Convergence Theorems for Right Bregman Strongly Nonexpansive Mappings in Reflexive Banach Spaces

Abstract

We prove a strong convergence theorem for a common fixed point of a finite family of right Bregman strongly nonexpansive mappings in the framework of real reflexive Banach spaces. Furthermore, we apply our method to approximate a common zero of a finite family of maximal monotone mappings and a solution of a finite family of convex feasibility problems in reflexive real Banach spaces. Our theorems complement some recent results that have been proved for this important class of nonlinear mappings.

1. Introduction

In this paper, without other specifications, let 𝐸 be a real reflexive Banach space and 𝐸^∗ as its dual, let ℝ be the set of real numbers, and let 𝐶 be a nonempty, closed, and convex subset 𝐸. Let 𝑓∶𝐸→(−∞,∞] be a proper convex and lower semicontinuous function. Denote the domain of 𝑓 by dom𝑓; that is, dom𝑓={𝑥∈𝐸∶𝑓(𝑥)<∞}. The Fenchel conjugate of 𝑓 is the function 𝑓^∗∶𝐸^∗→(−∞,+∞] defined by 𝑓^∗(𝑦)=sup{⟨𝑦,𝑥⟩−𝑓(𝑥)∶𝑥∈𝐸}. 𝑓 is called cofinite if dom𝑓^∗=𝐸^∗. For any 𝑥∈int(dom𝑓) and 𝑦∈𝐸, the right-hand derivative of 𝑓 at 𝑥 in the direction of 𝑦 is defined by 𝑓⁰(𝑥,𝑦)∶=lim_𝑡→0⁺(𝑓(𝑥+𝑡𝑦)−𝑓(𝑥))/𝑡.

The function 𝑓 is called Gâteaux differentiable at 𝑥 if lim_𝑡→0⁺(𝑓(𝑥+𝑡𝑦)−𝑓(𝑥))/𝑡 exists for any 𝑦. In this case, 𝑓⁰(𝑥,𝑦) coincides with ∇𝑓(𝑥), the value of the gradient ∇𝑓 of 𝑓 at 𝑥. The function 𝑓 is called Gâteaux differentiable if it is Gâteaux differentiable for any 𝑥∈ int(dom𝑓). The function 𝑓 is said to be Fréchet differentiable at 𝑥 if this limit is attained uniformly in ‖𝑦‖=1 and 𝑓 is said to be uniformly Fréchet differentiable on a subset 𝐶 of 𝐸 if the limit is attained uniformly for 𝑥∈𝐶 and ‖𝑦‖=1.

The function 𝑓 is said to be bounded if it maps bounded subsets of 𝐸 into bounded sets. We note that if 𝑓∶𝐸→ℝ is uniformly Fréchet differentiable and bounded, then ∇𝑓 is uniformly continuous on bounded subsets of 𝐸 from the strong topology of 𝐸 to the strong topology of 𝐸^∗ (Proposition 2.1, [1]) and 𝑓^∗ is uniformly Fréchet on bounded subsets of 𝐸^∗ (see [2]) and hence ∇𝑓^∗ is uniformly continuous on bounded subsets of 𝐸^∗ from the strong topology of 𝐸^∗ to the strong topology of 𝐸.

Let 𝑓∶𝐸→(−∞,+∞] be a Gâteaux differentiable function. The function 𝐷_𝑓∶dom𝑓×int(dom𝑓)→[0,+∞) defined by 𝐷_𝑓𝑥,𝑦∶=𝑓(𝑥)−𝑓𝑦−∇𝑓𝑦,𝑥−𝑦 (1) is called the Bregman distance with respect to 𝑓 [3].

A Bregman projection [4] of 𝑥∈int(dom𝑓) onto the nonempty closed and convex set 𝐶⊂int(dom𝑓) is the unique vector 𝑃^𝑓_𝐶(𝑥)∈𝐶 satisfying 𝐷_𝑓𝑃^𝑓_𝐶(𝑥),𝑥=inf𝐷_𝑓𝑦,𝑥∶𝑦∈𝐶. (2)

Remark 1. If 𝐸 is a smooth and strictly convex Banach space and 𝑓(𝑥)=‖𝑥‖² for all 𝑥∈𝐸, then we have that ∇𝑓(𝑥)=2𝐽𝑥 for all 𝑥∈𝐸, where 𝐽 is the normalized duality mapping from 𝐸 into 2^𝐸∗, and hence(i)𝐷_𝑓(𝑥,𝑦) reduces to 𝜙(𝑥,𝑦)=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+‖𝑦‖², for all 𝑥,𝑦∈𝐸, which is the Lyapunov function introduced by Alber [5] and(ii)𝑃^𝑓_𝐶(𝑥) reduces to the generalized projection Π_𝐶(𝑥) (see, e.g., [5]) which is defined by 𝜙Π_𝐶(𝑥),𝑥=min_𝑦∈𝐶𝜙𝑦,𝑥. (3)If 𝐸=𝐻, a Hilbert space, 𝐽 is the identity mapping and hence the Bregman distance becomes 𝐷_𝑓(𝑥,𝑦)=‖𝑥−𝑦‖², for 𝑥,𝑦∈𝐻, and the Bregman projection 𝑃^𝑓_𝐶(𝑥) reduces to the metric projection of 𝐻 onto 𝐶,𝑃_𝐶(𝑥).

Let 𝑇∶𝐶→𝐶 be a nonlinear mapping. Denote by 𝐹(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥} the set of fixed points of 𝑇. A mapping 𝑇 is said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖, for all 𝑥,𝑦∈𝐶, and 𝑇 is called quasinonexpansive if ‖𝑇𝑥−𝑝‖≤‖𝑥−𝑝‖, for all 𝑥∈𝐶 and 𝑝∈𝐹(𝑇). A point 𝑝∈𝐶 is called an asymptotic fixed point of 𝑇 (see [6]) if 𝐶 contains a sequence {𝑥_𝑛} which converges weakly to 𝑝 such that lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0. We denote by 𝐹(𝑇) the set of asymptotic fixed points of 𝑇.

