Applications of Bregman-Opial Property to Bregman Nonspreading Mappings in Banach Spaces

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Research Article

Applications of Bregman-Opial Property to Bregman Nonspreading Mappings in Banach Spaces

Eskandar Naraghirad,

¹

Ngai-Ching Wong,

²

and Jen-Chih Yao

³

1Department of Mathematics, Yasouj University, Yasouj 75918, Iran

2Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan

3Center for General Education, Kaohsiung Medical University, Kaohsiung 807, Taiwan

Correspondence should be addressed to Eskandar Naraghirad; [email protected] Received 29 October 2013; Accepted 6 December 2013; Published 16 January 2014 Academic Editor: Jinlu Li

Copyright © 2014 Eskandar Naraghirad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The Opial property of Hilbert spaces and some other special Banach spaces is a powerful tool in establishing fixed point theorems for nonexpansive and, more generally, nonspreading mappings. Unfortunately, not every Banach space shares the Opial property.

However, every Banach space has a similar Bregman-Opial property for Bregman distances. In this paper, using Bregman distances, we introduce the classes of Bregman nonspreading mappings and investigate the Mann and Ishikawa iterations for these mappings.

We establish weak and strong convergence theorems for Bregman nonspreading mappings.

1. Introduction

Let𝐸be a (real) Banach space with norm‖ ⋅ ‖and dual space 𝐸^∗. For any 𝑥 in 𝐸, we denote the value of 𝑥^∗ in𝐸^∗ at 𝑥 by⟨𝑥, 𝑥^∗⟩. When{𝑥_𝑛}_𝑛∈Nis a sequence in𝐸, we denote the strong convergence of{𝑥_𝑛}_𝑛∈Nto𝑥 ∈ 𝐸by𝑥_𝑛 → 𝑥and the weak convergence by𝑥_𝑛⇀ 𝑥. Let𝐶be a nonempty subset of 𝐸. Let𝑇 : 𝐶 → 𝐸be a map. We denote by𝐹(𝑇) = {𝑥 ∈ 𝐶 : 𝑇𝑥 = 𝑥}the set offixed pointsof𝑇. We call the map𝑇

(i)nonexpansiveif‖𝑇𝑥 − 𝑇𝑦‖ ≤ ‖𝑥 − 𝑦‖for all𝑥, 𝑦in𝐶, (ii)quasi-nonexpansiveif𝐹(𝑇) ̸= 0and‖𝑇𝑥−𝑦‖ ≤ ‖𝑥−𝑦‖

for all𝑥in𝐶and𝑦in𝐹(𝑇).

The nonexpansivity plays an important role in the study of theIshikawa iteration, given by

𝑦_𝑛= 𝛽_𝑛𝑇𝑥_𝑛+ (1 − 𝛽_𝑛) 𝑥_𝑛,

𝑥_𝑛+1= 𝛾_𝑛𝑇𝑦_𝑛+ (1 − 𝛾_𝑛) 𝑥_𝑛, (1) where the sequences{𝛽_𝑛}_𝑛∈Nand{𝛾_𝑛}_𝑛∈Nsatisfy some appro- priate conditions. When all 𝛽_𝑛 = 0, Ishikawa iteration (1) reduces to the classical Mann iteration. Construction of fixed points of nonexpansive mappings via Mann’s and

Ishikawa’s algorithms [1] has been extensively investigated in the literature (see, e.g., [2] and the references therein).

A powerful tool in deriving weak or strong convergence of iterative sequences is due to Opial [3]. A Banach space 𝐸is said to satisfy the Opial property[3] if for any weakly convergent sequence{𝑥_𝑛}_𝑛∈Nin𝐸with weak limit𝑥we have

lim sup

𝑛 → ∞ 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑥󵄩󵄩󵄩󵄩 <lim sup

𝑛 → ∞ 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑦󵄩󵄩󵄩󵄩 , (2) for all 𝑦in 𝐸with 𝑦 ̸= 𝑥. It is well known that all Hilbert spaces, all finite dimensional Banach spaces, and the Banach spaces𝑙^𝑝(1 ≤ 𝑝 < ∞) satisfy the Opial property. However, not every Banach space satisfies the Opial property; see, for example, [4,5].

Working with the Bregman distance 𝐷_𝑔, the following Bregman-Opial-like inequality holds for every Banach space 𝐸:

lim sup

𝑛 → ∞ 𝐷_𝑔(𝑥_𝑛, 𝑥) <lim sup

𝑛 → ∞ 𝐷_𝑔(𝑥_𝑛, 𝑦) , (3) whenever 𝑥_𝑛 ⇀ 𝑥 ̸= 𝑦. See Lemma 11 for details. The Bregman-Opial property suggests introducing the notions of Bregman nonexpansive-like mappings and developing fixed

Volume 2014, Article ID 272867, 14 pages http://dx.doi.org/10.1155/2014/272867

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point theorems and convergence results for the Ishikawa iterations for these mappings.

We recall the definition of Bregman distances. Let 𝑔 : 𝐸 → R be a strictly convex and Gˆateaux differentiable function on a Banach space𝐸. TheBregman distance[6] (see also [7,8]) corresponding to𝑔is the function𝐷_𝑔: 𝐸×𝐸 → R defined by

𝐷_𝑔(𝑥, 𝑦) = 𝑔 (𝑥) − 𝑔 (𝑦) − ⟨𝑥 − 𝑦, ∇𝑔 (𝑦)⟩ ,

∀𝑥, 𝑦 ∈ 𝐸. (4) It follows from the strict convexity of𝑔that𝐷_𝑔(𝑥, 𝑦) ≥0for all𝑥, 𝑦in𝐸. However,𝐷_𝑔might not be symmetric and𝐷_𝑔 might not satisfy the triangular inequality.

When𝐸is a smooth Banach space, setting𝑔(𝑥) = ‖𝑥‖² for all𝑥in𝐸, we have that∇𝑔(𝑥) = 2𝐽𝑥for all𝑥in𝐸. Here 𝐽is the normalized duality mapping from𝐸into𝐸^∗. Hence, 𝐷_𝑔(⋅, ⋅)reduces to the usual map𝜙(⋅, ⋅)as

𝐷_𝑔(𝑥, 𝑦) = 𝜙 (𝑥, 𝑦) := ‖𝑥‖²− 2 ⟨𝑥, 𝐽𝑦⟩ + 󵄩󵄩󵄩󵄩𝑦󵄩󵄩󵄩󵄩²,

∀𝑥, 𝑦 ∈ 𝐸. (5) If𝐸is a Hilbert space, then𝐷_𝑔(𝑥, 𝑦) = ‖𝑥 − 𝑦‖².

Let 𝑔 : 𝐸 → R be strictly convex and Gˆateaux differentiable, and let𝐶 ⊆ 𝐸be nonempty. A mapping𝑇 : 𝐶 → 𝐸is said to be

(i)Bregman nonexpansiveif

𝐷_𝑔(𝑇𝑥, 𝑇𝑦) ≤ 𝐷_𝑔(𝑥, 𝑦) , ∀𝑥, 𝑦 ∈ 𝐶; (6) (ii)Bregman quasi-nonexpansiveif𝐹(𝑇) ̸= 0and

𝐷_𝑔(𝑝, 𝑇𝑥) ≤ 𝐷_𝑔(𝑝, 𝑥) , ∀𝑥 ∈ 𝐶, ∀𝑝 ∈ 𝐹 (𝑇) ; (7) (iii)Bregman skew quasi-nonexpansiveif𝐹(𝑇) ̸= 0and

𝐷_𝑔(𝑇𝑥, 𝑝) ≤ 𝐷_𝑔(𝑥, 𝑝) , ∀𝑥 ∈ 𝐶, ∀𝑝 ∈ 𝐹 (𝑇) , (8) (iv)Bregman nonspreadingif

𝐷_𝑔(𝑇𝑥, 𝑇𝑦) + 𝐷_𝑔(𝑇𝑦, 𝑇𝑥)

≤ 𝐷_𝑔(𝑇𝑥, 𝑦) + 𝐷_𝑔(𝑇𝑦, 𝑥) , ∀𝑥, 𝑦 ∈ 𝐶. (9) It is obvious that every Bregman nonspreading map𝑇with 𝐹(𝑇) ̸= 0 is Bregman quasi-nonexpansive. Bregman nonspreading mappings include, in particular, the class of nonspreading functions studied by Takahashi and his coauthors (see, e.g., [9,10]), which is defined with the map𝜙in (5).

