Convergence of a Viscosity Iterative Method for Multivalued Nonself-Mappings in Banach Spaces

Abstract

Let 𝐸 be a reflexive Banach space having a weakly sequentially continuous duality mapping 𝐽_𝜑 with gauge function 𝜑, 𝐶 a nonempty closed convex subset of 𝐸, and 𝑇∶𝐶→𝒦(𝐸) a multivalued nonself-mapping such that 𝑃_𝑇 is nonexpansive, where 𝑃_𝑇(𝑥)={𝑢_𝑥∈𝑇𝑥∶‖𝑥−𝑢_𝑥‖=𝑑(𝑥,𝑇𝑥)}. Let 𝑓∶𝐶→𝐶 be a contraction with constant 𝑘. Suppose that, for each 𝑣∈𝐶 and 𝑡∈(0,1), the contraction defined by 𝑆_𝑡𝑥=𝑡𝑃_𝑇𝑥+(1−𝑡)𝑣 has a fixed point 𝑥_𝑡∈𝐶. Let {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} be three sequences in (0,1) satisfying approximate conditions. Then, for arbitrary 𝑥₀∈𝐶, the sequence {𝑥_𝑛} generated by 𝑥_𝑛∈𝛼_𝑛𝑓(𝑥_𝑛−1)+𝛽_𝑛𝑥_𝑛−1+𝛾_𝑛𝑃_𝑇(𝑥_𝑛) for all 𝑛≥1 converges strongly to a fixed point of 𝑇.

1. Introduction

Let 𝐸 be a Banach space and 𝐶 a nonempty closed subset of 𝐸. We will denote by ℱ(𝐸) the family of nonempty closed subsets of 𝐸, by 𝒞ℬ(𝐸) the family of nonempty closed bounded subsets of 𝐸, by 𝒦(𝐸) the family of nonempty compact subsets of 𝐸, and by 𝒦𝒞(𝐸) the family of nonempty compact convex subsets of 𝐸. Let 𝐻(⋅,⋅) be the Hausdorff distance on 𝒞ℬ(𝐸); that is, 𝐻(𝐴,𝐵)=maxsup_𝑎∈𝐴𝑑(𝑎,𝐵),sup_𝑏∈𝐵𝑑(𝑏,𝐴), (1) for all 𝐴,𝐵∈𝒞ℬ(𝐸), where 𝑑(𝑎,𝐵)=inf{‖𝑎−𝑏‖∶𝑏∈𝐵} is the distance from the point 𝑎 to the subset 𝐵. Recall that a mapping 𝑓∶𝐶→𝐶 is a contraction on 𝐶 if there exists a constant 𝑘∈(0,1) such that ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝑘‖𝑥−𝑦‖,𝑥,𝑦∈𝐶.

A multivalued mapping 𝑇∶𝐶→ℱ(𝐸) is said to be a contraction if there exists a constant 𝑘∈[0,1) such that 𝐻𝑇𝑥,𝑇𝑦≤𝑘𝑥−𝑦∀𝑥,𝑦∈𝐶. (2) If (2) is valid when 𝑘=1, the 𝑇 is called nonexpansive. A point 𝑥 is a fixed point for a multivalued mapping 𝑇 if 𝑥∈𝑇𝑥. Banach's contraction principle was extended to a multivalued contraction by Nadler [1] in 1969. The set of fixed points of 𝑇 is denoted by 𝐹(𝑇).

Given a contraction 𝑓 with constant 𝑘 and 𝑡∈(0,1), we can define a contraction 𝐺_𝑡∶𝐶→𝒦(𝐶) by 𝐺_𝑡𝑥∶=𝑡𝑇𝑥+(1−𝑡)𝑓(𝑥),𝑥∈𝐶. (3) Then 𝐺_𝑡 is a multivalued, and hence it has a (nonunique, in general) fixed point 𝑥_𝑡∶=𝑥^𝑓_𝑡∈𝐶 (see [1]); that is,𝑥_𝑡∈𝑡𝑇𝑥_𝑡+(1−𝑡)𝑓𝑥_𝑡. (4) If 𝑇 is single valued, we have 𝑥_𝑡=𝑡𝑇𝑥_𝑡+(1−𝑡)𝑓𝑥_𝑡, (5) which was studied by Moudafi [2] (see also Xu [3]). As a special case of (5), 𝑥_𝑡=𝑡𝑥_𝑡+(1−𝑡)u,forgiven𝑢∈𝐶, (6) has been considered by Browder [4], Halpern [5], Jung and Kim [6, 7], Kim and Takahashi [8], Reich [9], Singh and Watson [10], Takahashi and Kim [11], Xu [12], and Xu and Yin [13] in a Hilbert space and Banach spaces.

