New Iterative Algorithm for Two Infinite Families of Multivalued Quasi-Nonexpansive Mappings in Uniformly Convex Banach Spaces

Abstract

We introduce a new iterative scheme for finding a common fixed point of two countable families of multivalued quasi-nonexpansive mappings and prove a weak convergence theorem under the suitable control conditions in a uniformly convex Banach space. We also give a new proof method to the iteration in the paper of Abbas et al. (2011).

1. Introduction

A Banach space 𝐸 is said to be strictly convex if ‖𝑥−𝑦‖/2<1 for all 𝑥,𝑦∈𝐸 with ‖𝑥‖=‖𝑦‖=1 and 𝑥≠𝑦. A Banach space 𝐸 is called uniformly convex if for each 𝜀>0 there is a 𝛿>0 such that, for 𝑥, 𝑦∈𝐸 with ‖𝑥‖,‖𝑦‖≤1 and ‖𝑥−𝑦‖≥𝜀, ‖𝑥+𝑦‖≤2(1−𝛿) holds. The modulus of convexity of 𝐸 is defined by 𝛿_𝐸(𝜀)1=inf1−2𝑥+𝑦∶‖𝑥‖,𝑦≤1,𝑥−𝑦≥𝜀, (1) for all 𝜀∈[0,2]. 𝐸 is uniformly convex if 𝛿_𝐸(0)=0, and 𝛿_𝐸(𝜀)>0 for all 0<𝜀<2. It is known that every uniformly convex Banach space is strictly convex and reflexive. Let 𝑆(𝐸)={𝑥∈𝐸∶‖𝑥‖=1}. Then the norm of 𝐸 is said to be Gâteaux differentiable if lim_𝑡→0𝑥+𝑡𝑦−‖𝑥‖𝑡 (2) exists for each 𝑥,𝑦∈𝑆(𝐸). In this case 𝐸 is called smooth. The norm of 𝐸 is said to be Fréchet differentiable if for each 𝑥∈𝑆(𝐸) the limit is attained uniformly for 𝑦∈𝑆(𝐸). The norm of 𝐸 is called uniformly Fréchet differentiable if the limit is attained uniformly for 𝑥, 𝑦∈𝑆(𝐸). It is well known that (uniform) Fréchet differentiability of the norm of 𝐸 implies (uniform) Gâteaux differentiability of the norm of 𝐸.

Let 𝜌_𝐸∶[0,∞)→[0,∞) be the modulus of smoothness of 𝐸 defined by 𝜌_𝐸(𝑡)1=sup2𝑥+𝑦+𝑥−𝑦−1∶𝑥∈𝑆(𝐸),𝑦≤𝑡. (3)

A Banach space 𝐸 is said to be uniformly smooth if 𝜌_𝐸(𝑡)/𝑡→0 as 𝑡→0. Suppose that 𝑞>1; then 𝐸 is said to be 𝑞-uniformly smooth if there exists 𝑐>0 such that 𝜌_𝐸(𝑡)<𝑐𝑡^𝑞. It is easy to see that if 𝐸 is 𝑞-uniformly smooth, then 𝑞≤2 and 𝐸 is uniformly smooth. It is well known that 𝐸 is uniformly smooth if and only if the norm of 𝐸 is uniformly Fréchet differentiable, and hence the norm of 𝐸 is Fréchet differentiable. For more details, we refer the reader to Agarwal et al. [1] and Chidume et al. [2].

In recent years, many authors have done a lot of valuable research in Banach space. They extend some iterative schemes from Hilbert spaces to Banach spaces and obtain some convergence theorems which are very interesting and meaningful; see [3–5]. Moreover, a variety of mappings are considered by articles [3–9], including nonexpansive mappings, nonexpansive semigroup, strict pseudocontractions mappings, and total quasi-𝜙-asymptotically nonexpansive mappings.

It is easy to know that the theory of multivalued nonexpansive mappings is harder than the corresponding theory of single-valued nonexpansive mappings. In 1969, Nadler [10] introduced multivalued contraction mapping and proved the corresponding convergence theorem. In 1973, Markin [11] first used the Hausdorff metric to study the fixed points for multivalued contractions and nonexpansive mappings. Later in 1997, Hu et al. [12] proved the convergence theorems for finding common fixed point of two multivalued nonexpansive mappings that satisfied certain contractive condition.

