Convergence Theorems for Finite Family of Multivalued Maps in Uniformly Convex Banach Spaces

Abstract

We introduce a new iterative process to approximate a common fixed point of a finite family of multivalued maps in a uniformly convex real Banach space and establish strong convergence theorems for the proposed process. Furthermore, strong convergence theorems for finite family of quasi-nonexpansive multivalued maps are obtained. Our results extend important recent results.

1. Introduction

Let 𝐷 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻. The set 𝐷 is called proximinal if for each 𝑥∈𝐻, there exists 𝑦∈𝐷 such that ‖𝑥−𝑦‖=𝑑(𝑥,𝐷), where 𝑑(𝑥,𝐷)=inf{‖𝑥−𝑧‖∶𝑧∈𝐷}. Let 𝐶𝐵(𝐷),𝐾(𝐷), and 𝑃(𝐷) denote the families of nonempty, closed and bounded subsets, nonempty compact subsets, and nonempty proximinal bounded subsets of 𝐷, respectively. The Hausdorff metric on 𝐶𝐵(𝐷) is defined by𝐻(𝐴,𝐵)=maxsup_𝑥∈𝐴𝑑(𝑥,𝐵),sup_𝑦∈𝐵𝑑(𝑦,𝐴),(1.1) for 𝐴,𝐵∈𝐶𝐵(𝐷). A single-valued map 𝑇∶𝐷→𝐷 is called nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐷. A multivalued map 𝑇∶𝐷→𝐶𝐵(𝐷) is said to be nonexpansive if 𝐻(𝑇𝑥,𝑇𝑦)≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐷. An element 𝑝∈𝐷 is called a fixed point of 𝑇∶𝐷→𝐷 (resp., 𝑇∶𝐷→𝐶𝐵(𝐷)) if 𝑝=𝑇𝑝 (resp., 𝑝∈𝑇𝑝). The set of fixed points of 𝑇 is denoted by 𝐹(𝑇). A multivalued map 𝑇∶𝐷→𝐶𝐵(𝐷) is said to be quasi-nonexpansive if 𝐻(𝑇𝑥,𝑇𝑝)≤‖𝑥−𝑝‖ for all 𝑥∈𝐷 and for all 𝑝∈𝐹(𝑇).

A 𝑇∶𝐷→𝐶𝐵(𝐷) is said to satisfy Condition (I) if there is a nondecreasing function 𝑓∶[0,∞)→[0,∞) with 𝑓(0)=0,𝑓(𝑟)>0 for 𝑟∈(0,∞) such that 𝑑(𝑥,𝑇𝑥)≥𝑓(𝑑(𝑥,𝐹(𝑇))),(1.2) for all 𝑥∈𝐷.

The fixed point theory of multivalued nonexpansive mappings is much more complicated and difficult than the corresponding theory of single-valued nonexpansive mappings. However, some classical fixed point theorems for single-valued nonexpansive mappings have already been extended to multivalued mappings. The first results in this direction were established by Markin [1] in Hilbert spaces and by Browder [2] for spaces having weakly continuous duality mapping. Dozo [3] generalized these results to a Banach space satisfying Opial's condition.

In 1974, by using Edelstein's method of asymptotic centers, Lim [4] obtained a fixed point theorem for a multivalued nonexpansive self-mapping in a uniformly convex Banach space.

Theorem 1.1 (Lim [4]). Let 𝐷 be a nonempty, closed convex, and bounded subset of a uniformly convex Banach space 𝐸 and 𝑇∶𝐷→𝐶(𝐸) a multivalued nonexpansive mapping. Then, 𝑇 has a fixed point.

In 1990, Kirk and Massa [5] gave an extension of Lim's theorem proving the existence of a fixed point in a Banach space for which the asymptotic center of a bounded sequence in a closed bounded convex subset is nonempty and compact.

Theorem 1.2 (Kirk and Massa [5]). Let 𝐷 be a nonempty, closed convex, and bounded subset of a Banach space 𝐸 and 𝑇∶𝐷→𝐶𝐵(𝐸) a multivalued nonexpansive mapping. Suppose that the asymptotic center in 𝐸 of each bounded sequence of 𝐸 is nonempty and compact. Then, 𝑇 has a fixed point.

Banach contraction mapping principle was extended nicely multivalued mappings by Nadler [6] in 1969. (Below is stated in a Banach space setting).

Theorem 1.3 (Nadler [6]). Let 𝐷 be a nonempty closed subset of a Banach space 𝐸 and 𝑇∶𝐷→𝐶𝐵(𝐷) a multivalued contraction. Then, 𝑇 has a fixed point.

In 1953, Mann [7] introduced the following iterative scheme to approximate a fixed point of a nonexpansive mapping 𝑇 in a Hilbert space 𝐻: 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,∀𝑛≥1,(1.3) where the initial point 𝑥₀ is taken arbitrarily in 𝐷 and {𝛼_𝑛}^∞𝑛=1 is a sequence in [0,1]. However, we note that Mann's iteration has only weak convergence; see, for example, [8].

In 2005, Sastry and Babu [9] proved that the Mann and Ishikawa iteration schemes for a multivalued map 𝑇 with a fixed point 𝑝 converge to a fixed point 𝑞 of 𝑇 under certain conditions. They also claimed that the fixed point 𝑝 may be different from 𝑞.

In 2007, Panyanak [10] extended the results of Sastry and Babu to uniformly convex Banach spaces and proved the following theorems.

Theorem 1.4 (Panyanak [10]). Let 𝐸 be a uniformly convex Banach space, 𝐷 a nonempty closed bounded convex subset of 𝐸, and 𝑇∶𝐷→𝑃(𝐷) a multivalued nonexpansive mapping that satisfies condition (𝐼). Assume that (i)0≤𝛼_𝑛<1 and (ii)Σ^∞𝑛=1𝛼_𝑛=∞. Suppose that 𝐹(𝑇) a nonempty proximinal subset of 𝐷. Then, the Mann iterates {𝑥_𝑛} defined by 𝑥₀∈𝐷, 𝑥_𝑛+1=𝛼_𝑛𝑦_𝑛+1−𝛼_𝑛𝑥_𝑛,𝛼_𝑛∈[𝑎,𝑏],0<𝑎<𝑏<1,𝑛≥0,(1.4) where 𝑦_𝑛∈𝑇𝑥_𝑛 such that ‖𝑦_𝑛−𝑢_𝑛‖=𝑑(𝑢_𝑛,𝑇𝑥_𝑛) and 𝑢_𝑛∈𝐹(𝑇) such that ‖𝑥_𝑛−𝑢_𝑛‖=𝑑(𝑥_𝑛,𝐹(𝑇)), converges strongly to a fixed point of 𝑇.