In 2007, Jung [14] established the strong convergence of {𝑥_𝑡} defined by 𝑥_𝑡∈𝑡𝑇𝑥_𝑡+(1−𝑡)𝑢,𝑢∈𝐶 for the multivalued nonexpansive nonself-mapping 𝑇 in a reflexive Banach space having a uniformly Gâteaux differentiable norm under the assumption 𝑇𝑦={𝑦}.

In order to give a partial answer to Jung's open question [14] Can the assumption 𝑇𝑦={𝑦} be omitted?, in 2008, Shahzad and Zegeye [15] considered a class of multivalued mapping under some mild conditions as follows.

Let 𝐶 be a closed convex subset of a Banach space 𝐸. Let 𝑇∶𝐶→𝒦(𝐸) be a multivalued nonself-mapping and 𝑃_𝑇𝑥=𝑢_𝑥∈𝑇𝑥∶𝑥−𝑢_𝑥=𝑑(𝑥,𝑇𝑥). (7) Then 𝑃_𝑇∶𝐶→𝒦(𝐸) is multivalued, and 𝑃_𝑇𝑥 is nonempty and compact for every 𝑥∈𝐶. Instead of𝐺_𝑡𝑥=𝑡𝑇𝑥+(1−𝑡)𝑢,𝑢∈𝐶, (8) we consider, for 𝑡∈(0,1), that 𝑆_𝑡𝑥=𝑡𝑃_𝑇𝑥+(1−𝑡)𝑢,𝑢∈𝐶. (9)

It is clear that 𝑆_𝑡𝑥⊆𝐺_𝑡𝑥, and if 𝑃_𝑇 is nonexpansive and 𝑇 is weakly inward, then 𝑆_𝑡 is weakly inward contraction. Theorem 1 of Lim [16] guarantees that 𝑆_𝑡 has a fixed point, point 𝑥_𝑡; that is, 𝑥_𝑡∈𝑡𝑃_𝑇𝑥_𝑡+(1−𝑡)𝑢⊆𝑡𝑇𝑥_𝑡+(1−𝑡)𝑢. (10) If 𝑇 is single valued, then (10) is reduced to (6).

Shahzad and Zegeye [15] also gave the strong convergence result of {𝑥_𝑡} defined by (10) in a reflexive Banach space having a uniformly Gâteaux differentiable norm, which unified, extended, and complemented several known results including those of Jung [14], Jung and Kim [6, 7], Kim and Jung [17], López Acedo and Xu [18], Sahu [19], and Xu and Yin [13].

In 2009, motivated by the results of Rafiq [20] and Yao et al. [21], Ceng and Yao [22] considered the following iterative scheme.

Theorem CY (see [22, Theorem 3.1]). Let 𝐸 be a uniformly convex Banach space 𝐸 having a uniformly Gâteaux differentiable norm, 𝐶 a nonempty closed convex subset of 𝐸, and 𝑇∶𝐶→𝒦(𝐸) a multivalued nonself-mapping such that 𝑃_𝑇 is nonexpansive. Suppose that 𝐶 is a nonexpansive retract of 𝐸 and that for each 𝑣∈𝐶 and 𝑡∈(0,1) the contraction 𝑆_𝑡 defined by 𝑆_𝑡𝑥=𝑡𝑃_𝑇𝑥+(1−𝑡)𝑣 has a fixed point 𝑥_𝑡∈𝐶. Let {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} be three sequences in (0,1) satisfying the following conditions:(i)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1;(ii)lim_𝑛→∞𝛼_𝑛=0 and lim_𝑛→∞(𝛽_𝑛/𝛼_𝑛)=0. For arbitrary initial value 𝑥₀∈𝐶 and a fixed element 𝑢∈𝐶, let the sequence {𝑥_𝑛} be generated by 𝑥_𝑛∈𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛−1+𝛾_𝑛𝑃_𝑇𝑥_𝑛∀𝑛≥1. (11) Then {𝑥_𝑛} converges strongly to a fixed point of 𝑇.

Theorem CY also improves, develops, and complements the corresponding results in Jung [14], Jung and Kim [6, 7], Kim and Jung [17], López Acedo and Xu [18], Shahzad and Zegeye [15], and Xu and Yin [13] to the iterative scheme (11). For convergence of related iterative schemes for several nonlinear mappings, we can refer to [23–26] and the references therein.

In this paper, inspired and motivated by the above-mentioned results, we consider a viscosity iterative method for a multivalued nonself-mapping in a reflexive Banach space having a weakly sequentially continuous duality mapping and establish the strong convergence of the sequence generated by the proposed iterative method. Our results improve and develop the corresponding results of Ceng and Yao [22], as well as some known results in the earlier and recent literature, including those of Jung [14], Jung and Kim [6, 7], Kim and Jung [17], López Acedo and Xu [18], Sahu [19], Shahzad and Zegeye [15], Xu [12], and Xu and Yin [13], to the viscosity iterative scheme in different Banach space.