A mapping 𝑇∶𝐶→int(dom𝑓) is called(i)left quasi-Bregman nonexpansive [7] if 𝐹(𝑇)≠0/ and 𝐷_𝑓𝑝,𝑇𝑥≤𝐷_𝑓𝑝,𝑥,∀𝑥∈𝐶,𝑝∈𝐹(𝑇); (4)(ii)left Bregman relatively nonexpansive [7] if 𝐹(𝑇)≠0/ and 𝐷_𝑓𝑝,𝑇𝑥≤𝐷_𝑓,(𝑇),𝐹(𝑇)(𝑇);𝑝,𝑥∀𝑥∈𝐶,𝑝∈𝐹=𝐹 (5)(iii)left Bregman strongly nonexpansive (see [8, 9]), with respect to nonempty 𝐹(𝑇), if 𝐷_𝑓𝑝,𝑇𝑥≤𝐷_𝑓𝑝,𝑥,∀𝑥∈𝐶,𝑝∈𝐹(𝑇); (6) and, if, whenever {𝑥_𝑛}⊂𝐶 is bounded, 𝑝∈𝐹(𝑇) and lim_𝑛→∞𝐷_𝑓𝑝,𝑥_𝑛−𝐷_𝑓𝑝,𝑇𝑥_𝑛=0, (7) it follows that lim_𝑛→∞𝐷_𝑓𝑇𝑥_𝑛,𝑥_𝑛=0; (8)(iv)left Bregman firmly nonexpansive [10] if 𝐹(𝑇)≠0/ and for all 𝑥,𝑦∈𝐶, ()≤(𝑥)𝑦,∇𝑓𝑇𝑥−∇𝑓𝑇𝑦,𝑇𝑥−𝑇𝑦∇𝑓−∇𝑓,𝑇𝑥−𝑇𝑦 (9) or, equivalently, 𝐷_𝑓𝑇𝑥,𝑇𝑦+𝐷_𝑓𝑇𝑦,𝑇𝑥+𝐷_𝑓(𝑇𝑥,𝑥)+𝐷_𝑓𝑇𝑦,𝑦≤𝐷_𝑓𝑇𝑥,𝑦+𝐷_𝑓𝑇𝑦,𝑥. (10)If 𝑇 is left Bregman firmly nonexpansive and 𝑓 is Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of 𝐸, then it is known in [10] that 𝐹(𝑇)=𝐹(𝑇) and 𝐹(𝑇) is closed and convex (see [10]). It follows that every left Bregman firmly nonexpansive mapping is Bregman strongly nonexpansive with respect to a nonempty set 𝐹(𝑇)=𝐹(𝑇).

Existence and approximation of fixed points of nonexpansive and quasinonexpansive mappings have been intensively studied for almost fifty years or so by various authors (see e.g., [11–24] and the references therein) in Hilbert spaces. But most of the methods failed to give the same conclusion in Banach spaces more general than Hilbert spaces. One of the reasons is that a nonexpansive mapping in Hilbert spaces may not be nonexpansive in Banach spaces (e.g., the resolvent 𝑅_𝐴=(𝐼+𝐴)⁻¹ of a maximal monotone mapping 𝐴∶𝐻→2^𝐻 and the metric projection 𝑃_𝐾 onto a nonempty, closed, and convex subset 𝐶 of 𝐻).

To overcome this problem, researchers use the distance function 𝐷_𝑓(⋅,⋅) introduced by Bregman [4] instead of norm which opened a growing area of research in designing and analyzing iterative techniques for solving variational inequalities, approximating equilibria, computing fixed points of nonlinear mappings, and approximating solutions of convex feasibility problems (see, e.g., [4, 25–28] and the references therein).

In [29], Reich and Sabach proposed the following algorithm for finding a common fixed point of finitely many left Bregman firmly nonexpansive self-mappings 𝑇_𝑖(𝑖=1,2,…,𝑁) on 𝐸 satisfying ∩^𝑁_𝑖=1𝐹(𝑇_𝑖)≠0/. For 𝑥₁∈𝐸 let the sequence {𝑥_𝑛} be defined by 𝑄^𝑖₀𝑦=𝐸,_𝑖𝑛=𝑇_𝑖𝑥_𝑛+𝑒^𝑖_𝑛,𝑄^𝑖_𝑛+1=𝑧∈𝑄^𝑖_𝑛∶∇𝑓𝑥_𝑛+𝑒^𝑖_𝑛−∇𝑓𝑦^𝑖_𝑛,𝑧−𝑦^𝑖_𝑛,𝑄≤0_𝑛=^𝑁_𝑖=1𝑄^𝑖_𝑛,𝑥_𝑛+1=𝑃^𝑓_𝑄𝑛+1𝑥₀,∀𝑛≥1. (11) They proved that, under some suitable conditions, the sequence {𝑥_𝑛} generated by (11) converges strongly to a point in ∩^𝑁_𝑖=1𝐹(𝑇_𝑖) and applied it to the solution of convex feasibility and equilibrium problems.

Very recently, by using Bregman projection, Reich and Sabach [9] proposed an algorithm for finding a common fixed point of finitely many left Bregman strongly nonexpansive mappings 𝑇_𝑖∶𝐶→𝐶(𝑖=1,2,…,𝑁) satisfying ∩^𝑁_𝑖=1𝐹(𝑇_𝑖)≠0/ in a reflexive Banach space 𝐸 as follows: 𝑥₀𝑦∈𝐸,chosenarbitrarily,_𝑖𝑛=𝑇_𝑖𝑥_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^𝑖_𝑛=𝑧∈𝐸∶∇𝑓𝑥₀−∇𝑓𝑥_𝑛,𝑧−𝑥_𝑛,𝑥≤0_𝑛+1=𝑃^𝑓_{𝐶𝑛∩𝑄𝑛}𝑥₀,∀𝑛≥0. (12) Under some suitable conditions, they proved that the sequence {𝑥_𝑛} generated by (12) converges strongly to a point in ∩^𝑁_𝑖=1𝐹(𝑇_𝑖) and applied it to the solution of convex feasibility and equilibrium problems.

The above results naturally bring us to the following: a natural question arises whether we can establish analogous results for right Bregman strongly nonexpansive mappings or not.

