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 by𝐻(𝐴,𝐵)=maxsup𝑥∈𝐴𝑑(𝑥,𝐵),sup𝑦∈𝐵𝑑(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: 𝑥𝑛+1=𝛼𝑛𝑥𝑛+1−𝛼𝑛𝑇𝑥𝑛,∀𝑛≥1,(1.3) where the initial point 𝑥0 is taken arbitrarily in 𝐷 and {𝛼𝑛}âˆžğ‘›=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 ğ‘ž of 𝑇 under certain conditions. They also claimed that the fixed point 𝑝 may be different from ğ‘ž.

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≤𝛼𝑛<1 and (ii)Î£âˆžğ‘›=1𝛼𝑛=∞. Suppose that 𝐹(𝑇) a nonempty proximinal subset of 𝐷. Then, the Mann iterates {𝑥𝑛} defined by 𝑥0∈𝐷, 𝑥𝑛+1=𝛼𝑛𝑦𝑛+1−𝛼𝑛𝑥𝑛,𝛼𝑛∈[]ğ‘Ž,𝑏,0<ğ‘Ž<𝑏<1,𝑛≥0,(1.4) where 𝑦𝑛∈𝑇𝑥𝑛 such that ‖𝑦𝑛−𝑢𝑛‖=𝑑(𝑢𝑛,𝑇𝑥𝑛) and 𝑢𝑛∈𝐹(𝑇) such that ‖𝑥𝑛−𝑢𝑛‖=𝑑(𝑥𝑛,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≤𝛼𝑛,𝛽𝑛<1; (ii) 𝛽𝑛→0 and (iii) Î£âˆžğ‘›=1𝛼𝑛𝛽𝑛=∞. Then, the Ishikawa iterates {𝑥𝑛} defined by 𝑥0∈𝐷, 𝑦𝑛=𝛽𝑛𝑧𝑛+1−𝛽𝑛𝑥𝑛,𝛽𝑛∈[]0,1,𝑛≥0,(1.5)𝑧𝑛∈𝑇𝑥𝑛 such that ‖𝑧𝑛−𝑝‖=𝑑(𝑝,𝑇𝑥𝑛), and 𝑥𝑛+1=ğ›¼ğ‘›ğ‘§î…žğ‘›+1−𝛼𝑛𝑥𝑛,𝛼𝑛∈[]0,1,𝑛≠0,(1.6)ğ‘§î…žğ‘›âˆˆğ‘‡ğ‘¦ğ‘› such that â€–ğ‘§î…žğ‘›âˆ’ğ‘â€–=𝑑(𝑝,𝑇𝑦𝑛) 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 {𝑥𝑛}âˆžğ‘›=1 be a sequence defined iteratively by 𝑥0∈𝐷, 𝑦𝑛=𝛽𝑛𝑧𝑛+1−𝛽𝑛𝑥𝑛,𝛽𝑛∈[]0,1,𝑛≥0,(1.7)𝑧𝑛∈𝑇𝑥𝑛, and 𝑥𝑛+1=ğ›¼ğ‘›ğ‘§î…žğ‘›+1−𝛼𝑛𝑥𝑛,𝛼𝑛∈[]0,1,𝑛≠0,(1.8)ğ‘§î…žğ‘›âˆˆğ‘‡ğ‘¦ğ‘›. Assume that 𝑇 satisfies condition (I) and 𝛼𝑛,𝛽𝑛∈[ğ‘Ž,𝑏]⊂(0,1). Then, {𝑥𝑛}âˆžğ‘›=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 𝑃𝑇 is nonexpansive. Let {𝑥𝑛}âˆžğ‘›=1 be a sequence defined iteratively by 𝑥0∈𝐷, 𝑦𝑛=𝛽𝑛𝑧𝑛+1−𝛽𝑛𝑥𝑛,𝛽𝑛∈[]0,1,𝑛≥0,(1.9)𝑧𝑛∈𝑃𝑇(𝑥𝑛), and 𝑥𝑛+1=ğ›¼ğ‘›ğ‘§î…žğ‘›+1−𝛼𝑛𝑥𝑛,𝛼𝑛∈[]0,1,𝑛≠0,(1.10)ğ‘§î…žğ‘›âˆˆğ‘ƒğ‘‡(𝑦𝑛). Assume that 𝑇 satisfies condition (I) and 𝛼𝑛,𝛽𝑛∈[ğ‘Ž,𝑏]⊂(0,1). Then, {𝑥𝑛}âˆžğ‘›=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.𝑥1𝑥∈𝐷,𝑛+1=ğ‘Žğ‘›ğ‘¥ğ‘›+𝑏𝑛𝑦𝑛+𝑐𝑛𝑧𝑛,∀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 𝛿𝐸∶(0,2]→[0,1] defined by 𝛿𝐸||||||(𝜖)∶=inf1−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 𝛿𝐸(𝜖)>0 for all 𝜖∈(0,2].