In 2005, Sastry and Babu [13] defined Mann and Ishikawa iterates for a multivalued mapping with a fixed point and proved that these iterates converge to a fixed point of the multivalued mapping under certain conditions. They have given an example that the fixed point generalized by limit of the sequence is different from the point of initial choice.

In 2011, Abbas et al. [14] introduced a new one-step iterative process to approximate common fixed points of two multivalued nonexpansive mappings in a real uniformly convex Banach space. Let 𝑆,𝑇∶𝐾→𝐶𝐵(𝐾) be two multivalued nonexpansive mappings. They introduced iteration as follows: 𝑥₀𝑥∈𝐾chosenarbitrarily,_𝑛+1=𝑎_𝑛𝑥_𝑛+𝑏_𝑛𝑦_𝑛+𝑐_𝑛𝑧_𝑛,𝑛∈𝑁, (4) where 𝑦_𝑛∈𝑇𝑥_𝑛 and 𝑧_𝑛∈𝑆𝑥_𝑛 such that ‖𝑦_𝑛−𝑝‖≤𝑑(𝑝,𝑆𝑥_𝑛) and ‖𝑧_𝑛−𝑝‖≤𝑑(𝑝,𝑇𝑥_𝑛) for 𝑝∈𝐹(𝑇)∩𝐹(𝑆), and 𝑎_𝑛, 𝑏_𝑛, 𝑐_𝑛∈(0,1) satisfying 𝑎_𝑛+𝑏_𝑛+𝑐_𝑛≤1. Then they obtained strong convergence theorems for the proposed process under some basic boundary conditions.

In 2012, Chang et al. [15], for solving the convex feasibility problems for an infinite family of quasi-𝜙-asymptotically nonexpansive mappings, used the modified block iterative method and obtained some strong convergence theorems under suitable conditions in Banach space.

In the same year, Bunyawat and Suantai [16] introduced an iterative method for finding a common fixed point of a countable family of multivalued quasi-nonexpansive mapping in a uniformly convex Banach space. They proved that the iterative sequence generated by their proposed method is an approximating fixed point sequence of the multivalued quasi-nonexpansive mapping under certain control conditions, and they got some strong convergence theorems of their proposed method.

Phuangphoo and Kumam [9] extended and improved the above results in 2012. They introduced a new iterative procedure which was constructed by the shrinking hybrid projection method for solving the common solution of fixed point problems for two total quasi-𝜙-asymptotically nonexpansive multivalued mappings. Under suitable conditions, the strong convergence theorems were established in a uniformly smooth and strictly convex real Banach space with Kadec-Klee property.

Different iterative processes have been used to approximate fixed points of multivalued mappings. Many authors have intensively studied the fixed point theorems and got some results. At the same time, they extended this result to many discipline branches, such as control theory, convex optimization, variational inequalities, differential inclusion, and economics (see [3–9, 17–22]).

Motivated by the above authors, we generalize and modify the hybrid block iterative algorithm (see [23]) for two countable families of multivalued quasi-nonexpansive mappings in a uniformly convex Banach space. First we prove that the sequence generated by our iterative method is weak convergence under the property conditions. Then we use compact space to prove the strong convergence in a uniformly convex Banach space. As expected, we got some weak and strong convergence theorems about the common fixed point of two countable families of multivalued quasi-nonexpansive mappings in a uniformly convex Banach space.

2. Preliminaries

Let 𝐸 be a real Banach space. A subset 𝐶 of 𝐸 is called proximinal if for each 𝑥∈𝐸 there exists an element 𝑘∈𝐶 such that 𝑑(𝑥,𝑘)=𝑑(𝑥,𝐶), (5) where 𝑑(𝑥,𝐶)=inf_𝑦∈𝐶‖𝑥−𝑦‖ is the distance from the point 𝑥 to the set 𝐶. It is not hard to known that weakly compact convex subsets of a Banach space and closed convex subsets of a uniformly convex Banach space are proximinal.

A map 𝑇:𝐸→𝐶𝐵(𝐸) is called hemicompact if, for any sequence {𝑥_𝑛} in 𝐸 such that 𝑑(𝑥_𝑛,𝑇𝑥_𝑛)→0 as 𝑛→∞, there exists a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that 𝑥_𝑛𝑘→𝑝∈𝐸. We note that if 𝐸 is compact, then every multivalued mapping 𝑇 is hemicompact.

