International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 576108 | 13 pages | https://doi.org/10.5402/2011/576108

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

Academic Editor: O. Miyagaki
Received20 Apr 2011
Accepted09 Jun 2011
Published28 Aug 2011

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 π‘₯∈𝐻, there exists π‘¦βˆˆπ· such that β€–π‘₯βˆ’π‘¦β€–=𝑑(π‘₯,𝐷), where 𝑑(π‘₯,𝐷)=inf{β€–π‘₯βˆ’π‘§β€–βˆΆπ‘§βˆˆπ·}. Let 𝐢𝐡(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 𝐢𝐡(𝐷) is defined by𝐻(𝐴,𝐡)=maxsupπ‘₯βˆˆπ΄π‘‘(π‘₯,𝐡),supπ‘¦βˆˆπ΅ξƒ°π‘‘(𝑦,𝐴),(1.1) for 𝐴,𝐡∈𝐢𝐡(𝐷). A single-valued map π‘‡βˆΆπ·β†’π· is called nonexpansive if ‖𝑇π‘₯βˆ’π‘‡π‘¦β€–β‰€β€–π‘₯βˆ’π‘¦β€– for all π‘₯,π‘¦βˆˆπ·. A multivalued map π‘‡βˆΆπ·β†’πΆπ΅(𝐷) is said to be nonexpansive if 𝐻(𝑇π‘₯,𝑇𝑦)≀‖π‘₯βˆ’π‘¦β€– for all π‘₯,π‘¦βˆˆπ·. An element π‘βˆˆπ· is called a fixed point of π‘‡βˆΆπ·β†’π· (resp., π‘‡βˆΆπ·β†’πΆπ΅(𝐷)) if 𝑝=𝑇𝑝 (resp., π‘βˆˆπ‘‡π‘). The set of fixed points of 𝑇 is denoted by 𝐹(𝑇). A multivalued map π‘‡βˆΆπ·β†’πΆπ΅(𝐷) is said to be quasi-nonexpansive if 𝐻(𝑇π‘₯,𝑇𝑝)≀‖π‘₯βˆ’π‘β€– for all π‘₯∈𝐷 and for all π‘βˆˆπΉ(𝑇).

A π‘‡βˆΆπ·β†’πΆπ΅(𝐷) is said to satisfy Condition (I) if there is a nondecreasing function π‘“βˆΆ[0,∞)β†’[0,∞) with 𝑓(0)=0,𝑓(π‘Ÿ)>0 for π‘Ÿβˆˆ(0,∞) such that 𝑑(π‘₯,𝑇π‘₯)β‰₯𝑓(𝑑(π‘₯,𝐹(𝑇))),(1.2) for all π‘₯∈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 π‘‡βˆΆπ·β†’πΆ(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 π‘‡βˆΆπ·β†’πΆπ΅(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 π‘‡βˆΆπ·β†’πΆπ΅(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: π‘₯𝑛+1=𝛼𝑛π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡π‘₯𝑛,βˆ€π‘›β‰₯1,(1.3) where the initial point π‘₯0 is taken arbitrarily in 𝐷 and {𝛼𝑛}βˆžπ‘›=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 π‘ž of 𝑇 under certain conditions. They also claimed that the fixed point 𝑝 may be different from π‘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 π‘‡βˆΆπ·β†’π‘ƒ(𝐷) a multivalued nonexpansive mapping that satisfies condition (𝐼). Assume that (i)0≀𝛼𝑛<1 and (ii)Ξ£βˆžπ‘›=1𝛼𝑛=∞. Suppose that 𝐹(𝑇) a nonempty proximinal subset of 𝐷. Then, the Mann iterates {π‘₯𝑛} defined by π‘₯0∈𝐷, π‘₯𝑛+1=𝛼𝑛𝑦𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘₯𝑛,π›Όπ‘›βˆˆ[]π‘Ž,𝑏,0<π‘Ž<𝑏<1,𝑛β‰₯0,(1.4) where π‘¦π‘›βˆˆπ‘‡π‘₯𝑛 such that β€–π‘¦π‘›βˆ’π‘’π‘›β€–=𝑑(𝑒𝑛,𝑇π‘₯𝑛) and π‘’π‘›βˆˆπΉ(𝑇) such that β€–π‘₯π‘›βˆ’π‘’π‘›β€–=𝑑(π‘₯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 π‘‡βˆΆπ·β†’π‘ƒ(𝐷) a multivalued nonexpansive mapping with a fixed point 𝑝. Assume that (i) 0≀𝛼𝑛,𝛽𝑛<1; (ii) 𝛽𝑛→0 and (iii) Ξ£βˆžπ‘›=1𝛼𝑛𝛽𝑛=∞. Then, the Ishikawa iterates {π‘₯𝑛} defined by π‘₯0∈𝐷, 𝑦𝑛=𝛽𝑛𝑧𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ‘₯𝑛,π›½π‘›βˆˆ[]0,1,𝑛β‰₯0,(1.5)π‘§π‘›βˆˆπ‘‡π‘₯𝑛 such that β€–π‘§π‘›βˆ’π‘β€–=𝑑(𝑝,𝑇π‘₯𝑛), and π‘₯𝑛+1=π›Όπ‘›π‘§ξ…žπ‘›+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘₯𝑛,π›Όπ‘›βˆˆ[]0,1,𝑛≠0,(1.6)π‘§ξ…žπ‘›βˆˆπ‘‡π‘¦π‘› such that β€–π‘§ξ…žπ‘›βˆ’π‘β€–=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 π‘‡βˆΆπ·β†’πΆπ΅(𝐷) be a quasi-nonexpansive multivalued map with 𝐹(𝑇)β‰ βˆ… for which 𝑇𝑝={𝑝}, for allπ‘βˆˆπΉ(𝑇). Let {π‘₯𝑛}βˆžπ‘›=1 be a sequence defined iteratively by π‘₯0∈𝐷, 𝑦𝑛=𝛽𝑛𝑧𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ‘₯𝑛,π›½π‘›βˆˆ[]0,1,𝑛β‰₯0,(1.7)π‘§π‘›βˆˆπ‘‡π‘₯𝑛, and π‘₯𝑛+1=π›Όπ‘›π‘§ξ…žπ‘›+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘₯𝑛,π›Όπ‘›βˆˆ[]0,1,𝑛≠0,(1.8)π‘§ξ…žπ‘›βˆˆπ‘‡π‘¦π‘›. Assume that 𝑇 satisfies condition (I) and 𝛼𝑛,π›½π‘›βˆˆ[π‘Ž,𝑏]βŠ‚(0,1). Then, {π‘₯𝑛}βˆžπ‘›=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 π‘‡βˆΆπ·β†’π‘ƒ(𝐷) be a multivalued map with F(𝑇)β‰ βˆ… such that 𝑃𝑇 is nonexpansive. Let {π‘₯𝑛}βˆžπ‘›=1 be a sequence defined iteratively by π‘₯0∈𝐷, 𝑦𝑛=𝛽𝑛𝑧𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ‘₯𝑛,π›½π‘›βˆˆ[]0,1,𝑛β‰₯0,(1.9)π‘§π‘›βˆˆπ‘ƒπ‘‡(π‘₯𝑛), and π‘₯𝑛+1=π›Όπ‘›π‘§ξ…žπ‘›+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘₯𝑛,π›Όπ‘›βˆˆ[]0,1,𝑛≠0,(1.10)π‘§ξ…žπ‘›βˆˆπ‘ƒπ‘‡(𝑦𝑛). Assume that 𝑇 satisfies condition (I) and 𝛼𝑛,π›½π‘›βˆˆ[π‘Ž,𝑏]βŠ‚(0,1). Then, {π‘₯𝑛}βˆžπ‘›=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.π‘₯1π‘₯∈𝐷,𝑛+1=π‘Žπ‘›π‘₯𝑛+𝑏𝑛𝑦𝑛+𝑐𝑛𝑧𝑛,βˆ€π‘›β‰₯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 dim𝐸β‰₯2. The modulus of convexity of 𝐸 is the function π›ΏπΈβˆΆ(0,2]β†’[0,1] defined by 𝛿𝐸||||||(πœ–)∢=inf1βˆ’π‘₯+𝑦2||||||ξ‚‡βˆΆβ€–π‘₯β€–=‖𝑦‖=1;πœ–=β€–π‘₯βˆ’π‘¦β€–.(2.1)𝐸 is uniformly convex if for any πœ–βˆˆ(0,2], there exists a 𝛿=𝛿(πœ–)>0 such that if π‘₯,π‘¦βˆˆπΈ with β€–π‘₯‖≀1,‖𝑦‖≀1 and β€–π‘₯βˆ’π‘¦β€–β‰₯πœ–, then β€–(1/2)(π‘₯+𝑦)‖≀1βˆ’π›Ώ. Equivalently, 𝐸 is uniformly convex if and only if 𝛿𝐸(πœ–)>0 for all πœ–βˆˆ(0,2].