A family {𝑇𝑖∶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,𝑇𝑖𝑥𝑑≥𝑓𝑥,∩𝑚𝑖=1𝐹𝑇𝑖,(2.2) for all 𝑖=1,2,…,𝑚 and 𝑥∈𝐷.

The mapping 𝑇∶𝐷→𝐶𝐵(𝐷) is called hemicompact if for any sequence {𝑥𝑛} in 𝐷 such that 𝑑(𝑥𝑛,𝑇𝑥𝑛)→0 as ğ‘›â†’âˆž, there exists a subsequence {𝑥𝑛𝑘} of {𝑥𝑛} such that 𝑥𝑛𝑘→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 𝑃𝑇𝑢(𝑥)∶=𝑥‖‖∈𝑇𝑥∶𝑥−𝑢𝑥‖‖=𝑑(𝑥,𝑇𝑥).(2.3) Then, 𝑃𝑇(𝑥)∶𝐷→𝑃(𝐷) is nonempty and compact for every 𝑥∈𝐷. Furthermore, we observe that 𝑃𝑇(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 𝑢𝑥∈𝑇𝑥 with 𝑑(𝑥,𝑢𝑥)=inf{𝑑(𝑥,𝑧)∶𝑧∈𝑇𝑥}, there exists 𝑢𝑦∈𝑇𝑦 with 𝑑(𝑦,𝑢𝑦)=inf{𝑑(𝑦,𝑤)∶𝑤∈𝑇𝑦} such that 𝑑𝑢𝑥,𝑢𝑦≤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 𝑃𝑇 is nonexpansive.

Throughout this paper, we write 𝑥𝑛→𝑥 to indicate that the sequence {𝑥𝑛} 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<ğ‘â‰¤ğ‘¡ğ‘›â‰¤ğ‘ž<1 for all positive integers 𝑛. Also, suppose that {𝑥𝑛} and {𝑦𝑛} are two sequences of 𝐸 such that limsupğ‘›â†’âˆžâ€–ğ‘¥ğ‘›â€–â‰¤ğ‘Ÿ,limsupğ‘›â†’âˆžâ€–ğ‘¦ğ‘›â€–â‰¤ğ‘Ÿ and limğ‘›â†’âˆžâ€–ğ‘¡ğ‘›ğ‘¥ğ‘›+(1−𝑡𝑛)𝑦𝑛‖=𝑟 hold for some 𝑟>0. Then, limğ‘›â†’âˆžâ€–ğ‘¥ğ‘›âˆ’ğ‘¦ğ‘›â€–=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,…,𝑇𝑚 be multivalued maps of 𝐷 into 𝑃(𝐷) with 𝐹∶=∩𝑚𝑖=1𝐹(𝑇𝑖)≠∅ such that 𝑃𝑇1,𝑃𝑇2,…,𝑃𝑇𝑚 are nonexpansive and {𝛼𝑛𝑖}âˆžğ‘›=1,𝑖=0,1,…,𝑚 a sequence in [𝜖,1−𝜖],𝜖∈(0,1) such that ∑𝑚𝑖=0𝛼𝑛𝑖=1 for all 𝑛≥1. Let {𝑥𝑛}âˆžğ‘›=1 be a sequence defined iteratively by𝑥1𝑥∈𝐷,𝑛+1=𝛼𝑛0𝑥𝑛+𝛼𝑛1𝑦𝑛(1)+⋯+𝛼𝑛𝑚𝑦𝑛(𝑚),(3.1) where 𝑦𝑛(𝑖)∈𝑃𝑇𝑖𝑥𝑛,𝑖=1,2,…,𝑚.