2. Preliminaries

Let 𝐸 be a real Banach space with norm ‖⋅‖, and let 𝐸^∗ be its dual. The value of 𝑥^∗∈𝐸^∗ at 𝑥∈𝐸 will be denoted by ⟨𝑥,𝑥^∗⟩.

A Banach space 𝐸 is called uniformly convex if 𝛿(𝜀)>0 for every 𝜀>0, where the modulus 𝛿(𝜀) of convexity of 𝐸 is defined by 𝛿(𝜀)=inf1−𝑥+𝑦2∶‖𝑥‖≤1,𝑦≤1,𝑥−𝑦≥𝜀 (12) for every 𝜀 with 0≤𝜀≤2. It is well known that if 𝐸 is uniformly convex, then 𝐸 is reflexive and strictly convex (cf. [27]).

By a gauge function we mean a continuous strictly increasing function 𝜑 defined on ℝ⁺∶=[0,∞) such that 𝜑(0)=0 and lim_𝑟→∞𝜑(𝑟)=∞. The mapping 𝐽_𝜑∶𝐸→2^𝐸∗ defined by 𝐽_𝜑(𝑥)=𝑓∈𝐸^∗∶𝑥,𝑓=‖𝑥‖𝑓,𝑓=𝜑(‖𝑥‖)∀𝑥∈𝐸 (13) is called the duality mapping with gauge function 𝜑. In particular, the duality mapping with gauge function 𝜑(𝑡)=𝑡 denoted by 𝐽 is referred to as the normalized duality mapping. It is known that a Banach space 𝐸 is smooth if and only if the normalized duality mapping 𝐽 is single valued. The following property of duality mapping is also well known: 𝐽_𝜑()𝜑𝜆|⋅‖𝑥𝜆𝑥=sign𝜆(|‖)‖𝑥‖𝐽(𝑥)∀𝑥∈𝐸⧵{0},𝜆∈ℝ, (14) where ℝ is the set of all real numbers; in particular, 𝐽(−𝑥)=−𝐽(𝑥) for all 𝑥∈𝐸 ([28]).

We say that a Banach space 𝐸 has a weakly sequentially continuous duality mapping if there exists a gauge function 𝜑 such that the duality mapping 𝐽_𝜑 is single valued and continuous from the weak topology to the weak^∗ topology; that is, for any {𝑥_𝑛}∈𝐸 with 𝑥_𝑛⇀𝑥,𝐽_𝜑(𝑥_𝑛)⇀^∗𝐽_𝜑(𝑥). For example, every 𝑙^𝑝 space (1<𝑝<∞) has a weakly continuous duality mapping with gauge function 𝜑(𝑡)=𝑡^𝑝−1. It is well known that if 𝐸 is a Banach space having a weakly sequentially continuous duality mapping 𝐽_𝜑 with gauge function 𝜑, then 𝐸 has the opial condition [29]; this is, whenever a sequence {𝑥_𝑛} in 𝐸 converges weakly to 𝑥∈𝐸, then limsup_𝑛→∞𝑥_𝑛−𝑥<limsup_𝑛→∞𝑥_𝑛−𝑦∀𝑦∈𝐸,𝑦≠𝑥. (15) A mapping 𝑇∶𝐶→𝒞ℬ(𝐸) is ∗-nonexpansive [30] if, for all 𝑥,𝑦∈𝐶 and 𝑢_𝑥∈𝑇𝑥 with ‖𝑥−𝑢_𝑥‖=inf{‖𝑥−𝑧‖∶𝑧∈𝑇𝑥}, there exists 𝑢_𝑦∈𝑇𝑦 with ‖𝑦−𝑢_𝑦‖=inf{‖𝑦−𝑤‖∶𝑤∈𝑇𝑦} such that 𝑢_𝑥−𝑢_𝑦≤𝑥−𝑦. (16) It is known that ∗-nonexpansiveness is different from nonexpansiveness for multivalued mappings. There are some ∗-nonexpansiveness multivalued mappings which are not nonexpansive and some nonexpansive multivalued mappings which are not ∗-nonexpansive [31].