A family {π‘‡π‘–βˆΆπ·β†’πΆπ΅(𝐷),𝑖=1,2,…,π‘š} is said to satisfy Condition (II) if there is a nondecreasing function π‘“βˆΆ[0,∞)β†’[0,∞) with 𝑓(0)=0,𝑓(π‘Ÿ)>0 for π‘Ÿβˆˆ(0,∞) such that 𝑑π‘₯,𝑇𝑖π‘₯𝑑β‰₯𝑓π‘₯,βˆ©π‘šπ‘–=1𝐹𝑇𝑖,(2.2) for all 𝑖=1,2,…,π‘š and π‘₯∈𝐷.

The mapping π‘‡βˆΆπ·β†’πΆπ΅(𝐷) is called hemicompact if for any sequence {π‘₯𝑛} in 𝐷 such that 𝑑(π‘₯𝑛,𝑇π‘₯𝑛)β†’0 as π‘›β†’βˆž, there exists a subsequence {π‘₯π‘›π‘˜} of {π‘₯𝑛} such that π‘₯π‘›π‘˜β†’π‘βˆˆπ·. We note that if 𝐷 is compact, then every multivalued mapping π‘‡βˆΆπ·β†’πΆπ΅(𝐷) is hemicompact.

Let 𝐷 be a nonempty, closed, and convex subset of a real Banach space 𝐸. Let π‘‡βˆΆπ·β†’π‘ƒ(𝐷) be a multimap and 𝑃𝑇𝑒(π‘₯)∢=π‘₯β€–β€–βˆˆπ‘‡π‘₯∢π‘₯βˆ’π‘’π‘₯β€–β€–ξ€Ύ=𝑑(π‘₯,𝑇π‘₯).(2.3) Then, 𝑃𝑇(π‘₯)βˆΆπ·β†’π‘ƒ(𝐷) is nonempty and compact for every π‘₯∈𝐷. Furthermore, we observe that 𝑃𝑇(𝑦)={𝑦} if 𝑦 is a fixed point of 𝑇.

A mapping π‘‡βˆΆπ·β†’π‘ƒ(𝐷) is βˆ—-nonexpansive ([14]) if for all π‘₯,π‘¦βˆˆπ· and 𝑒π‘₯βˆˆπ‘‡π‘₯ with 𝑑(π‘₯,𝑒π‘₯)=inf{𝑑(π‘₯,𝑧)βˆΆπ‘§βˆˆπ‘‡π‘₯}, there exists π‘’π‘¦βˆˆπ‘‡π‘¦ with 𝑑(𝑦,𝑒𝑦)=inf{𝑑(𝑦,𝑀)βˆΆπ‘€βˆˆπ‘‡π‘¦} such that 𝑑𝑒π‘₯,𝑒𝑦≀𝑑(π‘₯,𝑦).(2.4)

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

By the definition of Hausdorff metric, we obtain that if a multimap π‘‡βˆΆπ·β†’π‘ƒ(𝐷) is βˆ—-nonexpansive, then 𝑃𝑇 is nonexpansive.

