Fixed Point and Weak Convergence Theorems for (𝛼,𝛽)-Hybrid Mappings in Banach Spaces

Abstract

We introduce the class of (𝛼,𝛽)-hybrid mappings relative to a Bregman distance 𝐷_𝑓 in a Banach space, and then we study the fixed point and weak convergence problem for such mappings.

1. Introduction

Let 𝐶 be a nonempty subset of a Hilbert space 𝐻. A mapping 𝑇∶𝐶→𝐻 is said to be (1.1) nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖,forall𝑥,𝑦∈𝐶;(1.2) nonspreading if 2‖𝑇𝑥−𝑇𝑦‖²≤‖𝑇𝑥−𝑦‖²+‖𝑇𝑦−𝑥‖²,forall𝑥,𝑦∈𝐶, cf. [1, 2];(1.3) hybrid if 3‖𝑇𝑥−𝑇𝑦‖²≤‖𝑥−𝑦‖²+‖𝑇𝑥−𝑦‖²+‖𝑇𝑦−𝑥‖²,forall𝑥,𝑦∈𝐶, cf. [3].

Recently, Kocourek et al. [4] introduced a new class of nonlinear mappings in a Hilbert space containing the classes of nonexpansive mappings, nonspreading mappings, and hybrid mappings and established some fixed point and ergodic theorems for mappings in this new class. For 𝛼,𝛽∈ℝ, they call a mapping 𝑇∶𝐶→𝐻(1.4)(𝛼,𝛽)-hybrid if 𝛼𝑇𝑥−𝑇𝑦²+(1−𝛼)𝑥−𝑇𝑦²≤𝛽𝑇𝑥−𝑦²+1−𝛽𝑥−𝑦² (1.1)

for all 𝑥,𝑦∈𝐶.

Obviously, 𝑇 is nonexpansive if and only if it is (1,0)-hybrid; 𝑇 is nonspreading if and only if it is (2,1)-hybrid; 𝑇 is hybrid if and only if it is (3/2,1/2)-hybrid.

Motivated by the above works, we extend the concept of (𝛼,𝛽)-hybrid from Hilbert spaces to Banach spaces. For a nonempty subset 𝐶 of a Banach space 𝑋, a Gâteaux differentiable convex function 𝑓∶𝑋→(−∞,∞] and 𝛼,𝛽∈ℝ, a mapping 𝑇∶𝐶→𝑋 is said to be(1.5)(𝛼,𝛽)-hybrid relative to 𝐷_𝑓 if there are 𝛼,𝛽∈ℝ such that 𝛼𝐷_𝑓𝑇𝑥,𝑇𝑦+(1−𝛼)𝐷_𝑓𝑥,𝑇𝑦≤𝛽𝐷_𝑓𝑇𝑥,𝑦+1−𝛽𝐷_𝑓𝑥,𝑦 (1.2)

for all 𝑥,𝑦∈𝐶, where 𝐷_𝑓 is the Bregman distance associated with 𝑓. Sections 3 and 4 are devoted to investigation of the fixed point and weak convergence problem for such type of mappings, respectively. Our fixed point theorem extends that of [4].

2. Preliminaries

In what follows, 𝑋 will be a real Banach space with topological dual 𝑋^∗, and 𝑓∶𝑋→(−∞,∞] will be a convex function. 𝒟 denotes the domain of 𝑓, that is, 𝒟=𝑥∈𝑋∶𝑓(𝑥)<∞, (2.1) and 𝒟^∘ denotes the algebraic interior of 𝒟, that is, the subset of 𝒟 consisting of all those points 𝑥∈𝒟 such that, for any 𝑦∈𝑋⧵{𝑥}, there is 𝑧 in the open segment (𝑥,𝑦) with [𝑥,𝑧]⊆𝒟. The topological interior of 𝒟, denoted by Int(𝒟), is contained in 𝒟^∘. 𝑓 is said to be proper provided that 𝒟≠∅. 𝑓 is called lower semicontinuous (l.s.c.) at 𝑥∈𝑋 if 𝑓(𝑥)≤liminf_𝑦→𝑥𝑓(𝑦).  𝑓 is strictly convex if 𝑓𝛼𝑥+(1−𝛼)𝑦<𝛼𝑓(𝑥)+(1−𝛼)𝑓𝑦 (2.2) for all 𝑥,𝑦∈𝑋 and 𝛼∈(0,1).

The function 𝑓∶𝑋→(−∞,∞] is said to be Gâteaux differentiable at 𝑥∈𝑋 if there is 𝑓′(𝑥)∈𝑋^∗ such that lim_𝑡→0𝑓𝑥+𝑡𝑦−𝑓(𝑥)𝑡=𝑦,𝑓^′(𝑥) (2.3) for all 𝑦∈𝑋.