A mapping 𝑇∶𝐶→int(dom𝑓) is called(i)right quasi-Bregman nonexpansive [30] if 𝐹(𝑇)≠0/ and 𝐷_𝑓𝑇𝑥,𝑝≤𝐷_𝑓𝑥,𝑝,∀𝑥∈𝐶,𝑝∈𝐹(𝑇); (13)(ii)right Bregman relatively nonexpansive [30] if 𝐹(𝑇)≠0/ and 𝐷_𝑓𝑇𝑥,𝑝≤𝐷_𝑓,(𝑇),𝐹(𝑇)(𝑇);𝑥,𝑝∀𝑥∈𝐶,𝑝∈𝐹=𝐹 (14)(iii)right Bregman strongly nonexpansive (see [8, 9]), with respect to nonempty 𝐹(𝑇), if 𝐷_𝑓𝑇𝑥,𝑝≤𝐷_𝑓𝑥,𝑝,∀𝑥∈𝐶,𝑝∈𝐹(𝑇); (15) and if, whenever {𝑥_𝑛}⊂𝐶 is bounded, 𝑝∈𝐹(𝑇) and lim_𝑛→∞𝐷_𝑓𝑥_𝑛,𝑝−𝐷_𝑓𝑇𝑥_𝑛,𝑝=0, (16) it follows that lim_𝑛→∞𝐷_𝑓𝑥_𝑛,𝑇𝑥_𝑛=0; (17)(iv)right Bregman firmly nonexpansive [10] if 𝐹(𝑇)≠0/ and for all 𝑥,𝑦∈𝐶, ()≤(),∇𝑓𝑇𝑥−∇𝑓𝑇𝑦,𝑇𝑥−𝑇𝑦∇𝑓𝑇𝑥−∇𝑓𝑇𝑦,𝑥−𝑦 (18) or, equivalently, 𝐷_𝑓𝑇𝑥,𝑇𝑦+𝐷_𝑓𝑇𝑦,𝑇𝑥+𝐷_𝑓(𝑥,𝑇𝑥)+𝐷_𝑓𝑦,𝑇𝑦≤𝐷_𝑓𝑥,𝑇𝑦+𝐷_𝑓𝑦,𝑇𝑥. (19)

Remark 2. It is shown in [10] that if 𝑇 is right Bregman firmly nonexpansive, then 𝐹(𝑇)=𝐹(𝑇) and hence it is right Bregman relatively nonexpansive mapping provided that the Legendre function 𝑓 is uniformly Fréchet differentiable and bounded on bounded sets of 𝐸.

The class of right Bregman firmly nonexpansive mappings associated with the Bregman distance induced by a convex function was introduced and studied by Martin-Marques et al. [30]. Examples of right Bregman firmly nonexpansive mappings are given in [30]. If 𝐶 is a nonempty and closed subset of int(dom𝑓), where 𝑓 is a Legendre and Fréchet differentiable function, and 𝑇∶𝐶→int(dom𝑓) is a right Bregman strongly nonexpansive mapping, it is proved that 𝐹(𝑇) is closed (see [30]). In addition, they have shown that this class of mappings is closed under composition and convex combination and proved weak convergence of the Picard iterative method to a fixed point of a mapping under suitable conditions (see [31]). However, Picard iteration process has only weak convergence.

In this paper, it is our purpose to introduce an iterative scheme which converges strongly to a common fixed point of a finite family of right Bregman strongly nonexpansive mappings. As a consequence, we use our results to approximate a common zero of a finite family of maximal monotone mappings and a solution of a finite family of convex feasibility problems in reflexive real Banach spaces. Our results complements the recent results due to Reich and Sabach [9], Suantai et al. [32], and Zhang and Cheng [33] in the sense that our scheme is applicable for right Bregman strongly nonexpansive self-mappings on 𝐶⊆𝐸.

2. Preliminaries

Let 𝑓∶𝐸→(−∞,+∞] be a convex and Gâteaux differentiable function. The modulus of total convexity of 𝑓 at 𝑥∈dom𝑓 is the function 𝜈_𝑓(𝑥,⋅)∶[0;+∞)→[0;+∞] defined by 𝜈_𝑓(𝑥,𝑡)∶=inf𝐷_𝑓𝑦,𝑥∶𝑦∈dom𝑓,𝑦−𝑥=𝑡. (20) The function 𝑓 is called totally convex at 𝑥 if 𝜈_𝑓(𝑥,𝑡)>0, whenever 𝑡>0. The function 𝑓 is called totally convex if it is totally convex at any point 𝑥∈int(dom𝑓) and is said to be totally convex on bounded sets if 𝜈_𝑓(𝐵,𝑡)>0 for any nonempty bounded subset 𝐵 of 𝐸 and 𝑡>0, where the modulus of total convexity of the function 𝑓 on the set 𝐵 is the function 𝜈_𝑓∶int(dom𝑓)×[0,+∞)→[0,+∞] defined by 𝜈_𝑓(𝐵,𝑡)∶=inf𝑉_𝑓(𝑥,𝑡)∶𝑥∈𝐵∩dom𝑓. (21) We know that 𝑓 is totally convex on bounded sets if and only if 𝑓 is uniformly convex on bounded sets (see [27], Theorem 2.10).

The function 𝑓 is called essentially smooth, if 𝜕𝑓 is both locally bounded and single-valued on its domain and it is called essentially strictly convex, if (𝜕𝑓)⁻¹ is locally bounded on its domain and 𝑓 is strictly convex on every convex subset of dom𝜕𝑓. 𝑓 is said to be Legendre, if it is both essentially smooth and essentially strictly convex. Since 𝐸 is reflexive, we know that (𝜕𝑓)⁻¹=𝜕𝑓^∗ (see [34]), 𝑓 is essentially smooth if and only if 𝑓^∗ is essentially strictly convex (see [35], Theorem 5.4), and 𝑓 is Legendre if and only if 𝑓^∗ is Legendre (see [35], Corollary 5.5); if 𝑓 is Legendre, then ∇𝑓 is a bijection satisfying ∇𝑓=(∇𝑓^∗)−1, ran ∇𝑓=dom∇𝑓^∗=intdom𝑓^∗, and ran ∇𝑓^∗=dom𝑓=intdom𝑓 (see [35], Theorem 5.10). From now on, we assume that the convex function 𝑓∶𝐸→(−∞,+∞] is Legendre.