Throughout this paper, we write π‘₯𝑛→π‘₯ to indicate that the sequence {π‘₯𝑛} converges strongly to π‘₯.

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<π‘β‰€π‘‘π‘›β‰€π‘ž<1 for all positive integers 𝑛. Also, suppose that {π‘₯𝑛} and {𝑦𝑛} are two sequences of 𝐸 such that limsupπ‘›β†’βˆžβ€–π‘₯π‘›β€–β‰€π‘Ÿ,limsupπ‘›β†’βˆžβ€–π‘¦π‘›β€–β‰€π‘Ÿ and limπ‘›β†’βˆžβ€–π‘‘π‘›π‘₯𝑛+(1βˆ’π‘‘π‘›)𝑦𝑛‖=π‘Ÿ hold for some π‘Ÿ>0. Then, limπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘¦π‘›β€–=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,…,π‘‡π‘š be multivalued maps of 𝐷 into 𝑃(𝐷) with 𝐹∢=βˆ©π‘šπ‘–=1𝐹(𝑇𝑖)β‰ βˆ… such that 𝑃𝑇1,𝑃𝑇2,…,π‘ƒπ‘‡π‘š are nonexpansive and {𝛼𝑛𝑖}βˆžπ‘›=1,𝑖=0,1,…,π‘š a sequence in [πœ–,1βˆ’πœ–],πœ–βˆˆ(0,1) such that βˆ‘π‘šπ‘–=0𝛼𝑛𝑖=1 for all 𝑛β‰₯1. Let {π‘₯𝑛}βˆžπ‘›=1 be a sequence defined iteratively byπ‘₯1π‘₯∈𝐷,𝑛+1=𝛼𝑛0π‘₯𝑛+𝛼𝑛1𝑦𝑛(1)+β‹―+π›Όπ‘›π‘šπ‘¦π‘›(π‘š),(3.1) where 𝑦𝑛(𝑖)βˆˆπ‘ƒπ‘‡π‘–π‘₯𝑛,𝑖=1,2,…,π‘š.

Lemma 3.1. Let 𝐸 be a real normed space and 𝐷 a nonempty subset of 𝐸. Let 𝑇1,𝑇2,…,π‘‡π‘š be multivalued maps of 𝐷 into 𝑃(𝐷) with 𝐹∢=βˆ©π‘šπ‘–=1𝐹(𝑇𝑖)β‰ βˆ… such that 𝑃𝑇1,𝑃𝑇2,…,π‘ƒπ‘‡π‘š are nonexpansive. Let {𝛼𝑛𝑖}βˆžπ‘›=1,𝑖=0,1,…,π‘š a sequence in [πœ–,1βˆ’πœ–],πœ–βˆˆ(0,1) such that βˆ‘π‘šπ‘–=0𝛼𝑛𝑖=1 for all 𝑛β‰₯1. Let {π‘₯𝑛}βˆžπ‘›=1 be a sequence defined iteratively by (3.1). Then, limπ‘›β†’βˆžπ‘‘ξ€·π‘₯𝑛,𝑇𝑖π‘₯𝑛=0,βˆ€π‘–=1,2,…,π‘š.(3.2)