The Bregman distance 𝐷_𝑓 associated with a proper convex function 𝑓 is the function 𝐷_𝑓∶𝒟×𝒟→[0,∞] defined by𝐷_𝑓𝑦,𝑥=𝑓𝑦−𝑓(𝑥)+𝑓^∘𝑥,𝑥−𝑦, (2.4) where 𝑓^∘(𝑥,𝑥−𝑦)=lim_𝑡→0+(𝑓(𝑥+𝑡(𝑥−𝑦))−𝑓(𝑥))/𝑡. 𝐷_𝑓(𝑦,𝑥) is finite valued if and only if 𝑥∈𝒟^∘, compare Proposition 1.1.2 (iv) of [5]. When 𝑓 is Gâteaux differentiable on 𝒟, (2.4) becomes 𝐷_𝑓𝑦,𝑥=𝑓𝑦−𝑓(𝑥)−𝑦−𝑥,𝑓^′(𝑥), (2.5) and then the modulus of total convexity is the function 𝜈_𝑓∶𝒟^∘×[0,∞)→[0,∞] defined by 𝜈_𝑓(𝑥,𝑡)=inf𝐷_𝑓𝑦,𝑥∶𝑦∈𝒟,𝑦−𝑥=𝑡. (2.6) It is known that𝜈_𝑓(𝑥,𝑐𝑡)≥𝑐𝜈_𝑓(𝑥,𝑡) (2.7) for all 𝑡≥0 and 𝑐≥1, compare Proposition 1.2.2 (ii) of [5]. By definition it follows that𝐷_𝑓𝑦,𝑥≥𝜈_𝑓𝑥,𝑦−𝑥. (2.8)

The modulus of uniform convexity of 𝑓 is the function 𝛿_𝑓∶[0,∞)→[0,∞] defined by 𝛿_𝑓(𝑡)=inf𝑓(𝑥)+𝑓𝑦−2𝑓𝑥+𝑦2∶𝑥,𝑦∈𝒟,𝑥−𝑦≥𝑡. (2.9)

The function 𝑓 is called uniformly convex if 𝛿_𝑓(𝑡)>0 for all 𝑡>0. If 𝑓 is uniformly convex then for any 𝜀>0 there is 𝛿>0 such that𝑓𝑥+𝑦2≤𝑓(𝑥)2+𝑓𝑦2−𝛿 (2.10) for all 𝑥,𝑦∈𝒟 with ‖𝑥−𝑦‖≥𝜀.

Note that for 𝑦∈𝒟 and 𝑥∈𝒟^∘, we have 𝑓(𝑥)+𝑓𝑦−2𝑓𝑥+𝑦2𝑦(𝑥)−𝑓(𝑥)=𝑓−𝑓𝑥+𝑦−𝑥/2−𝑓𝑦(𝑥)1/2≤𝑓−𝑓−𝑓^∘𝑥,𝑦−𝑥≤𝐷_𝑓𝑦,𝑥, (2.11) where the first inequality follows from the fact that the function 𝑡→(𝑓(𝑥+𝑡𝑧)−𝑓(𝑥))/𝑡 is nondecreasing on (0,∞). Therefore𝜈_𝑓(𝑥,𝑡)≥𝛿_𝑓(𝑡), (2.12) whenever 𝑥∈𝒟^∘ and 𝑡≥0. For other properties of the Bregman distance 𝐷_𝑓, we refer readers to [5].

The normalized duality mapping 𝐽 from 𝑋 to 2^𝑋∗ is defined by 𝐽𝑥=𝑥^∗∈𝑋^∗∶𝑥,𝑥^∗=‖𝑥‖²=𝑥^∗2 (2.13) for all 𝑥∈𝑋.

When 𝑓(𝑥)=‖𝑥‖² in a smooth Banach space, it is known that 𝑓^′(𝑥)=2𝐽(𝑥) for 𝑥∈𝑋, compare Corollaries 1.2.7 and 1.4.5 of [6]. Hence we have 𝐷_𝑓𝑦,𝑥=𝑦²−‖𝑥‖²−𝑦−𝑥,𝑓^′(𝑥=𝑦₂)−‖𝑥‖²=𝑦−2𝑦−𝑥,𝐽𝑥₂+‖𝑥‖²−2𝑦,𝐽𝑥. (2.14) Moreover, as the normalized duality mapping 𝐽 in a Hilbert space 𝐻 is the identity operator, we have 𝐷_𝑓𝑦,𝑥=𝑦²+‖𝑥‖²−2𝑦,𝑥=𝑦−𝑥². (2.15) Thus, in case 𝑓(𝑥)=‖𝑥‖² in a Hilbert space, (1.5) coincides with (1.4). However, in general they are different as the following example shows.

Example 2.1. Let 𝑓(𝑥)=|𝑥| for 𝑥∈ℝ. 𝑓 is a continuous convex function with 𝐷_𝑓𝑦,𝑥=0,if𝑥>0,𝑦≥0or𝑥<0,𝑦≤0,−2𝑦,if𝑥≥0,𝑦≤0,2𝑦,if𝑥≤0,𝑦≥0. (2.16) Let 𝐶=[0,1] and define 𝑇∶𝐶→𝐶 by 𝑇(𝑥)=𝑥,if(],1𝑥∈0,12,if𝑥=0. (2.17) If 𝑇 were (𝛼,𝛽)-hybrid for some 𝛼,𝛽∈ℝ, then we would have 𝛼‖𝑇𝑥−𝑇0‖²+(1−𝛼)‖𝑥−𝑇0‖²≤𝛽‖𝑇𝑥−0‖²+1−𝛽‖𝑥−0‖²,∀𝑥∈(0,1]. (2.18) Since ()12.18⟺𝛼𝑥−2²+(11−𝛼)𝑥−2²≤𝛽|𝑥|²+1−𝛽|𝑥|²(]⟺1,∀𝑥∈0,1𝑥−2²≤|𝑥|²(]⟺1,∀𝑥∈0,14≤𝑥,∀𝑥∈(0,1], (2.19) we see that 𝑇 is not (𝛼,𝛽)-hybrid for any 𝛼,𝛽∈ℝ. But some simple computations show that 𝑇 is (1,0)-hybrid relative to 𝐷_𝑓.