If 𝐸 is a smooth and strictly convex Banach space, then an important and interesting Legendre function is 𝑓(𝑥)∶=(1/𝑝)‖𝑥‖^𝑝(1<𝑝<∞). In this case, the gradient ∇𝑓 of 𝑓 coincides with the generalized duality mapping of 𝐸; that is, ∇𝑓=𝐽_𝑝(1<𝑝<∞). In particular, ∇𝑓=𝐼, the identity mapping in Hilbert spaces.

In the sequel, we shall use the following lemmas.

Lemma 3 (see [31]). Let 𝑓∶𝐸→ℝ be a bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of 𝐸. For each 𝑖=1,…,𝑁, let 𝑇_𝑖∶𝐾⊂𝐸→𝐾 be a right Bregman strongly nonexpansive mapping with respect to 𝐹(𝑇_𝑖)=𝐹(𝑇_𝑖), and let 𝑇∶=𝑇_𝑁∘𝑇_𝑁−1∘⋯∘𝑇₁. If ℱ∶=∩^𝑁_𝑖=1𝐹(𝑇_𝑖) is nonempty, then 𝑇 is also right Bregman strongly nonexpansive and 𝐹(𝑇)=∩^𝑁_𝑖=1𝐹(𝑇_𝑖).

Lemma 4 (see [30]). Let 𝑓∶𝐸→ℝ be a Fréchet differentiable function. Let 𝐶 be a nonempty closed convex subset of int(dom𝑓) and let 𝑇∶𝐶→int(dom𝑓) be a right quasi-Bregman nonexpansive mapping. Then 𝐹(𝑇) is closed.

Lemma 5 (see [36]). The function 𝑓∶𝐸→(−∞,+∞) is totally convex on bounded subsets of 𝐸 if and only if for any two sequences {𝑥_𝑛} and {𝑦_𝑛} in int(dom𝑓) and dom𝑓, respectively, such that the first one is bounded and lim_𝑛→∞𝐷_𝑓𝑦_𝑛,𝑥_𝑛=0⟹lim_𝑛→∞𝑦_𝑛−𝑥_𝑛=0. (22)

Lemma 6 (see [27]). Let 𝐶 be a nonempty, closed, and convex subset of 𝐸. Let 𝑓∶𝐸→ℝ be a Gâteaux differentiable and totally convex function and let 𝑥∈𝐸. Then(i)𝑧=𝑃^𝑓_𝐶(𝑥) if and only if ⟨∇𝑓(𝑥)−∇𝑓(𝑧),𝑦−𝑧⟩≤0,∀𝑦∈𝐶;(ii)𝐷_𝑓(𝑦,𝑃^𝑓_𝐶(𝑥))+𝐷_𝑓(𝑃^𝑓_𝐶(𝑥),𝑥)≤𝐷_𝑓(𝑦,𝑥),∀𝑦∈𝐶.

Lemma 7 (see [37]). If 𝑓∶𝐸→(−∞,+∞] is a proper, lower semicontinuous, and convex function, then 𝑓^∗∶𝐸^∗→(−∞,+∞] is a proper, 𝑤𝑒𝑎𝑘^∗ lower semicontinuous and convex function. Thus, for all 𝑧∈𝐸, we have 𝐷_𝑓𝑧,∇𝑓^∗𝑁_𝑖=1𝑡_𝑖∇𝑓𝑥_𝑖≤^𝑁_𝑖=1𝑡_𝑖𝐷_𝑓𝑧,𝑥_𝑖. (23)

Lemma 8 (see [31]). Let 𝑓∶𝐸→ℝ be admissible and totally bounded at a point 𝑥∈int(dom𝑓). Let {𝑥_𝑛}⊂dom(𝑓). If {𝐷_𝑓(𝑥_𝑛,𝑥)} is bounded, then so is the sequence {𝑥_𝑛}.

Let 𝑓∶𝐸→ℝ be a Gâteaux differentiable function. Following [3, 5], we make use of the function 𝑉_𝑓∶𝐸×𝐸^∗→[0,+∞) associated with 𝑓, which is defined by 𝑉_𝑓𝑥,𝑥^∗=𝑓(𝑥)−𝑥,𝑥^∗+𝑓^∗𝑥^∗,∀𝑥∈𝐸,𝑥^∗∈𝐸^∗. (24)

Then, 𝑉_𝑓 is nonnegative and 𝑉_𝑓𝑥,𝑥^∗=𝐷_𝑓𝑥,∇𝑓^∗𝑥^∗∀𝑥∈𝐸,𝑥^∗∈𝐸^∗. (25)

Moreover, by the subdifferential inequality, 𝑉_𝑓𝑥,𝑥^∗+𝑦^∗,∇𝑓^∗𝑥^∗−𝑥≤𝑉_𝑓𝑥,𝑥^∗+𝑦^∗, (26)

for all 𝑥∈𝐸 and 𝑥^∗,𝑦^∗∈𝐸^∗ (see [38]).

Lemma 9 (see [39]). Let {𝑎_𝑛} be a sequence of nonnegative real numbers satisfying the following relation: 𝑎_𝑛+1≤1−𝛼_𝑛𝑎_𝑛+𝛼_𝑛𝛿_𝑛,𝑛≥𝑛₀, (27) where {𝛼_𝑛}⊂(0,1) and {𝛿_𝑛}⊂𝑅 satisfying the following conditions: lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=1𝛼_𝑛=∞, and limsup_𝑛→∞𝛿_𝑛≤0. Then, lim_𝑛→∞𝑎_𝑛=0.

Lemma 10 (see [40]). Let {𝑎_𝑛} be sequences of real numbers such that there exists a subsequence {𝑛_𝑖} of {𝑛} such that 𝑎_𝑛𝑖<𝑎_𝑛𝑖+1 for all 𝑖∈ℕ. Then there exists an increasing sequence {𝑚_𝑘}⊂ℕ such that 𝑚_𝑘→∞ and the following properties are satisfied by all (sufficiently large) numbers 𝑘∈ℕ: 𝑎_𝑚𝑘≤𝑎_𝑚𝑘+1,𝑎_𝑘≤𝑎_𝑚𝑘+1. (28) In fact, 𝑚_𝑘 is the largest number 𝑛 in the set {1,2,…,𝑘} such that the condition 𝑎_𝑛≤𝑎_𝑛+1 holds.