Theorem 1.5 (Panyanak [10]). Let 𝐸 be a uniformly convex Banach space, 𝐷 a nonempty compact convex subset of 𝐸, and 𝑇∶𝐷→𝑃(𝐷) a multivalued nonexpansive mapping with a fixed point 𝑝. Assume that (i) 0≤𝛼_𝑛,𝛽_𝑛<1; (ii) 𝛽_𝑛→0 and (iii) Σ^∞𝑛=1𝛼_𝑛𝛽_𝑛=∞. Then, the Ishikawa iterates {𝑥_𝑛} defined by 𝑥₀∈𝐷, 𝑦_𝑛=𝛽_𝑛𝑧_𝑛+1−𝛽_𝑛𝑥_𝑛,𝛽_𝑛∈[0,1],𝑛≥0,(1.5)𝑧_𝑛∈𝑇𝑥_𝑛 such that ‖𝑧_𝑛−𝑝‖=𝑑(𝑝,𝑇𝑥_𝑛), and 𝑥_𝑛+1=𝛼_𝑛𝑧^𝑛+1−𝛼_𝑛𝑥_𝑛,𝛼_𝑛∈[0,1],𝑛≠0,(1.6)𝑧^𝑛∈𝑇𝑦_𝑛 such that ‖𝑧^𝑛−𝑝‖=𝑑(𝑝,𝑇𝑦_𝑛) converges strongly to a fixed point of 𝑇.

Later, Song and Wang [11] noted there was a gap in the proofs of Theorem 1.5 above and of [9, Theorem 5]. They further solved/revised the gap and also gave the affirmative answer Panyanak [10] question using the Ishikawa iterative scheme. In the main results, the domain of 𝑇 is still compact, which is a strong condition (see [11, Theorem 1]) and 𝑇 satisfies condition (𝐼) (see [11, Theorem 1]).

Recently, Shahzad and Zegeye [12] proved the following theorems for quasi-nonexpansive multivalued map and multivalued map in uniformly convex Banach space.

Theorem 1.6 (Shahzad and Zegeye [12]). Let 𝐸 be a uniformly convex real Banach space and 𝐷 a nonempty, closed and convex subset of 𝐸. Let 𝑇∶𝐷→𝐶𝐵(𝐷) be a quasi-nonexpansive multivalued map with 𝐹(𝑇)≠∅ for which 𝑇𝑝={𝑝}, for all𝑝∈𝐹(𝑇). Let {𝑥_𝑛}^∞𝑛=1 be a sequence defined iteratively by 𝑥₀∈𝐷, 𝑦_𝑛=𝛽_𝑛𝑧_𝑛+1−𝛽_𝑛𝑥_𝑛,𝛽_𝑛∈[0,1],𝑛≥0,(1.7)𝑧_𝑛∈𝑇𝑥_𝑛, and 𝑥_𝑛+1=𝛼_𝑛𝑧^𝑛+1−𝛼_𝑛𝑥_𝑛,𝛼_𝑛∈[0,1],𝑛≠0,(1.8)𝑧^𝑛∈𝑇𝑦_𝑛. Assume that 𝑇 satisfies condition (I) and 𝛼_𝑛,𝛽_𝑛∈[𝑎,𝑏]⊂(0,1). Then, {𝑥_𝑛}^∞𝑛=1 converges strongly to a fixed point of 𝑇.

Theorem 1.7 (Shahzad and Zegeye [12]). Let 𝐸 be a uniformly convex real Banach space and 𝐷 a nonempty, closed, and convex subset of 𝐸. Let 𝑇∶𝐷→𝑃(𝐷) be a multivalued map with F(𝑇)≠∅ such that 𝑃_𝑇 is nonexpansive. Let {𝑥_𝑛}^∞𝑛=1 be a sequence defined iteratively by 𝑥₀∈𝐷, 𝑦_𝑛=𝛽_𝑛𝑧_𝑛+1−𝛽_𝑛𝑥_𝑛,𝛽_𝑛∈[0,1],𝑛≥0,(1.9)𝑧_𝑛∈𝑃_𝑇(𝑥_𝑛), and 𝑥_𝑛+1=𝛼_𝑛𝑧^𝑛+1−𝛼_𝑛𝑥_𝑛,𝛼_𝑛∈[0,1],𝑛≠0,(1.10)𝑧^𝑛∈𝑃_𝑇(𝑦_𝑛). Assume that 𝑇 satisfies condition (I) and 𝛼_𝑛,𝛽_𝑛∈[𝑎,𝑏]⊂(0,1). Then, {𝑥_𝑛}^∞𝑛=1 converges strongly to a fixed point of 𝑇.

More recently, Abbas et al. [13] introduced the following one-step iterative process to compute common fixed points of two multivalued nonexpansive mappings.𝑥𝑥₁∈𝐷,_𝑛+1=𝑎_𝑛𝑥_𝑛+𝑏_𝑛𝑦_𝑛+𝑐_𝑛𝑧_𝑛,∀𝑛≥1.(1.11) Using (1.11), Abbas et al. [13] proved weak and strong convergence theorems for approximation of common fixed point of two multivalued nonexpansive mappings in Banach spaces.

Motivated by the ongoing research and the above mentioned results, we introduce a new iterative scheme for approximation of common fixed points of finite family of multivalued maps in a real Banach space. Furthermore, we prove strong convergence theorems for approximation of common fixed points of finite family of multivalued maps in a uniformly convex real Banach space. Next, we prove a necessary and sufficient condition for strong convergence of our new iterative process to a common fixed point of finite family of multivalued maps. Finally, we introduce a new iterative scheme and prove strong convergence theorems for finite family of quasi-nonexpansive multivalued maps in a uniformly convex real Banach space. Our results extend the results of Sastry and Babu [9], Panyanak [10], Shahzad and Zegeye [12], and Song and Wang [11].