Let 𝐸 be a real Banach space. We denote 𝐶𝐵(𝐸) as the class of all nonempty bounded and closed subsets of 𝐸 (see [10]), and denote 𝑃(𝐸) as the family of nonempty bounded proximinal subsets of 𝐸. The Hausdorff metric induced by the metric 𝑑 of 𝐸 is defined by 𝐻(𝐴,𝐵)=maxsup_𝑥∈𝐴𝑑(𝑥,𝐵),sup_𝑦∈𝐵𝑑𝑦,𝐴, (6) for all 𝐴, 𝐵∈𝐶𝐵(𝐸).

For a single-valued mapping 𝑇, a point 𝑝 is called a fixed point of 𝑇 if 𝑝=𝑇𝑝. For a multivalued mapping 𝑇, a point 𝑝 is called a fixed point of 𝑇 if 𝑝∈𝑇𝑝. The set of fixed points of 𝑇 is denoted by 𝐹(𝑇).

Let 𝐸 be a uniformly convex real Banach space, and let 𝐾 be a nonempty closed convex subset of 𝐸. A multivalued mapping 𝑇∶𝐾→𝐶𝐵(𝐾) is said to be(i)a contraction if there exists a constant 𝑘∈[0,1) such that 𝐻𝑇𝑥,𝑇𝑦≤𝑘𝑥−𝑦,∀𝑥,𝑦∈𝐾; (7)(ii)nonexpansive if 𝐻𝑇𝑥,𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐾; (8)(iii)quasi-nonexpansive if 𝐹(𝑇)≠∅ and 𝐻𝑇𝑥,𝑇𝑝≤𝑥−𝑝,∀𝑥∈𝐾,∀𝑝∈𝐹(𝑇). (9)

It is well known that every nonexpansive multivalued mapping 𝑇 with 𝐹(𝑇)≠∅ is multivalued quasi-nonexpansive. But there exist multivalued quasi-nonexpansive mappings that are not multivalued nonexpansive. It is clear that if 𝑇 is a quasi-nonexpansive multivalued mapping, then 𝐹(𝑇) is closed.

A Banach space 𝐸 is said to satisfy Opial’s condition if whenever {𝑥_𝑛} is a sequence in 𝐸 which converges weakly to 𝑥, then limsup_𝑛→∞𝑥_𝑛−𝑥<limsup_𝑛→∞𝑥_𝑛−𝑦,∀𝑦∈𝐸,𝑥≠𝑦. (10)

Lemma 1 (see Chang et al. [15]). Let E be a uniformly convex Banach space, 𝑟>0 a positive number, and 𝐵_𝑟(0) a closed ball of 𝐸. Then, for any given sequence {𝑥_𝑖}^∞_𝑖=1⊂𝐵_𝑟(0) and for any given sequence {𝜆_𝑖}^∞_𝑖=1 of positive number with ∑^∞_𝑛=1𝜆_𝑛=1, there exists a continuous, strictly increasing, and convex function 𝑔∶[0,2𝑟)→[0,∞) with 𝑔(0)=0 such that, for any positive integer 𝑖, 𝑗 with 𝑖≠𝑗, ^∞_𝑛=1𝜆_𝑛𝑥_𝑛²≤^∞_𝑛=1𝜆_𝑛𝑥_𝑛²−𝜆_𝑖𝜆_𝑗𝑔𝑥_𝑖−𝑥_𝑗. (11)

Lemma 2 (see Schu [24]). Suppose that 𝐸 is a uniformly convex Banach space and 0<𝑝≤𝑡_𝑛≤𝑞<1 for all positive integers 𝑛. Also suppose that {𝑥_𝑛} and {𝑦_𝑛} are two sequences of 𝐸 such that limsup_𝑛→∞‖𝑥_𝑛‖≤𝑟, limsup_𝑛→∞‖𝑦_𝑛‖≤𝑟, and lim_𝑛→∞‖𝑡_𝑛𝑥_𝑛+(1−𝑡_𝑛)𝑦_𝑛‖=𝑟 hold for some 𝑟≥0. Then lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0.

Lemma 3 (see Liu [25] and Xu [26]). Let {𝑎_𝑛} be a sequence of nonnegative real numbers satisfying the following property: 𝑎_𝑛+1≤1−𝑡_𝑛𝑎_𝑛+𝑏_𝑛+𝑡_𝑛𝑐_𝑛, (12) where {𝑡_𝑛}, {𝑏_𝑛}, and {𝑐_𝑛} satisfy the following restrictions:(i)∑^∞_𝑛=0𝑡_𝑛=∞;(ii)∑^∞_𝑛=0𝑏_𝑛<∞;(iii)limsup_𝑛→∞𝑐_𝑛≤0.
Then, {𝑎_𝑛} converges to zero as 𝑛→∞.