We introduce some terminology for boundary conditions for nonself-mappings. The inward set of 𝐶 at 𝑥 is defined by 𝐼_𝐶(𝑥)=𝑧∈𝐸∶𝑧=𝑥+𝜆𝑦−𝑥∶𝑦∈𝐶,𝜆≥0. (17) Let 𝐼_𝐶(𝑥)=𝑥+𝑇_𝐶(𝑥) with 𝑇_𝐶(𝑥)=𝑦∈𝐸∶liminf_𝜆→0+𝑑𝑥+𝜆𝑦,𝐶𝜆=0 (18) for any 𝑥∈𝐶. Note that, for a convex set 𝐶, we have 𝐼_𝐶(𝑥)=𝐼_𝐶(𝑥), the closure of 𝐼_𝐶(𝑥). A multivalued mapping 𝑇∶𝐶→ℱ(𝐸) is said to satisfy the weak inwardness condition if 𝑇𝑥⊂𝐼_𝐶(𝑥) for all 𝑥∈𝐶.

Finally, the following lemma was given by Xu [32] (also see Xu [33]).

Lemma 1. If 𝐶 is a closed bounded convex subset of a uniformly convex Banach space 𝐸 and 𝑇∶𝐶→𝒦(𝐸) is a nonexpansive mapping satisfying the weak inwardness condition, then 𝑇 has a fixed point.

3. Main Results

Now, we establish strong convergence of a viscosity iterative scheme for a multivalued nonself-mapping.

Theorem 2. Let 𝐸 be a reflexive Banach space having a weakly sequentially continuous duality mapping 𝐽_𝜑 with gauge function 𝜑. Let 𝐶 be a nonempty closed convex subset of 𝐸 and 𝑇∶𝐶→𝒦(𝐸) a multivalued nonself-mapping such that 𝐹(𝑇)≠∅ and 𝑃_𝑇 is nonexpansive. Let 𝑓∶𝐶→𝐶 be a contraction with constant 𝑘. Suppose that for each 𝑣∈𝐶 and 𝑡∈(0,1), the contraction 𝑆_𝑡 defined by 𝑆_𝑡𝑥=𝑡𝑃_𝑇𝑥+(1−𝑡)𝑣 has a fixed point 𝑥_𝑡∈𝐶. Let {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} be three sequences in (0,1) satisfying the following conditions:(i)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1;(ii)lim_𝑛→∞𝛼_𝑛=0 and lim_𝑛→∞(𝛽_𝑛/𝛼_𝑛)=0. For arbitrary initial value 𝑥₀∈𝐶, let the sequence {𝑥_𝑛} be defined by 𝑥_𝑛∈𝛼_𝑛𝑓𝑥_𝑛−1+𝛽_𝑛𝑥_𝑛−1+𝛾_𝑛𝑃_𝑇𝑥_𝑛∀𝑛≥1. (19) Then {𝑥_𝑛} converges strongly to a fixed point of 𝑇.