2. Preliminaries

Let 𝐸 be Banach space and dim𝐸≥2. The modulus of convexity of 𝐸 is the function 𝛿_𝐸∶(0,2]→[0,1] defined by 𝛿_𝐸(𝜖)∶=inf1−||||||𝑥+𝑦2||||||∶‖𝑥‖=‖𝑦‖=1;𝜖=‖𝑥−𝑦‖.(2.1)𝐸 is uniformly convex if for any 𝜖∈(0,2], there exists a 𝛿=𝛿(𝜖)>0 such that if 𝑥,𝑦∈𝐸 with ‖𝑥‖≤1,‖𝑦‖≤1 and ‖𝑥−𝑦‖≥𝜖, then ‖(1/2)(𝑥+𝑦)‖≤1−𝛿. Equivalently, 𝐸 is uniformly convex if and only if 𝛿_𝐸(𝜖)>0 for all 𝜖∈(0,2].

A family {𝑇_𝑖∶𝐷→𝐶𝐵(𝐷),𝑖=1,2,…,𝑚} is said to satisfy Condition (II) if there is a nondecreasing function 𝑓∶[0,∞)→[0,∞) with 𝑓(0)=0,𝑓(𝑟)>0 for 𝑟∈(0,∞) such that 𝑑𝑥,𝑇_𝑖𝑥≥𝑓𝑑𝑥,∩^𝑚𝑖=1𝐹𝑇_𝑖,(2.2) for all 𝑖=1,2,…,𝑚 and 𝑥∈𝐷.

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

Let 𝐷 be a nonempty, closed, and convex subset of a real Banach space 𝐸. Let 𝑇∶𝐷→𝑃(𝐷) be a multimap and 𝑃_𝑇(𝑥)∶=𝑢_𝑥∈𝑇𝑥∶‖‖𝑥−𝑢_𝑥‖‖=𝑑(𝑥,𝑇𝑥).(2.3) Then, 𝑃_𝑇(𝑥)∶𝐷→𝑃(𝐷) is nonempty and compact for every 𝑥∈𝐷. Furthermore, we observe that 𝑃_𝑇(𝑦)={𝑦} if 𝑦 is a fixed point of 𝑇.

A mapping 𝑇∶𝐷→𝑃(𝐷) is ∗-nonexpansive ([14]) if for all 𝑥,𝑦∈𝐷 and 𝑢_𝑥∈𝑇𝑥 with 𝑑(𝑥,𝑢_𝑥)=inf{𝑑(𝑥,𝑧)∶𝑧∈𝑇𝑥}, there exists 𝑢_𝑦∈𝑇𝑦 with 𝑑(𝑦,𝑢_𝑦)=inf{𝑑(𝑦,𝑤)∶𝑤∈𝑇𝑦} such that 𝑑𝑢_𝑥,𝑢_𝑦≤𝑑(𝑥,𝑦).(2.4)

It is known that ∗-nonexpansiveness is different from nonexpansiveness for multimaps. There are some ∗-nonexpansive multimaps which are not nonexpansive and some nonexpansive multimaps which are not ∗-nonexpansive ([15, 16]).

By the definition of Hausdorff metric, we obtain that if a multimap 𝑇∶𝐷→𝑃(𝐷) is ∗-nonexpansive, then 𝑃_𝑇 is nonexpansive.

Throughout this paper, we write 𝑥_𝑛→𝑥 to indicate that the sequence {𝑥_𝑛} converges strongly to 𝑥.

Also, this following lemma will be used in the sequel.

Lemma 2.1 (Schu [17]). 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.

3. Main Results

We now introduce the following iteration scheme. Let 𝐸 be a real normed space and 𝐷 a nonempty subset of 𝐸. Let 𝑇₁,𝑇₂,…,𝑇_𝑚 be multivalued maps of 𝐷 into 𝑃(𝐷) with 𝐹∶=∩^𝑚𝑖=1𝐹(𝑇_𝑖)≠∅ such that 𝑃_𝑇1,𝑃_𝑇2,…,𝑃_𝑇𝑚 are nonexpansive and {𝛼_𝑛𝑖}^∞𝑛=1,𝑖=0,1,…,𝑚 a sequence in [𝜖,1−𝜖],𝜖∈(0,1) such that ∑^𝑚𝑖=0𝛼_𝑛𝑖=1 for all 𝑛≥1. Let {𝑥_𝑛}^∞𝑛=1 be a sequence defined iteratively by𝑥𝑥₁∈𝐷,_𝑛+1=𝛼_𝑛0𝑥_𝑛+𝛼_𝑛1𝑦^𝑛(1)+⋯+𝛼_𝑛𝑚𝑦^𝑛(𝑚),(3.1) where 𝑦^𝑛(𝑖)∈𝑃_𝑇𝑖𝑥_𝑛,𝑖=1,2,…,𝑚.

Lemma 3.1. Let 𝐸 be a real normed space and 𝐷 a nonempty subset of 𝐸. Let 𝑇₁,𝑇₂,…,𝑇_𝑚 be multivalued maps of 𝐷 into 𝑃(𝐷) with 𝐹∶=∩^𝑚𝑖=1𝐹(𝑇_𝑖)≠∅ such that 𝑃_𝑇1,𝑃_𝑇2,…,𝑃_𝑇𝑚 are nonexpansive. Let {𝛼_𝑛𝑖}^∞𝑛=1,𝑖=0,1,…,𝑚 a sequence in [𝜖,1−𝜖],𝜖∈(0,1) such that ∑^𝑚𝑖=0𝛼_𝑛𝑖=1 for all 𝑛≥1. Let {𝑥_𝑛}^∞𝑛=1 be a sequence defined iteratively by (3.1). Then, lim_𝑛→∞𝑑𝑥_𝑛,𝑇_𝑖𝑥_𝑛=0,∀𝑖=1,2,…,𝑚.(3.2)