A function 𝑔∶𝑋→(−∞,∞] is said to be subdifferentiable at a point 𝑥∈𝑋 if there exists a linear functional 𝑥^∗∈𝑋^∗ such that 𝑔𝑦−𝑔(𝑥)≥𝑦−𝑥,𝑥^∗,∀𝑦∈𝑋. (2.20) We call such 𝑥^∗ the subgradient of 𝑔 at 𝑥. The set of all subgradients of 𝑔 at 𝑥 is denoted by 𝜕𝑔(𝑥), and the mapping 𝜕𝑔∶𝑋→2^𝑋∗ is called the subdifferential of 𝑔. For a l.s.c. convex function 𝑓,𝜕𝑓 is bounded on bounded subsets of Int(𝒟) if and only if 𝑓 is bounded on bounded subsets there, compare Proposition 1.1.11 of [5]. A proper convex l.s.c. function 𝑓 is Gâteaux differentiable at 𝑥∈Int(𝒟) if and only if it has a unique subgradient at 𝑥; in such case 𝜕𝑓(𝑥)=𝑓^′(𝑥), compare Corollary 1.2.7 of [6].

The following lemma will be quoted in the sequel.

Lemma 2.2 (see Proposition 1.1.9 of [5]). If a proper convex function 𝑓∶𝑋→(−∞,∞] is Gâteaux differentiable on Int(𝒟) in a Banach space 𝑋, then the following statements are equivalent.(a)The function 𝑓 is strictly convex on Int(𝒟).(b)For any two distinct points 𝑥,𝑦∈Int(𝒟), one has 𝐷_𝑓(𝑦,𝑥)>0.(c)For any two distinct points 𝑥,𝑦∈Int(𝒟), one has 𝑥−𝑦,𝑓^′(𝑥)−𝑓^′𝑦>0. (2.21)

Throughout this paper, 𝐹(𝑇) will denote the set of all fixed points of a mapping 𝑇.

3. Fixed Point Theorems

In this section, we apply Lemma 2.2 to study the fixed point problem for mappings satisfying (1.5).

Theorem 3.1. Let 𝑋 be a reflexive Banach space and let 𝑓∶𝑋→(−∞,∞] be a l.s.c. strictly convex function so that it is Gâteaux differentiable on Int(𝒟) and is bounded on bounded subsets of Int(𝒟). Suppose 𝐶⊆Int(𝒟) is a nonempty closed convex subset of 𝑋 and 𝑇∶𝐶→𝐶 is (𝛼,𝛽)-hybrid relative to 𝐷_𝑓. Then the following two statements are equivalent.(a)There is a point 𝑥∈𝐶 such that {𝑇^𝑛𝑥}_𝑛∈ℕ is bounded.(b)𝐹(𝑇)≠∅.