Proof. Let π‘₯βˆ—βˆˆβˆ©π‘šπ‘–=1𝐹(𝑇𝑖). Then, from (3.1), we have the following estimates: β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–β‰€π›Όπ‘›0β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–+𝛼𝑛1‖‖𝑦𝑛(1)βˆ’π‘₯βˆ—β€–β€–+β‹―+π›Όπ‘›π‘šβ€–β€–π‘¦π‘›(π‘š)βˆ’π‘₯βˆ—β€–β€–β‰€π›Όπ‘›0β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–+𝛼𝑛1𝑑𝑦𝑛(1),𝑃𝑇1π‘₯βˆ—ξ‚+β‹―+π›Όπ‘›π‘šπ‘‘ξ‚€π‘¦π‘›(π‘š),π‘ƒπ‘‡π‘šπ‘₯βˆ—ξ‚β‰€π›Όπ‘›0β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–+𝛼𝑛1𝐻𝑃𝑇1π‘₯𝑛,𝑃𝑇1π‘₯βˆ—ξ€Έ+β‹―+π›Όπ‘›π‘šπ»ξ€·π‘ƒπ‘‡π‘šπ‘₯𝑛,π‘ƒπ‘‡π‘šπ‘₯βˆ—ξ€Έβ‰€π›Όπ‘›0β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–+𝛼𝑛1β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–+β‹―+π›Όπ‘›π‘šβ€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–=β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–.(3.3) Thus, limπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘₯βˆ—β€– exists. Let limπ‘›β†’βˆžβ€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–=𝑐,(3.4) for some 𝑐β‰₯0. Then, 𝑐=limπ‘›β†’βˆžβ€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–=limπ‘›β†’βˆžβ€–β€–π›Όπ‘›0ξ€·π‘₯π‘›βˆ’π‘₯βˆ—ξ€Έ+𝛼𝑛1𝑦𝑛(1)βˆ’π‘₯βˆ—ξ‚+β‹―+π›Όπ‘›π‘šξ‚€π‘¦π‘›(π‘š)βˆ’π‘₯βˆ—ξ‚β€–β€–=limπ‘›β†’βˆžβ€–β€–β€–ξ€·1βˆ’π›Όπ‘›0𝛼𝑛11βˆ’π›Όπ‘›0𝑦𝑛(1)βˆ’π‘₯βˆ—ξ‚π›Ό+β‹―+π‘›π‘š1βˆ’π›Όπ‘›0𝑦𝑛(π‘š)βˆ’π‘₯βˆ—ξ‚ξ‚Ή+𝛼𝑛0ξ€·π‘₯π‘›βˆ’π‘₯βˆ—ξ€Έβ€–β€–β€–.(3.5) Since 𝑃𝑇𝑖,𝑖=1,2,…,π‘š is nonexpansive mapping and βˆ©π‘šπ‘–=1𝐹(𝑇𝑖)β‰ βˆ…, we have ‖‖𝑦𝑛(𝑖)βˆ’π‘₯βˆ—β€–β€–ξ‚€π‘¦β‰€π‘‘π‘›(𝑖),𝑃𝑇𝑖π‘₯βˆ—ξ‚ξ€·π‘ƒβ‰€π»π‘‡π‘–π‘₯𝑛,𝑃𝑇𝑖π‘₯βˆ—ξ€Έβ‰€β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–,(3.6) for each π‘₯βˆ—βˆˆβˆ©π‘šπ‘–=1𝐹(𝑇𝑖). Taking limsup on both sides, we obtain limsupπ‘›β†’βˆžβ€–β€–π‘¦π‘›(𝑖)βˆ’π‘₯βˆ—β€–β€–β‰€π‘,𝑖=1,2,…,π‘š.(3.7) Next, limsupπ‘›β†’βˆžβ€–β€–β€–π›Όπ‘›11βˆ’π›Όπ‘›0𝑦𝑛(1)βˆ’π‘₯βˆ—ξ‚+𝛼𝑛21βˆ’π›Όπ‘›0𝑦𝑛(2)βˆ’π‘₯βˆ—ξ‚π›Ό+β‹―+π‘›π‘š1βˆ’π›Όπ‘›0𝑦𝑛(π‘š)βˆ’π‘₯βˆ—ξ‚β€–β€–β€–β‰€limsupπ‘›β†’βˆžξ‚Έπ›Όπ‘›11βˆ’π›Όπ‘›0‖‖𝑦𝑛(1)βˆ’π‘₯βˆ—β€–β€–+𝛼𝑛21βˆ’π›Όπ‘›0‖‖𝑦𝑛(2)βˆ’π‘₯βˆ—β€–β€–π›Ό+β‹―+π‘›π‘š1βˆ’π›Όπ‘›0‖‖𝑦𝑛(π‘š)βˆ’π‘₯βˆ—β€–β€–ξ‚Ήβ‰€limsupπ‘›β†’βˆžπ›Όπ‘›1+𝛼𝑛2+β‹―+π›Όπ‘›π‘š1βˆ’π›Όπ‘›0β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–=𝑐.(3.8) Using (3.5), (3.8), and Lemma 2.1, we obtain limπ‘›β†’βˆžβ€–β€–β€–π›Όπ‘›11βˆ’π›Όπ‘›0𝑦𝑛(1)βˆ’π‘₯βˆ—ξ‚+𝛼𝑛21βˆ’π›Όπ‘›0𝑦𝑛(2)βˆ’π‘₯βˆ—ξ‚π›Ό+β‹―+π‘›π‘š1βˆ’π›Όπ‘›0𝑦𝑛(π‘š)βˆ’π‘₯βˆ—ξ‚βˆ’ξ€·π‘₯π‘›βˆ’π‘₯βˆ—ξ€Έβ€–β€–β€–=0.(3.9) This yields 0=limπ‘›β†’βˆžβ€–β€–β€–π›Όπ‘›11βˆ’π›Όπ‘›0𝑦𝑛(1)+𝛼𝑛21βˆ’π›Όπ‘›0𝑦𝑛(2)𝛼+β‹―+π‘›π‘š1βˆ’π›Όπ‘›0𝑦𝑛(π‘š)βˆ’π‘₯𝑛‖‖‖=limπ‘›β†’βˆžξ‚΅11βˆ’π›Όπ‘›0‖‖𝛼𝑛1𝑦𝑛(1)+𝛼𝑛2𝑦𝑛(2)+β‹―+π›Όπ‘›π‘šπ‘¦π‘›(π‘š)βˆ’ξ€·1βˆ’π›Όπ‘›0ξ€Έπ‘₯𝑛‖‖=limπ‘›β†’βˆžξ‚΅11βˆ’π›Όπ‘›0ξ‚Άβ€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖.(3.10) Thus, limπ‘›β†’βˆžβ€–π‘₯𝑛+1βˆ’π‘₯𝑛‖=0. Furthermore, 𝑐=limπ‘›β†’βˆžβ€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–=limπ‘›β†’βˆžβ€–β€–π›Όπ‘›0ξ€·π‘₯π‘›βˆ’π‘₯βˆ—ξ€Έ+𝛼𝑛1𝑦𝑛(1)βˆ’π‘₯βˆ—ξ‚+β‹―+π›Όπ‘›π‘šξ‚€π‘¦π‘›(π‘š)βˆ’π‘₯βˆ—ξ‚β€–β€–=limπ‘›β†’βˆžβ€–β€–β€–ξ€·1βˆ’π›Όπ‘›1𝛼𝑛01βˆ’π›Όπ‘›1ξ€·π‘₯π‘›βˆ’π‘₯βˆ—ξ€Έ+𝛼𝑛21βˆ’π›Όπ‘›1𝑦𝑛(2)βˆ’π‘₯βˆ—ξ‚π›Ό+β‹―+π‘›π‘š1βˆ’π›Όπ‘›1𝑦𝑛(π‘š)βˆ’π‘₯βˆ—ξ‚ξ‚Ή+𝛼𝑛1𝑦𝑛(1)βˆ’π‘₯βˆ—ξ‚β€–β€–β€–,limsupπ‘›β†’βˆžβ€–β€–β€–π›Όπ‘›01βˆ’π›Όπ‘›1ξ€·π‘₯π‘›βˆ’π‘₯βˆ—ξ€Έ+𝛼𝑛21βˆ’π›Όπ‘›1𝑦𝑛(2)βˆ’π‘₯βˆ—ξ‚π›Ό+β‹―+π‘›π‘š1βˆ’π›Όπ‘›1𝑦𝑛(π‘š)βˆ’π‘₯βˆ—ξ‚β€–β€–β€–β‰€limsupπ‘›β†’βˆžξ‚Έπ›Όπ‘›01βˆ’π›Όπ‘›1β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–+𝛼𝑛21βˆ’π›Όπ‘›1‖‖𝑦𝑛(2)βˆ’π‘₯βˆ—β€–β€–π›Ό+β‹―+π‘›π‘š1βˆ’π›Όπ‘›1‖‖𝑦𝑛(π‘š)βˆ’π‘₯βˆ—β€–β€–ξ‚Ήβ‰€limsupπ‘›β†’βˆžπ›Όπ‘›0+𝛼𝑛2+β‹―+π›Όπ‘›π‘š1βˆ’π›Όπ‘›1β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–=𝑐.(3.11) Using (3.11) and Lemma 2.1, we obtain limπ‘›β†’βˆžβ€–β€–β€–π›Όπ‘›01βˆ’π›Όπ‘›1ξ€·π‘₯π‘›βˆ’π‘₯βˆ—ξ€Έ+𝛼𝑛21βˆ’π›Όπ‘›1𝑦𝑛(2)βˆ’π‘₯βˆ—ξ‚π›Ό+β‹―+π‘›π‘š1βˆ’π›Όπ‘›1𝑦𝑛(π‘š)βˆ’π‘₯βˆ—ξ‚βˆ’ξ‚€π‘¦π‘›(1)βˆ’π‘₯βˆ—ξ‚β€–β€–β€–=0.(3.12) This yields 0=limπ‘›β†’βˆžβ€–β€–β€–π›Όπ‘›01βˆ’π›Όπ‘›1π‘₯𝑛+𝛼𝑛21βˆ’π›Όπ‘›1𝑦𝑛(2)𝛼+β‹―+π‘›π‘š1βˆ’π›Όπ‘›1𝑦𝑛(π‘š)βˆ’π‘¦π‘›(1)β€–β€–β€–=limπ‘›β†’βˆžξ‚΅11βˆ’π›Όπ‘›1‖‖𝛼𝑛0π‘₯𝑛+𝛼𝑛2𝑦𝑛(2)+β‹―+π›Όπ‘›π‘šπ‘¦π‘›(π‘š)βˆ’ξ€·1βˆ’π›Όπ‘›1𝑦𝑛(1)β€–β€–=limπ‘›β†’βˆžξ‚΅11βˆ’π›Όπ‘›1ξ‚Άβ€–β€–π‘₯𝑛+1βˆ’π‘¦π‘›(1)β€–β€–.(3.13) Thus, limπ‘›β†’βˆžβ€–π‘₯𝑛+1βˆ’π‘¦π‘›(1)β€–=0. So, β€–β€–π‘₯π‘›βˆ’π‘¦π‘›(1)‖‖≀‖‖π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+β€–β€–π‘₯𝑛+1βˆ’π‘¦π‘›(1)β€–β€–βŸΆ0,π‘›βŸΆβˆž.(3.14) Then, 𝑑π‘₯𝑛,𝑇1π‘₯𝑛π‘₯≀𝑑𝑛,𝑃𝑇1π‘₯𝑛≀‖‖π‘₯π‘›βˆ’π‘¦π‘›(1)β€–β€–βŸΆ0,π‘›βŸΆβˆž.(3.15) In a similar way, we can show that limπ‘›β†’βˆžπ‘‘ξ€·π‘₯𝑛,𝑇𝑖π‘₯𝑛=0,𝑖=2,3,…,π‘š.(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,…,π‘‡π‘š be multivalued maps of 𝐷 into 𝑃(𝐷) with 𝐹∢=βˆ©π‘šπ‘–=1𝐹(𝑇𝑖)β‰ βˆ… such that 𝑃𝑇1,𝑃𝑇2,…,π‘ƒπ‘‡π‘š are nonexpansive and {𝑇𝑖}π‘ši=1 satisfying condition (II). Let {𝛼𝑛𝑖}βˆžπ‘›=1,𝑖=0,1,…,π‘š a sequence in [πœ–,1βˆ’πœ–],πœ–βˆˆ(0,1) such that βˆ‘π‘šπ‘–=0𝛼𝑛𝑖=1 for all 𝑛β‰₯1. Let {π‘₯𝑛}βˆžπ‘›=1 be a sequence defined iteratively by (3.1). Then, {π‘₯𝑛}βˆžπ‘›=1 converges strongly to a common fixed point of {𝑇𝑖}π‘šπ‘–=1.