Lemma 3.1. Let 𝐸 be a real normed space and 𝐷 a nonempty subset of 𝐸. Let 𝑇1,𝑇2,…,𝑇𝑚 be multivalued maps of 𝐷 into 𝑃(𝐷) with 𝐹∶=∩𝑚𝑖=1𝐹(𝑇𝑖)≠∅ such that 𝑃𝑇1,𝑃𝑇2,…,𝑃𝑇𝑚 are nonexpansive. Let {𝛼𝑛𝑖}âˆžğ‘›=1,𝑖=0,1,…,𝑚 a sequence in [𝜖,1−𝜖],𝜖∈(0,1) such that ∑𝑚𝑖=0𝛼𝑛𝑖=1 for all 𝑛≥1. Let {𝑥𝑛}âˆžğ‘›=1 be a sequence defined iteratively by (3.1). Then, limğ‘›â†’âˆžğ‘‘î€·ğ‘¥ğ‘›,𝑇𝑖𝑥𝑛=0,∀𝑖=1,2,…,𝑚.(3.2)

Proof. Let 𝑥∗∈∩𝑚𝑖=1𝐹(𝑇𝑖). Then, from (3.1), we have the following estimates: ‖‖𝑥𝑛+1−𝑥∗‖‖≤𝛼𝑛0‖‖𝑥𝑛−𝑥∗‖‖+𝛼𝑛1‖‖𝑦𝑛(1)−𝑥∗‖‖+⋯+𝛼𝑛𝑚‖‖𝑦𝑛(𝑚)−𝑥∗‖‖≤𝛼𝑛0‖‖𝑥𝑛−𝑥∗‖‖+𝛼𝑛1𝑑𝑦𝑛(1),𝑃𝑇1𝑥∗+⋯+𝛼𝑛𝑚𝑑𝑦𝑛(𝑚),𝑃𝑇𝑚𝑥∗≤𝛼𝑛0‖‖𝑥𝑛−𝑥∗‖‖+𝛼𝑛1𝐻𝑃𝑇1𝑥𝑛,𝑃𝑇1𝑥∗+⋯+𝛼𝑛𝑚𝐻𝑃𝑇𝑚𝑥𝑛,𝑃𝑇𝑚𝑥∗≤𝛼𝑛0‖‖𝑥𝑛−𝑥∗‖‖+𝛼𝑛1‖‖𝑥𝑛−𝑥∗‖‖+⋯+𝛼𝑛𝑚‖‖𝑥𝑛−𝑥∗‖‖=‖‖𝑥𝑛−𝑥∗‖‖.(3.3) Thus, limğ‘›â†’âˆžâ€–ğ‘¥ğ‘›âˆ’ğ‘¥âˆ—â€– exists. Let limğ‘›â†’âˆžâ€–â€–ğ‘¥ğ‘›âˆ’ğ‘¥âˆ—â€–â€–=𝑐,(3.4) for some 𝑐≥0. Then, 𝑐=limğ‘›â†’âˆžâ€–â€–ğ‘¥ğ‘›+1−𝑥∗‖‖=limğ‘›â†’âˆžâ€–â€–ğ›¼ğ‘›0𝑥𝑛−𝑥∗+𝛼𝑛1𝑦𝑛(1)−𝑥∗+⋯+𝛼𝑛𝑚𝑦𝑛(𝑚)−𝑥∗‖‖=limğ‘›â†’âˆžâ€–â€–â€–î€·1−𝛼𝑛0𝛼𝑛11−𝛼𝑛0𝑦𝑛(1)−𝑥∗𝛼+⋯+𝑛𝑚1−𝛼𝑛0𝑦𝑛(𝑚)−𝑥∗+𝛼𝑛0𝑥𝑛−𝑥∗‖‖‖.(3.5) Since 𝑃𝑇𝑖,𝑖=1,2,…,𝑚 is nonexpansive mapping and ∩𝑚𝑖=1𝐹(𝑇𝑖)≠∅, we have ‖‖𝑦𝑛(𝑖)−𝑥∗‖‖𝑦≤𝑑𝑛(𝑖),𝑃𝑇𝑖𝑥∗𝑃≤𝐻𝑇𝑖𝑥𝑛,𝑃𝑇𝑖𝑥∗≤‖‖𝑥𝑛−𝑥∗‖‖,(3.6) for each 𝑥∗∈∩𝑚𝑖=1𝐹(𝑇𝑖). Taking limsup on both sides, we obtain limsupğ‘›â†’âˆžâ€–â€–ğ‘¦ğ‘›(𝑖)−𝑥∗‖‖≤𝑐,𝑖=1,2,…,𝑚.(3.7) Next, limsupğ‘›â†’âˆžâ€–â€–â€–ğ›¼ğ‘›11−𝛼𝑛0𝑦𝑛(1)−𝑥∗+𝛼𝑛21−𝛼𝑛0𝑦𝑛(2)−𝑥∗𝛼+⋯+𝑛𝑚1−𝛼𝑛0𝑦𝑛(𝑚)−𝑥∗‖‖‖≤limsupğ‘›â†’âˆžî‚¸ğ›¼ğ‘›11−𝛼𝑛0‖‖𝑦𝑛(1)−𝑥∗‖‖+𝛼𝑛21−𝛼𝑛0‖‖𝑦𝑛(2)−𝑥∗‖‖𝛼+⋯+𝑛𝑚1−𝛼𝑛0‖‖𝑦𝑛(𝑚)−𝑥∗‖‖≤limsupğ‘›â†’âˆžğ›¼ğ‘›1+𝛼𝑛2+⋯+𝛼𝑛𝑚1−𝛼𝑛0‖‖𝑥𝑛−𝑥∗‖‖=𝑐.(3.8) Using (3.5), (3.8), and Lemma 2.1, we obtain limğ‘›â†’âˆžâ€–â€–â€–ğ›¼ğ‘›11−𝛼𝑛0𝑦𝑛(1)−𝑥∗+𝛼𝑛21−𝛼𝑛0𝑦𝑛(2)−𝑥∗𝛼+⋯+𝑛𝑚1−𝛼𝑛0𝑦𝑛(𝑚)−𝑥∗−𝑥𝑛−𝑥∗‖‖‖=0.