Let us give an example of a Bregman nonspreading mapping with nonempty fixed point set, which is not quasi- nonexpansive.

Example 1. Let𝑔 : R → Rbe defined by𝑔(𝑥) = 𝑥⁴. The associated Bregman distance is given by

𝐷_𝑔(𝑥, 𝑦) = 𝑥⁴− 𝑦⁴− 4 (𝑥 − 𝑦) 𝑦³

= 𝑥⁴+ 3𝑦⁴− 4𝑥𝑦³, ∀𝑥, 𝑦 ∈ 𝑅. (10)

Define𝑇 : [0, 2] → [0, 2]by

𝑇𝑥 = {0 if 𝑥 ∈ [0, 2) ,

1 if 𝑥 = 2. (11)

We have𝐹(𝑇) = {0}. Plainly,𝑇is neither nonexpansive nor continuous.

However, 𝑇 is Bregman nonspreading. To see this, we define𝑓 : [0, 2] × [0, 2] → Rby

𝑓 (𝑥, 𝑦) = 𝐷_𝑔(𝑇𝑥, 𝑇𝑦) + 𝐷_𝑔(𝑇𝑦, 𝑇𝑥)

− 𝐷_𝑔(𝑇𝑥, 𝑦) − 𝐷_𝑔(𝑇𝑦, 𝑥) , ∀𝑥, 𝑦 ∈ [0, 2] . (12) Consider the following three possible cases.

Case 1. If𝑥 = 𝑦 = 2, then we have𝑇𝑥 = 𝑇𝑦 = 1and hence 𝑓 (2, 2) = 0 + 0 − 17 − 17 = −34 < 0. (13) Case 2. If𝑥 = 2and𝑦 ∈ [0, 2), then we have𝑇𝑥 = 1,𝑇𝑦 = 0, and hence

𝑓 (2, 𝑦) = 1 + 3 − 1 − 3𝑦⁴+ 4𝑦³− 48

= − 3𝑦⁴+ 4𝑦³− 45 < 0. (14) Case 3. If𝑥, 𝑦 ∈ [0, 2), then we have𝑇𝑥 = 𝑇𝑦 = 0and hence 𝑓 (𝑥, 𝑦) = −3 (𝑥⁴+ 𝑦⁴) ≤ 0. (15) Thus we have𝑓(𝑥, 𝑦) ≤ 0for all𝑥, 𝑦in[0, 2]and hence𝑇is a Bregman nonspreading mapping.

In Section 2, we collect and study some basic ties of Bregman distances. InSection 3, utilizing the Bregman-Opial property, we present some fixed point theorems. In Sections 4 and 5, we investigate weak and strong convergence of the Ishikawa and Bregman-Ishikawa iterations for Bregman nonspreading mappings. Our results improve and generalize some known results in the current literature; see, for example, [11].

2. Bregman Functions and Bregman Distances

Let𝐸be a (real) Banach space, and let𝑔 : 𝐸 → R. For any𝑥 in𝐸, thegradient ∇𝑔(𝑥)is defined to be the linear functional in𝐸^∗such that

⟨𝑦, ∇𝑔 (𝑥)⟩ =lim

𝑡 → 0

𝑔 (𝑥 + 𝑡𝑦) − 𝑔 (𝑥)

𝑡 , ∀𝑦 ∈ 𝐸. (16)

The function 𝑔 is said to be Gˆateaux differentiable at𝑥 if

∇𝑔(𝑥)is well defined, and𝑔isGâteaux differentiableif it is Gâteaux differentiable everywhere on𝐸. We call𝑔Fréchet differentiable at𝑥 (see, e.g., [12, page 13] or [13, page 508]) if, for all𝜖 > 0, there exists𝛿 > 0such that

󵄨󵄨󵄨󵄨𝑔(𝑦) − 𝑔(𝑥) − ⟨𝑦 − 𝑥,∇𝑔(𝑥)⟩󵄨󵄨󵄨󵄨 ≤ 𝜖󵄩󵄩󵄩󵄩𝑦 − 𝑥󵄩󵄩󵄩󵄩, whenever󵄩󵄩󵄩󵄩𝑦 − 𝑥󵄩󵄩󵄩󵄩 ≤ 𝛿. (17)

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The function𝑔is said to beFr´echet differentiableif it is Fr´echet differentiable everywhere.

Let 𝐵 be the closed unit ball of a Banach space 𝐸. A function𝑔 : 𝐸 → Ris said to be

(i)strongly coerciveif

‖𝑥_𝑛‖ → +∞lim 𝑔 (𝑥_𝑛)

󵄩󵄩󵄩󵄩𝑥^𝑛󵄩󵄩󵄩󵄩 = +∞; (18) (ii)locally boundedif𝑔(𝑟𝐵)is bounded for all𝑟 > 0;

(iii)locally uniformly smoothon𝐸([14, pp. 207, 221]) if the function𝜎_𝑟: [0, +∞) → [0, +∞], defined by 𝜎_𝑟(𝑡)

= sup

𝑥∈𝑟𝐵,𝑦∈𝑆_𝐸,𝛼∈(0,1)(𝛼𝑔 (𝑥 + (1 − 𝛼) 𝑡𝑦)

+ (1 − 𝛼) 𝑔 (𝑥 − 𝛼𝑡𝑦) − 𝑔 (𝑥))

× (𝛼 (1 − 𝛼))^−1/2,

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satisfies lim𝑡↓0

𝜎_𝑟(𝑡)

𝑡 = 0, ∀𝑟 > 0; (20)

(iv)locally uniformly convex on𝐸 (or uniformly convex on bounded subsetsof 𝐸 ([14, pp. 203, 221])) ifthe gauge𝜌_𝑟 : [0, +∞) → [0, +∞]ofuniform convexity of𝑔, defined by

𝜌_𝑟(𝑡)

= inf

𝑥,𝑦∈𝑟𝐵,‖𝑥−𝑦‖=𝑡,𝛼∈(0,1)(𝛼𝑔 (𝑥) + (1 − 𝛼) 𝑔 (𝑦)

− 𝑔 (𝛼𝑥 + (1 − 𝛼) 𝑦))

× (𝛼 (1 − 𝛼))^−1/2,

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satisfies

𝜌_𝑟(𝑡) > 0, ∀𝑟, 𝑡 > 0. (22) For a locally uniformly convex map𝑔 : 𝐸 → R, we have

𝑔 (𝛼𝑥 + (1 − 𝛼) 𝑦) ≤ 𝛼 (𝑥) 𝑔 + (1 − 𝛼) 𝑔 (𝑦)

− 𝛼 (1 − 𝛼) 𝜌_𝑟(󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩), (23) for all𝑥, 𝑦in𝑟𝐵and for all𝛼in(0, 1).

Let𝐸be a Banach space and𝑔 : 𝐸 → Ra strictly convex and Gˆateaux differentiable function. By (4), the Bregman distance satisfies that [6]

𝐷_𝑔(𝑥, 𝑧) = 𝐷_𝑔(𝑥, 𝑦) + 𝐷_𝑔(𝑦, 𝑧)

+ ⟨𝑥 − 𝑦, ∇𝑔 (𝑦) − ∇𝑔 (𝑧)⟩ , ∀𝑥, 𝑦, 𝑧 ∈ 𝐸.

(24) In particular,

𝐷_𝑔(𝑥, 𝑦) = − 𝐷_𝑔(𝑦, 𝑥)

+ ⟨𝑦 − 𝑥, ∇𝑔 (𝑦) − ∇𝑔 (𝑥)⟩ , ∀𝑥, 𝑦 ∈ 𝐸. (25)

Lemma 2 (see [15]). Let𝐸be a Banach space and𝑔 : 𝐸 → Ra Gˆateaux differentiable function which is locally uniformly convex on𝐸. Let{𝑥_𝑛}_𝑛∈Nand{𝑦_𝑛}_𝑛∈Nbe 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.

The following Bregman-Opial-like inequality has been proved in [16].

Lemma 3 (see [16]). Let𝐸be a Banach space and let𝑔 : 𝐸 → R be a strictly convex and Gˆateaux differentiable function.