Proof. If 𝐹(𝑇)≠∅, then {𝑇^𝑛𝑣}_𝑛∈ℕ={𝑣} is bounded for any 𝑣∈𝐹(𝑇). Now assume that {𝑇^𝑛𝑥}_𝑛∈ℕ is bounded for some 𝑥∈𝐶, and for any 𝑛∈ℕ, let 𝑆_𝑛(𝑥)=(1/𝑛)∑^𝑛−1_𝑘=0𝑇^𝑘𝑥. Then {𝑆_𝑛𝑥}_𝑛∈ℕ is bounded, and so, in view of 𝑋 being reflexive, it has a subsequence {𝑆_𝑛𝑖𝑥}_𝑖∈ℕ so that 𝑆_𝑛𝑖𝑥 converges weakly to some 𝑣∈𝐶 as 𝑛_𝑖→∞. Since 𝑇 is (𝛼,𝛽)-hybrid relative to 𝐷_𝑓, we have, for any 𝑦∈𝐶 and 𝑘∈ℕ∪{0}, 0≤𝛽𝐷_𝑓𝑇^𝑘+1𝑥,𝑦+1−𝛽𝐷_𝑓𝑇^𝑘𝑥,𝑦−𝛼𝐷_𝑓𝑇^𝑘+1𝑥,𝑇𝑦+(1−𝛼)𝐷_𝑓𝑇^𝑘𝑥,𝑇𝑦. (3.1) Rewrite 𝛼𝐷_𝑓(𝑇^𝑘+1𝑥,𝑇𝑦)+(1−𝛼)𝐷_𝑓(𝑇^𝑘𝑥,𝑇𝑦) as 𝛼𝐷_𝑓𝑇^𝑘+1𝑥,𝑇𝑦+(1−𝛼)𝐷_𝑓𝑇^𝑘𝑥,𝑇𝑦=𝛼𝑓𝑇^𝑘+1𝑥−𝑓𝑇𝑦−𝑇^𝑘+1𝑥−𝑇𝑦,𝑓^′+(𝑓𝑇𝑇𝑦1−𝛼)^𝑘𝑥−𝑓𝑇𝑦−𝑇^𝑘𝑇𝑥−𝑇𝑦,𝑓′𝑇𝑦=𝛼𝑓^𝑘+1𝑥+(1−𝛼)𝑓𝑇^𝑘𝑥−𝛼𝑇−𝑓𝑇𝑦_𝑘+1𝑥−𝑇𝑦+(1−𝛼)𝑇^𝑘𝑥−𝑇𝑦,𝑓^′𝑇𝑇𝑦=𝛼𝑓^𝑘+1𝑥+(1−𝛼)𝑓𝑇^𝑘𝑥−−𝑓𝑇𝑦𝛼𝑇^𝑘+1𝑥+(1−𝛼)𝑇^𝑘𝑥−𝑇𝑦,𝑓^′𝑇𝑦. (3.2) Similarly, we also have 𝛽𝐷_𝑓𝑇^𝑘+1𝑥,𝑦+1−𝛽𝐷_𝑓𝑇^𝑘𝑇𝑥,𝑦=𝛽𝑓^𝑘+1𝑥+1−𝛽𝑓𝑇^𝑘𝑥−𝑓𝑦−𝛽𝑇^𝑘+1𝑥+1−𝛽𝑇^𝑘𝑥−𝑦,𝑓^′𝑦. (3.3) Consequently, we obtain from (3.1) that 0≤𝛽−𝛼𝑓𝑇^𝑘+1𝑥−𝑓𝑇^𝑘𝑥+𝑓𝑦+𝑇𝑇𝑦−𝑓_𝑘𝑥+𝛼𝑇^𝑘+1𝑥−𝑇^𝑘𝑥−𝑇𝑦,𝑓^′𝑇𝑦−𝑇^𝑘𝑥+𝛽𝑇^𝑘+1𝑥−𝑇^𝑘𝑥−𝑦,𝑓^′𝑦. (3.4) Summing up these inequalities with respect to 𝑘=0,1,…,𝑛−1, we get 0≤𝛽−𝛼𝑓𝑇^𝑛𝑥(𝑥𝑓𝑦+−𝑓)+𝑛𝑇𝑦−𝑓𝑛𝑆_𝑛𝑥+𝛼𝑇^𝑛𝑥−𝑥−𝑛𝑇𝑦,𝑓^′𝑇𝑦−𝑛𝑆_𝑛𝑥+𝛽𝑇^𝑛𝑥−𝑥−𝑛𝑦,𝑓^′𝑦. (3.5) Dividing the above inequality by 𝑛, we have 0≤𝛽−𝛼𝑛𝑓𝑇^𝑛𝑥−𝑓(𝑥)+𝑓𝑇𝑦−𝑓𝑦+𝑆_𝑛𝛼𝑥+𝑛𝑇^𝑛𝑥−𝑥−𝑇𝑦,𝑓^′−𝑆𝑇𝑦_𝑛𝛽𝑥+𝑛𝑇^𝑛𝑥−𝑥−𝑦,𝑓^′𝑦. (3.6) Replacing 𝑛 by 𝑛_𝑖 and letting 𝑛_𝑖→∞, we obtain from the fact that {𝑓(𝑇^𝑛𝑥)}_𝑛∈ℕ is bounded that 0≤𝑓𝑇𝑦−𝑓𝑦+𝑣−𝑇𝑦,𝑓^′𝑇𝑦−𝑣−𝑦,𝑓^′𝑦. (3.7) Putting 𝑦=𝑣 in (3.7), we get 0≤𝑓(𝑇𝑣)−𝑓(𝑣)+𝑣−𝑇𝑣,𝑓^′(𝑇𝑣), (3.8) that is, 0≤−𝐷_𝑓(𝑣,𝑇𝑣), (3.9) from which follows that 𝐷_𝑓(𝑣,𝑇𝑣)=0. Therefore 𝑇𝑣=𝑣 by Lemma 2.2.

Since the function 𝑓(𝑥)=‖𝑥‖² in a Hilbert space 𝐻 satisfies all the requirements of Theorem 3.1, the corollary below follows immediately.

Corollary 3.2 (see [4]). Let 𝐶 be a nonempty closed convex subset of Hilbert space 𝐻 and Suppose 𝑇∶𝐶→𝐶 is (𝛼,𝛽)-hybrid. Then the following two statements are equivalent.(a)There exists 𝑥∈𝐶 such that {𝑇^𝑛(𝑥)}_𝑛∈ℕ is bounded.(b)𝑇 has a fixed point.

We now show that the fixed point set 𝐹(𝑇) is closed and convex under the assumptions of Theorem 3.1.