(3.9) This yields 0=limğ‘›â†’âˆžâ€–â€–â€–ğ›¼ğ‘›11−𝛼𝑛0𝑦𝑛(1)+𝛼𝑛21−𝛼𝑛0𝑦𝑛(2)𝛼+⋯+𝑛𝑚1−𝛼𝑛0𝑦𝑛(𝑚)−𝑥𝑛‖‖‖=limğ‘›â†’âˆžî‚µ11−𝛼𝑛0‖‖𝛼𝑛1𝑦𝑛(1)+𝛼𝑛2𝑦𝑛(2)+⋯+𝛼𝑛𝑚𝑦𝑛(𝑚)−1−𝛼𝑛0𝑥𝑛‖‖=limğ‘›â†’âˆžî‚µ11−𝛼𝑛0‖‖𝑥𝑛+1−𝑥𝑛‖‖.(3.10) Thus, limğ‘›â†’âˆžâ€–ğ‘¥ğ‘›+1−𝑥𝑛‖=0. Furthermore, 𝑐=limğ‘›â†’âˆžâ€–â€–ğ‘¥ğ‘›+1−𝑥∗‖‖=limğ‘›â†’âˆžâ€–â€–ğ›¼ğ‘›0𝑥𝑛−𝑥∗+𝛼𝑛1𝑦𝑛(1)−𝑥∗+⋯+𝛼𝑛𝑚𝑦𝑛(𝑚)−𝑥∗‖‖=limğ‘›â†’âˆžâ€–â€–â€–î€·1−𝛼𝑛1𝛼𝑛01−𝛼𝑛1𝑥𝑛−𝑥∗+𝛼𝑛21−𝛼𝑛1𝑦𝑛(2)−𝑥∗𝛼+⋯+𝑛𝑚1−𝛼𝑛1𝑦𝑛(𝑚)−𝑥∗+𝛼𝑛1𝑦𝑛(1)−𝑥∗‖‖‖,limsupğ‘›â†’âˆžâ€–â€–â€–ğ›¼ğ‘›01−𝛼𝑛1𝑥𝑛−𝑥∗+𝛼𝑛21−𝛼𝑛1𝑦𝑛(2)−𝑥∗𝛼+⋯+𝑛𝑚1−𝛼𝑛1𝑦𝑛(𝑚)−𝑥∗‖‖‖≤limsupğ‘›â†’âˆžî‚¸ğ›¼ğ‘›01−𝛼𝑛1‖‖𝑥𝑛−𝑥∗‖‖+𝛼𝑛21−𝛼𝑛1‖‖𝑦𝑛(2)−𝑥∗‖‖𝛼+⋯+𝑛𝑚1−𝛼𝑛1‖‖𝑦𝑛(𝑚)−𝑥∗‖‖≤limsupğ‘›â†’âˆžğ›¼ğ‘›0+𝛼𝑛2+⋯+𝛼𝑛𝑚1−𝛼𝑛1‖‖𝑥𝑛−𝑥∗‖‖=𝑐.(3.11) Using (3.11) and Lemma 2.1, we obtain limğ‘›â†’âˆžâ€–â€–â€–ğ›¼ğ‘›01−𝛼𝑛1𝑥𝑛−𝑥∗+𝛼𝑛21−𝛼𝑛1𝑦𝑛(2)−𝑥∗𝛼+⋯+𝑛𝑚1−𝛼𝑛1𝑦𝑛(𝑚)−𝑥∗−𝑦𝑛(1)−𝑥∗‖‖‖=0.(3.12) This yields 0=limğ‘›â†’âˆžâ€–â€–â€–ğ›¼ğ‘›01−𝛼𝑛1𝑥𝑛+𝛼𝑛21−𝛼𝑛1𝑦𝑛(2)𝛼+⋯+𝑛𝑚1−𝛼𝑛1𝑦𝑛(𝑚)−𝑦𝑛(1)‖‖‖=limğ‘›â†’âˆžî‚µ11−𝛼𝑛1‖‖𝛼𝑛0𝑥𝑛+𝛼𝑛2𝑦𝑛(2)+⋯+𝛼𝑛𝑚𝑦𝑛(𝑚)−1−𝛼𝑛1𝑦𝑛(1)‖‖=limğ‘›â†’âˆžî‚µ11−𝛼𝑛1‖‖𝑥𝑛+1−𝑦𝑛(1)‖‖.(3.13) Thus, limğ‘›â†’âˆžâ€–ğ‘¥ğ‘›+1−𝑦𝑛(1)‖=0. So, ‖‖𝑥𝑛−𝑦𝑛(1)‖‖≤‖‖𝑥𝑛+1−𝑥𝑛‖‖+‖‖𝑥𝑛+1−𝑦𝑛(1)‖‖⟶0,ğ‘›âŸ¶âˆž.(3.14) Then, 𝑑𝑥𝑛,𝑇1𝑥𝑛𝑥≤𝑑𝑛,𝑃𝑇1𝑥𝑛≤‖‖𝑥𝑛−𝑦𝑛(1)‖‖⟶0,ğ‘›âŸ¶âˆž.(3.15) In a similar way, we can show that limğ‘›â†’âˆžğ‘‘î€·ğ‘¥ğ‘›,𝑇𝑖𝑥𝑛=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,…,𝑇𝑚 be multivalued maps of 𝐷 into 𝑃(𝐷) with 𝐹∶=∩𝑚𝑖=1𝐹(𝑇𝑖)≠∅ such that 𝑃𝑇1,𝑃𝑇2,…,𝑃𝑇𝑚 are nonexpansive and {𝑇𝑖}𝑚i=1 satisfying condition (II). Let {𝛼𝑛𝑖}âˆžğ‘›=1,𝑖=0,1,…,𝑚 a sequence in [𝜖,1−𝜖],𝜖∈(0,1) such that ∑𝑚𝑖=0𝛼𝑛𝑖=1 for all 𝑛≥1. Let {𝑥𝑛}âˆžğ‘›=1 be a sequence defined iteratively by (3.1). Then, {𝑥𝑛}âˆžğ‘›=1 converges strongly to a common fixed point of {𝑇𝑖}𝑚𝑖=1.