Proof. Since {𝑇𝑖}π‘šπ‘–=1 satisfies condition (II), we have that 𝑑(π‘₯𝑛,βˆ©π‘šπ‘–=1𝐹(𝑇𝑖))β†’0 as π‘›β†’βˆž. Thus, there is a subsequence {π‘₯π‘›π‘˜} of {π‘₯𝑛} and a sequence {π‘π‘˜}βŠ‚βˆ©π‘šπ‘–=1𝐹(𝑇𝑖) such that β€–β€–π‘₯π‘›π‘˜βˆ’π‘π‘˜β€–β€–<12π‘˜,(3.17) for all π‘˜. By Lemma 3.1, we obtain β€–β€–π‘₯π‘›π‘˜+1βˆ’π‘π‘˜β€–β€–β‰€β€–β€–π‘₯π‘›π‘˜βˆ’π‘π‘˜β€–β€–<12π‘˜.(3.18) We now show that {π‘π‘˜} is a Cauchy sequence in 𝐷. Observe that β€–β€–π‘π‘˜+1βˆ’π‘π‘˜β€–β€–β‰€β€–β€–π‘π‘˜+1βˆ’π‘₯π‘›π‘˜+1β€–β€–+β€–β€–π‘₯π‘›π‘˜+1βˆ’π‘π‘˜β€–β€–<12π‘˜+1+12π‘˜<12π‘˜βˆ’1.(3.19) This shows that {π‘π‘˜} is a Cauchy sequence in 𝐷, and thus converges to π‘βˆˆπ·. Since π‘‘ξ€·π‘π‘˜,π‘‡π‘–π‘ξ€Έβ€–β€–ξ€·π‘β‰€π‘‘π‘˜,𝑃𝑇𝑖𝑝𝑃≀𝐻𝑇𝑖𝑝,π‘ƒπ‘‡π‘–π‘π‘˜ξ€Έβ‰€β€–β€–π‘βˆ’π‘π‘˜β€–β€–,(3.20) and π‘π‘˜β†’π‘ as π‘˜β†’βˆž, it follows that 𝑑(𝑝,𝑇𝑖𝑝)=0, and thus π‘βˆˆβˆ©π‘šπ‘–=1𝐹(𝑇𝑖), and {π‘₯nπ‘˜} converges strongly to 𝑝. Since limπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘β€– exists, it follows that {π‘₯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,…,π‘‡π‘š be βˆ—-nonexpansive multimaps of 𝐷 into 𝑃(𝐷) with 𝐹∢=βˆ©π‘šπ‘–=1𝐹(𝑇𝑖)β‰ βˆ… and {𝑇𝑖}π‘šπ‘–=1 satisfying condition (II). Let {𝛼𝑛𝑖}βˆžπ‘›=1,𝑖=0,1,…,π‘š a sequence in [πœ–,1βˆ’πœ–],πœ–βˆˆ(0,1) such that βˆ‘π‘šπ‘–=0𝛼𝑛𝑖=1 for all 𝑛β‰₯1. Let {π‘₯𝑛}βˆžπ‘›=1 be a sequence defined iteratively by (3.1). Then, {π‘₯𝑛}βˆžπ‘›=1 converges strongly to a common fixed point of {𝑇𝑖}π‘šπ‘–=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,…,π‘‡π‘š be multivalued maps of 𝐷 into 𝑃(𝐷) with 𝐹∢=βˆ©π‘šπ‘–=1𝐹(𝑇𝑖)β‰ βˆ… such that 𝑃𝑇1,𝑃𝑇2,…,π‘ƒπ‘‡π‘š are nonexpansive and 𝑇𝑖 is hemicompact and continuous for each 𝑖=1,2,…,π‘š. Let {𝛼𝑛𝑖}βˆžπ‘›=1,𝑖=0,1,…,π‘š a sequence in [πœ–,1βˆ’πœ–],πœ–βˆˆ(0,1) such that βˆ‘π‘šπ‘–=0𝛼𝑛𝑖=1 for all 𝑛β‰₯1. Let {π‘₯𝑛}βˆžπ‘›=1 be a sequence defined iteratively by (3.1). Then, {π‘₯𝑛}βˆžπ‘›=1 converges strongly to a common fixed point of {𝑇𝑖}π‘šπ‘–=1.