3. Main Results

Theorem 11. Let 𝑓∶𝐸→ℝ be a cofinite function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of 𝐸. Let 𝐶 be a nonempty, closed, and convex subset of int(dom𝑓) and let 𝑇_𝑖∶𝐶→𝐶, for 𝑖=1,2,…,𝑁, be a finite family of right Bregman strongly nonexpansive mappings such that 𝐹(𝑇_𝑖)=𝐹(𝑇_𝑖), for each 𝑖∈{1,2,…,𝑁}. Assume that ℱ∶=∩^𝑁_𝑖=1𝐹(𝑇_𝑖) is nonempty. For 𝑢,𝑥₁∈𝐶, let {𝑥_𝑛} be a sequence generated by 𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑥_𝑛,𝑛=1,2,…, (29) where 𝑇=𝑇_𝑁∘𝑇_𝑁−1∘⋯∘𝑇₁, {𝛼_𝑛}⊂(0,1) satisfy lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞. Then, {𝑥_𝑛} converges strongly to a point 𝑝 in ℱ.

Proof. Note that from Lemma 3 we have ℱ=𝐹(𝑇)=∩^𝑁_𝑖=1𝐹(𝑇_𝑖) and 𝑇 is right Bregman strongly nonexpansive mapping. Let 𝑝∈ℱ. Then, using (29), the convexity of 𝑓, and property of 𝑇 we get that 𝐷_𝑓𝑥_𝑛+1,𝑝=𝐷_𝑓𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑥_𝑛,𝑝≤𝛼_𝑛𝐷_𝑓𝑢,𝑝+1−𝛼_𝑛𝐷_𝑓𝑇𝑥_𝑛,𝑝≤𝛼_𝑛𝐷_𝑓𝑢,𝑝+1−𝛼_𝑛𝐷_𝑓𝑥_𝑛,𝑝. (30) Thus, by induction we obtain that 𝐷_𝑓𝑥_𝑛+1,𝑝≤max𝐷_𝑓𝑢,𝑝,𝐷_𝑓𝑥₁,𝑝∀𝑛≥1, (31) which implies that {𝐷_𝑓(𝑥_𝑛,𝑝)} and hence 𝐷_𝑓(𝑇𝑥_𝑛,𝑝) are bounded. Thus, from Lemma 8 we get that {𝑥_𝑛} and {𝑇𝑥_𝑛} are bounded. Now, let 𝑦_𝑛=∇𝑓(𝑥_𝑛). Then, iteration process (29) becomes 𝑦_𝑛+1=∇𝑓𝛼_𝑛∇𝑓^∗∇𝑓(𝑢)+1−𝛼_𝑛∇𝑓^∗𝑇^∗𝑦_𝑛,𝑛=1,2,…, (32) where 𝑇^∗∶=∇𝑓𝑇∇𝑓^∗, a conjugate of 𝑇. Since ∇𝑓 and ∇𝑓^∗ are uniformly continuous on bounded subsets of int(dom𝑓) and int(dom𝑓^∗), respectively, we get that {𝑦_𝑛} and {𝑇^∗𝑦_𝑛} are bounded and by Section 6 of Martin-Marquez et al. [31] we have that 𝑇^∗ is left Bergman strongly nonexpansive with respect to ∇𝑓(𝐹(𝑇)). In addition, by Proposition 3.3 of [30] we have that ∇𝑓(𝐹(𝑇))=𝐹(𝑇^∗)=𝐹(𝑇^∗)∶=ℱ^′ is closed and convex. Let 𝑝^′=𝑃^𝑓∗_ℱ′(∇𝑓(𝑢)). Now, from (32), (25), (26), and Lemma 6 we obtain that 𝐷_𝑓^∗𝑝^′,𝑦_𝑛+1=𝐷_𝑓^∗𝑝^′,∇𝑓𝛼_𝑛∇𝑓^∗∇𝑓(𝑢)+1−𝛼_𝑛∇𝑓^∗𝑇^∗𝑦_𝑛=𝑉_𝑓^∗𝑝^′,𝛼_𝑛∇𝑓^∗∇𝑓(𝑢)+1−𝛼_𝑛∇𝑓^∗𝑇^∗𝑦_𝑛≤𝑉_𝑓^∗𝑝^′,𝛼_𝑛∇𝑓^∗∇𝑓(𝑢)+1−𝛼_𝑛∇𝑓^∗𝑇^∗𝑦_𝑛−𝛼_𝑛∇𝑓^∗∇𝑓(𝑢)−∇𝑓^∗𝑝^′+𝛼_𝑛∇𝑓^∗∇𝑓(𝑢)−∇𝑓^∗𝑝^′,𝑦_𝑛+1−𝑝^′=𝑉_𝑓^∗𝑝^′,𝛼_𝑛∇𝑓^∗𝑝^′+1−𝛼_𝑛∇𝑓^∗𝑇^∗𝑦_𝑛+𝛼_𝑛∇𝑓^∗∇𝑓(𝑢)−∇𝑓^∗𝑝^′,𝑦_𝑛+1−𝑝^′≤𝛼_𝑛𝑉_𝑓^∗𝑝^′,∇𝑓^∗𝑝^′+1−𝛼_𝑛𝑉_𝑓^∗𝑝^′,∇𝑓^∗𝑇^∗𝑦_𝑛+𝛼_𝑛∇𝑓^∗∇𝑓(𝑢)−∇𝑓^∗𝑝^′,𝑦_𝑛+1−𝑝^′=1−𝛼_𝑛𝐷_𝑓^∗𝑝^′,𝑇^∗𝑦_𝑛+𝛼_𝑛∇𝑓^∗∇𝑓(𝑢)−∇𝑓^∗𝑝^′,𝑦_𝑛+1−𝑝^′≤1−𝛼_𝑛𝐷_𝑓^∗𝑝^′,𝑦_𝑛+𝛼_𝑛∇𝑓^∗∇𝑓(𝑢)−∇𝑓^∗𝑝^′,𝑦_𝑛+1−𝑝^′. (33) Now, we consider two cases.