Proof. Since {𝑇𝑖}𝑚𝑖=1 satisfies condition (II), we have that 𝑑(𝑥𝑛,∩𝑚𝑖=1𝐹(𝑇𝑖))→0 as ğ‘›â†’âˆž. Thus, there is a subsequence {𝑥𝑛𝑘} of {𝑥𝑛} and a sequence {𝑝𝑘}⊂∩𝑚𝑖=1𝐹(𝑇𝑖) such that ‖‖𝑥𝑛𝑘−𝑝𝑘‖‖<12𝑘,(3.17) for all 𝑘. By Lemma 3.1, we obtain ‖‖𝑥𝑛𝑘+1−𝑝𝑘‖‖≤‖‖𝑥𝑛𝑘−𝑝𝑘‖‖<12𝑘.(3.18) We now show that {𝑝𝑘} is a Cauchy sequence in 𝐷. Observe that ‖‖𝑝𝑘+1−𝑝𝑘‖‖≤‖‖𝑝𝑘+1−𝑥𝑛𝑘+1‖‖+‖‖𝑥𝑛𝑘+1−𝑝𝑘‖‖<12𝑘+1+12𝑘<12𝑘−1.(3.19) This shows that {𝑝𝑘} is a Cauchy sequence in 𝐷, and thus converges to 𝑝∈𝐷. Since 𝑑𝑝𝑘,𝑇𝑖𝑝‖‖𝑝≤𝑑𝑘,𝑃𝑇𝑖𝑝𝑃≤𝐻𝑇𝑖𝑝,𝑃𝑇𝑖𝑝𝑘≤‖‖𝑝−𝑝𝑘‖‖,(3.20) and 𝑝𝑘→𝑝 as ğ‘˜â†’âˆž, it follows that 𝑑(𝑝,𝑇𝑖𝑝)=0, and thus 𝑝∈∩𝑚𝑖=1𝐹(𝑇𝑖), and {𝑥n𝑘} converges strongly to 𝑝. Since limğ‘›â†’âˆžâ€–ğ‘¥ğ‘›âˆ’ğ‘â€– exists, it follows that {𝑥𝑛} 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,…,𝑇𝑚 be ∗-nonexpansive multimaps of 𝐷 into 𝑃(𝐷) with 𝐹∶=∩𝑚𝑖=1𝐹(𝑇𝑖)≠∅ and {𝑇𝑖}𝑚𝑖=1 satisfying condition (II). Let {𝛼𝑛𝑖}âˆžğ‘›=1,𝑖=0,1,…,𝑚 a sequence in [𝜖,1−𝜖],𝜖∈(0,1) such that ∑𝑚𝑖=0𝛼𝑛𝑖=1 for all 𝑛≥1. Let {𝑥𝑛}âˆžğ‘›=1 be a sequence defined iteratively by (3.1). Then, {𝑥𝑛}âˆžğ‘›=1 converges strongly to a common fixed point of {𝑇𝑖}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,…,𝑇𝑚 be multivalued maps of 𝐷 into 𝑃(𝐷) with 𝐹∶=∩𝑚𝑖=1𝐹(𝑇𝑖)≠∅ such that 𝑃𝑇1,𝑃𝑇2,…,𝑃𝑇𝑚 are nonexpansive and 𝑇𝑖 is hemicompact and continuous for each 𝑖=1,2,…,𝑚. Let {𝛼𝑛𝑖}âˆžğ‘›=1,𝑖=0,1,…,𝑚 a sequence in [𝜖,1−𝜖],𝜖∈(0,1) such that ∑𝑚𝑖=0𝛼𝑛𝑖=1 for all 𝑛≥1. Let {𝑥𝑛}âˆžğ‘›=1 be a sequence defined iteratively by (3.1). Then, {𝑥𝑛}âˆžğ‘›=1 converges strongly to a common fixed point of {𝑇𝑖}𝑚𝑖=1.