Proof. First, observe that, for each 𝑛≥1, 𝛼_𝑛𝑓𝑥_𝑛−1+𝛽_𝑛𝑥_𝑛−1=1−𝛾_𝑛𝛼_𝑛1−𝛾_𝑛𝑓𝑥_𝑛−1+𝛽_𝑛1−𝛾_𝑛𝑥_𝑛−1. (20) From 𝑥_𝑛−1, 𝑓(𝑥_𝑛−1)∈𝐶, it follows that (𝛼_𝑛/(1−𝛾_𝑛))𝑓(𝑥_𝑛−1)+(𝛽_𝑛/(1−𝛾_𝑛))𝑥_𝑛−1∈𝐶. Also, notice that, for each 𝑣∈𝐶 and 𝑡∈(0,1), the contraction 𝑆_𝑡 defined by 𝑆_𝑡𝑥=𝑡𝑃_𝑇𝑥+(1−𝑡)𝑣 has a fixed point 𝑥_𝑡∈𝐶. Thus, for (𝛼_𝑛/(1−𝛾_𝑛))𝑓(𝑥_𝑛−1)+(𝛽_𝑛/(1−𝛾_𝑛))𝑥_𝑛−1∈𝐶 and 𝛾_𝑛∈(0,1), there exists 𝑥_𝑛∈𝐶 such that 𝑥_𝑛∈1−𝛾_𝑛𝛼_𝑛1−𝛾_𝑛𝑓𝑥_𝑛−1+𝛽_𝑛1−𝛾_𝑛𝑥_𝑛−1+𝛾_𝑛𝑃_𝑇𝑥_𝑛. (21) This shows that the sequence {𝑥_𝑛} can be defined well via the following: 𝑥_𝑛∈𝛼_𝑛𝑓𝑥_𝑛−1+𝛽_𝑛𝑥_𝑛−1+𝛾_𝑛𝑃_𝑇𝑥_𝑛∀𝑛≥1. (22) Therefore, for any 𝑛≥1, we can find some 𝑧_𝑛∈𝑃_𝑇(𝑥_𝑛) such that 𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛−1+𝛽_𝑛𝑥_𝑛−1+𝛾_𝑛𝑧_𝑛. (23) Next, we divide the proof into several steps.
Step 1. We show that {𝑥_𝑛} is bounded. Indeed, notice that 𝑃_𝑇(𝑦)={𝑦} whenever 𝑦 is a fixed point of 𝑇. Let 𝑝∈𝐹(𝑇). Then 𝑝∈𝑃_𝑇(𝑝) and we have 𝑧_𝑛−𝑝=𝑑𝑧_𝑛,𝑃_𝑇𝑝≤𝐻𝑃_𝑇𝑥_𝑛,𝑃_𝑇𝑝≤𝑥_𝑛−𝑝. (24) It follows that 𝑥_𝑛−𝑝=𝛼_𝑛𝑓𝑥_𝑛−1+𝛽_𝑛𝑥_𝑛−1+𝛾_𝑛𝑧_𝑛−𝑝≤𝛼_𝑛𝑓𝑥_𝑛−1−𝑓𝑝+𝛼_𝑛𝑓𝑝−𝑝+𝛽_𝑛𝑥_𝑛−1−𝑝+𝛾_𝑛𝑧_𝑛−𝑝≤𝛼_𝑛𝑘𝑥_𝑛−1−𝑝+𝑓𝑝−𝑝+𝛽_𝑛𝑥_𝑛−1−𝑝+𝛾_𝑛𝑥_𝑛−𝑝, (25) and so 𝑥_𝑛≤𝛼−𝑝^𝑛1−𝛾_𝑛𝑘𝑥_𝑛−1+𝑓𝑝+𝛽−𝑝−𝑝_𝑛1−𝛾_𝑛𝑥_𝑛−1=−𝑝𝑘𝛼_𝑛+𝛽_𝑛1−𝛾_𝑛𝑥_𝑛−1+()𝛼−𝑝1−𝑘_𝑛1−𝛾_𝑛𝑓𝑝−𝑝𝑥1−𝑘≤max_𝑛−1,𝑓𝑝−𝑝−𝑝.1−𝑘 (26) By induction, we have 𝑥_𝑛−𝑝≤max𝑥₀,1−𝑝𝑓𝑝1−𝑘−𝑝for𝑛≥1. (27) Hence {𝑥_𝑛} is bounded and so are {𝑧_𝑛} and {𝑓(𝑥_𝑛)}.
Step 2. We show that lim_𝑛→∞‖𝑥_𝑛−𝑧_𝑛‖=0. In fact, since 𝑥_𝑛=𝛼_𝑛𝑓(𝑥_𝑛−1)+𝛽_𝑛𝑥_𝑛−1+𝛾_𝑛𝑧_𝑛 for some 𝑧_𝑛∈𝑃_𝑇(𝑥_𝑛), by conditions (i) and (ii), we have 𝑥_𝑛−𝑧_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−1−𝑧_𝑛+𝛽_𝑛𝑥_𝑛−1−𝑧_𝑛→0as𝑛→∞. (28)Step 3. We show that there exists 𝑝∈𝑃_𝑇(𝑝)⊂𝑇𝑝. In fact, since {𝑥_𝑛} and {𝑧_𝑛} are bounded and 𝐸 is reflexive, there exists a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that 𝑥_𝑛𝑘⇀𝑝. For this 𝑝, by compactness of 𝑃_𝑇(𝑝), we can find 𝑤_𝑛∈𝑃_𝑇(𝑝), ∀𝑛≥1, such that 𝑧_𝑛−𝑤_𝑛=𝑑𝑧_𝑛,𝑃_𝑇𝑝≤𝐻𝑃_𝑇𝑥_𝑛,𝑃_𝑇𝑝≤𝑥_𝑛−𝑝. (29) Now suppose that 𝑥_𝑛∶=𝑥_𝑛𝑘 and 𝑥_𝑛⇀𝑝. The sequence {𝑤_𝑛} has a convergent subsequence, which is denoted again by {𝑤_𝑛} with 𝑤_𝑛→𝑤∈𝑃_𝑇(𝑝). Assume that 𝑤≠𝑝. Since a Banach space having the weakly sequentially continuous duality mapping satisfies the opial condition [29], by Step 2, we have limsup_𝑛→∞𝑥_𝑛−𝑤≤limsup_𝑛→∞𝑥_𝑛−𝑧_𝑛+𝑧_𝑛−𝑤_𝑛+𝑤_𝑛−𝑤≤limsup_𝑛→∞𝑧_𝑛−𝑤_𝑛≤limsup_𝑛→∞𝑥_𝑛−𝑝<limsup_𝑛→∞𝑥_𝑛−𝑤, (30) which is a contradiction. Hence we have 𝑤=𝑝, and so 𝑝=𝑤∈𝑃_𝑇(𝑝)⊂𝑇𝑝.