3. Weak Convergence Theorem

In this section, we prove the weak convergence theorems for finding a common element in common set of fixed sets of two infinite families of multivalued quasi-nonexpansive mappings in a Banach space.

Theorem 4. Let 𝐶 be a nonempty closed convex subset of a uniformly convex Banach space 𝐸 with Opial's condition. For 𝑖∈𝑁, let {𝑇_𝑖} and {𝑆_𝑖} be two sequences of multivalued quasi-nonexpansive mappings from 𝐶 into 𝑃(𝐶) with 𝐹∶=[⋂^∞_𝑖=1𝐹(𝑇_𝑖)]∩[⋂^∞_𝑖=1𝐹(𝑆_𝑖)]≠∅ and 𝑝∈𝐹. Let {𝑥_𝑛} be a sequence defined by 𝑥₀𝑥∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛+1=𝛼_𝑛,0𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝑦_𝑛,𝑖,𝑦_𝑛=𝛽_𝑛,0𝑥_𝑛+^∞_𝑖=1𝛽_𝑛,𝑖𝑥_𝑛,𝑖, (13) where 𝑦_𝑛,𝑖∈𝑇_𝑖𝑦_𝑛 such that 𝑑(𝑝,𝑦_𝑛,𝑖)=𝑑(𝑝,𝑇_𝑖𝑦_𝑛) and 𝑥_𝑛,𝑖∈𝑆_𝑖𝑥_𝑛 such that 𝑑(𝑝,𝑥_𝑖,𝑛)=𝑑(𝑝,𝑆_𝑖𝑥_𝑛), {𝛼_𝑛,𝑖}^∞_𝑖=0 and {𝛽_𝑛,𝑖}^∞_𝑖=0 are sequences in [0,1] which satisfies ∑^∞_𝑖=0𝛼_𝑛,𝑖=1 and ∑^∞_𝑖=0𝛽_𝑛,𝑖=1, limsup_𝑛→∞𝛼_𝑛,0<1, liminf_𝑛→∞𝛼_𝑛,0𝛼_𝑛,𝑗>0, and liminf_𝑛→∞𝛽_𝑛,0𝛽_𝑛,𝑖>0 for 𝑖∈𝑁. Then {𝑥_𝑛} converges weakly to a point 𝑢∈𝐹.