Proof. Since limğ‘›â†’âˆžğ‘‘(𝑥𝑛,𝑇𝑖𝑥𝑛)=0,for all𝑖=1,2,…,𝑚 and 𝑇𝑖 is hemicompact for each 𝑖=1,2,…,𝑚, there is a subsequence {𝑥𝑛𝑘} of {𝑥𝑛} such that 𝑥𝑛𝑘→𝑝 as ğ‘˜â†’âˆž for some 𝑝∈𝐷. Since 𝑇𝑖 is continuous for each 𝑖=1,2,…,𝑚, we have 𝑑(𝑥𝑛𝑘,𝑇𝑖𝑥𝑛𝑘)→𝑑(𝑝,𝑇𝑖𝑝). As a result, we have that 𝑑(𝑝,𝑇𝑖𝑝)=0,for all𝑖=1,2,…,𝑚, and so, 𝑝∈∩𝑚𝑖=1𝐹(𝑇𝑖). Since limğ‘›â†’âˆžâ€–ğ‘¥ğ‘›âˆ’ğ‘â€– exists, it follows that {𝑥𝑛} 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,…,𝑇𝑚 be ∗-nonexpansive multimaps of 𝐷 into 𝑃(𝐷) with 𝐹∶=∩𝑚𝑖=1𝐹(𝑇𝑖)≠∅ and 𝑇𝑖 is hemicompact and continuous for each i=1,2,…,𝑚. Let {𝛼𝑛𝑖}âˆžğ‘›=1,𝑖=0,1,…,𝑚 a sequence in [𝜖,1−𝜖],𝜖∈(0,1) such that ∑𝑚𝑖=0𝛼𝑛𝑖=1 for all 𝑛≥1. Let {𝑥𝑛}âˆžğ‘›=1 be a sequence defined iteratively by (3.1). Then, {𝑥𝑛}âˆžğ‘›=1 converges strongly to a common fixed point of {𝑇𝑖}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,…,𝑇𝑚 be multivalued maps of 𝐷 into 𝑃(𝐷) with 𝐹∶=∩𝑚𝑖=1𝐹(𝑇𝑖)≠∅ such that 𝑃𝑇1,𝑃𝑇2,…,𝑃T𝑚 are nonexpansive. Let {𝛼𝑛𝑖}âˆžğ‘›=1,𝑖=0,1,…,𝑚 a sequence in [𝜖,1−𝜖],𝜖∈(0,1) such that ∑𝑚𝑖=0𝛼𝑛𝑖=1 for all 𝑛≥1. Let {𝑥𝑛}âˆžğ‘›=1 be a sequence defined iteratively by (3.1). Then, {𝑥𝑛}âˆžğ‘›=1 converges strongly to a common fixed point of {𝑇𝑖}𝑚𝑖=1.

Proof. From the compactness of 𝐷, there exists a subsequence {𝑥𝑛𝑘}âˆžğ‘›=𝑘 of {𝑥𝑛}âˆžğ‘›=1 such that limğ‘˜â†’âˆžâ€–ğ‘¥ğ‘›ğ‘˜âˆ’ğ‘žâ€–=0 for some ğ‘žâˆˆğ·. Thus, ğ‘‘î€·ğ‘ž,ğ‘‡ğ‘–ğ‘žî€¸î€·â‰¤ğ‘‘ğ‘ž,ğ‘ƒğ‘‡ğ‘–ğ‘žî€¸â‰¤â€–â€–ğ‘¥ğ‘›ğ‘˜â€–â€–î€·ğ‘¥âˆ’ğ‘ž+𝑑𝑛𝑘,𝑃𝑇𝑖𝑥𝑛𝑘𝑃+𝐻𝑇𝑖𝑥𝑛𝑘,ğ‘ƒğ‘‡ğ‘–ğ‘žî€¸â€–â€–ğ‘¥â‰¤2ğ‘›ğ‘˜â€–â€–î€·ğ‘¥âˆ’ğ‘ž+𝑑𝑛𝑘,𝑃𝑇𝑖𝑥𝑛𝑘⟶0asğ‘˜âŸ¶âˆž.(3.21) Hence, ğ‘žâˆˆâˆ©ğ‘šğ‘–=1𝐹(𝑇𝑖). Now, on taking ğ‘ž in place of 𝑥∗, we get that limğ‘›â†’âˆžâ€–ğ‘¥ğ‘›âˆ’ğ‘žâ€– 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 {𝑇𝑖}𝑚𝑖=1.

Theorem 3.7. Let 𝐷 be a nonempty, closed, and convex subset of a real Banach space 𝐸. Let 𝑇1,𝑇2,…,𝑇𝑚 be multivalued maps of 𝐷 into 𝑃(𝐷) with 𝐹∶=∩𝑚𝑖=1𝐹(𝑇𝑖)≠∅ such that 𝑃𝑇1,𝑃𝑇2,…,𝑃𝑇𝑚 are nonexpansive. Let {𝛼𝑛𝑖}âˆžğ‘›=1,𝑖=0,1,…,𝑚 a sequence in [𝜖,1−𝜖],𝜖∈(0,1) such that ∑𝑚𝑖=0𝛼𝑛𝑖=1 for all 𝑛≥1. Let {𝑥𝑛}âˆžğ‘›=1 be a sequence defined iteratively by (3.1). Then, {𝑥𝑛}âˆžğ‘›=1 converges strongly to a common fixed point of {𝑇𝑖}𝑚𝑖=1 if and only if liminfğ‘›â†’âˆžğ‘‘(𝑥𝑛,𝐹)=0.