Proof. Let 𝑥^∗∈∩^𝑚𝑖=1𝐹(𝑇_𝑖). Then, from (3.1), we have the following estimates: ‖‖𝑥_𝑛+1−𝑥^∗‖‖≤≤𝛼𝛼_𝑛0𝑛0‖‖𝑥_𝑛−𝑥^∗‖‖+𝛼_𝑛1‖‖𝑦^𝑛(1)−𝑥^∗‖‖+⋯+𝛼_𝑛𝑚‖‖𝑦^𝑛(𝑚)−𝑥^∗‖‖≤𝛼_𝑛0‖‖𝑥_𝑛−𝑥^∗‖‖+𝛼_𝑛1𝑑𝑦^𝑛(1),𝑃_𝑇1𝑥^∗+⋯+𝛼^𝑛𝑚𝑑𝑦^𝑛(𝑚),𝑃_𝑇𝑚𝑥^∗≤𝛼_𝑛0‖‖𝑥_𝑛−𝑥^∗‖‖+𝛼_𝑛1𝐻𝑃_𝑇1𝑥_𝑛,𝑃_𝑇1𝑥^∗+⋯+𝛼_𝑛𝑚𝐻𝑃_𝑇𝑚𝑥_𝑛,𝑃_𝑇𝑚𝑥^∗=‖‖𝑥_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖+𝛼_𝑛1‖‖𝑥_𝑛−𝑥^∗‖‖+⋯+𝛼_𝑛𝑚‖‖𝑥_𝑛−𝑥^∗‖‖−𝑥^∗‖‖.(3.3) Thus, lim_𝑛→∞‖𝑥_𝑛−𝑥^∗‖ exists. Let lim_𝑛→∞‖‖𝑥_𝑛−𝑥^∗‖‖=𝑐,(3.4) for some 𝑐≥0. Then, 𝑐==lliimm_{𝑛→∞𝑛→∞}‖‖𝑥_𝑛+1−𝑥^∗‖‖‖‖𝛼_𝑛0=lim_𝑛→∞𝑥_𝑛−𝑥^∗+𝛼_𝑛1𝑦^𝑛(1)−𝑥^∗+⋯+𝛼^𝑛𝑚𝑦^𝑛(𝑚)−𝑥^∗‖‖‖‖‖1−𝛼_𝑛01−𝛼𝛼^𝑛1_𝑛0𝑦^𝑛(1)−𝑥^∗+⋯+1−𝛼𝛼^𝑛𝑚_𝑛0𝑦^𝑛(𝑚)−𝑥^∗+𝛼^𝑛0𝑥_𝑛−𝑥^∗‖‖‖.(3.5) Since 𝑃_𝑇𝑖,𝑖=1,2,…,𝑚 is nonexpansive mapping and ∩^𝑚𝑖=1𝐹(𝑇_𝑖)≠∅, we have ‖‖𝑦^𝑛(𝑖)−𝑥^∗‖‖≤𝑑𝑦^𝑛(𝑖),𝑃_𝑇𝑖𝑥^∗≤𝐻𝑃^𝑇_𝑖𝑥_𝑛,𝑃_𝑇𝑖𝑥^∗≤‖‖𝑥_𝑛−𝑥^∗‖‖,(3.6) for each 𝑥^∗∈∩^𝑚𝑖=1𝐹(𝑇_𝑖). Taking limsup on both sides, we obtain limsup_𝑛→∞‖‖𝑦^𝑛(𝑖)−𝑥^∗‖‖≤𝑐,𝑖=1,2,…,𝑚.(3.7) Next, limsup_𝑛→∞‖‖‖𝛼_𝑛11−𝛼_𝑛0𝑦^𝑛(1)−𝑥^∗+𝛼^𝑛21−𝛼_𝑛0𝑦^𝑛(2)−𝑥^∗+⋯+𝛼^𝑛𝑚1−𝛼_𝑛0𝑦^𝑛(𝑚)≤limsup_𝑛→∞−𝑥^∗‖‖‖1−𝛼𝛼^𝑛1_𝑛0‖‖𝑦^𝑛(1)−𝑥^∗‖‖+𝛼^𝑛21−𝛼_𝑛0‖‖𝑦^𝑛(2)−𝑥^∗‖‖+⋯+𝛼^𝑛𝑚1−𝛼_𝑛0≤limsup_𝑛→∞‖‖𝑦^𝑛(𝑚)−𝑥^∗‖‖𝛼_𝑛1+𝛼_𝑛2+⋯+𝛼_𝑛𝑚1−𝛼_𝑛0‖‖𝑥_𝑛−𝑥^∗‖‖=𝑐.(3.8) Using (3.5), (3.8), and Lemma 2.1, we obtain lim_𝑛→∞‖‖‖𝛼_𝑛11−𝛼_𝑛0𝑦^𝑛(1)−𝑥^∗+𝛼^𝑛21−𝛼_𝑛0𝑦^𝑛(2)−𝑥^∗+⋯+𝛼^𝑛𝑚1−𝛼_𝑛0𝑦^𝑛(𝑚)−𝑥^∗−𝑥^𝑛−𝑥^∗‖‖‖=0.(3.9) This yields 0=lim_𝑛→∞‖‖‖𝛼_𝑛11−𝛼_𝑛0𝑦^𝑛(1)+𝛼^𝑛21−𝛼_𝑛0𝑦^𝑛(2)+⋯+𝛼^𝑛𝑚1−𝛼_𝑛0𝑦^𝑛(𝑚)−𝑥_𝑛=lim_𝑛→∞‖‖‖1−1𝛼_𝑛0‖‖𝛼^𝑛1=lim_𝑛→∞𝑦^𝑛(1)+𝛼_𝑛2𝑦^𝑛(2)+⋯+𝛼_𝑛𝑚𝑦^𝑛(𝑚)−1−𝛼_𝑛0𝑥_𝑛‖‖1−1𝛼_𝑛0‖‖𝑥^𝑛+1−𝑥_𝑛‖‖.(3.10) Thus, lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. Furthermore, 𝑐==lliimm_{𝑛→∞𝑛→∞}‖‖𝑥_𝑛+1−𝑥^∗‖‖‖‖𝛼_𝑛0=lim_𝑛→∞𝑥_𝑛−𝑥^∗+𝛼_𝑛1𝑦^𝑛(1)−𝑥^∗+⋯+𝛼^𝑛𝑚𝑦^𝑛(𝑚)−𝑥^∗‖‖‖‖‖1−𝛼_𝑛1+𝛼_𝑛11−𝛼𝛼^𝑛0_𝑛1𝑥_𝑛−𝑥^∗+1−𝛼𝛼^𝑛2_𝑛1𝑦^𝑛(2)−𝑥^∗+⋯+1−𝛼𝛼^𝑛𝑚_𝑛1𝑦^𝑛(𝑚)−𝑥^∗limsup_𝑛→∞𝑦^𝑛(1)−𝑥^∗‖‖‖,‖‖‖𝛼_𝑛01−𝛼_𝑛1𝑥_𝑛−𝑥^∗+𝛼^𝑛21−𝛼_𝑛1𝑦^𝑛(2)−𝑥^∗+⋯+𝛼^𝑛𝑚1−𝛼_𝑛1𝑦^𝑛(𝑚)≤limsup_𝑛→∞−𝑥^∗‖‖‖1−𝛼𝛼^𝑛0_𝑛1‖‖𝑥_𝑛−𝑥^∗‖‖+𝛼^𝑛21−𝛼_𝑛1‖‖𝑦^𝑛(2)−𝑥^∗‖‖+⋯+𝛼^𝑛𝑚1−𝛼_𝑛1≤limsup_𝑛→∞‖‖𝑦^𝑛(𝑚)−𝑥^∗‖‖𝛼_𝑛0+𝛼_𝑛2+⋯+𝛼_𝑛𝑚1−𝛼_𝑛1‖‖𝑥_𝑛−𝑥^∗‖‖=𝑐.(3.11) Using (3.11) and Lemma 2.1, we obtain lim_𝑛→∞‖‖‖𝛼_𝑛01−𝛼_𝑛1𝑥_𝑛−𝑥^∗+𝛼^𝑛21−𝛼_𝑛1𝑦^𝑛(2)−𝑥^∗+⋯+𝛼^𝑛𝑚1−𝛼_𝑛1𝑦^𝑛(𝑚)−𝑥^∗−𝑦^𝑛(1)−𝑥^∗‖‖‖=0.(3.12) This yields 0=lim_𝑛→∞‖‖‖𝛼_𝑛01−𝛼_𝑛1𝑥_𝑛+𝛼^𝑛21−𝛼_𝑛1𝑦^𝑛(2)+⋯+𝛼^𝑛𝑚1−𝛼_𝑛1𝑦^𝑛(𝑚)−𝑦^𝑛(1)=lim_𝑛→∞‖‖‖1−1𝛼_𝑛1‖‖𝛼^𝑛0=lim_𝑛→∞𝑥_𝑛+𝛼_𝑛2𝑦^𝑛(2)+⋯+𝛼_𝑛𝑚𝑦^𝑛(𝑚)−1−𝛼_𝑛1𝑦^𝑛(1)‖‖1−1𝛼_𝑛1‖‖𝑥^𝑛+1−𝑦^𝑛(1)‖‖.(3.13) Thus, lim_𝑛→∞‖𝑥_𝑛+1−𝑦^𝑛(1)‖=0. So, ‖‖𝑥_𝑛−𝑦^𝑛(1)‖‖≤‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+‖‖𝑥_𝑛+1−𝑦^𝑛(1)‖‖⟶0,𝑛⟶∞.(3.14) Then, 𝑑𝑥_𝑛,𝑇₁𝑥_𝑛≤𝑑𝑥_𝑛,𝑃_𝑇1𝑥_𝑛≤‖‖𝑥_𝑛−𝑦^𝑛(1)‖‖⟶0,𝑛⟶∞.(3.15) In a similar way, we can show that lim_𝑛→∞𝑑𝑥_𝑛,𝑇_𝑖𝑥_𝑛=0,𝑖=2,3,…,𝑚.(3.16) This completes the proof.