Suppose{𝑥_𝑛}_𝑛∈Nis a sequence in𝐸such that𝑥_𝑛⇀ 𝑥for some 𝑥in𝐸. Then

lim sup

𝑛 → ∞ 𝐷_𝑔(𝑥_𝑛, 𝑥) <lim sup

𝑛 → ∞ 𝐷_𝑔(𝑥_𝑛, 𝑦) , (26) for all𝑦in the interior of dom𝑔with𝑦 ̸= 𝑥.

We call a function𝑔 : 𝐸 → (−∞, +∞]lower semicon- tinuousif{𝑥 ∈ 𝐸 : 𝑔(𝑥) ≤ 𝑟}is closed for all𝑟inR. For a lower semicontinuous convex function𝑔 : 𝐸 → R, the subdifferential𝜕𝑔of𝑔is defined by

𝜕𝑔 (𝑥) = {𝑥^∗∈ 𝐸^∗: 𝑔 (𝑥)

+ ⟨𝑦 − 𝑥, 𝑥^∗⟩ ≤ 𝑔 (𝑦) , ∀𝑦 ∈ 𝐸} , (27) for all𝑥in𝐸. It is well known that𝜕𝑔 ⊂ 𝐸 × 𝐸^∗is maximal monotone [17, 18]. For any lower semicontinuous convex function𝑔 : 𝐸 → (−∞, +∞], theconjugate function𝑔^∗of𝑔 is defined by

𝑔^∗(𝑥^∗) =sup

𝑥∈𝐸{⟨𝑥, 𝑥^∗⟩ − 𝑔 (𝑥)} , ∀𝑥^∗∈ 𝐸^∗. (28) It is well known that

𝑔 (𝑥) + 𝑔^∗(𝑥^∗) ≥ ⟨𝑥, 𝑥^∗⟩ , ∀ (𝑥, 𝑥^∗) ∈ 𝐸 × 𝐸^∗, (29) (𝑥, 𝑥^∗) ∈ 𝜕𝑔is equivalent to 𝑔 (𝑥) + 𝑔^∗(𝑥^∗) = ⟨𝑥, 𝑥^∗⟩ .

(30) We also know that if 𝑔 : 𝐸 → (−∞, +∞] is a proper lower semicontinuous convex function, then 𝑔^∗ : 𝐸^∗ → (−∞, +∞]is a proper weak^∗ lower semicontinuous convex function. Here, saying𝑔is properwe mean that dom𝑔 :=

{𝑥 ∈ 𝐸 : 𝑔(𝑥) < +∞} ̸= 0.

The following definition is slightly different from that in Butnariu and Iusem [12].

Definition 4(see [13]). Let𝐸be a Banach space. A function 𝑔 : 𝐸 → Ris said to be a Bregman function if the following conditions are satisfied:

(1)𝑔is continuous, strictly convex, and Gˆateaux differentiable;

(2) the set{𝑦 ∈ 𝐸 : 𝐷_𝑔(𝑥, 𝑦) ≤ 𝑟}is bounded for all𝑥in 𝐸and𝑟 > 0.

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The following lemma follows from Butnariu and Iusem [12] and Zˇalinescu [14].

Lemma 5. Let𝐸be a reflexive Banach space and𝑔 : 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)⟩ = 0if and only if𝑥 = 𝑦;

(3){𝑥 ∈ 𝐸 : 𝐷_𝑔(𝑥, 𝑦) ≤ 𝑟}is bounded for all𝑦in𝐸and 𝑟 > 0;

(4) dom𝑔^∗= 𝐸^∗, 𝑔^∗is Gˆateaux differentiable, and∇𝑔^∗= (∇𝑔)⁻¹.

The following two results follow from [14, Proposition 3.6.4].

Proposition 6. Let𝐸be a reflexive Banach space and let𝑔 : 𝐸 → Rbe a convex function which is locally bounded. The following assertions are equivalent:

(1)𝑔is strongly coercive and locally uniformly convex on 𝐸;

(2) dom𝑔^∗ = 𝐸^∗ and𝑔^∗ is locally bounded and locally uniformly smooth on𝐸;

(3) dom𝑔^∗ = 𝐸^∗, 𝑔^∗ is Fr´echet differentiable, and∇𝑔^∗ is uniformly norm-to-norm continuous on bounded subsets of𝐸^∗.

Proposition 7. Let𝐸be a reflexive Banach space and𝑔 : 𝐸 → Ra continuous convex function which is strongly coercive. The following assertions are equivalent:

(1)𝑔is locally bounded and locally uniformly smooth on 𝐸;

(2)𝑔^∗is Fr´echet differentiable and∇𝑔^∗is uniformly norm- to-norm continuous on bounded subsets of𝐸;

(3) dom𝑔^∗ = 𝐸^∗ and𝑔^∗ is strongly coercive and locally uniformly convex on𝐸.

Lemma 8 (see [13, 19]). Let 𝐸 be a reflexive Banach space, 𝑔 : 𝐸 → Ra strongly coercive Bregman function, and𝑉the function defined by

𝑉 (𝑥, 𝑥^∗)=𝑔 (𝑥) −⟨𝑥, 𝑥^∗⟩+𝑔^∗(𝑥^∗) , ∀𝑥 ∈ 𝐸, ∀𝑥^∗∈ 𝐸^∗. (31) The following assertions hold:

(1)𝐷_𝑔(𝑥, ∇𝑔^∗(𝑥^∗)) = 𝑉(𝑥, 𝑥^∗)for all𝑥in𝐸and𝑥^∗in𝐸^∗, (2)𝑉(𝑥, 𝑥^∗) + ⟨∇𝑔^∗(𝑥^∗) − 𝑥, 𝑦^∗⟩ ≤ 𝑉(𝑥, 𝑥^∗+ 𝑦^∗)for all

𝑥in𝐸and𝑥^∗, 𝑦^∗in𝐸^∗.

It also follows from the definition that𝑉is convex in the second variable𝑥^∗, and

𝑉 (𝑥, ∇𝑔 (𝑦)) = 𝐷_𝑔(𝑥, 𝑦) . (32)

Let𝐸be a Banach space and let𝐶be a nonempty convex subset of𝐸. Let𝑔 : 𝐸 → Rbe a strictly convex and Gˆateaux differentiable function. Then, we know from [20] that, for𝑥 in𝐸and𝑥₀in𝐶, we have

𝐷_𝑔(𝑥₀, 𝑥) =min

𝑦∈𝐶𝐷_𝑔(𝑦, 𝑥) if and only if

⟨𝑦 − 𝑥₀, ∇𝑔 (𝑥) − ∇𝑔 (𝑥₀)⟩ ≤ 0, ∀𝑦 ∈ 𝐶.

(33)

Further, if𝐶is a nonempty, closed, and convex subset of a reflexive Banach space 𝐸 and 𝑔 : 𝐸 → R is a strongly coercive Bregman function, then, for each𝑥in𝐸, there exists a unique𝑥₀in𝐶such that

𝐷_𝑔(𝑥₀, 𝑥) =min

𝑦∈𝐶𝐷_𝑔(𝑦, 𝑥) . (34) The Bregman projection proj^𝑔_𝐶 from 𝐸 onto 𝐶 defined by proj^𝑔_𝐶(𝑥) = 𝑥₀has the following property:

𝐷_𝑔(𝑦,proj^𝑔_𝐶 𝑥) + 𝐷_𝑔(proj^𝑔_𝐶 𝑥, 𝑥)

≤ 𝐷_𝑔(𝑦, 𝑥) , ∀𝑦 ∈ 𝐶, ∀𝑥 ∈ 𝐸. (35) See [12] for details.

Let 𝐸 be a reflexive Banach space and let 𝑔 : 𝐸 → Rbe a lower semicontinuous, strictly convex, and Gˆateaux differentiable function. Let 𝐶 be a nonempty, closed, and convex subset of𝐸and let{𝑥_𝑛}_𝑛∈N be a bounded sequence in𝐸. For any𝑥in𝐸, we set

Br(𝑥, {𝑥_𝑛}) =lim sup

𝑛 → ∞ 𝐷_𝑔(𝑥_𝑛, 𝑥) . (36) The Bregman asymptotic radius of {𝑥_𝑛}_𝑛∈N relative to 𝐶 is defined by

Br(𝐶, {𝑥_𝑛}) =inf{Br(𝑥, {𝑥_𝑛}) : 𝑥 ∈ 𝐶} . (37) TheBregman asymptotic centerof{𝑥_𝑛}_𝑛∈Nrelative to𝐶is the set

BA(𝐶, {𝑥_𝑛}) = {𝑥 ∈ 𝐶 :Br(𝑥, {𝑥_𝑛}) =Br(𝐶, {𝑥_𝑛})} . (38) Proposition 9. Let𝐶be a nonempty, closed, and convex subset of a reflexive Banach space𝐸, and let𝑔 : 𝐸 → Rbe strictly convex, Gˆateaux differentiable, and locally bounded on𝐸. If {𝑥_𝑛}_𝑛∈Nis a bounded sequence of𝐶, then𝐵𝐴(𝐶, {𝑥_𝑛}_𝑛∈N)is a singleton.