Case  1. Suppose that there exists 𝑛₀∈ℕ such that {𝐷_𝑓^∗(𝑝^′,𝑦_𝑛)} is decreasing for all 𝑛≥𝑛₀. Then, we get that {𝐷_𝑓^∗(𝑝^′,𝑦_𝑛)} is convergent and hence 𝐷_𝑓^∗𝑝^′,𝑦_𝑛−𝐷_𝑓^∗𝑝^′,𝑦_𝑛+1⟶0as𝑛⟶∞. (34) In addition, from (32) and Lemma 7 we have that 𝐷_𝑓𝑝^′,𝑦_𝑛+1=𝐷_𝑓^∗𝑝^′,∇𝑓𝛼_𝑛∇𝑓^∗∇𝑓(𝑢)+1−𝛼_𝑛∇𝑓^∗𝑇^∗𝑦_𝑛≤𝛼_𝑛𝐷_𝑓^∗𝑝^′,∇𝑓(𝑢)+1−𝛼_𝑛𝐷_𝑓^∗𝑝^′,𝑇^∗𝑦_𝑛. (35) Following from (35), (34), and the fact that 𝛼_𝑛→0, as 𝑛→∞, we get that 𝐷_𝑓^∗𝑝^′,𝑦_𝑛−𝐷_𝑓^∗𝑝^′,𝑇^∗𝑦_𝑛=𝐷_𝑓^∗𝑝^′,𝑦_𝑛−𝐷_𝑓^∗𝑝^′,𝑦_𝑛+1+𝐷_𝑓^∗𝑝^′,𝑦_𝑛+1−𝐷_𝑓^∗𝑝^′,𝑇^∗𝑦_𝑛≤𝐷_𝑓^∗𝑝^′,𝑦_𝑛−𝐷_𝑓^∗𝑝^′,𝑦_𝑛+1+𝛼_𝑛𝐷_𝑓^∗𝑝^′,𝑢−𝐷_𝑓^∗𝑝^′,𝑇^∗𝑦_𝑛⟶0as𝑛⟶∞. (36) This with the fact that 𝑇^∗ is left Bregman strongly nonexpansive implies that lim_𝑛→∞𝐷_𝑓^∗𝑇^∗𝑦_𝑛,𝑦_𝑛=0. (37) Then, by Lemma 5 we obtain that lim_𝑛→∞𝑇^∗𝑦_𝑛−𝑦_𝑛=0. (38) Now, since 𝐸^∗ is reflexive and {𝑦_𝑛+1} is bounded, there exists a subsequence {𝑦_𝑛𝑘+1} of {𝑦_𝑛+1} such that 𝑦_𝑛𝑘+1⇀𝑦∈𝐸^∗,limsup_𝑛→∞∇𝑓^∗∇𝑓(𝑢)−∇𝑓^∗𝑝^′,𝑦_𝑛+1−𝑝^′=limsup_𝑘→∞∇𝑓^∗∇𝑓(𝑢)−∇𝑓^∗𝑝^′,𝑦_𝑛𝑘+1−𝑝^′. (39)(40) Thus, from (39), (38), the fact that 𝑇^∗ is left Bregman strongly nonexpansive mapping with 𝐹(𝑇^∗)=𝐹(𝑇^∗), and Lemma 6 we get that 𝑦∈𝐹(𝑇^∗)=ℱ^′ and limsup_𝑛→∞∇𝑓^∗∇𝑓(𝑢)−∇𝑓^∗𝑝^′,𝑦_𝑛+1−𝑝^′=limsup_𝑘→∞∇𝑓^∗∇𝑓(𝑢)−∇𝑓^∗𝑝^′,𝑦_𝑛𝑘+1−𝑝^′=∇𝑓^∗∇𝑓(𝑢)−∇𝑓^∗𝑝^′,𝑦−𝑝^′≤0. (41) Therefore, it follows from (33), (41), and Lemma 9 that 𝐷_𝑓(𝑝^′,𝑦_𝑛)→0 as 𝑛→∞. Consequently, by Lemma 5 we obtain that 𝑦_𝑛→𝑝^′=𝑃^𝑓∗_ℱ′(∇𝑓(𝑢)) and hence 𝑥_𝑛=∇𝑓^∗(𝑦_𝑛)→∇𝑓^∗(𝑝^′)=𝑝∈ℱ.
Case  2. Suppose that there exists a subsequence {𝑛_𝑖} of {𝑛} such that 𝐷_𝑓^∗𝑝^′,𝑦_𝑛𝑖<𝐷_𝑓^∗𝑝^′,𝑦_𝑛𝑖+1, (42) for all 𝑖∈ℕ. Then, by Lemma 10, there exist a nondecreasing sequence {𝑚_𝑘}⊂ℕ such that 𝑚_𝑘→∞ and 𝐷_𝑓^∗𝑝^′,𝑦_𝑚𝑘≤𝐷_𝑓^∗𝑝^′,𝑦_𝑚𝑘+1,𝐷_𝑓^∗𝑝^′,𝑦_𝑘≤𝐷_𝑓^∗𝑝^′,𝑦_𝑚𝑘+1, (43) for all 𝑘∈ℕ. Thus, we get that 𝐷_𝑓^∗𝑝^′,𝑦_𝑚𝑘−𝐷_𝑓^∗𝑝^′,𝑇^∗𝑦_𝑚𝑘≤𝐷_𝑓^∗𝑝^′,𝑦_𝑚𝑘−𝐷_𝑓^∗𝑝^′,𝑦_𝑚𝑘+1+𝐷_𝑓^∗𝑝^′,𝑦_𝑚𝑘+1−𝐷_𝑓^∗𝑝^′,𝑇^∗𝑦_𝑚𝑘≤𝐷_𝑓^∗𝑝^′,𝑦_𝑚𝑘−𝐷_𝑓^∗𝑝^′,𝑦_𝑚𝑘+1+𝛼_𝑚𝑘𝐷_𝑓^∗𝑝^′,∇𝑓(𝑢)−𝐷_𝑓^∗𝑝^′,𝑇^∗𝑦_𝑚𝑘⟶0. (44) This implies that 𝐷_𝑓^∗(𝑇^∗𝑦_𝑛𝑘,𝑦_𝑛𝑘)→0 as 𝑘→∞. Now, following the method in Case 1 we obtain that limsup_𝑘→∞∇𝑓^∗∇𝑓(𝑢)−∇𝑓^∗𝑝^′,𝑦_𝑚𝑘+1−𝑝^′≤0. (45) Now, from (33) we have that 𝐷_𝑓^∗𝑝^′,𝑦_𝑚𝑘+1≤1−𝛼_𝑚𝑘𝐷_𝑓^∗𝑝^′,𝑦_𝑚𝑘+𝛼_𝑚𝑘∇𝑓^∗∇𝑓(𝑢)−∇𝑓^∗𝑝^′,𝑦_𝑚𝑘+1−𝑝^′. (46) But (43) and (46) imply that 𝛼_𝑚𝑘𝐷_𝑓^∗𝑝^′,𝑦_𝑚𝑘≤𝐷_𝑓^∗𝑝^′,𝑦_𝑚𝑘−𝐷_𝑓^∗𝑝^′,𝑦_𝑚𝑘+1+𝛼_𝑚𝑘∇𝑓^∗∇𝑓(𝑢)−∇𝑓^∗𝑝^′,𝑦_𝑚𝑘+1−𝑝^′≤𝛼_𝑚𝑘∇𝑓^∗∇𝑓(𝑢)−∇𝑓^∗𝑝^′,𝑦_𝑚𝑘+1−𝑝^′, (47) and noting that 𝛼_𝑚𝑘>0, we get that 𝐷_𝑓^∗𝑝^′,𝑦_𝑚𝑘≤∇𝑓^∗∇𝑓(𝑢)−∇𝑓^∗𝑝^′,𝑦_𝑚𝑘+1−𝑝^′. (48) Thus, using (45) we get that 𝐷_𝑓^∗(𝑝^′,𝑦_𝑚𝑘)→0 and hence from (46) we have that 𝐷_𝑓^∗(𝑝^′,𝑦_𝑚𝑘+1)→0 as 𝑘→∞. But 𝐷_𝑓^∗(𝑝^′,𝑦_𝑘)≤𝐷_𝑓^∗(𝑝^′,𝑦_𝑚𝑘+1), for all 𝑘∈ℕ, implies that 𝐷_𝑓^∗(𝑝^′,𝑦_𝑘)→0 and hence by Lemma 5 we obtain that 𝑦_𝑘→𝑝^′ and 𝑥_𝑘=∇𝑓^∗(𝑦_𝑘)→𝑝=∇𝑓^∗(𝑝^′)∈ℱ. Therefore, from the above two cases, we can conclude that {𝑥_𝑛} converges strongly to 𝑝∈ℱ and the proof is complete.