Theorem 3.2. Let 𝐸 be a uniformly convex real Banach space and 𝐷 a nonempty, closed, and convex subset of 𝐸. Let 𝑇₁,𝑇₂,…,𝑇_𝑚 be multivalued maps of 𝐷 into 𝑃(𝐷) with 𝐹∶=∩^𝑚𝑖=1𝐹(𝑇_𝑖)≠∅ such that 𝑃_𝑇1,𝑃_𝑇2,…,𝑃_𝑇𝑚 are nonexpansive and {𝑇_𝑖}^𝑚_i=1 satisfying condition (II). Let {𝛼_𝑛𝑖}^∞𝑛=1,𝑖=0,1,…,𝑚 a sequence in [𝜖,1−𝜖],𝜖∈(0,1) such that ∑^𝑚𝑖=0𝛼_𝑛𝑖=1 for all 𝑛≥1. Let {𝑥_𝑛}^∞𝑛=1 be a sequence defined iteratively by (3.1). Then, {𝑥_𝑛}^∞𝑛=1 converges strongly to a common fixed point of {𝑇_𝑖}^𝑚𝑖=1.

Proof. Since {𝑇_𝑖}^𝑚𝑖=1 satisfies condition (II), we have that 𝑑(𝑥_𝑛,∩^𝑚𝑖=1𝐹(𝑇_𝑖))→0 as 𝑛→∞. Thus, there is a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} and a sequence {𝑝_𝑘}⊂∩^𝑚𝑖=1𝐹(𝑇_𝑖) such that ‖‖𝑥_𝑛𝑘−𝑝_𝑘‖‖<12^𝑘,(3.17) for all 𝑘. By Lemma 3.1, we obtain ‖‖𝑥_𝑛𝑘+1−𝑝_𝑘‖‖≤‖‖𝑥_𝑛𝑘−𝑝_𝑘‖‖<12^𝑘.(3.18) We now show that {𝑝_𝑘} is a Cauchy sequence in 𝐷. Observe that ‖‖𝑝_𝑘+1−𝑝_𝑘‖‖≤<‖‖𝑝1_𝑘+1−𝑥_𝑛𝑘+1‖‖+‖‖𝑥_𝑛𝑘+1−𝑝_𝑘‖‖2^𝑘+1+12^𝑘<12^𝑘−1.(3.19) This shows that {𝑝_𝑘} is a Cauchy sequence in 𝐷, and thus converges to 𝑝∈𝐷. Since 𝑑𝑝_𝑘,𝑇_𝑖𝑝‖‖≤≤𝑑‖‖𝑝𝑝−_𝑘,𝑝𝑃_𝑇𝑖𝑘𝑝≤𝐻𝑃_𝑇𝑖𝑝,𝑃_𝑇𝑖𝑝_𝑘‖‖,(3.20) and 𝑝_𝑘→𝑝 as 𝑘→∞, it follows that 𝑑(𝑝,𝑇_𝑖𝑝)=0, and thus 𝑝∈∩^𝑚𝑖=1𝐹(𝑇_𝑖), and {𝑥_n𝑘} converges strongly to 𝑝. Since lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists, it follows that {𝑥_𝑛} converges strongly to 𝑝. This completes the proof.

