(𝑇) arbitrarily, by Lemma 2.18, 𝐷_𝑓𝑝,𝑦_𝑛=𝐷_𝑓𝑝,∇𝑓^∗𝛼_𝑛∇𝑓𝑧_𝑛+1−𝛼_𝑛∇𝑓𝑥_𝑛+𝑒_𝑛≤𝛼_𝑛𝐷_𝑓𝑝,𝑧_𝑛+1−𝛼_𝑛𝐷_𝑓𝑝,𝑥_𝑛+𝑒_𝑛=1−𝛼_𝑛𝐷_𝑓𝑝,𝑥_𝑛+𝑒_𝑛+𝛼_𝑛𝐷_𝑓𝑝,∇𝑓^∗𝛽_𝑛∇𝑓𝑥₀+1−𝛽_𝑛∇𝑓𝑇𝑥_𝑛+𝑒_𝑛≤1−𝛼_𝑛𝐷_𝑓𝑝,𝑥_𝑛+𝑒_𝑛+𝛼_𝑛𝛽_𝑛𝐷_𝑓𝑝,𝑥₀+𝛼_𝑛1−𝛽_𝑛𝐷_𝑓𝑝,𝑇𝑥_𝑛+𝑒_𝑛≤1−𝛼_𝑛𝐷_𝑓𝑝,𝑥_𝑛+𝑒_𝑛+𝛼_𝑛𝛽_𝑛𝐷_𝑓𝑝,𝑥₀+𝛼_𝑛1−𝛽_𝑛𝐷_𝑓𝑝,𝑥_𝑛+𝑒_𝑛=1−𝛼_𝑛𝛽_𝑛𝐷_𝑓𝑝,𝑥_𝑛+𝑒_𝑛+𝛼_𝑛𝛽_𝑛𝐷_𝑓𝑝,𝑥₀, (3.31) that is, 𝑝∈𝐶_𝑛, and so, 𝐹(𝑇)⊂𝐶_𝑛 for all 𝑛≥0. As in the proof of Theorem 3.1, we get 𝐹(𝑇)⊂𝑄_𝑛 for all 𝑛≥0, {𝑥_𝑛} is a Cauchy sequence, {𝑥_𝑛},{𝑦_𝑛}, and {𝑧_𝑛} are also bounded, and thus, lim_𝑛→∞𝐷_𝑓𝑥_𝑛+1,𝑥_𝑛+𝑒_𝑛=0,lim_𝑛→∞𝑥_𝑛+1+𝑒_𝑛+1−𝑥_𝑛+𝑒_𝑛=0,lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛+𝑒_𝑛=0. (3.32)(3.33) Consequently, 𝐹(𝑇)⊂𝐶_𝑛∩𝑄_𝑛 and 𝐶_𝑛∩𝑄_𝑛 is nonempty closed and convex for all 𝑛≥0. Moreover, {𝑥_𝑛} is well defined. Set 𝑥_𝑛→𝜔∈𝐶.
Secondly, we show that {𝑥_𝑛} converges strongly to a point of 𝐹(𝑇). Since 𝑓 is uniformly Fréchet differentiable on bounded subsets of 𝐸, from Lemma 2.12, ∇𝑓 is norm-to-norm uniformly continuous on bounded subsets of 𝐸. So, by (3.33), lim_𝑛→∞∇𝑓𝑥_𝑛+1−∇𝑓𝑥_𝑛+𝑒_𝑛=0. (3.34) In view of 𝑥_𝑛+1=proj^𝑓_{(𝐶𝑛∩𝑄𝑛)}(𝑥₀)∈𝐶_𝑛, we have 𝐷_𝑓𝑥_𝑛+1,𝑦_𝑛≤1−𝛼_𝑛𝛽_𝑛𝐷_𝑓𝑥_𝑛+1,𝑥_𝑛+𝑒_𝑛+𝛼_𝑛𝛽_𝑛𝐷_𝑓𝑥_𝑛+1,𝑥₀. (3.35) Due to lim_𝑛→∞𝛽_𝑛=0, from (3.32), one has lim_𝑛→∞𝐷_𝑓𝑥_𝑛+1,𝑦