Proof. Since limπ‘›β†’βˆžπ‘‘(π‘₯𝑛,𝑇𝑖π‘₯𝑛)=0,for all𝑖=1,2,…,π‘š and 𝑇𝑖 is hemicompact for each 𝑖=1,2,…,π‘š, there is a subsequence {π‘₯π‘›π‘˜} of {π‘₯𝑛} such that π‘₯π‘›π‘˜β†’π‘ as π‘˜β†’βˆž for some π‘βˆˆπ·. Since 𝑇𝑖 is continuous for each 𝑖=1,2,…,π‘š, we have 𝑑(π‘₯π‘›π‘˜,𝑇𝑖π‘₯π‘›π‘˜)→𝑑(𝑝,𝑇𝑖𝑝). As a result, we have that 𝑑(𝑝,𝑇𝑖𝑝)=0,for all𝑖=1,2,…,π‘š, and so, π‘βˆˆβˆ©π‘šπ‘–=1𝐹(𝑇𝑖). Since limπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘β€– exists, it follows that {π‘₯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,…,π‘‡π‘š be βˆ—-nonexpansive multimaps of 𝐷 into 𝑃(𝐷) with 𝐹∢=βˆ©π‘šπ‘–=1𝐹(𝑇𝑖)β‰ βˆ… and 𝑇𝑖 is hemicompact and continuous for each i=1,2,…,π‘š. Let {𝛼𝑛𝑖}βˆžπ‘›=1,𝑖=0,1,…,π‘š a sequence in [πœ–,1βˆ’πœ–],πœ–βˆˆ(0,1) such that βˆ‘π‘šπ‘–=0𝛼𝑛𝑖=1 for all 𝑛β‰₯1. Let {π‘₯𝑛}βˆžπ‘›=1 be a sequence defined iteratively by (3.1). Then, {π‘₯𝑛}βˆžπ‘›=1 converges strongly to a common fixed point of {𝑇𝑖}π‘šπ‘–=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,…,π‘‡π‘š be multivalued maps of 𝐷 into 𝑃(𝐷) with 𝐹∢=βˆ©π‘šπ‘–=1𝐹(𝑇𝑖)β‰ βˆ… such that 𝑃𝑇1,𝑃𝑇2,…,𝑃Tπ‘š are nonexpansive. Let {𝛼𝑛𝑖}βˆžπ‘›=1,𝑖=0,1,…,π‘š a sequence in [πœ–,1βˆ’πœ–],πœ–βˆˆ(0,1) such that βˆ‘π‘šπ‘–=0𝛼𝑛𝑖=1 for all 𝑛β‰₯1. Let {π‘₯𝑛}βˆžπ‘›=1 be a sequence defined iteratively by (3.1). Then, {π‘₯𝑛}βˆžπ‘›=1 converges strongly to a common fixed point of {𝑇𝑖}π‘šπ‘–=1.

Proof. From the compactness of 𝐷, there exists a subsequence {π‘₯π‘›π‘˜}βˆžπ‘›=π‘˜ of {π‘₯𝑛}βˆžπ‘›=1 such that limπ‘˜β†’βˆžβ€–π‘₯π‘›π‘˜βˆ’π‘žβ€–=0 for some π‘žβˆˆπ·. Thus, π‘‘ξ€·π‘ž,π‘‡π‘–π‘žξ€Έξ€·β‰€π‘‘π‘ž,π‘ƒπ‘‡π‘–π‘žξ€Έβ‰€β€–β€–π‘₯π‘›π‘˜β€–β€–ξ€·π‘₯βˆ’π‘ž+π‘‘π‘›π‘˜,𝑃𝑇𝑖π‘₯π‘›π‘˜ξ€Έξ€·π‘ƒ+𝐻𝑇𝑖π‘₯π‘›π‘˜,π‘ƒπ‘‡π‘–π‘žξ€Έβ€–β€–π‘₯≀2π‘›π‘˜β€–β€–ξ€·π‘₯βˆ’π‘ž+π‘‘π‘›π‘˜,𝑃𝑇𝑖π‘₯π‘›π‘˜ξ€ΈβŸΆ0asπ‘˜βŸΆβˆž.(3.21) Hence, π‘žβˆˆβˆ©π‘šπ‘–=1𝐹(𝑇𝑖). Now, on taking π‘ž in place of π‘₯βˆ—, we get that limπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘žβ€– 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}π‘šπ‘–=1.