A mapping 𝑇∶𝐶→𝑋 is said to be quasi-nonexpansive with respect to 𝐷_𝑓 if 𝐹(𝑇)≠∅ and 𝐷_𝑓(𝑣,𝑇𝑥)≤𝐷_𝑓(𝑣,𝑥) for all 𝑥∈𝐶 and all 𝑣∈𝐹(𝑇).

The following lemma is shown in Huang et al. [7].

Lemma 3.3. Let 𝑓∶𝑋→(−∞,∞] be a proper strictly convex function on a Banach space 𝑋 so that it is Gâteaux differentiable on Int(𝒟), and let 𝐶⊆Int(𝒟) be a nonempty closed convex subset of 𝑋. If 𝑇∶𝐶→𝐶 is quasi-nonexpansive with respect to 𝐷_𝑓, then 𝐹(𝑇) is a closed convex subset.

Proposition 3.4. Let 𝑓∶𝑋→(−∞,∞] be a proper strictly convex function on a reflexive Banach space 𝑋 so that it is Gâteaux differentiable on Int(𝒟) and is bounded on bounded subsets of Int(𝒟), and let 𝐶⊆Int(𝒟) be a nonempty closed convex subset of 𝑋. Suppose 𝑇∶𝐶→𝐶 is (𝛼,𝛽)-hybrid relative to 𝐷_𝑓 and has a point 𝑥₀∈𝐶 such that {𝑇^𝑛(𝑥₀)}_𝑛∈ℕ is bounded. Then 𝑇 is quasi-nonexpansive with respect to 𝐷_𝑓, and therefore 𝐹(𝑇) is a nonempty closed convex subset of 𝐶.

Proof. In view of Theorem 3.1, 𝐹(𝑇)≠∅. Now, for any 𝑣∈𝐹(𝑇) and any 𝑦∈𝐶, as 𝑇 is (𝛼,𝛽)-hybrid relative to 𝐷_𝑓, we have 𝐷_𝑓𝑣,𝑇𝑦=𝛼𝐷_𝑓𝑣,𝑇𝑦+(1−𝛼)𝐷_𝑓𝑣,𝑇𝑦=𝛼𝐷_𝑓𝑇𝑣,𝑇𝑦+(1−𝛼)𝐷_𝑓𝑣,𝑇𝑦≤𝛽𝐷_𝑓𝑇𝑣,𝑦+1−𝛽𝐷_𝑓𝑣,𝑦=𝛽𝐷_𝑓𝑣,𝑦+1−𝛽𝐷_𝑓𝑣,𝑦=𝐷_𝑓𝑣,𝑦, (3.10) so 𝑇 is quasi-nonexpansive with respect to 𝐷_𝑓, and hence 𝐹(𝑇) is a nonempty closed convex subset of 𝐶 by Lemma 3.3.

For the remainder of this section, we establish a common fixed point theorem for a commutative family of (𝛼,𝛽)-hybrid mappings relative to 𝐷_𝑓.

Lemma 3.5. Let 𝑋 be a reflexive Banach space and let 𝑓∶𝑋→(−∞,∞] be a l.s.c. strictly convex function so that it is Gâteaux differentiable on Int(𝒟) and is bounded on bounded subsets of Int(𝒟). Suppose 𝐶⊆Int(𝒟) is a nonempty bounded closed convex subset of 𝑋 and {𝑇₁,𝑇₂,…,𝑇_𝑁} is a commutative finite family of (𝛼,𝛽)-hybrid mappings relative to 𝐷_𝑓 from 𝐶 into itself. Then {𝑇₁,𝑇₂,…,𝑇_𝑁} has a common fixed point.

Proof. We prove this lemma by induction with respect to 𝑁. To begin with, we deal with the case that 𝑁=2. By Proposition 3.4, we see that 𝐹(𝑇₁) and 𝐹(𝑇₂) are nonempty bounded closed convex subsets of 𝑋. Moreover, 𝐹(𝑇₁) is 𝑇₂-invariant. Indeed, for any 𝑣∈𝐹(𝑇₁), it follows from 𝑇₁𝑇₂=𝑇₂𝑇₁ that 𝑇₁𝑇₂𝑣=𝑇₂𝑇₁𝑣=𝑇₂𝑣, which shows that 𝑇₂𝑣∈𝐹(𝑇₁). Consequently, the restriction of 𝑇₂ to 𝐹(𝑇₁) is (𝛼,𝛽)-hybrid relative to 𝐷_𝑓, and so by Theorem 3.1, 𝑇₂ has a fixed point 𝑢∈𝐹(𝑇₁), that is, 𝑢∈𝐹(𝑇₁)∩𝐹(𝑇₂).
By induction hypothesis, assume that for some 𝑛≥2, 𝐸=∩^𝑛_𝑘=1𝐹(𝑇_𝑘) is nonempty. Then 𝐸 is a nonempty closed convex subset of 𝑋, and the restriction of 𝑇_𝑛+1 to 𝐸 is a (𝛼,𝛽)-hybrid mapping relative to 𝐷_𝑓 from 𝐸 into itself. By Theorem 3.1, 𝑇_𝑛+1 has a fixed point in 𝑋. This shows that 𝐸∩𝐹(𝑇_𝑛+1)≠∅, that is, ∩^𝑛+1_𝑘=1𝐹(𝑇_𝑘)≠∅, completing the proof.