Proof. In view of the definition of Bregman asymptotic radius, we may assume that{𝑥_𝑛}_𝑛∈Nconverges weakly to𝑧in 𝐶. ByLemma 3, we conclude that BA(𝐶, {𝑥_𝑛}_𝑛∈N) = {𝑧}.

3. Fixed Point Theorems

Lemma 10 (see [21]). Let𝐶be a nonempty, closed, and convex subset of a reflexive Banach space𝐸. Let𝑔 : 𝐸 → Rbe strictly convex, continuous, strongly coercive, Gˆateaux differentiable, and locally bounded on𝐸. Let𝑇 : 𝐶 → 𝐸be a Bregman quasi- nonexpansive mapping. Then𝐹(𝑇)is closed and convex.

(5)

Lemma 11. Let𝐶be a nonempty, closed, and convex subset of a reflexive Banach space𝐸. Let𝑔 : 𝐸 → Rbe a strictly convex and Gˆateaux differentiable function. Let𝑇 : 𝐶 → 𝐸be a Bregman nonspreading mapping. Then

𝐷_𝑔(𝑥, 𝑇𝑦) ≤ 𝐷_𝑔(𝑥, 𝑦) + 𝐷_𝑔(𝑇𝑥, 𝑥) + ⟨𝑥 − 𝑇𝑥, ∇𝑔 (𝑦) − ∇𝑔 (𝑇𝑦)⟩

+ ⟨𝑇𝑥 − 𝑇𝑦, ∇𝑔 (𝑥) − ∇𝑔 (𝑇𝑥)⟩ ,

∀𝑥, 𝑦 ∈ 𝐶.

(39)

Proof. Let𝑥, 𝑦 ∈ 𝐶. In view of (24), we have

𝐷_𝑔(𝑇𝑥, 𝑇𝑦) ≤ 𝐷_𝑔(𝑇𝑥, 𝑦) + 𝐷_𝑔(𝑇𝑦, 𝑥) − 𝐷_𝑔(𝑇𝑦, 𝑇𝑥)

= 𝐷_𝑔(𝑇𝑥, 𝑥) + 𝐷_𝑔(𝑥, 𝑦) + ⟨𝑇𝑥 − 𝑥, ∇𝑔 (𝑥) − ∇𝑔 (𝑦)⟩

+ 𝐷_𝑔(𝑇𝑦, 𝑇𝑥) + 𝐷_𝑔(𝑇𝑥, 𝑥)

+⟨𝑇𝑦−𝑇𝑥, ∇𝑔 (𝑇𝑥)−∇𝑔 (𝑥)⟩−𝐷_𝑔(𝑇𝑦, 𝑇𝑥)

= 𝐷_𝑔(𝑥, 𝑦) + 2𝐷_𝑔(𝑇𝑥, 𝑥) + ⟨𝑇𝑥 − 𝑥, ∇𝑔 (𝑥) − ∇𝑔 (𝑦)⟩

+ ⟨𝑇𝑥 − 𝑇𝑦, ∇𝑔 (𝑥) − ∇𝑔 (𝑇𝑥)⟩ .

(40) This, together with (24), implies that

𝐷_𝑔(𝑥, 𝑇𝑦) = 𝐷_𝑔(𝑥, 𝑇𝑥) + 𝐷_𝑔(𝑇𝑥, 𝑇𝑦) + ⟨𝑥 − 𝑇𝑥, ∇𝑔 (𝑇𝑥) − ∇𝑔 (𝑇𝑦)⟩

≤ 𝐷_𝑔(𝑥, 𝑇𝑥) + 𝐷_𝑔(𝑥, 𝑦) + 2𝐷_𝑔(𝑇𝑥, 𝑥) + ⟨𝑇𝑥 − 𝑥, ∇𝑔 (𝑥) − ∇𝑔 (𝑦)⟩

+ ⟨𝑇𝑥 − 𝑇𝑦, ∇𝑔 (𝑥) − ∇𝑔 (𝑇𝑥)⟩

+ ⟨𝑥 − 𝑇𝑥, ∇𝑔 (𝑇𝑥) − ∇𝑔 (𝑇𝑦)⟩

= 𝐷_𝑔(𝑥, 𝑦) + 𝐷_𝑔(𝑇𝑥, 𝑥) + ⟨𝑥 − 𝑇𝑥, ∇𝑔 (𝑥) − ∇𝑔 (𝑇𝑥)⟩

+ ⟨𝑇𝑥 − 𝑇𝑦, ∇𝑔 (𝑥) − ∇𝑔 (𝑇𝑥)⟩

+ ⟨𝑇𝑥 − 𝑥, ∇𝑔 (𝑥) − ∇𝑔 (𝑦)⟩

+ ⟨𝑥 − 𝑇𝑥, ∇𝑔 (𝑇𝑥) − ∇𝑔 (𝑇𝑦)⟩

= 𝐷_𝑔(𝑥, 𝑦) + 𝐷_𝑔(𝑇𝑥, 𝑥) + ⟨𝑥 − 𝑇𝑥, ∇𝑔 (𝑦) − ∇𝑔 (𝑇𝑦)⟩

+ ⟨𝑇𝑥 − 𝑇𝑦, ∇𝑔 (𝑥) − ∇𝑔 (𝑇𝑥)⟩ .

(41)

Proposition 12 (demiclosedness principle). Let 𝐶 be a nonempty subset of a reflexive Banach space𝐸. Let𝑔 : 𝐸 → R be a strictly convex, Gˆateaux differentiable, and locally bounded function. Let𝑇 : 𝐶 → 𝐸be a Bregman nonspreading mapping. If𝑥_𝑛 ⇀ 𝑧in𝐶andlim_{𝑛 → ∞}‖𝑇𝑥_𝑛− 𝑥_𝑛‖ = 0, then 𝑇𝑧 = 𝑧. That is,𝐼 − 𝑇 is demiclosed at zero, where𝐼is the identity mapping on𝐸.

Proof. Since {𝑥_𝑛}_𝑛∈N converges weakly to 𝑧 and lim_{𝑛 → ∞}‖𝑇𝑥_𝑛 − 𝑥_𝑛‖ = 0, both the sequences {𝑥_𝑛}_𝑛∈N and{𝑇𝑥_𝑛}_𝑛∈N are bounded. Since∇𝑔is uniformly norm-to- norm continuous on bounded subsets of𝐸 (see, e.g., [14]), we arrive at

𝑛 → ∞lim 󵄩󵄩󵄩󵄩∇𝑔(𝑥^𝑛) − ∇𝑔 (𝑇𝑥_𝑛)󵄩󵄩󵄩󵄩 = 0. (42) In view ofLemma 2, we deduce that lim_{𝑛 → ∞}𝐷_𝑔(𝑥_𝑛, 𝑇𝑥_𝑛) = 0. Set

𝑀₁= sup{󵄩󵄩󵄩󵄩𝑇𝑥^𝑛󵄩󵄩󵄩󵄩,‖𝑇𝑧‖,󵄩󵄩󵄩󵄩∇𝑔(𝑧)󵄩󵄩󵄩󵄩,󵄩󵄩󵄩󵄩∇𝑔(𝑇𝑧)󵄩󵄩󵄩󵄩 : 𝑛 ∈N}

< + ∞.