Remark 12. We note that the sequence {𝑥_𝑛} in Theorem 11 converges strongly to a point 𝑝∈ℱ such that 𝑝=∇𝑓^∗(𝑝^′), where 𝑝^′=𝑃_ℱ′(∇𝑓(𝑢)).

If, in Theorem 11, we consider a single right Bregman strongly nonexpansive mapping, we get the following corollary.

Corollary 13. Let 𝐸 be a reflexive Banach space and let 𝑓∶𝐸→ℝ be a cofinite function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of 𝐸. Let 𝐶 be a nonempty, closed, and convex subset of int(dom𝑓) and let 𝑇∶𝐶→𝐶 be a right Bregman strongly nonexpansive mapping such that 𝐹(𝑇)=𝐹(𝑇)≠0/. For 𝑢,𝑥₁∈𝐶 let {𝑥_𝑛} be a sequence generated by 𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑥_𝑛,𝑛=1,2,…, (49) where {𝛼_𝑛}⊂(0,1) satisfy lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞. Then, {𝑥_𝑛} converges strongly to some 𝑝 in 𝐹(𝑇).

If, in Theorem 11, we assume that each 𝑇_𝑖,(𝑖=1,2,…,𝑁) is right Bregman firmly nonexpansive, then we have that 𝑇=𝑇_𝑁∘𝑇_𝑁−1∘⋯∘𝑇₁ is right Bregman firmly nonexpansive with 𝐹(𝑇)=𝐹(𝑇)=∩^𝑁_𝑖=1𝐹(𝑇_𝑖) (see [10]) and hence it is right Bregman strongly nonexpansive mapping. Thus, we have the following.

Corollary 14. Let 𝑓∶𝐸→ℝ be a cofinite function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of 𝐸. Let 𝐶 be a nonempty, closed, and convex subset of int(dom𝑓) and let 𝑇_𝑖∶𝐶→𝐶, for 𝑖=1,2,…,𝑁, be a finite family of right Bregman firmly nonexpansive mappings with ℱ∶=∩^𝑁_𝑖=1𝐹(𝑇_𝑖)≠0/. For 𝑢,𝑥₁∈𝐶 let {𝑥_𝑛} be a sequence generated by 𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑥_𝑛,𝑛=1,2,…, (50) where 𝑇=𝑇_𝑁∘𝑇_𝑁−1∘⋯∘𝑇₁, {𝛼_𝑛}⊂(0,1) satisfying lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞. Then, {𝑥_𝑛} converges strongly to 𝑝 in ℱ.

4. Applications

4.1. Zeroes of Maximal Mappings

Let 𝐴∶𝐸→2^𝐸∗ be a maximal monotone mapping. Recently, many authors studied zero points of monotone mappings using different methods (see e.g., [13, 25, 28, 30, 31, 38]). In this section we use Halpern’s type scheme to find common zeros of a finite family of maximal monotone set-valued mappings.