Here we would like to remark that in the above lemma, the assumption 𝑓 is bounded on bounded subsets of Int(𝒟) can be dropped by checking the proof of Theorem 3.1 and noting that we have assumed 𝐶 is bounded.

Theorem 3.6. Let 𝑋 be a reflexive Banach space and let 𝑓∶𝑋→(−∞,∞] be a l.s.c. strictly convex function so that it is Gâteaux differentiable on Int(𝒟). Suppose 𝐶⊆Int(𝒟) is a nonempty bounded closed convex subset of 𝑋 and {𝑇_𝑖}_𝑖∈𝐼 is a commutative family of (𝛼,𝛽)-hybrid mappings relative to 𝐷_𝑓 from 𝐶 into itself. Then {𝑇_𝑖}_𝑖∈𝐼 has a common fixed point.

Proof. Since 𝐶 is a nonempty bounded closed convex subset of the reflexive Banach space 𝑋, it is weakly compact. By Proposition 3.4, each 𝐹(𝑇_𝑖) is a nonempty weakly compact subset of 𝐶. Therefore, the conclusion follows once we note that {𝐹(𝑇_𝑖)}_𝑖∈𝐼 has the finite intersection property by Lemma 3.5.

4. Weak Convergence Theorems

In this section, we discuss the demiclosedness and the weak convergence problem of (𝛼,𝛽)-hybrid mappings relative to 𝐷_𝑓. We denote the weak convergence and strong convergence of a sequence {𝑥_𝑛} to 𝑣 in a Banach space 𝑋 by 𝑥_𝑛⇀𝑣 and 𝑥_𝑛→𝑣, respectively. For a nonempty closed convex subset 𝐶 of a Banach space 𝑋, a mapping 𝑇∶𝐶→𝑋 is demiclosed if for any sequence {𝑥_𝑛} in 𝐶 with 𝑥_𝑛⇀𝑣 and 𝑥_𝑛−𝑇𝑥_𝑛→0, one has 𝑇𝑣=𝑣.

The following Opial-like inequality for the Bregman distance is proved in [7]. For the Opial’s inequality we refer readers to Lemma 1 of [8].

Lemma 4.1. Suppose 𝑓∶𝑋→(−∞,∞] is a proper strictly convex function so that it is Gâteaux differentiable on Int(𝒟) in a Banach space 𝑋 and {𝑥_𝑛}_𝑛∈ℕ is a sequence in 𝒟 such that 𝑥_𝑛⇀𝑣 for some 𝑣∈Int(𝒟). Then liminf_𝑛→∞𝐷_𝑓𝑥_𝑛,𝑣<liminf_𝑛→∞𝐷_𝑓𝑥_𝑛,𝑦,∀𝑦∈Int(𝒟)with𝑦≠𝑣. (4.1)

Proposition 4.2. Suppose 𝑓∶𝑋→(−∞,∞] is a strictly convex function so that it is Gâteaux differentiable on Int(𝒟) and is bounded on bounded subsets of Int(𝒟) in a Banach space 𝑋, and suppose 𝐶 is a closed convex subset of Int(𝒟). If 𝑇∶𝐶→𝑋 is (𝛼,𝛽)-hybrid relative to 𝐷_𝑓, then 𝑇 is demiclosed.