Corollary 3.3. Let 𝐸 be a uniformly convex real Banach space and 𝐷 a nonempty, closed, and convex subset of 𝐸. Let 𝑇₁,𝑇₂,…,𝑇_𝑚 be ∗-nonexpansive multimaps of 𝐷 into 𝑃(𝐷) with 𝐹∶=∩^𝑚𝑖=1𝐹(𝑇_𝑖)≠∅ and {𝑇_𝑖}^𝑚𝑖=1 satisfying condition (II). Let {𝛼_𝑛𝑖}^∞𝑛=1,𝑖=0,1,…,𝑚 a sequence in [𝜖,1−𝜖],𝜖∈(0,1) such that ∑^𝑚𝑖=0𝛼_𝑛𝑖=1 for all 𝑛≥1. Let {𝑥_𝑛}^∞𝑛=1 be a sequence defined iteratively by (3.1). Then, {𝑥_𝑛}^∞𝑛=1 converges strongly to a common fixed point of {𝑇_𝑖}^𝑚𝑖=1.

Theorem 3.4. Let 𝐸 be a uniformly convex real Banach space and 𝐷 a nonempty, closed and convex subset of 𝐸. Let 𝑇₁,𝑇₂,…,𝑇_𝑚 be multivalued maps of 𝐷 into 𝑃(𝐷) with 𝐹∶=∩^𝑚𝑖=1𝐹(𝑇_𝑖)≠∅ such that 𝑃_𝑇1,𝑃_𝑇2,…,𝑃_𝑇𝑚 are nonexpansive and 𝑇_𝑖 is hemicompact and continuous for each 𝑖=1,2,…,𝑚. Let {𝛼_𝑛𝑖}^∞𝑛=1,𝑖=0,1,…,𝑚 a sequence in [𝜖,1−𝜖],𝜖∈(0,1) such that ∑^𝑚𝑖=0𝛼_𝑛𝑖=1 for all 𝑛≥1. Let {𝑥_𝑛}^∞𝑛=1 be a sequence defined iteratively by (3.1). Then, {𝑥_𝑛}^∞𝑛=1 converges strongly to a common fixed point of {𝑇_𝑖}^𝑚𝑖=1.

Proof. Since lim_𝑛→∞𝑑(𝑥_𝑛,𝑇_𝑖𝑥_𝑛)=0,for all𝑖=1,2,…,𝑚 and 𝑇_𝑖 is hemicompact for each 𝑖=1,2,…,𝑚, there is a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that 𝑥_𝑛𝑘→𝑝 as 𝑘→∞ for some 𝑝∈𝐷. Since 𝑇_𝑖 is continuous for each 𝑖=1,2,…,𝑚, we have 𝑑(𝑥_𝑛𝑘,𝑇_𝑖𝑥_𝑛𝑘)→𝑑(𝑝,𝑇_𝑖𝑝). As a result, we have that 𝑑(𝑝,𝑇_𝑖𝑝)=0,for all𝑖=1,2,…,𝑚, and so, 𝑝∈∩^𝑚𝑖=1𝐹(𝑇_𝑖). Since lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists, it follows that {𝑥_𝑛} converges strongly to 𝑝. This completes the proof.

Corollary 3.5. Let 𝐸 be a uniformly convex real Banach space and 𝐷 a nonempty, closed, and convex subset of 𝐸. Let 𝑇₁,𝑇₂,…,𝑇_𝑚 be ∗-nonexpansive multimaps of 𝐷 into 𝑃(𝐷) with 𝐹∶=∩^𝑚𝑖=1𝐹(𝑇_𝑖)≠∅ and 𝑇_𝑖 is hemicompact and continuous for each i=1,2,…,𝑚. Let {𝛼_𝑛𝑖}^∞𝑛=1,𝑖=0,1,…,𝑚 a sequence in [𝜖,1−𝜖],𝜖∈(0,1) such that ∑^𝑚𝑖=0𝛼_𝑛𝑖=1 for all 𝑛≥1. Let {𝑥_𝑛}^∞𝑛=1 be a sequence defined iteratively by (3.1). Then, {𝑥_𝑛}^∞𝑛=1 converges strongly to a common fixed point of {𝑇_𝑖}^𝑚𝑖=1.

Theorem 3.6. Let 𝐸 be a uniformly convex real Banach space and 𝐷 a nonempty compact convex subset of 𝐸. Let 𝑇₁,𝑇₂,…,𝑇_𝑚 be multivalued maps of 𝐷 into 𝑃(𝐷) with 𝐹∶=∩^𝑚𝑖=1𝐹(𝑇_𝑖)≠∅ such that 𝑃_𝑇1,𝑃_𝑇2,…,𝑃_T𝑚 are nonexpansive. Let {𝛼_𝑛𝑖}^∞𝑛=1,𝑖=0,1,…,𝑚 a sequence in [𝜖,1−𝜖],𝜖∈(0,1) such that ∑^𝑚𝑖=0𝛼_𝑛𝑖=1 for all 𝑛≥1. Let {𝑥_𝑛}^∞𝑛=1 be a sequence defined iteratively by (3.1). Then, {𝑥_𝑛}^∞𝑛=1 converges strongly to a common fixed point of {𝑇_𝑖}^𝑚𝑖=1.

Proof. From the compactness of 𝐷, there exists a subsequence {𝑥_𝑛𝑘}^∞𝑛=𝑘 of {𝑥_𝑛}^∞𝑛=1 such that lim_𝑘→∞‖𝑥_𝑛𝑘−𝑞‖=0 for some 𝑞∈𝐷. Thus, 𝑑𝑞,𝑇_𝑖𝑞≤≤2𝑑‖‖𝑥𝑞,_𝑛𝑘𝑃_𝑇𝑖𝑞≤‖‖𝑥_𝑛𝑘−𝑞‖‖+𝑑𝑥_𝑛𝑘,𝑃_𝑇𝑖𝑥_𝑛𝑘+𝐻𝑃_𝑇𝑖𝑥_𝑛𝑘,𝑃_𝑇𝑖𝑞−𝑞‖‖+𝑑𝑥_𝑛𝑘,𝑃_𝑇𝑖𝑥_𝑛𝑘⟶0as𝑘⟶∞.(3.21) Hence, 𝑞∈∩^𝑚𝑖=1𝐹(𝑇_𝑖). Now, on taking 𝑞 in place of 𝑥^∗, we get that lim_𝑛→∞‖𝑥_𝑛−𝑞‖ exists. This completes the proof.