(43) ByLemma 11, for all𝑛inN,

𝐷_𝑔(𝑥_𝑛, 𝑇𝑧)

≤ 𝐷_𝑔(𝑥_𝑛, 𝑧) + 𝐷_𝑔(𝑇𝑥_𝑛, 𝑥_𝑛) + ⟨𝑥_𝑛− 𝑇𝑥_𝑛, ∇𝑔 (𝑧) − ∇𝑔 (𝑇𝑧)⟩

+ ⟨𝑇𝑥_𝑛− 𝑇𝑧, ∇𝑔 (𝑥_𝑛) − ∇𝑔 (𝑇𝑥_𝑛)⟩

≤ 𝐷_𝑔(𝑥_𝑛, 𝑧) + 𝐷_𝑔(𝑇𝑥_𝑛, 𝑥_𝑛) + 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑇𝑥_𝑛󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩∇𝑔(𝑧) − ∇𝑔(𝑇𝑧)󵄩󵄩󵄩󵄩

+ 󵄩󵄩󵄩󵄩𝑇𝑥^𝑛− 𝑇𝑧󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩∇𝑔(𝑥^𝑛) − ∇𝑔 (𝑇𝑥_𝑛)󵄩󵄩󵄩󵄩

≤ 𝐷_𝑔(𝑥_𝑛, 𝑧) + 𝐷_𝑔(𝑇𝑥_𝑛, 𝑥_𝑛)

+ 2𝑀₁󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑇𝑥_𝑛󵄩󵄩󵄩󵄩 + 2𝑀¹󵄩󵄩󵄩󵄩∇𝑔(𝑥^𝑛) − ∇𝑔 (𝑇𝑥_𝑛)󵄩󵄩󵄩󵄩 . (44) This implies

lim sup

𝑛 → ∞ 𝐷_𝑔(𝑥_𝑛, 𝑇𝑧) ≤lim sup

𝑛 → ∞ 𝐷_𝑔(𝑥_𝑛, 𝑧) . (45) From the Bregman-Opial-like property, we obtain𝑇𝑧 = 𝑧.

Letℓ^∞be the Banach lattice of bounded real sequences with the supremum norm. It is well known that there exists a bounded linear functional𝜇onℓ^∞such that the following three conditions hold:

(1) if{𝑡_𝑛}_𝑛∈N ∈ ℓ^∞ and𝑡_𝑛 ≥ 0for every𝑛inN, then 𝜇({𝑡_𝑛}) ≥0;

(2) if𝑡_𝑛= 1for every𝑛inN, then𝜇({𝑡_𝑛}) = 1;

(3)𝜇({𝑡_𝑛+1}) = 𝜇({𝑡_𝑛})for all{𝑡_𝑛}_𝑛∈Ninℓ^∞.

(6)

Here, {𝑡_𝑛+1} denotes the sequence (𝑡₂, 𝑡₃, 𝑡₄, . . . , 𝑡_𝑛+1, . . .) in ℓ^∞. Such a functional𝜇is called aBanach limitand the value of𝜇at{𝑡_𝑛}_𝑛∈Ninℓ^∞is denoted by𝜇_𝑛𝑡_𝑛. Therefore, condition (3)means𝜇_𝑛𝑡_𝑛 = 𝜇_𝑛𝑡_𝑛+1. If𝜇satisfies conditions(1)and(2), we call𝜇ameanonℓ^∞. See, for example, [22].

To see some examples of those mappings𝑇satisfying all the stated hypotheses in the following result, we refer the reader to [23].

Theorem 13 (see [23]). Let 𝐶 be a nonempty, closed, and convex subset of a reflexive Banach space𝐸. Let 𝑔 : 𝐸 → R be strictly convex, continuous, strongly coercive, Gˆateaux differentiable, locally bounded and locally uniformly convex on 𝐸. Let𝑇 : 𝐶 → 𝐶be a mapping. Let{𝑥_𝑛}_𝑛∈Nbe a bounded sequence of𝐶and let𝜇be a mean onℓ^∞. Suppose that

𝜇_𝑛𝐷_𝑔(𝑥_𝑛, 𝑇𝑦) ≤ 𝜇_𝑛𝐷_𝑔(𝑥_𝑛, 𝑦) , ∀𝑦 ∈ 𝐶. (46) Then𝑇has a fixed point in𝐶.

Corollary 14. Let 𝐶 be a nonempty, bounded, closed, and convex subset of a reflexive Banach space𝐸. Let 𝑔 : 𝐸 → R be strictly convex, continuous, strongly coercive, Gˆateaux differentiable function, locally bounded, and locally uniformly convex on𝐸. Let𝑇 : 𝐶 → 𝐶 be a Bregman nonspreading mapping. Then𝑇has a fixed point.

Proof. Let𝜇a Banach limit onℓ^∞ and𝑥 ∈ 𝐶be such that {𝑇^𝑛𝑥}_𝑛∈Nis bounded. For any𝑛inNwe have

𝐷_𝑔(𝑇^𝑛𝑥, 𝑇𝑦) + 𝐷_𝑔(𝑇𝑦, 𝑇^𝑛𝑥)

≤ 𝐷_𝑔(𝑇^𝑛𝑥, 𝑦) + 𝐷_𝑔(𝑇𝑦, 𝑇^𝑛−1𝑥) , ∀𝑦 ∈ 𝐶. (47) This implies that

𝜇_𝑛𝐷_𝑔(𝑇^𝑛𝑥, 𝑇𝑦) + 𝜇_𝑛𝐷_𝑔(𝑇𝑦, 𝑇^𝑛𝑥)

≤ 𝜇_𝑛𝐷_𝑔(𝑇^𝑛𝑥, 𝑦) + 𝜇_𝑛𝐷_𝑔(𝑇𝑦, 𝑇^𝑛−1𝑥) , ∀𝑦 ∈ 𝐶. (48) Thus we have

𝜇_𝑛𝐷_𝑔(𝑇^𝑛𝑥, 𝑇𝑦) ≤ 𝜇_𝑛𝐷_𝑔(𝑇^𝑛𝑥, 𝑦) , ∀𝑦 ∈ 𝐶. (49) It follows fromTheorem 13that𝐹(𝑇) ̸= 0.

4. Weak and Strong Convergence Theorems for Bregman Nonspreading Mappings

In this section, we prove weak and strong convergence theorems concerning Bregman nonspreading mappings in a reflexive Banach space.

Lemma 15. Let𝐶be a nonempty, closed, and convex subset of a reflexive Banach space𝐸. Let𝑔 : 𝐸 → Rbe a strictly convex and Gˆateaux differentiable function. Let𝑇 : 𝐶 → 𝐶be a Bregman skew quasi-nonexpansive mapping with a nonempty fixed point set𝐹(𝑇). Let{𝑥_𝑛}_𝑛∈Nand{𝑦_𝑛}_𝑛∈Nbe two sequences defined by (1) such that {𝛽_𝑛}_𝑛∈N and {𝛾_𝑛}_𝑛∈N are arbitrary

sequences in[0, 1]. Then the following assertions hold:

(1) max{𝐷_𝑔(𝑥_𝑛+1, 𝑧),𝐷_𝑔(𝑦_𝑛, 𝑧)} ≤ 𝐷_𝑔(𝑥_𝑛, 𝑧)for all𝑧in 𝐹(𝑇)and𝑛 = 1, 2, . . .,

(2) lim_{𝑛 → ∞}𝐷_𝑔(𝑥_𝑛, 𝑧)exists for any𝑧in𝐹(𝑇).

Proof. Let𝑧 ∈ 𝐹(𝑇). In view of (23), we have 𝐷_𝑔(𝑦_𝑛, 𝑧) = 𝐷_𝑔(𝛽_𝑛𝑇𝑥_𝑛+ (1 − 𝛽_𝑛) 𝑥_𝑛, 𝑧)

≤ 𝛽_𝑛𝐷_𝑔(𝑇𝑥_𝑛, 𝑧) + (1 − 𝛽_𝑛) 𝐷_𝑔(𝑥_𝑛, 𝑧)

≤ 𝛽_𝑛𝐷_𝑔(𝑥_𝑛, 𝑧) + (1 − 𝛽_𝑛) 𝐷_𝑔(𝑥_𝑛, 𝑧)

= 𝐷_𝑔(𝑥_𝑛, 𝑧) .