Step 4. We show that lim_𝑛→∞⟨𝑧_𝑛−𝑓(𝑥_𝑛−1),𝐽_𝜑(𝑥_𝑛−𝑦)⟩≤0 for 𝑦∈𝐹(𝑇). Indeed, for 𝑦∈𝐹(𝑇), by (24), we have 𝑥_𝑛−𝑧_𝑛,𝐽_𝜑𝑥_𝑛−𝑦=𝑥_𝑛−𝑦+𝑦−𝑧_𝑛,𝐽_𝜑𝑥_𝑛≥𝑥−𝑦_𝑛−𝑦𝜑𝑥_𝑛−𝑦−𝑧_𝑛−𝑦𝜑𝑥_𝑛≥𝑥−𝑦_𝑛−𝑦𝜑𝑥_𝑛−𝑦−𝑥_𝑛−𝑦𝜑𝑥_𝑛−𝑦=0. (31) So, 0≤𝑥_𝑛−𝑧_𝑛,𝐽_𝜑𝑥_𝑛−𝑦=𝛼_𝑛𝑓𝑥_𝑛−1−𝑧_𝑛,𝐽_𝜑𝑥_𝑛−𝑦+𝛽_𝑛𝑥_𝑛−1−𝑧_𝑛,𝐽_𝜑𝑥_𝑛−𝑦≤𝛼_𝑛𝑓𝑥_𝑛−1−𝑧_𝑛,𝐽_𝜑𝑥_𝑛+𝛽−𝑦_𝑛𝛼_𝑛𝑥_𝑛−1−𝑧_𝑛𝜑𝑥_𝑛−𝑦. (32) Thus it follows that 𝑧_𝑛−𝑓𝑥_𝑛−1,𝐽_𝜑𝑥_𝑛≤𝛽−𝑦^𝑛𝛼_𝑛𝑥_𝑛−1−𝑧_𝑛𝜑𝑥_𝑛−𝑦. (33) Hence, from condition (ii), we conclude that lim_𝑛→∞𝑧_𝑛−𝑓𝑥_𝑛−1,𝐽_𝜑𝑥_𝑛−𝑦≤0. (34)Step 5. We show that lim_𝑛→∞‖𝑥_𝑛−𝑝‖=0, where 𝑝 is defined as in Step 3. Indeed, for 𝑦∈𝐹(𝑇), 𝑥_𝑛−𝑦𝜑𝑥_𝑛−𝑦=𝛼_𝑛𝑓𝑥_𝑛−1−𝑦,𝐽_𝜑𝑥_𝑛−𝑦+𝛽_𝑛𝑥_𝑛−1−𝑦,𝐽_𝜑𝑥_𝑛−𝑦+𝛾_𝑛𝑧_𝑛−𝑦,𝐽_𝜑𝑥_𝑛=𝛼−𝑦_𝑛+𝛾_𝑛𝑓𝑥_𝑛−1−𝑦,𝐽_𝜑𝑥_𝑛+𝛽−𝑦_𝑛𝛼_𝑛𝑥_𝑛−1−𝑦,𝐽_𝜑𝑥_𝑛−𝑦𝛼_𝑛+𝛾_𝑛𝑧_𝑛−𝑓𝑥_𝑛−1,𝐽_𝜑𝑥_𝑛−𝑦. (35) Interchanging 𝑝 and 𝑦 in (35), we obtain 𝑥_𝑛−𝑝𝜑𝑥_𝑛=𝛼−𝑝_𝑛+𝛾_𝑛𝑓𝑥_𝑛−1−𝑓𝑝,𝐽_𝜑𝑥_𝑛+𝛼−𝑝_𝑛+𝛾_𝑛𝑓𝑝−𝑝,𝐽_𝜑𝑥_𝑛+𝛽−𝑝_𝑛𝛼_𝑛𝑥_𝑛−1−𝑝𝜑𝑥_𝑛−𝑝+𝛾_𝑛𝑧_𝑛−𝑓𝑥_𝑛−1,𝐽_𝜑𝑥_𝑛−𝑝, (36) and so 𝑥_𝑛−𝑝−𝑘𝑥_𝑛−1−𝑝𝜑𝑥_𝑛−𝑝≤𝛽_𝑛𝑘𝑥_𝑛−1−𝑝𝜑𝑥_𝑛+−𝑝1−𝛽_𝑛𝑓𝑝−𝑝,𝐽_𝜑𝑥_𝑛+𝛽−𝑝_𝑛𝛼_𝑛𝑥_𝑛−1−𝑝𝜑𝑥_𝑛−𝑝+𝛾_𝑛𝑧_𝑛−𝑓𝑥_𝑛−1,𝐽_𝜑𝑥_𝑛−𝑝. (37) Using the fact that 𝐽_𝜑 is weakly sequentially continuous, Step 4, and condition (ii), we have 𝑥_𝑛−𝑝−𝑘𝑥_𝑛−1−𝑝𝜑𝑥_𝑛−𝑝→0as𝑛→∞. (38) This implies that 𝑥_𝑛→𝑝 as 𝑛→∞. In fact, if lim_𝑛→∞‖𝑥_𝑛−𝑝‖=𝜂≠0, then lim_𝑛→∞𝜑(‖𝑥_𝑛−𝑝‖)≠0, and by (38) lim_𝑛→∞(‖𝑥_𝑛−𝑝‖−𝑘‖𝑥_𝑛−1−𝑝‖)=0. This means that lim_𝑛→∞‖𝑥_𝑛−𝑝‖=𝜂=𝑘𝜂, which is a contradiction. Thus we proved that there exists a subsequence {𝑥_𝑛𝑘} of {𝑥_n} which converges strongly to a fixed point 𝑝 of 𝑇.
Step 6. We show that the entire sequence {𝑥_𝑛} converges strongly to 𝑝∈𝐹(𝑇). Suppose that there exists another subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} such that 𝑥_𝑛𝑗→𝑞 as 𝑗→∞. Since 𝑑(𝑥_𝑛𝑗,𝑃_𝑇(𝑥_𝑛𝑗))≤‖𝑥_𝑛𝑗−𝑧_𝑛𝑗‖→0 as 𝑗→∞, it follows that 𝑑(𝑞,𝑃_𝑇(𝑞))=0 and so 𝑞∈𝑃_𝑇(𝑞)⊂𝑇(𝑞); that is, 𝑞∈𝐹(𝑇). Notice that 𝑃_𝑇(𝑞)={𝑞}. Since {𝑥_𝑛} is bounded and the duality mapping 𝐽_𝜑 is single valued and weakly sequentially continuous from 𝐸 to 𝐸^∗, we have 𝑥_𝑛𝑗−𝑓𝑞,𝐽_𝜑𝑥_𝑛𝑗−𝑝−𝑞−𝑓𝑞,𝐽_𝜑=𝑥𝑞−𝑝_𝑛𝑗−𝑞,𝐽_𝜑𝑥_𝑛𝑗+𝑞−𝑝𝑞−𝑓,𝐽_𝜑𝑥_𝑛𝑗−𝑝−𝐽_𝜑≤𝑥𝑞−𝑝_𝑛𝑗−𝑞𝜑𝑥_𝑛𝑗+𝑞−𝑝𝑞−𝑓,𝐽_𝜑𝑥_𝑛𝑗−𝑝−𝐽_𝜑𝑞−𝑝→0as𝑗→∞. (39)
Thus, from Steps 2 and 4, it follows that 𝑞−𝑓𝑞,𝐽_𝜑𝑞−𝑝=lim_𝑗→∞𝑥_𝑛𝑗−𝑓𝑞,𝐽_𝜑𝑥_𝑛𝑗−𝑝=lim_𝑗→∞𝑥_𝑛𝑗−𝑧_𝑛𝑗,𝐽_𝜑𝑥_𝑛𝑗−𝑝+lim_𝑗→∞𝑧_𝑛𝑗−𝑓𝑥_𝑛𝑗−1,𝐽_𝜑𝑥_𝑛𝑗−𝑝+lim_𝑗→∞𝑓𝑥_𝑛𝑗−1−𝑓𝑞,𝐽_𝜑𝑥_𝑛𝑗≤𝑥−𝑝_𝑛𝑗−𝑧_𝑛𝑗𝜑𝑥_𝑛𝑗−𝑝+𝑘lim_𝑗→∞𝑥_𝑛𝑗−1−𝑞𝜑𝑥_𝑛𝑗−𝑝+lim_𝑗→∞𝑧_𝑛𝑗−𝑓𝑥_𝑛𝑗−1,𝐽_𝜑𝑥_𝑛𝑗−𝑝≤0. (40)
By the same argument, we also have 𝑝−𝑓𝑝,𝐽_𝜑𝑝−𝑞≤0. (41)
Therefore, from (40) and (41), we obtain 𝜑=𝑝−𝑞𝑝−𝑞𝑝−𝑞,𝐽_𝜑=𝑝𝑝−𝑞𝑝−𝑓,𝐽_𝜑+𝑓𝑝𝑞𝑝−𝑞−𝑓,𝐽_𝜑+𝑞𝑝−𝑞𝑞−𝑓,𝐽_𝜑≤𝑓𝑝𝑞𝑞−𝑝−𝑓,𝐽_𝜑𝜑,𝑝−𝑞≤𝑘𝑝−𝑞𝑝−𝑞 (42) and so (1−𝑘)‖𝑝−𝑞‖𝜑(‖𝑝−𝑞‖)≤0. Thus 𝑝=𝑞. This completes the proof.