Proof. The necessity is obvious. Conversely, suppose that liminfğ‘›â†’âˆžğ‘‘(𝑥𝑛,F)=0. By (3.3), we have ‖‖𝑥𝑛+1−𝑥∗‖‖≤‖‖𝑥𝑛−𝑥∗‖‖.(3.22) This gives 𝑑𝑥𝑛+1𝑥,𝐹≤𝑑𝑛,𝐹.(3.23) Hence, limğ‘›â†’âˆžğ‘‘(𝑥𝑛,𝐹) exists. By hypothesis, liminfğ‘›â†’âˆžğ‘‘(𝑥𝑛,𝐹)=0, so we must have limğ‘›â†’âˆžğ‘‘(𝑥𝑛,𝐹)=0.
Next, we show that {𝑥𝑛}âˆžğ‘›=1 is a Cauchy sequence in 𝐷. Let 𝜖>0 be given, and since liminfğ‘›â†’âˆžğ‘‘(𝑥𝑛,𝐹)=0, there exists 𝑛0 such that for all 𝑛≥𝑛0, we have 𝑑𝑥𝑛<𝜖,𝐹4.(3.24) In particular, inf{‖𝑥𝑛0−𝑝‖∶𝑝∈𝐹}<𝜖/4 so that there must exist a 𝑝∗∈𝐹 such that ‖‖𝑥𝑛0−𝑝∗‖‖<𝜖2.(3.25) Now, for 𝑚,𝑛≥𝑛0, we have ‖‖𝑥𝑛+𝑚−𝑥𝑛‖‖≤||‖‖𝑥𝑛+𝑚−𝑝∗‖‖+‖‖𝑥𝑛−𝑝∗‖‖‖‖𝑥≤2𝑛0−𝑝∗‖‖𝜖<22=𝜖.(3.26) Hence, {𝑥𝑛} is a Cauchy sequence in a closed subset 𝐷 of a Banach space 𝐸, and therefore, it must converge in 𝐷. Let limğ‘›â†’âˆžğ‘¥ğ‘›=𝑝. Now, for each 𝑖=1,2,…,𝑚, we obtain 𝑑𝑝,𝑇𝑖𝑝≤𝑑𝑝,𝑃𝑇𝑖𝑝≤𝑑𝑝,𝑥𝑛𝑥+𝑑𝑛,𝑃𝑇𝑖𝑥𝑛𝑃+𝐻𝑇𝑖𝑥𝑛,𝑃𝑇𝑖𝑝≤𝑑𝑝,𝑥𝑛𝑥+𝑑𝑛,𝑃𝑇𝑖𝑥𝑛𝑥+𝑑𝑛,𝑝⟶0asğ‘›âŸ¶âˆž(3.27) gives that 𝑑(𝑝,𝑇𝑖𝑝)=0,𝑖=1,2,…,𝑚 which implies that 𝑝∈𝑇𝑖𝑝. Consequently, 𝑝∈𝐹=∩𝑚𝑖=1𝐹(𝑇𝑖)≠∅.

All the results we have obtained so far can be established for finite family of quasi-nonexpansive multivalued maps. Let 𝑇1,𝑇2,…,𝑇𝑚 be quasi-nonexpansive multivalued maps of 𝐷 into 𝐶𝐵(𝐷) such that 𝐹∶=∩𝑚𝑖=1𝐹(𝑇𝑖)≠∅ for which 𝑇𝑖𝑝={𝑝}, for all 𝑝∈∩𝑚𝑖=1𝐹(𝑇𝑖). Let {𝛼𝑛𝑖}âˆžğ‘›=1,𝑖=0,1,…,𝑚 a sequence in [𝜖,1−𝜖],𝜖∈(0,1) such that ∑𝑚𝑖=0𝛼𝑛𝑖=1 for all 𝑛≥1. Let {𝑥𝑛}âˆžğ‘›=1 be a sequence defined iteratively by𝑥1𝑥∈𝐷,𝑛+1=𝛼𝑛0𝑥𝑛+𝛼n1𝑦𝑛(1)+⋯+𝛼𝑛𝑚𝑦𝑛(𝑚),(3.28) where 𝑦𝑛(𝑖)∈𝑇𝑖𝑥𝑛, 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,…,𝑇𝑚 be quasi-nonexpansive multivalued maps of 𝐷 into 𝐶𝐵(𝐷) such that 𝐹∶=∩𝑚𝑖=1𝐹(𝑇𝑖)≠∅ for which 𝑇𝑖𝑝={𝑝}, for all 𝑝∈∩𝑚𝑖=1𝐹(𝑇𝑖) and {𝑇𝑖}𝑚𝑖=1 satisfying condition (II). Let {𝛼𝑛𝑖}âˆžğ‘›=1,𝑖=0,1,…,𝑚 a sequence in [𝜖,1−𝜖],𝜖∈(0,1) such that ∑𝑚𝑖=0𝛼𝑛𝑖=1 for all 𝑛≥1. Let {𝑥𝑛}âˆžğ‘›=1 be a sequence defined iteratively by (3.28). Then, {𝑥𝑛}âˆžğ‘›=1 