Theorem 3.7. Let 𝐷 be a nonempty, closed, and convex subset of a real Banach space 𝐸. Let 𝑇1,𝑇2,…,π‘‡π‘š be multivalued maps of 𝐷 into 𝑃(𝐷) with 𝐹∢=βˆ©π‘šπ‘–=1𝐹(𝑇𝑖)β‰ βˆ… such that 𝑃𝑇1,𝑃𝑇2,…,π‘ƒπ‘‡π‘š are nonexpansive. Let {𝛼𝑛𝑖}βˆžπ‘›=1,𝑖=0,1,…,π‘š a sequence in [πœ–,1βˆ’πœ–],πœ–βˆˆ(0,1) such that βˆ‘π‘šπ‘–=0𝛼𝑛𝑖=1 for all 𝑛β‰₯1. Let {π‘₯𝑛}βˆžπ‘›=1 be a sequence defined iteratively by (3.1). Then, {π‘₯𝑛}βˆžπ‘›=1 converges strongly to a common fixed point of {𝑇𝑖}π‘šπ‘–=1 if and only if liminfπ‘›β†’βˆžπ‘‘(π‘₯𝑛,𝐹)=0.

Proof. The necessity is obvious. Conversely, suppose that liminfπ‘›β†’βˆžπ‘‘(π‘₯𝑛,F)=0. By (3.3), we have β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–β‰€β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–.(3.22) This gives 𝑑π‘₯𝑛+1ξ€Έξ€·π‘₯,𝐹≀𝑑𝑛,𝐹.(3.23) Hence, limπ‘›β†’βˆžπ‘‘(π‘₯𝑛,𝐹) exists. By hypothesis, liminfπ‘›β†’βˆžπ‘‘(π‘₯𝑛,𝐹)=0, so we must have limπ‘›β†’βˆžπ‘‘(π‘₯𝑛,𝐹)=0.
Next, we show that {π‘₯𝑛}βˆžπ‘›=1 is a Cauchy sequence in 𝐷. Let πœ–>0 be given, and since liminfπ‘›β†’βˆžπ‘‘(π‘₯𝑛,𝐹)=0, there exists 𝑛0 such that for all 𝑛β‰₯𝑛0, we have 𝑑π‘₯𝑛<πœ–,𝐹4.(3.24) In particular, inf{β€–π‘₯𝑛0βˆ’π‘β€–βˆΆπ‘βˆˆπΉ}<πœ–/4 so that there must exist a π‘βˆ—βˆˆπΉ such that β€–β€–π‘₯𝑛0βˆ’π‘βˆ—β€–β€–<πœ–2.(3.25) Now, for π‘š,𝑛β‰₯𝑛0, we have β€–β€–π‘₯𝑛+π‘šβˆ’π‘₯𝑛‖‖≀||β€–β€–π‘₯𝑛+π‘šβˆ’π‘βˆ—β€–β€–+β€–β€–π‘₯π‘›βˆ’π‘βˆ—β€–β€–β€–β€–π‘₯≀2𝑛0βˆ’π‘βˆ—β€–β€–ξ‚€πœ–<22=πœ–.(3.26) Hence, {π‘₯𝑛} is a Cauchy sequence in a closed subset 𝐷 of a Banach space 𝐸, and therefore, it must converge in 𝐷. Let limπ‘›β†’βˆžπ‘₯𝑛=𝑝. Now, for each 𝑖=1,2,…,π‘š, we obtain 𝑑𝑝,𝑇𝑖𝑝≀𝑑𝑝,𝑃𝑇𝑖𝑝≀𝑑𝑝,π‘₯𝑛π‘₯+𝑑𝑛,𝑃𝑇𝑖π‘₯𝑛𝑃+𝐻𝑇𝑖π‘₯𝑛,𝑃𝑇𝑖𝑝≀𝑑𝑝,π‘₯𝑛π‘₯+𝑑𝑛,𝑃𝑇𝑖π‘₯𝑛π‘₯+𝑑𝑛,π‘βŸΆ0asπ‘›βŸΆβˆž(3.27) gives that 𝑑(𝑝,𝑇𝑖𝑝)=0,𝑖=1,2,…,π‘š which implies that π‘βˆˆπ‘‡π‘–π‘. Consequently, π‘βˆˆπΉ=βˆ©π‘šπ‘–=1𝐹(𝑇𝑖)β‰ βˆ….