Proof. First we prove that {𝑥_𝑛} is bounded and lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists for each 𝑝∈𝐹. In fact, from Lemma 2, (13), and 𝑇_𝑖 and 𝑆_𝑖 both being multivalued quasi-nonexpansive mappings, we have that 𝑦_𝑛−𝑝²=𝛽_𝑛,0(𝑥_𝑛−𝑝)+^∞_𝑖=1𝛽_𝑛,𝑖(𝑥_𝑛,𝑖−𝑝)²≤𝛽_𝑛,0𝑥_𝑛−𝑝²+^∞_𝑖=1𝛽_𝑛,𝑖𝑥_𝑛,𝑖−𝑝²−𝛽_𝑛,0𝛽_𝑛,𝑗𝑔𝑥_𝑛−𝑥_𝑛,𝑗=𝛽_𝑛,0𝑥_𝑛−𝑝²+^∞_𝑖=1𝛽_𝑛,𝑖𝑑𝑆_𝑖𝑥_𝑛,𝑝²−𝛽_𝑛,0𝛽_𝑛,𝑗𝑔𝑥_𝑛−𝑥_𝑛,𝑗≤𝛽_𝑛,0𝑥_𝑛−𝑝²+^∞_𝑖=1𝛽_𝑛,𝑖𝐻𝑆_𝑖𝑥_𝑛,𝑝²−𝛽_𝑛,0𝛽_𝑛,𝑗𝑔𝑥_𝑛−𝑥_𝑛,𝑗≤𝛽_𝑛,0𝑥_𝑛−𝑝²+^∞_𝑖=1𝛽_𝑛,𝑖𝑥_𝑛−𝑝²−𝛽_𝑛,0𝛽_𝑛,𝑗𝑔𝑥_𝑛−𝑥_𝑛,𝑗=𝑥_𝑛−𝑝²−𝛽_𝑛,0𝛽_𝑛,𝑗𝑔𝑥_𝑛−𝑥_𝑛,𝑗,𝑥_𝑛+1−𝑝²=𝛼_𝑛,0(𝑥_𝑛−𝑝)+^∞_𝑖=1𝛼_𝑛,𝑖(𝑦_𝑛,𝑖−𝑝)²≤𝛼_𝑛,0𝑥_𝑛−𝑝²+^∞_𝑖=1𝛼_𝑛,𝑖𝑦_𝑛,𝑖−𝑝²=𝛼_𝑛,0𝑥_𝑛−𝑝²+^∞_𝑖=1𝛼_𝑛,𝑖𝑑𝑇_𝑖𝑦_𝑛,𝑝²≤𝛼_𝑛,0𝑥_𝑛−𝑝²+^∞_𝑖=1𝛼_𝑛,𝑖𝐻𝑇_𝑖𝑦_𝑛,𝑝²≤𝛼_𝑛,0𝑥_𝑛−𝑝²+^∞_𝑖=1𝛼_𝑛,𝑖𝑦_𝑛−𝑝²≤𝛼_𝑛,0𝑥_𝑛−𝑝²+^∞_𝑖=1𝛼_𝑛,𝑖𝑥_𝑛−𝑝²−𝛽_𝑛,0𝛽_𝑛,𝑗𝑔𝑥_𝑛−𝑥_𝑛,𝑗=𝑥_𝑛−𝑝²−^∞_𝑖=1𝛼_𝑛,𝑖𝛽_𝑛,0𝛽_𝑛,𝑗𝑔𝑥_𝑛−𝑥_𝑛,𝑗=𝑥_𝑛−𝑝²−1−𝛼_𝑛,0𝛽_𝑛,0𝛽_𝑛,𝑗𝑔𝑥_𝑛−𝑥_𝑛,𝑗. (14)(15) From (15) we have that {‖𝑥_𝑛−𝑝‖} is nondecreasing and bounded by induction, so {𝑥_𝑛} is bounded and lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists.
Next we prove that lim_𝑛→∞‖𝑥_𝑛−𝑥_𝑛,𝑗‖=0 and lim_𝑛→∞‖𝑦_𝑛−𝑦_𝑛,𝑗‖=0. Indeed, we can rewrite (15) as follows: 1−𝛼_𝑛,0𝛽_𝑛,0𝛽_𝑛,𝑗𝑔𝑥_𝑛−𝑥_𝑛,𝑗≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝². (16) Since limsup_𝑛→∞𝛼_𝑛,0<1, liminf_𝑛→∞𝛽_𝑛,0𝛽_𝑛,𝑗>0, and lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists, we have lim_𝑛→∞𝑔(‖𝑥_𝑛−𝑥_𝑛,𝑗‖)=0; by the property of 𝑔, we also get that lim_𝑛→∞𝑥_𝑛−𝑥_𝑛,𝑗=0, (17) for each 𝑗∈𝑁. From (13) we have 𝑦_𝑛−𝑥_𝑛=𝛽_𝑛,0𝑥_𝑛+^∞_𝑖=1𝛽_𝑛,𝑖𝑥_𝑛,𝑖−𝑥_𝑛≤^∞_𝑖=1𝛽_𝑛,𝑖𝑥_𝑛,𝑖−𝑥_𝑛. (18) Together with (17), we have lim_𝑛→∞𝑦_𝑛−𝑥_𝑛=0. (19) On the other hand, from (14) and Lemma 1, we have 𝑥_𝑛+1−𝑝²=𝛼_𝑛,0𝑥_𝑛−𝑝+^∞_𝑖=1𝛼_𝑛,𝑖𝑦_𝑛,𝑖−𝑝²≤𝛼_𝑛,0𝑥_𝑛−𝑝²+^∞_𝑖=1𝛼_𝑛,𝑖𝑦_𝑛,𝑖−𝑝²−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔𝑥_𝑛−𝑦_𝑛,𝑗=𝛼_𝑛,0𝑥_𝑛−𝑝²+^∞_𝑖=1𝛼_𝑛,𝑖𝑑𝑇_𝑖𝑦_𝑛,𝑝²−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔𝑥_𝑛−𝑦_𝑛,𝑗≤𝛼_𝑛,0𝑥_𝑛−𝑝²+^∞_𝑖=1𝛼_𝑛,𝑖𝐻𝑇_𝑖𝑦_𝑛,𝑝²−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔𝑥_𝑛−𝑦_𝑛,𝑗≤𝛼_𝑛,0𝑥_𝑛−𝑝²+^∞_𝑖=1𝛼_𝑛,𝑖𝑦_𝑛−𝑝²−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔𝑥_𝑛−𝑦_𝑛,𝑗≤𝛼_𝑛,0𝑥_𝑛−𝑝²+^∞_𝑖=1𝛼_𝑛,𝑖𝑥_𝑛−𝑝²−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔𝑥_𝑛−𝑦_𝑛,𝑗=𝑥_𝑛−𝑝²−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔𝑥_𝑛−𝑦_𝑛,𝑗. (20) Then, we can rewrite (20) as follows: 