Proof. Let {𝑥_𝑛} be any sequence in 𝐶 with 𝑥_𝑛⇀𝑣 and 𝑥_𝑛−𝑇𝑥_𝑛→0. We have to show that 𝑇𝑣=𝑣. Since 𝑓 is bounded, by Proposition 1.1.11 of [5] there exists a constant 𝑀>0 such that maxsup𝑓^′𝑥_𝑛∶𝑛∈ℕ,𝑓^′(𝑇𝑣),𝑓^′(𝑣)≤𝑀. (4.2) Note that, for 𝛼∈ℝ, 𝛼𝐷_𝑓𝑇𝑥_𝑛,𝑇𝑣+(1−𝛼)𝐷_𝑓𝑥_𝑛,𝑇𝑣=𝛼𝑓𝑇𝑥_𝑛−𝑓(𝑇𝑣)−𝑇𝑥_𝑛−𝑇𝑣,𝑓^′(+(𝑓𝑥𝑇𝑣)1−𝛼)_𝑛−𝑓(𝑇𝑣)−𝑥_𝑛−𝑇𝑣,𝑓^′(𝑓𝑇𝑣)=𝛼𝑇𝑥_𝑛−𝑓(𝑇𝑣)+𝑓(𝑇𝑣)−𝑓𝑥_𝑛−𝑇𝑥_𝑛−𝑇𝑣−𝑥_𝑛+𝑇𝑣,𝑓^′(𝑥𝑇𝑣)+𝑓_𝑛−𝑓(𝑇𝑣)−𝑥_𝑛−𝑇𝑣,𝑓^′(𝑓𝑇𝑣)=𝛼𝑇𝑥_𝑛−𝑓𝑥_𝑛−𝑇𝑥_𝑛−𝑥_𝑛,𝑓^′(𝑇𝑣)+𝐷_𝑓𝑥_𝑛,𝑇𝑣. (4.3) Similarly, for 𝛽∈ℝ, we have 𝛽𝐷_𝑓𝑇𝑥_𝑛,𝑣+1−𝛽𝐷_𝑓𝑥_𝑛,𝑣=𝛽𝑓𝑇𝑥_𝑛−𝑓𝑥_𝑛−𝑇𝑥_𝑛−𝑥_𝑛,𝑓^′(𝑣)+𝐷_𝑓𝑥_𝑛,𝑣. (4.4) Thus we obtain from the (𝛼,𝛽)-hybrid of 𝑇 that 𝛼𝑓𝑇𝑥_𝑛−𝑓𝑥_𝑛−𝑇𝑥_𝑛−𝑥_𝑛,𝑓^′(𝑇𝑣)+𝐷_𝑓𝑥_𝑛𝑓,𝑇𝑣≤𝛽𝑇𝑥_𝑛−𝑓𝑥_𝑛−𝑇𝑥_𝑛−𝑥_𝑛,𝑓^′(𝑣)+𝐷_𝑓𝑥_𝑛,𝑣, (4.5) which implies that 𝐷_𝑓𝑥_𝑛,𝑇𝑣≤𝐷_𝑓𝑥_𝑛,𝑣+𝛼−𝛽𝑓𝑥_𝑛−𝑓𝑇𝑥_𝑛+𝑇𝑥_𝑛−𝑥_𝑛,𝛼𝑓^′(𝑇𝑣)−𝛽𝑓^′(𝑣)≤𝐷_𝑓𝑥_𝑛,𝑣+𝛼−𝛽𝑥_𝑛−𝑇𝑥_𝑛,𝑓^′𝑥_𝑛+𝑇𝑥_𝑛−𝑥_𝑛,𝛼𝑓′(𝑇𝑣)−𝛽𝑓^′(𝑣)≤𝐷_𝑓𝑥_𝑛,𝑣+3𝑀|𝛼|+𝛽𝑥_𝑛−𝑇𝑥_𝑛. (4.6) Consequently, if 𝑇𝑣≠𝑣, then Lemma 4.1 implies that liminf_𝑛→∞𝐷_𝑓𝑥_𝑛,𝑣<liminf_𝑛→∞𝐷_𝑓𝑥_𝑛,𝑇𝑣=liminf_𝑛→∞𝐷_𝑓𝑥_𝑛,𝑣+3𝑀|𝛼|+𝛽𝑥_𝑛−𝑇𝑥_𝑛=liminf_𝑛→∞𝐷_𝑓𝑥_𝑛,𝑣, (4.7) a contradiction. This completes the proof.

A mapping 𝑇∶𝐶→𝐶 is said to be asymptotically regular if, for any 𝑥∈𝐶, the sequence {𝑇^𝑛+1𝑥−𝑇^𝑛𝑥} tends to zero as 𝑛→∞.

Theorem 4.3. Suppose the following conditions hold. (4.3.1)𝑓∶𝑋→(−∞,∞] is l.s.c. uniformly convex function so that it is Gâteaux differentiable on Int(𝒟) and is bounded on bounded subsets of Int(𝒟) in a reflexive Banach space 𝑋.(4.3.2)𝐶⊆Int(𝒟) is a closed convex subset of 𝑋.(4.3.3)𝑇∶𝐶→𝐶 is (𝛼,𝛽)-hybrid relative to 𝐷_𝑓 and is asymptotically regular with a bounded sequence {𝑇^𝑛𝑥₀}_𝑛∈ℕ for some 𝑥₀∈𝐶.(4.3.4) The mapping 𝑥→𝑓′(𝑥) for 𝑥∈𝑋 is weak-to-weak* continuous.Then for any 𝑥∈𝐶, {𝑇^𝑛𝑥}_𝑛∈ℕ is weakly convergent to an element 𝑣∈𝐹(𝑇).