Definition 15 (see [31]). Let 𝑓∶𝐸→(−∞,+∞] be an admissible function and let 𝐴∶𝐸→2^𝐸∗ be a set-valued mapping such that int(dom𝑓)∩dom𝐴≠0/. The conjugate resolvent of 𝐴 with respect to 𝑓, or the conjugate ∇𝑓-resolvent, is the operator CRes^𝑓_𝐴∶𝐸^∗→2^𝐸∗ defined by CRes^𝑓_𝐴∶=𝐼+𝐴∘∇𝑓^∗−1. (51)

Remark 16. If, in addition, 𝐴 is monotone and 𝑓_int(dom𝑓) is strictly convex, then it is shown in [31] that CRes^𝑓_𝐴 is right Bregman firmly nonexpansive and ∇𝑓^∗(𝐹(CRes^𝑓_𝐴))=int(dom𝑓)∩𝐴⁻¹(0^∗). Moreover, we know that if 𝑓 is Legendre, bounded, and uniformly continuous on bounded subsets of 𝐸, then, for every right Bregman firmly nonexpansive operator 𝑇, 𝐹(𝑇)=𝐹(𝑇)  (see [10]). Thus, under these assumptions on 𝐴 and 𝑓, the operator CRes^𝑓_𝐴 is right Bregman strongly nonexpansive mapping.

We shall need the following lemma.

Lemma 17 (see [30]). Let 𝑓∶𝐸→ℝ be a strictly convex, cofinite, and admissible function, and let 𝐴∶𝐸→2^𝐸∗ be a set-valued monotone mapping. Then 𝐴 is maximal monotone if and only if dom(𝐶𝑅𝑒𝑠^𝑓_𝐴)=𝐸^∗.

Theorem 18. Let 𝑓∶𝐸→ℝ be a cofinite function such that 𝑓^∗ is uniformly Fréchet differentiable and totally convex on bounded subsets of 𝐸^∗. Let 𝐴_𝑖∶𝐸→2^𝐸∗, 𝑖=1,2,…,𝑁, be maximal monotone mappings such that ℱ∶=∩^𝑁_𝑖=1𝐴_𝑖⁻¹(0^∗)≠0/. For each 𝑤,𝑧₁∈𝐸^∗, consider the sequence {𝑧_𝑛} generated iteratively by 𝑧_𝑛+1=𝛼_𝑛𝑤+1−𝛼_𝑛𝑇𝑧_𝑛,𝑛=1,2,…, (52) where 𝑇=𝐶𝑅𝑒𝑠^𝑓_𝐴𝑁∘𝐶𝑅𝑒𝑠^𝑓_𝑁−1∘⋯∘𝐶𝑅𝑒𝑠^𝑓_𝐴1, {𝛼_𝑛}⊂(0,1) satisfy lim_𝑛→∞𝛼_𝑛=0, and ∑^∞_𝑛=1𝛼_𝑛=∞. Then, {𝑧_𝑛} converges strongly to 𝑝^′ in 𝐹(𝑇)=∩^𝑁_𝑖=1𝐹(𝐶𝑅𝑒𝑠^𝑓_𝐴𝑖), and ∇𝑓^∗(𝑝^′)=𝑝∈ℱ.

Proof. From Lemma 17 we know that each 𝑇_𝑖=CRes^𝑓_𝐴𝑖, 𝑖=1,2,…,𝑁 is a mapping from 𝐸^∗ into itself, since ^𝑁_𝑖=1𝐹CRes^𝑓_𝐴𝑖=^𝑁_𝑖=1∇𝑓𝐴_𝑖⁻¹0^∗=∇𝑓(ℱ)≠0/. (53) Remark 16 guarantees that each 𝑇_𝑖,𝑖=1,2,…,𝑁 is right Bregman strongly nonexpansive mapping with respect to 𝐹(𝑇_𝑖)=𝐹(𝑇_𝑖). Now the result follows immediately from Theorem 11 applied to 𝐸^∗.

4.2. Convex Feasibility Problems

The convex feasibility problem (CFP) is finding an element 𝑥^∗∈∩^𝑁_𝑖=1𝐾_𝑖, where 𝐾_𝑖 for 𝑖=1,2,…,𝑁, are nonempty, closed, and convex subsets of 𝐸. Let 𝐾⊂int(dom𝑓). The right Bregman projection [30] onto 𝐾 is the operator 𝑃^𝑓_𝐾∶int(dom𝑓)→𝐾 defined by 𝑃^𝑓_𝐾(𝑥)∶=arglim_𝑦∈𝐾𝐷_𝑓=𝑥,𝑦𝑧∈𝐾∶𝐷_𝑓(𝑥,𝑧)≤𝐷_𝑓𝑥,𝑦∀𝑦∈𝐾. (54) If 𝑓∶𝐸→ℝ is Legendre and uniformly continuous on bounded subsets of 𝐸 and 𝑓 is weakly sequentially continuous, then the right Bregman projection 𝑃^𝑓_𝐾𝑖 is right Bregman strongly nonexpansive mapping with 𝐹(𝑃^𝑓_𝐾𝑖)=𝐹(𝑃^𝑓_𝐾𝑖) (see [30]). Therefore, if we take 𝑇_𝑖=𝑃^𝑓_𝐾𝑖 for each 𝑖∈{1,2,…,𝑁}, then we get an algorithm for solving convex feasibility problems. More precisely, we have the following result.

Theorem 19. Let 𝑓∶𝐸→ℝ be a cofinite function which is bounded, uniformly continuous, and totally convex on bounded subsets of 𝐸. Assume that ∇𝑓 is weakly sequentially continuous. Let 𝐾_𝑖,𝑖=1,…,𝑁 be 𝑁 nonempty, closed, and convex subsets of 𝐸 such that ℱ∶=∩^𝑁_𝑖=1𝐾_𝑖≠0/. For each 𝑥₁∈𝐾, let the sequence {𝑥_𝑛} be generated iteratively by 𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑧_𝑛,𝑛=1,2,…, (55) where 𝑇=𝑃^𝑓_𝐾𝑁∘𝑃^𝑓_𝐾𝑁−1∘⋯∘𝑃^𝑓_𝐾1, {𝛼_𝑛}⊂(0,1) satisfy lim_𝑛→∞𝛼_𝑛=0, and ∑^∞_𝑛=1𝛼_𝑛=∞. Then, {𝑥_𝑛} converges strongly to 𝑝 in ℱ.

Remark 20. Theorem 11 complements the results due to Reich and Sabach [9], Suantai et al. [32], and Zhang and Cheng [33] in the sense that our scheme is applicable for right Bregman strongly nonexpansive self-mappings on 𝐶, where 𝐶 is nonempty, closed, and convex subset of 𝐸.

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The second author acknowledges with thanks DSR for financial support.