(50)

Consequently,

𝐷_𝑔(𝑥_𝑛+1, 𝑧) = 𝐷_𝑔(𝛾_𝑛𝑇𝑦_𝑛+ (1 − 𝛾_𝑛) 𝑥_𝑛, 𝑧)

≤ 𝛾_𝑛𝐷_𝑔(𝑇𝑦_𝑛, 𝑧) + (1 − 𝛾_𝑛) 𝐷_𝑔(𝑥_𝑛, 𝑧)

≤ 𝛾_𝑛𝐷_𝑔(𝑦_𝑛, 𝑧) + (1 − 𝛾_𝑛) 𝐷_𝑔(𝑥_𝑛, 𝑧)

≤ 𝛾_𝑛𝐷_𝑔(𝑥_𝑛, 𝑧) + (1 − 𝛾_𝑛) 𝐷_𝑔(𝑥_𝑛, 𝑧)

= 𝐷_𝑔(𝑥_𝑛, 𝑧) .

(51)

This implies that {𝐷_𝑔(𝑥_𝑛, 𝑧)}_𝑛∈N is a bounded and nonin- creasing sequence for all 𝑧 in 𝐹(𝑇). Thus we have that lim_{𝑛 → ∞}𝐷_𝑔(𝑥_𝑛, 𝑧)exists for any𝑧in𝐹(𝑇).

Theorem 16. Let𝐶be a nonempty, closed, and convex subset of a reflexive Banach space𝐸. Let𝑔 : 𝐸 → Rbe strictly convex, Gˆateaux differentiable, locally bounded, and locally uniformly convex on𝐸. Let𝑇 : 𝐶 → 𝐶be a Bregman nonspreading and Bregman skew quasi-nonexpansive mapping. Let{𝛽_𝑛}_𝑛∈N and {𝛾_𝑛}_𝑛∈N be sequences in[0, 1], and let{𝑥_𝑛}_𝑛∈N be a sequence with𝑥₁in𝐶defined by(1).

(a)If{𝑥_𝑛}_𝑛∈Nis bounded andlim inf_{𝑛 → ∞}‖𝑇𝑥_𝑛− 𝑥_𝑛‖ = 0, then the fixed point set𝐹(𝑇) ̸= 0.

(b)Assume𝐹(𝑇) ̸= 0. Then{𝑥_𝑛}_𝑛∈Nis bounded.

(i) lim_{𝑛 → ∞}‖𝑇𝑥_𝑛− 𝑥_𝑛‖ = 0whenlim inf_{𝑛 → ∞}𝛾_𝑛(1 − 𝛾_𝑛) > 0andlim_{𝑛 → ∞}𝛽_𝑛= 1.

(ii) lim inf_{𝑛 → ∞}‖𝑇𝑥_𝑛− 𝑥_𝑛‖ = 0when either

(1) lim sup_{𝑛 → ∞}𝛾_𝑛(1 − 𝛾_𝑛) > 0 and lim_{𝑛 → ∞}𝛽_𝑛= 1or

(2) lim inf_{𝑛 → ∞}𝛾_𝑛(1 − 𝛾_𝑛) > 0 and lim sup_{𝑛 → ∞}𝛽_𝑛 = 1.

Proof. Assume that {𝑥_𝑛}_𝑛∈N is bounded and lim inf_{𝑛 → ∞}‖𝑇𝑥_𝑛 − 𝑥_𝑛‖ = 0. Consequently, there is a bounded subsequence {𝑇𝑥_𝑛_𝑘}_𝑘∈N of {𝑇𝑥_𝑛}_𝑛∈N such that lim_{𝑘 → ∞}‖𝑇𝑥_𝑛_𝑘 − 𝑥_𝑛_𝑘‖ = 0. Since ∇𝑔 is uniformly norm- to-norm continuous on bounded subsets of 𝐸 (see, e.g., [14]),

𝑘 → ∞lim 󵄩󵄩󵄩󵄩󵄩∇𝑔 (𝑇𝑥^𝑛^𝑘) − ∇𝑔 (𝑥_𝑛_𝑘)󵄩󵄩󵄩󵄩󵄩 = 0. (52)

(7)

In view ofProposition 9, we conclude that BA(𝐶, {𝑥_𝑛_𝑘}) = {𝑧}

for some𝑧in𝐶. Let

𝑀₂=sup{‖𝑇 (𝑧)‖ ,󵄩󵄩󵄩󵄩󵄩𝑇𝑥^𝑛^𝑘󵄩󵄩󵄩󵄩󵄩 ,󵄩󵄩󵄩󵄩∇𝑔(𝑧)󵄩󵄩󵄩󵄩,

󵄩󵄩󵄩󵄩∇𝑔(𝑇𝑧)󵄩󵄩󵄩󵄩 : 𝑘 ∈N} < +∞. (53) It follows fromLemma 11that

𝐷_𝑔(𝑥_𝑛_𝑘, 𝑇𝑧)

≤ 𝐷_𝑔(𝑥_𝑛_𝑘, 𝑧) + 𝐷_𝑔(𝑇𝑥_𝑛_𝑘, 𝑥_𝑛_𝑘) + ⟨𝑥_𝑛_𝑘− 𝑇𝑥_𝑛_𝑘, ∇𝑔 (𝑧) − ∇𝑔 (𝑇𝑧)⟩

+ ⟨𝑇𝑥_𝑛_𝑘− 𝑇𝑧, ∇𝑔 (𝑥_𝑛_𝑘) − ∇𝑔 (𝑇𝑥_𝑛_𝑘)⟩

≤ 𝐷_𝑔(𝑥_𝑛_𝑘, 𝑧) + 𝐷_𝑔(𝑇𝑥_𝑛_𝑘, 𝑥_𝑛_𝑘) + 󵄩󵄩󵄩󵄩󵄩𝑥^𝑛^𝑘− 𝑇𝑥_𝑛_𝑘󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩∇𝑔(𝑧) − ∇𝑔(𝑇𝑧)󵄩󵄩󵄩󵄩

+ 󵄩󵄩󵄩󵄩󵄩𝑇𝑥^𝑛^𝑘− 𝑇𝑧󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩∇𝑔 (𝑥^𝑛^𝑘) − ∇𝑔 (𝑇𝑥_𝑛_𝑘)󵄩󵄩󵄩󵄩󵄩

≤ 𝐷_𝑔(𝑥_𝑛_𝑘, 𝑧) + 𝐷_𝑔(𝑇𝑥_𝑛_𝑘, 𝑥_𝑛_𝑘) + 2𝑀₂󵄩󵄩󵄩󵄩󵄩𝑥^𝑛^𝑘− 𝑇𝑥_𝑛_𝑘󵄩󵄩󵄩󵄩󵄩

+ 2𝑀₂󵄩󵄩󵄩󵄩󵄩∇𝑔 (𝑥^𝑛^𝑘) − ∇𝑔 (𝑇𝑥_𝑛_𝑘)󵄩󵄩󵄩󵄩󵄩, 𝑘 = 1,2,....

(54)

This implies lim sup

𝑘 → ∞ 𝐷_𝑔(𝑥_𝑛_𝑘, 𝑇𝑧) ≤lim sup

𝑘 → ∞ 𝐷_𝑔(𝑥_𝑛_𝑘, 𝑧) . (55) From the Bregman-Opial-like property, we obtain𝑇𝑧 = 𝑧.

Let𝐹(𝑇) ̸= 0and let𝑧 ∈ 𝐹(𝑇). It follows fromLemma 15 that lim_{𝑛 → ∞}‖𝑥_𝑛 − 𝑧‖exists and hence{𝑥_𝑛}_𝑛∈N is bounded.

This implies that the sequence{𝑇𝑦_𝑛}_𝑛∈N is bounded too. Let 𝑠₁ = sup{‖𝑥_𝑛‖, ‖𝑇𝑦_𝑛‖ : 𝑛 ∈ N} < ∞. In view of (23), we obtain a continuous, strictly increasing, and convex function 𝜌_𝑠₁: [0, +∞) → [0, +∞)with𝜌_𝑠₁(0) = 0such that

𝐷_𝑔(𝑥_𝑛+1, 𝑧) = 𝐷_𝑔(𝛾_𝑛𝑇𝑦_𝑛+ (1 − 𝛾_𝑛) 𝑥_𝑛, 𝑧)

≤ 𝛾_𝑛𝐷_𝑔(𝑇𝑦_𝑛, 𝑧) + (1 − 𝛾_𝑛) 𝐷_𝑔(𝑥_𝑛, 𝑧)

− 𝛾_𝑛(1 − 𝛾_𝑛) 𝜌_𝑠₁(󵄩󵄩󵄩󵄩𝑇𝑦^𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩)

≤ 𝛾_𝑛𝐷_𝑔(𝑦_𝑛, 𝑧) + (1 − 𝛾_𝑛) 𝐷_𝑔(𝑥_𝑛, 𝑧)

− 𝛾_𝑛(1 − 𝛾_𝑛) 𝜌_𝑠₁(󵄩󵄩󵄩󵄩𝑇𝑦^𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩)

≤ 𝛾_𝑛𝐷_𝑔(𝑥_𝑛, 𝑧) + (1 − 𝛾_𝑛) 𝐷_𝑔(𝑥_𝑛, 𝑧)

− 𝛾_𝑛(1 − 𝛾_𝑛) 𝜌_𝑠₁(󵄩󵄩󵄩󵄩𝑇𝑦^𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩)

= 𝐷_𝑔(𝑥_𝑛, 𝑧) − 𝛾_𝑛(1 − 𝛾_𝑛) 𝜌_𝑠₁(󵄩󵄩󵄩󵄩𝑇𝑦^𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩).