converges strongly to a common fixed point of {𝑇𝑖}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,…,𝑇𝑚 be quasi-nonexpansive multivalued maps of 𝐷 into 𝐶𝐵(𝐷) such that 𝐹∶=∩𝑚𝑖=1𝐹(𝑇𝑖)≠∅ for which 𝑇𝑖𝑝={𝑝}, for all 𝑝∈∩𝑚𝑖=1𝐹(𝑇𝑖) and 𝑇𝑖 is hemicompact and continuous for each 𝑖=1,2,…,𝑚. Let {𝛼𝑛𝑖}âˆžğ‘›=1,𝑖=0,1,…,𝑚 a sequence in [𝜖,1−𝜖],𝜖∈(0,1) such that ∑𝑚𝑖=0𝛼𝑛𝑖=1 for all 𝑛≥1. Let {𝑥𝑛}âˆžğ‘›=1 be a sequence defined iteratively by (3.28). Then, {𝑥𝑛}âˆžğ‘›=1 converges strongly to a common fixed point of {𝑇𝑖}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,…,𝑇𝑚 be quasi-nonexpansive multivalued maps of 𝐷 into 𝐶𝐵(𝐷) such that 𝐹∶=∩𝑚𝑖=1𝐹(𝑇𝑖)≠∅ for which 𝑇𝑖𝑝={𝑝}, for all 𝑝∈∩𝑚𝑖=1𝐹(𝑇𝑖). {𝛼𝑛𝑖}âˆžğ‘›=1,𝑖=0,1,…,𝑚 a sequence in [𝜖,1−𝜖],𝜖∈(0,1) such that ∑𝑚𝑖=0𝛼𝑛𝑖=1 for all 𝑛≥1. Let {𝑥𝑛}âˆžğ‘›=1 be a sequence defined iteratively by (3.28). Then, {𝑥𝑛}âˆžğ‘›=1 converges strongly to a common fixed point of {𝑇𝑖}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 {ğ‘Žğ‘›}âˆžğ‘›=1,{𝑏𝑛}âˆžğ‘›=1, and {𝑐𝑛}âˆžğ‘›=1 be sequence in (0,1) satisfying ğ‘Žğ‘›+𝑏𝑛+𝑐𝑛≤1. Let {𝑥𝑛}âˆžğ‘›=1 be a sequence defined iteratively by 𝑥1𝑥∈𝐷,𝑛+1=ğ‘Žğ‘›ğ‘¥ğ‘›+𝑏𝑛𝑦𝑛+𝑐𝑛𝑧𝑛,∀𝑛≥1,(3.29) where 𝑦𝑛∈𝑇𝑥𝑛,𝑧𝑛∈𝑆𝑥𝑛 such that ‖𝑦𝑛−𝑝‖≤𝑑(𝑝,𝑇𝑥𝑛) and ‖𝑧𝑛−𝑝‖≤𝑑(𝑝,𝑆𝑥𝑛) whenever 𝑝 is a fixed point of any one of mappings 𝑇 and 𝑆. Then, {𝑥𝑛}âˆžğ‘›=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 {ğ‘Žğ‘›}âˆžğ‘›=1,{𝑏𝑛}âˆžğ‘›=1 and {𝑐𝑛}âˆžğ‘›=1 be sequence in (0,1) satisfying ğ‘Žğ‘›+𝑏𝑛+𝑐𝑛≤1. Let {𝑥𝑛}âˆžğ‘›=1 be a sequence defined iteratively by 𝑥1𝑥∈𝐷,𝑛+1=ğ‘Žğ‘›ğ‘¥ğ‘›+𝑏𝑛𝑦𝑛+𝑐𝑛𝑧𝑛,∀𝑛≥1,(3.30) where 𝑦𝑛∈𝑇𝑥𝑛, 𝑧𝑛∈𝑆𝑥𝑛 such that ‖𝑦𝑛−𝑝‖≤𝑑(𝑝,𝑇𝑥𝑛) and ‖𝑧𝑛−𝑝‖≤𝑑(𝑝,𝑆𝑥𝑛) whenever 𝑝 is a fixed point of any one of mappings 𝑇 and 𝑆. Then, {𝑥𝑛}âˆžğ‘›=1 converges strongly to a common fixed point of 𝐹(𝑇)∩𝐹(𝑆) if and only if liminfğ‘›â†’âˆžğ‘‘(𝑥𝑛,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.