Remark 3. (1) In Theorem 2, if 𝑓(𝑥)=𝑢∈𝐶,𝑥∈𝐸, is a constant mapping, then the iterative scheme (19) is reduced to the iterative scheme (11) in Theorem CY of Ceng and Yao [22] in the Introduction section. Therefore Theorem 2 improves Theorem CY to the viscosity iterative scheme in different Banach space.
(2) In Theorem 2, we remove the assumption that 𝐶 is a nonexpansive retract of 𝐸 in Theorem CY.
(3) In Theorem 2, if 𝛽_𝑛=0 for 𝑛≥0, then the iterative scheme (19) becomes the following scheme: 𝑥_𝑛∈𝛼_𝑛𝑓𝑥_𝑛−1+1−𝛼_𝑛𝑃_𝑇𝑥_𝑛, (43) which is a viscosity iterative scheme for those in Shahzad and Zegeye [15]. Therefore Theorem 2 develops Theorem 3.1 of Shahzad and Zegeye [15], as well as Theorem 1 of Jung [14], to the viscosity iterative method in different Banach space.
(4) Theorem 2 also improves and complements the corresponding results of Kim and Jung [17] and Sahu [19] as well as Jung and Kim [6, 7], López Acedo and Xu [18], and Xu and Yin [13].