𝛼_𝑛,0𝛼_𝑛,𝑗𝑔𝑥_𝑛−𝑦_𝑛,𝑗≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝². (21) Since liminf_𝑛→∞𝛼_𝑛,0𝛼_𝑛,𝑗>0 and lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists, we have lim_𝑛→∞𝑔(‖𝑥_𝑛−𝑦_𝑛,𝑗‖)=0; by the property of 𝑔, we also get that lim_𝑛→∞𝑥_𝑛−𝑦_𝑛,𝑗=0, (22) for each 𝑗∈𝑁. By the triangle inequality, we have 𝑦_𝑛−𝑦_𝑛,𝑗≤𝑦_𝑛−𝑥_𝑛+𝑥_𝑛−𝑦_𝑛,𝑗. (23) Together with (19) and (22), we get that lim_𝑛→∞𝑦_𝑛−𝑦_𝑛,𝑗=0. (24)
Now we prove that {𝑥_𝑛} converges weakly to a point 𝑢∈𝐹. Since {𝑥_𝑛} is bounded, so there exists a subsequence {𝑥_𝑛𝑘}^∞_𝑘=1 of {𝑥_𝑛}^∞_𝑛=1 such that 𝑥_𝑛𝑘⇀𝑢 for some 𝑢∈𝐶; from (19) we have 𝑦_𝑛𝑘⇀𝑢. Suppose that there exists 𝑖∈𝑁 such that 𝑇_𝑖𝑢≠𝑢 and 𝑆_𝑖𝑢≠𝑢. By Opial's condition, we have that limsup_𝑘→∞𝑥_𝑛𝑘−𝑢<limsup_𝑘→∞𝑥_𝑛𝑘−𝑆_𝑖𝑢,limsup_𝑘→∞𝑦_𝑛𝑘−𝑢<limsup_𝑘→∞𝑦_𝑛𝑘−𝑇_𝑖𝑢. (25)(26) From 𝑇_𝑖 and 𝑆_𝑖 both being two multivalued quasi-nonexpansive mappings, we have that 𝑥_𝑛𝑘−𝑆_𝑖𝑢≤𝑥_𝑛𝑘−𝑥_{𝑛𝑘,𝑖}+𝑥_{𝑛𝑘,𝑖}−𝑆_𝑖𝑢=𝑥_𝑛𝑘−𝑥_{𝑛𝑘,𝑖}+𝑑𝑆_𝑖𝑥_𝑛𝑘,𝑆_𝑖𝑢≤𝑥_𝑛𝑘−𝑥_{𝑛𝑘,𝑖}+𝐻𝑆_𝑖𝑥_𝑛𝑘,𝑆_𝑖𝑢≤𝑥_𝑛𝑘−𝑥_{𝑛𝑘,𝑖}+𝑥_𝑛𝑘,𝑦−𝑢_𝑛𝑘−𝑇_𝑖𝑢≤𝑦_𝑛𝑘−𝑦_{𝑛𝑘,𝑖}+𝑦_{𝑛𝑘,𝑖}−𝑇_𝑖𝑢=𝑦_𝑛𝑘−𝑦_{𝑛𝑘,𝑖}+𝑑𝑇_𝑖𝑦_𝑛𝑘,𝑇_𝑖𝑢≤𝑦_𝑛𝑘−𝑦_{𝑛𝑘,𝑖}+𝐻𝑇_𝑖𝑦_𝑛𝑘,𝑇_𝑖𝑢≤𝑦_𝑛𝑘−𝑦_{𝑛𝑘,𝑖}+𝑦_𝑛𝑘−𝑢. (27)(28) Taking limsup_𝑘→∞ on both sides of (27) and (28), respectively, from (17) and (24), we get that limsup_𝑘→∞𝑥_𝑛𝑘−𝑆_𝑖𝑢≤limsup_𝑘→∞𝑥_𝑛𝑘−𝑥_{𝑛𝑘,𝑖}+𝑥_𝑛𝑘−𝑢≤limsup_𝑘→∞𝑥_𝑛𝑘−𝑥_{𝑛𝑘,𝑖}+limsup_𝑘→∞𝑥_𝑛𝑘−𝑢=limsup_𝑘→∞𝑥_𝑛𝑘−𝑢,limsup_𝑘→∞𝑦_𝑛𝑘−𝑇_𝑖𝑢≤limsup_𝑘→∞𝑦_𝑛𝑘−𝑦_{𝑛𝑘,𝑖}+𝑦_𝑛𝑘−𝑢≤limsup_𝑘→∞𝑦_𝑛𝑘−𝑦_{𝑛𝑘,𝑖}+limsup_𝑘→∞𝑦_𝑛𝑘−𝑢=limsup_𝑘→∞𝑦_𝑛𝑘−𝑢. (29)(30) Combining (25) with (29) and combining (26) with (30), respectively, we get that limsup_𝑘→∞𝑥_𝑛𝑘−𝑢<limsup_𝑘→∞𝑥_𝑛𝑘−𝑢,limsup_𝑘→∞𝑦_𝑛𝑘−𝑢<limsup_𝑘→∞𝑦_𝑛𝑘−𝑢, (31) a contradiction. So we have 𝑢=𝑇_𝑖𝑢 and 𝑢=𝑆_𝑖𝑢 for all 𝑖∈𝑁 by the proof of contradiction; this means that 𝑢∈𝐹. Next we prove that {𝑥_𝑛} converges weakly to 𝑢. Take another subsequence {𝑥_𝑚𝑘}^∞_𝑘=1 of {𝑥_𝑛} such that 𝑥_𝑚𝑘⇀𝑣, for some 𝑣∈𝐶. Again, as above, we conclude that 𝑣∈𝐹. Now we show that 𝑢=𝑣. We assume that 𝑢≠𝑣, since lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists for every 𝑝∈𝐹. By (10) we have that lim_𝑛→∞𝑥_𝑛−𝑢=limsup_𝑘→∞𝑥_𝑛𝑘−𝑢<limsup_𝑘→∞𝑥_𝑛𝑘−𝑣=limsup_𝑘→∞𝑥_𝑚𝑘−𝑣<limsup_𝑘→∞𝑥_𝑚𝑘−𝑢=lim_𝑛→∞𝑥_𝑛−𝑢, (32) which means that lim_𝑛→∞‖𝑥_𝑛−𝑢‖<lim_𝑛→∞‖𝑥_𝑛−𝑢‖, a contradiction. So we get that 𝑢=𝑣. Thus 𝑥_𝑛⇀𝑢 as 𝑛→∞. The proof is complete.