(56)

Consequently, we conclude that

𝛾_𝑛(1 − 𝛾_𝑛) 𝜌_𝑠₁(󵄩󵄩󵄩󵄩𝑇𝑦^𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩)

≤ 𝐷_𝑔(𝑥_𝑛, 𝑧) − 𝐷_𝑔(𝑥_𝑛+1, 𝑧)

󳨀→ 0, as𝑛 󳨀→ ∞.

(57)

It follows that

lim inf_{𝑛 → ∞}𝜌_𝑠₁(󵄩󵄩󵄩󵄩𝑇𝑦^𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩)= 0whenever lim sup

𝑛 → ∞ 𝛾_𝑛(1 − 𝛾_𝑛) > 0.

(58) From the property of𝜌_𝑠₁we deduce that

lim inf

𝑛 → ∞ 󵄩󵄩󵄩󵄩𝑇𝑦^𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩 = 0whenever lim sup

𝑛 → ∞ 𝛾_𝑛(1 − 𝛾_𝑛) > 0.

(59) In the same manner, we also obtain that

𝑛 → ∞lim 󵄩󵄩󵄩󵄩𝑇𝑦^𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩 = 0whenever lim inf

𝑛 → ∞ 𝛾_𝑛(1 − 𝛾_𝑛) > 0.

(60) Since ∇𝑔 is uniformly norm-to-norm continuous on bounded subsets of𝐸(see, e.g., [14]), we arrive at

𝑛 → ∞lim 󵄩󵄩󵄩󵄩∇𝑔(𝑇𝑦^𝑛) − ∇𝑔 (𝑥_𝑛)󵄩󵄩󵄩󵄩 = 0. (61) On the other hand, from (1) we get

𝑇𝑥_𝑛− 𝑦_𝑛 = (1 − 𝛽_𝑛) (𝑇𝑥_𝑛− 𝑥_𝑛) ,

𝑥_𝑛− 𝑦_𝑛 = 𝛽_𝑛(𝑥_𝑛− 𝑇𝑥_𝑛) . (62) Assuming first lim inf_{𝑛 → ∞}𝛾_𝑛(1−𝛾_𝑛) > 0. By (60) we see that

𝑀₃:=sup{󵄩󵄩󵄩󵄩∇𝑔 (𝑥^𝑛)󵄩󵄩󵄩󵄩 ,󵄩󵄩󵄩󵄩∇𝑔(𝑇𝑥^𝑛)󵄩󵄩󵄩󵄩 ,

󵄩󵄩󵄩󵄩∇𝑔(𝑇𝑦^𝑛)󵄩󵄩󵄩󵄩 : 𝑛 ∈N} < +∞. (63) Since𝑇is Bregman nonspreading, in view of (24), (25), and (62), we obtain

𝐷_𝑔(𝑥_𝑛, 𝑇𝑥_𝑛)

= 𝐷_𝑔(𝑥_𝑛, 𝑇𝑦_𝑛) + 𝐷_𝑔(𝑇𝑦_𝑛, 𝑇𝑥_𝑛) + ⟨𝑥_𝑛− 𝑇𝑦_𝑛, ∇𝑔 (𝑇𝑦_𝑛) − ∇𝑔 (𝑇𝑥_𝑛)⟩

≤ 𝐷_𝑔(𝑥_𝑛, 𝑇𝑦_𝑛)

+ [𝐷_𝑔(𝑇𝑦_𝑛, 𝑥_𝑛) + 𝐷_𝑔(𝑇𝑥_𝑛, 𝑦_𝑛) − 𝐷_𝑔(𝑇𝑥_𝑛, 𝑇𝑦_𝑛)]

+ 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑇𝑦_𝑛󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩∇𝑔(𝑇𝑦^𝑛) − ∇𝑔 (𝑇𝑥_𝑛)󵄩󵄩󵄩󵄩

(8)

≤ 𝐷_𝑔(𝑥_𝑛, 𝑇𝑦_𝑛)

+ [−𝐷_𝑔(𝑥_𝑛, 𝑇𝑦_𝑛) + ⟨𝑥_𝑛− 𝑇𝑦_𝑛, ∇𝑔 (𝑥_𝑛) − ∇𝑔 (𝑇𝑦_𝑛)⟩]

+ [−𝐷_𝑔(𝑦_𝑛, 𝑇𝑥_𝑛) + ⟨𝑦_𝑛− 𝑇𝑥_𝑛, ∇𝑔 (𝑦_𝑛) − ∇𝑔 (𝑇𝑥_𝑛)⟩]

+ 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑇𝑦_𝑛󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩∇𝑔(𝑇𝑦^𝑛) − ∇𝑔 (𝑇𝑥_𝑛)󵄩󵄩󵄩󵄩

≤ 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑇𝑦_𝑛󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩∇𝑔(𝑥^𝑛) − ∇𝑔 (𝑇𝑦_𝑛)󵄩󵄩󵄩󵄩

+ 󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑇𝑥_𝑛󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩∇𝑔(𝑦^𝑛) − ∇𝑔 (𝑇𝑥_𝑛)󵄩󵄩󵄩󵄩

+ 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑇𝑦_𝑛󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩∇𝑔(𝑇𝑦^𝑛) − ∇𝑔 (𝑇𝑥_𝑛)󵄩󵄩󵄩󵄩

= (1 − 𝛽_𝑛) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑇𝑥_𝑛󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩∇𝑔(𝑦^𝑛) − ∇𝑔 (𝑇𝑥_𝑛)󵄩󵄩󵄩󵄩

+ 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑇𝑦_𝑛󵄩󵄩󵄩󵄩[󵄩󵄩󵄩󵄩∇𝑔(𝑥^𝑛) − ∇𝑔 (𝑇𝑦_𝑛) + 󵄩󵄩󵄩󵄩∇𝑔 (𝑇𝑦^𝑛) − ∇𝑔 (𝑇𝑥_𝑛)󵄩󵄩󵄩󵄩]

≤ 2 (1 − 𝛽_𝑛) 𝑀₃󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑇𝑥_𝑛󵄩󵄩󵄩󵄩 + 4𝑀³󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑇𝑦_𝑛󵄩󵄩󵄩󵄩.

(64) When lim_{𝑛 → ∞}𝛽_𝑛= 1, we conclude that

𝑛 → ∞lim𝐷_𝑔(𝑥_𝑛, 𝑇𝑥_𝑛) = 0. (65) In view ofLemma 2, we have that

𝑛 → ∞lim 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑇𝑥_𝑛󵄩󵄩󵄩󵄩 = 0. (66) Finally, we assume lim sup_{𝑛 → ∞}𝛾_𝑛(1 − 𝛾_𝑛) > 0 and lim_{𝑛 → ∞}𝛽_𝑛 = 1 instead. By (59) we have subsequences {𝑥_𝑛_𝑘}_𝑘∈N and {𝑦_𝑛_𝑘}_𝑘∈N of {𝑥_𝑛}_𝑛∈N and {𝑦_𝑛}_𝑛∈N, respectively, such that

𝑘 → ∞lim 󵄩󵄩󵄩󵄩󵄩𝑇𝑦^𝑛^𝑘− 𝑥_𝑛_𝑘󵄩󵄩󵄩󵄩󵄩 = 0. (67) Replacing 𝑀₃ with the finite number sup{‖∇𝑔(𝑥_𝑛_𝑘)‖,

‖∇𝑔(𝑇𝑥_𝑛_𝑘)‖, ‖∇𝑔(𝑇𝑦_𝑛_𝑘)‖ : 𝑘 ∈N} < +∞, and dealing with the subsequences{𝑥_𝑛_𝑘}_𝑘∈Nand{𝑦_𝑛_𝑘}_𝑘∈Nin (60) and (62). Passing to a further subsequence if necessary, we will arrive at the desired conclusion with (66) that lim_{𝑘 → ∞}‖𝑇𝑥_𝑛_𝑘 − 𝑥_𝑛_𝑘‖ = 0.