Proof. Let 𝑣∈𝐹(𝑇) and 𝑥∈𝐶. If {𝑇^𝑛𝑥}_𝑛∈ℕ is not bounded, then there is a subsequence {𝑇^𝑛𝑖𝑥}_𝑖∈ℕ such that ‖𝑣−𝑇^𝑛𝑖𝑥‖≥1 for all 𝑖∈ℕ and ‖𝑣−𝑇^𝑛𝑖𝑥‖→∞ as 𝑖→∞. From (4.3.3), for any 𝑛∈ℕ∪{0} we have 𝐷_𝑓𝑣,𝑇^𝑛+1𝑥=𝛼𝐷_𝑓𝑇𝑣,𝑇^𝑛+1𝑥+(1−𝛼)𝐷_𝑓𝑣,𝑇^𝑛+1𝑥≤𝛽𝐷_𝑓𝑇𝑣,𝑇^𝑛𝑥+1−𝛽𝐷_𝑓𝑣,𝑇^𝑛𝑥=𝐷_𝑓𝑣,𝑇^𝑛𝑥≤𝐷_𝑓(𝑣,𝑥), (4.8) which in conjunction with (2.7), (2.8), and (2.12) implies that 𝐷_𝑓(𝑣,𝑥)≥𝐷_𝑓𝑣,𝑇^𝑛𝑖𝑥≥𝜈_𝑓𝑇^𝑛𝑖𝑥,𝑣−𝑇^𝑛𝑖𝑥≥𝑣−𝑇^𝑛𝑖𝑥𝜈_𝑓𝑇^𝑛𝑖≥𝑥,1𝑣−𝑇^𝑛𝑖𝑥𝛿_𝑓(1)⟶∞,as𝑖⟶∞, (4.9) a contradiction. Therefore, for any 𝑥∈𝑋, {𝑇^𝑛𝑥}_𝑛∈ℕ is bounded, and so it has a subsequence {𝑇^𝑛𝑗𝑥}_𝑗∈ℕ which is weakly convergent to 𝑤 for some 𝑤∈𝐶. As 𝑇^𝑛𝑗𝑥−𝑇^𝑛𝑗+1𝑥→0, it follows from the demiclosedness of 𝑇 that 𝑤∈𝐹(𝑇). It remains to show that 𝑇^𝑛𝑥⇀𝑤 as 𝑛→∞. Let {𝑇^𝑛𝑘𝑥}_𝑘∈ℕ be any subsequence of {𝑇^𝑛𝑥}_𝑛∈ℕ so that 𝑇^𝑛𝑘𝑥⇀𝑢 for some 𝑢∈𝐶. Then 𝑢∈𝐹(𝑇). Since both of {𝐷_𝑓(𝑤,𝑇^𝑛𝑥)}_𝑛∈ℕ and {𝐷_𝑓(𝑢,𝑇^𝑛𝑥)}_𝑛∈ℕ are decreasing, we have lim_𝑛→∞𝐷_𝑓𝑤,𝑇^𝑛𝑥−𝐷_𝑓𝑢,𝑇^𝑛𝑥=lim_𝑛→∞𝑓(𝑤)−𝑓(𝑢)−𝑤−𝑢,𝑓^′𝑇^𝑛𝑥=𝑎 (4.10) for some 𝑎∈ℝ. Particularly, from (4.3.4) we obtain 𝑎=lim_{𝑛𝑗→∞}𝑓(𝑤)−𝑓(𝑢)−𝑤−𝑢,𝑓^′𝑇^𝑛𝑗𝑥=𝑓(𝑤)−𝑓(𝑢)−𝑤−𝑢,𝑓^′(𝑤),𝑎=lim_{𝑛𝑘→∞}𝑓(𝑤)−𝑓(𝑢)−𝑤−𝑢,𝑓^′𝑇^𝑛𝑘𝑥=𝑓(𝑤)−𝑓(𝑢)−𝑤−𝑢,𝑓^′(𝑢). (4.11) Consequently, ⟨𝑤−𝑢,𝑓^′(𝑤)−𝑓^′(𝑢)⟩=0, and hence 𝑤=𝑢 by the strict convexity of 𝑓. This shows that 𝑇^𝑛𝑥⇀𝑤 for some 𝑤∈𝐹(𝑇).

5. Conclusion

In this paper, we have introduced the Bregman distance 𝐷_𝑓 and a new class of mappings, (𝛼,𝛽)-hybrid mappings relative to 𝐷_𝑓 in Banach spaces. We also have given and proved a necessary and sufficient condition for the existence of fixed points of the introduced mappings and some properties of the mappings. In fact, our result properly extends the Kocourek-Takahashi-Yao fixed point theorems for (𝛼,𝛽)-hybrid mappings in Hilbert spaces in 2010 [4]. Since the Kocourek-Takahashi-Yao fixed point theorems can be applied to study the nonexpansive mappings, the nonspreading mappings, and the hybrid mappings, our theorems in this paper are also good for these famous mappings in the field of fixed point theory.

According to [7], the fixed point theorems in this paper can be expected to discuss in a wider class of mappings, called point-dependent (𝛼,𝛽)-hybrid mappings relative to 𝐷_𝑓 in Banach spaces. In a point-dependent (𝛼,𝛽)-hybrid mapping, the 𝛼 and the 𝛽 are not constant again but two functions from a nonempty subset of a Banach space to real numbers. Therefore, inequality (1.4) for point-dependent (𝛼,𝛽)-hybrid mappings becomes 𝛼𝑦𝑇𝑥−𝑇𝑦²+1−𝛼𝑦𝑥−𝑇𝑦²≤𝛽𝑦𝑇𝑥−𝑦²+1−𝛽𝑦𝑥−𝑦² (5.1) for all 𝑥,𝑦∈𝐶.

In addition, Noor [9–11] provides algorithms to search the fixed points of nonexpansive mappings and then combines the result with general variational inequalities to study applied mathematical problems. We are motivated by that and expect to develop algorithms from the theorems of this paper to approach the fixed points of the introduced mappings. Through the combination of the fixed point theorems and the corresponding algorithms, the introduced mappings of this paper would be able to be applied to more fields of applied mathematics.