Remark 5. Motivated by [6, 7, 9, 14, 23], Theorem 4 improves and extends the following senses.(1)For the mappings, we extend the mappings from nonexpansive mappings to quasi-nonexpansive multi-valued mappings.(2)We extend all single-valued mappings to multi-valued mappings.(3)We extend two nonexpansive mappings or a countable family of mappings to two countable families of multivalued quasi-nonexpansive mappings.
If  𝑇_𝑖=𝑇,  𝑆_𝑖=𝑆  for each 𝑖∈𝑁, then Theorem 4 is reduced to the following corollary.

Corollary 6. Let 𝐶 be a nonempty closed convex subset of a uniformly convex Banach space 𝐸 with Opial’s condition. Let 𝑇 and 𝑆 be two multivalued quasi-nonexpansive mappings from 𝐶 into 𝑃(𝐶) with 𝐹∶=𝐹(𝑇)∩𝐹(𝑆)≠∅ and 𝑝∈𝐹. Let {𝑥_𝑛} be a sequence defined by 𝑥₀𝑥∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑦_𝑛,𝑦_𝑛=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆𝑥_𝑛, (33) where 𝑦_𝑛∈𝑇𝑦_𝑛 such that 𝑑(𝑝,𝑦_𝑛)=𝑑(𝑝,𝑇𝑦_𝑛) and 𝑥_𝑛∈𝑆𝑥_𝑛 such that 𝑑(𝑝,𝑥_𝑛)=𝑑(𝑝,𝑆𝑥_𝑛), {𝛼_𝑛} and {𝛽_𝑛} are sequences in [0,1] which satisfies limsup_𝑛→∞𝛼_𝑛<1, liminf_𝑛→∞𝛼_𝑛(1−𝛼_𝑛)>0, and liminf_𝑛→∞𝛽_𝑛(1−𝛽_𝑛)>0. Then {𝑥_𝑛} converges weakly to a point 𝑢∈𝐹.