Hence, lim inf_{𝑛 → ∞}‖𝑇𝑥_𝑛 − 𝑥_𝑛‖ = 0. The other case can be argued similarly.

Theorem 17. Let𝐶be a nonempty, closed, and convex subset of a reflexive Banach space𝐸. Let 𝑔 : 𝐸 → Rbe strictly convex, Gˆateaux differentiable, locally bounded, and locally uniformly convex on 𝐸. Let 𝑇 : 𝐶 → 𝐶 be a Bregman nonspreading and Bregman skew quasi-nonexpansive mapping with𝐹(𝑇) ̸= 0. Let{𝛽_𝑛}_𝑛∈Nand{𝛾_𝑛}_𝑛∈Nbe sequences in[0, 1], and let {𝑥_𝑛}_𝑛∈N be a sequence with 𝑥₁ in𝐶 defined by (1).

Assume thatlim inf_{𝑛 → ∞}𝛾_𝑛(1 − 𝛾_𝑛) > 0andlim_{𝑛 → ∞}𝛽_𝑛 = 1.

Then{𝑥_𝑛}_𝑛∈Nconverges weakly to a fixed point of𝑇.

Proof. It follows fromTheorem 16that {𝑥_𝑛}_𝑛∈𝑁is bounded and lim_{𝑛 → ∞}‖𝑇𝑥_𝑛− 𝑥_𝑛‖ = 0. Since𝐸is reflexive, then there exists a subsequence{𝑥_𝑛_𝑖}_𝑖∈Nof{𝑥_𝑛}_𝑛∈Nsuch that𝑥_𝑛_𝑖 ⇀ 𝑝 ∈ 𝐶 as𝑖 → ∞. ByProposition 12, 𝑝 ∈ 𝐹(𝑇). We claim that 𝑥_𝑛 ⇀ 𝑝as𝑛 → ∞. If not, then there exists a subsequence

{𝑥_𝑛_𝑗}_𝑗∈N of{𝑥_𝑛}_𝑛∈𝑁 such that{𝑥_𝑛_𝑗}_𝑗∈N converges weakly to some𝑞in𝐶with𝑝 ̸= 𝑞. In view ofProposition 12again, we conclude that 𝑞 ∈ 𝐹(𝑇). ByLemma 15, lim_{𝑛 → ∞}𝐷_𝑔(𝑥_𝑛, 𝑧) exists for all𝑧in𝐹(𝑇). Thus we obtain by the Bregman-Opial- like property that

𝑛 → ∞lim𝐷_𝑔(𝑥_𝑛, 𝑝)

= lim

𝑖 → ∞𝐷_𝑔(𝑥_𝑛_𝑖, 𝑝) < lim

𝑖 → ∞𝐷_𝑔(𝑥_𝑛_𝑖, 𝑞)

=_{𝑛 → ∞}lim𝐷_𝑔(𝑥_𝑛, 𝑞) = lim

𝑗 → ∞𝐷_𝑔(𝑥_𝑛_𝑗, 𝑞)

< lim

𝑗 → ∞𝐷_𝑔(𝑥_𝑛_𝑗, 𝑝) = lim

𝑛 → ∞𝐷_𝑔(𝑥_𝑛, 𝑝) .

(68)

This is a contradiction. Thus we have𝑝 = 𝑞, and the desired assertion follows.

Theorem 18. Let𝐶be a nonempty, compact, and convex subset of a reflexive Banach space𝐸. Let 𝑔 : 𝐸 → Rbe strictly convex, Gˆateaux differentiable, locally bounded, and uniformly convex on bounded sets. Let 𝑇 : 𝐶 → 𝐶 be a Bregman nonspreading and Bregman skew quasi-nonexpansive map- ping. Let{𝛽_𝑛}_𝑛∈N and{𝛾_𝑛}_𝑛∈N be sequences in[0, 1]. Assume that eitherlim sup_{𝑛 → ∞}𝛾_𝑛(1 − 𝛾_𝑛) > 0andlim_{𝑛 → ∞}𝛽_𝑛 = 1 orlim inf_{𝑛 → ∞}𝛾_𝑛(1 − 𝛾_𝑛) > 0andlim sup_{𝑛 → ∞}𝛽_𝑛 = 1. Let {𝑥_𝑛}_𝑛∈Nbe a sequence with𝑥₁in𝐶defined by(1). Then{𝑥_𝑛}_𝑛∈N converges strongly to a fixed point𝑧of𝑇.

Proof. ByCorollary 14, we see that the fixed point set𝐹(𝑇)of 𝑇is nonempty. In view ofTheorem 16, we obtain that{𝑥_𝑛}_𝑛∈N is bounded and lim inf_{𝑛 → ∞}‖𝑇𝑥_𝑛− 𝑥_𝑛‖ = 0. By the compact- ness of𝐶, there exists a subsequence{𝑥_𝑛_𝑘}_𝑘∈Nof{𝑥_𝑛}_𝑛∈Nsuch that{𝑥_𝑛_𝑘}_𝑘∈Nconverges strongly to some𝑧in𝐶. In view of Lemma 2we deduce that lim_{𝑘 → ∞}𝐷_𝑔(𝑥_𝑛_𝑘, 𝑧) = 0. We can even assume that lim_{𝑘 → ∞}‖𝑇𝑥_𝑛_𝑘− 𝑥_𝑛_𝑘‖ = 0, and in particular, {𝑇𝑥_𝑛_𝑘}_𝑘∈Nis bounded. Since∇𝑔is uniformly norm-to-norm continuous on bounded subsets of𝐸(see, e.g., [14]),

𝑘 → ∞lim 󵄩󵄩󵄩󵄩󵄩∇𝑔 (𝑇𝑥^𝑛^𝑘) − ∇𝑔 (𝑥_𝑛_𝑘)󵄩󵄩󵄩󵄩󵄩 = 0. (69) Let𝑀₄ = sup{‖𝑇𝑧‖, ‖𝑇𝑥_𝑛_𝑘‖, ‖∇𝑔(𝑧)‖, ‖∇𝑔(𝑇𝑧)‖ : 𝑘 ∈ N} <

+∞. In view ofLemma 11, we obtain 𝐷_𝑔(𝑥_𝑛_𝑘, 𝑇𝑧)

≤ 𝐷_𝑔(𝑥_𝑛_𝑘, 𝑧) + 𝐷_𝑔(𝑇𝑥_𝑛_𝑘, 𝑥_𝑛_𝑘) + ⟨𝑥_𝑛_𝑘− 𝑇𝑥_𝑛_𝑘, ∇𝑔 (𝑧) − ∇𝑔 (𝑇𝑧)⟩

+ ⟨𝑇𝑥_𝑛_𝑘− 𝑇𝑧, ∇𝑔 (𝑥_𝑛_𝑘) − ∇𝑔 (𝑇𝑥_𝑛_𝑘)⟩

≤ 𝐷_𝑔(𝑥_𝑛_𝑘, 𝑧) + 𝐷_𝑔(𝑇𝑥_𝑛_𝑘, 𝑥_𝑛_𝑘)

+2𝑀₄[󵄩󵄩󵄩󵄩󵄩𝑥^𝑛^𝑘−𝑇𝑥_𝑛_𝑘󵄩󵄩󵄩󵄩󵄩+󵄩󵄩󵄩󵄩󵄩∇𝑔 (𝑥^𝑛^𝑘)−∇𝑔 (𝑇𝑥_𝑛_𝑘)󵄩󵄩󵄩󵄩󵄩]

(70)

for all𝑘inN.

It follows that lim_{𝑘 → ∞}‖𝑥_𝑛_𝑘 − 𝑇𝑧‖ = 0. Thus we have 𝑇𝑧 = 𝑧. In view of Lemmas 15 and 2, we conclude that lim_{𝑛 → ∞}‖𝑥_𝑛− 𝑧‖ = 0. Therefore,𝑧is the strong limit of the sequence{𝑥_𝑛}_𝑛∈N.