All the results we have obtained so far can be established for finite family of quasi-nonexpansive multivalued maps. Let 𝑇1,𝑇2,…,π‘‡π‘š be quasi-nonexpansive multivalued maps of 𝐷 into 𝐢𝐡(𝐷) such that 𝐹∢=βˆ©π‘šπ‘–=1𝐹(𝑇𝑖)β‰ βˆ… for which 𝑇𝑖𝑝={𝑝}, for all π‘βˆˆβˆ©π‘šπ‘–=1𝐹(𝑇𝑖). Let {𝛼𝑛𝑖}βˆžπ‘›=1,𝑖=0,1,…,π‘š a sequence in [πœ–,1βˆ’πœ–],πœ–βˆˆ(0,1) such that βˆ‘π‘šπ‘–=0𝛼𝑛𝑖=1 for all 𝑛β‰₯1. Let {π‘₯𝑛}βˆžπ‘›=1 be a sequence defined iteratively byπ‘₯1π‘₯∈𝐷,𝑛+1=𝛼𝑛0π‘₯𝑛+𝛼n1𝑦𝑛(1)+β‹―+π›Όπ‘›π‘šπ‘¦π‘›(π‘š),(3.28) where 𝑦𝑛(𝑖)βˆˆπ‘‡π‘–π‘₯𝑛, 𝑖=1,2,…,π‘š. 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,…,π‘‡π‘š be quasi-nonexpansive multivalued maps of 𝐷 into 𝐢𝐡(𝐷) such that 𝐹∢=βˆ©π‘šπ‘–=1𝐹(𝑇𝑖)β‰ βˆ… for which 𝑇𝑖𝑝={𝑝}, for all π‘βˆˆβˆ©π‘šπ‘–=1𝐹(𝑇𝑖) and {𝑇𝑖}π‘šπ‘–=1 satisfying condition (II). Let {𝛼𝑛𝑖}βˆžπ‘›=1,𝑖=0,1,…,π‘š a sequence in [πœ–,1βˆ’πœ–],πœ–βˆˆ(0,1) such that βˆ‘π‘šπ‘–=0𝛼𝑛𝑖=1 for all 𝑛β‰₯1. Let {π‘₯𝑛}βˆžπ‘›=1 be a sequence defined iteratively by (3.28). Then, {π‘₯𝑛}βˆžπ‘›=1 converges strongly to a common fixed point of {𝑇𝑖}π‘šπ‘–=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,…,π‘‡π‘š be quasi-nonexpansive multivalued maps of 𝐷 into 𝐢𝐡(𝐷) such that 𝐹∢=βˆ©π‘šπ‘–=1𝐹(𝑇𝑖)β‰ βˆ… for which 𝑇𝑖𝑝={𝑝}, for all π‘βˆˆβˆ©π‘šπ‘–=1𝐹(𝑇𝑖) and 𝑇𝑖 is hemicompact and continuous for each 𝑖=1,2,…,π‘š. Let {𝛼𝑛𝑖}βˆžπ‘›=1,𝑖=0,1,…,π‘š a sequence in [πœ–,1βˆ’πœ–],πœ–βˆˆ(0,1) such that βˆ‘π‘šπ‘–=0𝛼𝑛𝑖=1 for all 𝑛β‰₯1. Let {π‘₯𝑛}βˆžπ‘›=1 be a sequence defined iteratively by (3.28). Then, {π‘₯𝑛}βˆžπ‘›=1 converges strongly to a common fixed point of {𝑇𝑖}π‘šπ‘–=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,…,π‘‡π‘š be quasi-nonexpansive multivalued maps of 𝐷 into 𝐢𝐡(𝐷) such that 𝐹∢=βˆ©π‘šπ‘–=1𝐹(𝑇𝑖)β‰ βˆ… for which 𝑇𝑖𝑝={𝑝}, for all π‘βˆˆβˆ©π‘šπ‘–=1𝐹(𝑇𝑖). {𝛼𝑛𝑖}βˆžπ‘›=1,𝑖=0,1,…,π‘š a sequence in [πœ–,1βˆ’πœ–],πœ–βˆˆ(0,1) such that βˆ‘π‘šπ‘–=0𝛼𝑛𝑖=1 for all 𝑛β‰₯1. Let {π‘₯𝑛}βˆžπ‘›=1 be a sequence defined iteratively by (3.28). Then, {π‘₯𝑛}βˆžπ‘›=1 converges strongly to a common fixed point of {𝑇𝑖}π‘šπ‘–=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)β‰ βˆ…. Let {π‘Žπ‘›}βˆžπ‘›=1,{𝑏𝑛}βˆžπ‘›=1, and {𝑐𝑛}βˆžπ‘›=1 be sequence in (0,1) satisfying π‘Žπ‘›+𝑏𝑛+𝑐𝑛≀1. Let {π‘₯𝑛}βˆžπ‘›=1 be a sequence defined iteratively by π‘₯1π‘₯∈𝐷,𝑛+1=π‘Žπ‘›π‘₯𝑛+𝑏𝑛𝑦𝑛+𝑐𝑛𝑧𝑛,βˆ€π‘›β‰₯1,(3.29) where π‘¦π‘›βˆˆπ‘‡π‘₯𝑛,π‘§π‘›βˆˆπ‘†π‘₯𝑛 such that β€–π‘¦π‘›βˆ’π‘β€–β‰€π‘‘(𝑝,𝑇π‘₯𝑛) and β€–π‘§π‘›βˆ’π‘β€–β‰€π‘‘(𝑝,𝑆π‘₯𝑛) whenever 𝑝 is a fixed point of any one of mappings 𝑇 and 𝑆. Then, {π‘₯𝑛}βˆžπ‘›=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)β‰ βˆ…. Let {π‘Žπ‘›}βˆžπ‘›=1,{𝑏𝑛}βˆžπ‘›=1 and {𝑐𝑛}βˆžπ‘›=1 be sequence in (0,1) satisfying π‘Žπ‘›+𝑏𝑛+𝑐𝑛≀1. Let {π‘₯𝑛}βˆžπ‘›=1 be a sequence defined iteratively by π‘₯1π‘₯∈𝐷,𝑛+1=π‘Žπ‘›π‘₯𝑛+𝑏𝑛𝑦𝑛+𝑐𝑛𝑧𝑛,βˆ€π‘›β‰₯1,(3.30) where π‘¦π‘›βˆˆπ‘‡π‘₯𝑛, π‘§π‘›βˆˆπ‘†π‘₯𝑛 such that β€–π‘¦π‘›βˆ’π‘β€–β‰€π‘‘(𝑝,𝑇π‘₯𝑛) and β€–π‘§π‘›βˆ’π‘β€–β‰€π‘‘(𝑝,𝑆π‘₯𝑛) whenever 𝑝 is a fixed point of any one of mappings 𝑇 and 𝑆. Then, {π‘₯𝑛}βˆžπ‘›=1 converges strongly to a common fixed point of 𝐹(𝑇)∩𝐹(𝑆) if and only if liminfπ‘›β†’βˆžπ‘‘(π‘₯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.

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Copyright © 2011 Yekini Shehu. 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.

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