Theorem 7. Let 𝐶 be a nonempty compact convex subset of a uniformly convex Banach space 𝐸. For 𝑖∈𝑁, let {𝑆_𝑖} be a sequence of multivalued quasi-nonexpansive mappings from 𝐶 into 𝑃(𝐶) with 𝐹∶=⋂^∞_𝑖=1𝐹(𝑆_𝑖)≠∅ and 𝑝∈𝐹. Let {𝑥_𝑛} be a sequence defined by 𝑥₀𝑥∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛+1=𝛽_𝑛,0𝑥_𝑛+^∞_𝑖=1𝛽_𝑛,𝑖𝑥_𝑛,𝑖, (34) where 𝑥_𝑛,𝑖∈𝑆_𝑖𝑥_𝑛 such that 𝑑(𝑝,𝑥_𝑖,𝑛)=𝑑(𝑝,𝑆_𝑖𝑥_𝑛), {𝛽_𝑛,𝑖}^∞_𝑖=0 is a sequence in [0,1] which satisfies ∑^∞_𝑖=0𝛽_𝑛,𝑖=1, and liminf_𝑛→∞𝛽_𝑛,0𝛽_𝑛,𝑖>0 for 𝑖∈𝑁. Then {𝑥_𝑛} converges strongly to a point 𝑢∈𝐹.

Proof. Put 𝑇_𝑖=𝐼 and 𝛼_𝑛,0=0 for all 𝑛,𝑖∈𝑁 in Theorem 4; then the conclusion of Theorem 7 is obtained.

4. Strong Convergence Theorem

In this section, we give a new proof method to iteration (4) in the basis of [14], and this proof method is easier than the one in [14].

Lemma 8 (see Abbas et al. [14]). Let 𝑋 be a uniformly convex Banach space and 𝐾 a nonempty closed convex subset of 𝑋. Let 𝑆,𝑇∶𝐾→𝐶(𝐾) be multivalued nonexpansive mappings and 𝐹≠∅. Then for the sequence {𝑥_𝑛} in (4), one has that lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists for each 𝑝∈𝐹, and lim_𝑛→∞𝑑(𝑥_𝑛,𝑆𝑥_𝑛)=0=lim_𝑛→∞𝑑(𝑥_𝑛,𝑇𝑥_𝑛).

Theorem 9. Let 𝐾 be a closed convex subset of a real Banach space 𝑋, and let 𝑆 and 𝑇 assume bounded values, and let {𝑥_𝑛} be as in Lemma 8. Then {𝑥_𝑛} converges strongly to a common fixed point of 𝐹.

Proof. Since 𝐶(𝐾) is a nonempty compact subset of 𝐾 and {𝑥_𝑛}⊂𝐶(𝐾), so there exists a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that lim_𝑘→∞‖𝑥_𝑛𝑘−𝑞‖=0 for some 𝑞∈𝐾. Thus, it follows by Lemma 8 we have that 𝑑𝑞,𝑇𝑞≤𝑑𝑞,𝑥_𝑛𝑘+𝑑𝑥_𝑛𝑘,𝑇𝑥_𝑛𝑘+𝐻𝑇𝑥_𝑛𝑘𝑥,𝑇𝑞≤2_𝑛𝑘−𝑞+𝑑𝑥_𝑛𝑘,𝑇𝑥_𝑛𝑘𝑑⟶0as𝑘⟶∞,𝑞,𝑆𝑞≤𝑑𝑞,𝑥_𝑛𝑘+𝑑𝑥_𝑛𝑘,𝑆𝑥_𝑛𝑘+𝐻𝑆𝑥_𝑛𝑘𝑥,𝑆𝑞≤2_𝑛𝑘−𝑞+𝑑𝑥_𝑛𝑘,𝑆𝑥_𝑛𝑘⟶0as𝑘⟶∞. (35) Hence 𝑞∈𝐹. By Lemma 8, lim_𝑛→∞‖𝑥_𝑛−𝑞‖ exists. Hence lim_𝑛→∞𝑥_𝑛=𝑞. The proof is complete.

Acknowledgment

The authors are grateful for the reviewers for their careful reading of the paper and for their suggestions which improved the quality of this work.