The following result gives a necessary and sufficient condition for strong convergence of the sequence in (3.1) to a common fixed point of {𝑇_𝑖}^𝑚𝑖=1.

Theorem 3.7. Let 𝐷 be a nonempty, closed, and convex subset of a real Banach space 𝐸. Let 𝑇₁,𝑇₂,…,𝑇_𝑚 be multivalued maps of 𝐷 into 𝑃(𝐷) with 𝐹∶=∩^𝑚𝑖=1𝐹(𝑇_𝑖)≠∅ such that 𝑃_𝑇1,𝑃_𝑇2,…,𝑃_𝑇𝑚 are nonexpansive. Let {𝛼_𝑛𝑖}^∞𝑛=1,𝑖=0,1,…,𝑚 a sequence in [𝜖,1−𝜖],𝜖∈(0,1) such that ∑^𝑚𝑖=0𝛼_𝑛𝑖=1 for all 𝑛≥1. Let {𝑥_𝑛}^∞𝑛=1 be a sequence defined iteratively by (3.1). Then, {𝑥_𝑛}^∞𝑛=1 converges strongly to a common fixed point of {𝑇_𝑖}^𝑚𝑖=1 if and only if liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0.

Proof. The necessity is obvious. Conversely, suppose that liminf_𝑛→∞𝑑(𝑥_𝑛,F)=0. By (3.3), we have ‖‖𝑥_𝑛+1−𝑥^∗‖‖≤‖‖𝑥_𝑛−𝑥^∗‖‖.(3.22) This gives 𝑑𝑥_𝑛+1,𝐹≤𝑑𝑥_𝑛,𝐹.(3.23) Hence, lim_𝑛→∞𝑑(𝑥_𝑛,𝐹) exists. By hypothesis, liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0, so we must have lim_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0.
Next, we show that {𝑥_𝑛}^∞𝑛=1 is a Cauchy sequence in 𝐷. Let 𝜖>0 be given, and since liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0, there exists 𝑛₀ such that for all 𝑛≥𝑛₀, we have 𝑑𝑥_𝑛,𝐹<𝜖4.(3.24) In particular, inf{‖𝑥_𝑛0−𝑝‖∶𝑝∈𝐹}<𝜖/4 so that there must exist a 𝑝^∗∈𝐹 such that ‖‖𝑥_𝑛0−𝑝^∗‖‖<𝜖2.(3.25) Now, for 𝑚,𝑛≥𝑛₀, we have ‖‖𝑥_𝑛+𝑚−𝑥_𝑛‖‖≤≤2||‖‖‖‖𝑥𝑥_{𝑛+𝑚𝑛0}−𝑝^∗‖‖+‖‖𝑥_𝑛−𝑝^∗‖‖<2𝜖−𝑝^∗‖‖2=𝜖.(3.26) Hence, {𝑥_𝑛} is a Cauchy sequence in a closed subset 𝐷 of a Banach space 𝐸, and therefore, it must converge in 𝐷. Let lim_𝑛→∞𝑥_𝑛=𝑝. Now, for each 𝑖=1,2,…,𝑚, we obtain 𝑑𝑝,𝑇_𝑖𝑝≤≤𝑑𝑑𝑝𝑝,,𝑃𝑥_𝑇𝑖𝑛𝑝≤𝑑𝑝,𝑥_𝑛+𝑑𝑥_𝑛,𝑃_𝑇𝑖𝑥_𝑛+𝐻𝑃_𝑇𝑖𝑥_𝑛,𝑃_𝑇𝑖𝑝+𝑑𝑥_𝑛,𝑃_𝑇𝑖𝑥_𝑛+𝑑𝑥_𝑛,𝑝⟶0as𝑛⟶∞(3.27) gives that 𝑑(𝑝,𝑇_𝑖𝑝)=0,𝑖=1,2,…,𝑚 which implies that 𝑝∈𝑇_𝑖𝑝. Consequently, 𝑝∈𝐹=∩^𝑚𝑖=1𝐹(𝑇_𝑖)≠∅.

All the results we have obtained so far can be established for finite family of quasi-nonexpansive multivalued maps. Let 𝑇₁,𝑇₂,…,𝑇_𝑚 be quasi-nonexpansive multivalued maps of 𝐷 into 𝐶𝐵(𝐷) such that 𝐹∶=∩^𝑚𝑖=1𝐹(𝑇_𝑖)≠∅ for which 𝑇_𝑖𝑝={𝑝}, for all 𝑝∈∩^𝑚𝑖=1𝐹(𝑇_𝑖). Let {𝛼_𝑛𝑖}^∞𝑛=1,𝑖=0,1,…,𝑚 a sequence in [𝜖,1−𝜖],𝜖∈(0,1) such that ∑^𝑚𝑖=0𝛼_𝑛𝑖=1 for all 𝑛≥1. Let {𝑥_𝑛}^∞𝑛=1 be a sequence defined iteratively by𝑥𝑥₁∈𝐷,_𝑛+1=𝛼_𝑛0𝑥_𝑛+𝛼_n1𝑦^𝑛(1)+⋯+𝛼_𝑛𝑚𝑦^𝑛(𝑚),(3.28) where 𝑦^𝑛(𝑖)∈𝑇_𝑖𝑥_𝑛, 𝑖=1,2,…,𝑚. Thus, we obtain the following theorems using iterative process (3.28).

Theorem 3.8. Let 𝐸 be a uniformly convex real Banach space and 𝐷 a nonempty, closed, and convex subset of 𝐸. Let 𝑇₁,𝑇₂,…,𝑇_𝑚 be quasi-nonexpansive multivalued maps of 𝐷 into 𝐶𝐵(𝐷) such that 𝐹∶=∩^𝑚𝑖=1𝐹(𝑇_𝑖)≠∅ for which 𝑇_𝑖𝑝={𝑝}, for all 𝑝∈∩^𝑚𝑖=1𝐹(𝑇_𝑖) and {𝑇_𝑖}^𝑚𝑖=1 satisfying condition (II). Let {𝛼_𝑛𝑖}^∞𝑛=1,𝑖=0,1,…,𝑚 a sequence in [𝜖,1−𝜖],𝜖∈(0,1) such that ∑^𝑚𝑖=0𝛼_𝑛𝑖=1 for all 𝑛≥1. Let {𝑥_𝑛}^∞𝑛=1 be a sequence defined iteratively by (3.28). Then, {𝑥_𝑛}^∞𝑛=1 converges strongly to a common fixed point of {𝑇_𝑖}^𝑚𝑖=1.

Theorem 3.9. Let 𝐸 be a uniformly convex real Banach space and 𝐷 a nonempty, closed, and convex subset of 𝐸. Let 𝑇₁,𝑇₂,…,𝑇_𝑚 be quasi-nonexpansive multivalued maps of 𝐷 into 𝐶𝐵(𝐷) such that 𝐹∶=∩^𝑚𝑖=1𝐹(𝑇_𝑖)≠∅ for which 𝑇_𝑖𝑝={𝑝}, for all 𝑝∈∩^𝑚𝑖=1𝐹(𝑇_𝑖) and 𝑇_𝑖 is hemicompact and continuous for each 𝑖=1,2,…,𝑚. Let {𝛼_𝑛𝑖}^∞𝑛=1,𝑖=0,1,…,𝑚 a sequence in [𝜖,1−𝜖],𝜖∈(0,1) such that ∑^𝑚𝑖=0𝛼_𝑛𝑖=1 for all 𝑛≥1. Let {𝑥_𝑛}^∞𝑛=1 be a sequence defined iteratively by (3.28). Then, {𝑥_𝑛}^∞𝑛=1 converges strongly to a common fixed point of {𝑇_𝑖}^𝑚𝑖=1.

Theorem 3.10. Let 𝐸 be a uniformly convex real Banach space and 𝐷 a nonempty compact convex subset of 𝐸. Let 𝑇₁,𝑇₂,…,𝑇_𝑚 be quasi-nonexpansive multivalued maps of 𝐷 into 𝐶𝐵(𝐷) such that 𝐹∶=∩^𝑚𝑖=1𝐹(𝑇_𝑖)≠∅ for which 𝑇_𝑖𝑝={𝑝}, for all 𝑝∈∩^𝑚𝑖=1𝐹(𝑇_𝑖). {𝛼_𝑛𝑖}^∞𝑛=1,𝑖=0,1,…,𝑚 a sequence in [𝜖,1−𝜖],𝜖∈(0,1) such that ∑^𝑚𝑖=0𝛼_𝑛𝑖=1 for all 𝑛≥1. Let {𝑥_𝑛}^∞𝑛=1 be a sequence defined iteratively by (3.28). Then, {𝑥_𝑛}^∞𝑛=1 converges strongly to a common fixed point of {𝑇_𝑖}^𝑚𝑖=1.

Corollary 3.11 (Abbas et al. [13]). Let 𝐸 be a uniformly convex real Banach space satisfying Opial's condition. Let 𝐷 be a nonempty, closed, and convex of 𝐸. Let 𝑇,𝑆 be multivalued nonexpansive mappings of 𝐷 into 𝐾(𝐷) such that 𝐹∶=𝐹(𝑇)∩𝐹(𝑆)≠∅. Let {𝑎_𝑛}^∞𝑛=1,{𝑏_𝑛}^∞𝑛=1, and {𝑐_𝑛}^∞𝑛=1 be sequence in (0,1) satisfying 𝑎_𝑛+𝑏_𝑛+𝑐_𝑛≤1. Let {𝑥_𝑛}^∞𝑛=1 be a sequence defined iteratively by 𝑥𝑥₁∈𝐷,_𝑛+1=𝑎_𝑛𝑥_𝑛+𝑏_𝑛𝑦_𝑛+𝑐_𝑛𝑧_𝑛,∀𝑛≥1,(3.29) where 𝑦_𝑛∈𝑇𝑥_𝑛,𝑧_𝑛∈𝑆𝑥_𝑛 such that ‖𝑦_𝑛−𝑝‖≤𝑑(𝑝,𝑇𝑥_𝑛) and ‖𝑧_𝑛−𝑝‖≤𝑑(𝑝,𝑆𝑥_𝑛) whenever 𝑝 is a fixed point of any one of mappings 𝑇 and 𝑆. Then, {𝑥_𝑛}^∞𝑛=1 converges weakly to a common fixed point of 𝐹(𝑇)∩𝐹(𝑆).

Corollary 3.12 (Abbas et al. [13]). Let 𝐸 be a real Banach space and 𝐷 a nonempty, closed, and convex subset of 𝐸. Let 𝑇,𝑆 be multivalued nonexpansive mappings of 𝐷 into 𝐾(D) such that 𝐹∶=𝐹(𝑇)∩𝐹(𝑆)≠∅. Let {𝑎_𝑛}^∞𝑛=1,{𝑏_𝑛}^∞𝑛=1 and {𝑐_𝑛}^∞𝑛=1 be sequence in (0,1) satisfying 𝑎_𝑛+𝑏_𝑛+𝑐_𝑛≤1. Let {𝑥_𝑛}^∞𝑛=1 be a sequence defined iteratively by 𝑥𝑥₁∈𝐷,_𝑛+1=𝑎_𝑛𝑥_𝑛+𝑏_𝑛𝑦_𝑛+𝑐_𝑛𝑧_𝑛,∀𝑛≥1,(3.30) where 𝑦_𝑛∈𝑇𝑥_𝑛, 𝑧_𝑛∈𝑆𝑥_𝑛 such that ‖𝑦_𝑛−𝑝‖≤𝑑(𝑝,𝑇𝑥_𝑛) and ‖𝑧_𝑛−𝑝‖≤𝑑(𝑝,𝑆𝑥_𝑛) whenever 𝑝 is a fixed point of any one of mappings 𝑇 and 𝑆. Then, {𝑥_𝑛}^∞𝑛=1 converges strongly to a common fixed point of 𝐹(𝑇)∩𝐹(𝑆) if and only if liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0.

Remark 3.13. Our results extend the results of Sastry and Babu [9], Panyanak [10], and Song and Wang [11] from approximation of a fixed point of a single multivaued nonexpansive mapping to approximation of common fixed point of a finite family of quasi-nonexpansive multivaued mappings.

Remark 3.14. Our results extend the results of Shahzad and Zegeye [12] from approximation of a fixed point of a single quasi-nonexpansive multivaued mapping and single multivalued map to approximation of common fixed point of a finite family of quasi-nonexpansive multivaued mappings and a finite family of multivalued maps.