By definition of the Hausdorff metric, we obtain that if 𝑇 is ∗-nonexpansive, then 𝑃_𝑇 is nonexpansive. Hence, as a direct consequence of Theorem 2, we have the following result.

Corollary 4. Let 𝐸 be a reflexive Banach space having a weakly sequentially continuous duality mapping 𝐽_𝜑 with gauge function 𝜑. Let 𝐶 be a nonempty closed convex subset of 𝐸 and 𝑇∶𝐶→𝒦(𝐸) a multivalued ∗-nonexpansive nonself-mapping such that 𝐹(𝑇)≠∅. Let 𝑓∶𝐶→𝐶 be a contraction with constant 𝑘. Suppose that, for each 𝑣∈𝐶 and 𝑡∈(0,1), the contraction 𝑆_𝑡 defined by 𝑆_𝑡𝑥=𝑡𝑃_𝑇𝑥+(1−𝑡)𝑣 has a fixed point 𝑥_𝑡∈𝐶. Let {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} be three sequences in (0,1) satisfying the following conditions:(i)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1;(ii)lim_𝑛→∞𝛼_𝑛=0 and lim_𝑛→∞(𝛽_𝑛/𝛼_𝑛)=0. For arbitrary initial value 𝑥₀∈𝐶, let the sequence {𝑥_𝑛} be generated by (19). Then {𝑥_𝑛} converges strongly to a fixed point of 𝑇.

It is well known that every nonempty closed convex subset 𝐶 of a strictly convex and reflexive Banach space 𝐸 is Chebyshev; that is, for any 𝑥∈𝐸, there is a unique element 𝑢∈𝐶 such that ‖𝑥−𝑢‖=inf{‖𝑥−𝑣‖∶𝑣∈𝐶}. Thus, we have the following corollary.

Corollary 5. Let 𝐸 be a strictly convex and reflexive Banach space having a weakly sequentially continuous duality mapping 𝐽_𝜑 with gauge function 𝜑. Let 𝐶 be a nonempty closed convex subset of 𝐸 and 𝑇∶𝐶→𝒦𝒞(𝐸) a multivalued nonself-mapping such that 𝐹(𝑇)≠∅ and 𝑃_𝑇 is nonexpansive. Let 𝑓∶𝐶→𝐶 be a contraction with constant 𝑘. Suppose that, for each 𝑣∈𝐶 and 𝑡∈(0,1), the contraction 𝑆_𝑡 defined by 𝑆_𝑡𝑥=𝑡𝑃_𝑇𝑥+(1−𝑡)𝑣 has a fixed point 𝑥_𝑡∈𝐶. Let {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} be three sequences in (0,1) satisfying the following conditions:(i)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1; (ii)lim_𝑛→∞𝛼_𝑛=0 and lim_𝑛→∞(𝛽_𝑛/𝛼_𝑛)=0. For arbitrary initial value 𝑥₀∈𝐶, let the sequence {𝑥_𝑛} be generated by (19). Then {𝑥_𝑛} converges strongly to a fixed point of 𝑇.

Proof. In this case, 𝑇𝑥 is Chebyshev for each 𝑥∈𝐶. So 𝑃_𝑇 is a selector of 𝑇 and 𝑃_𝑇 is single valued. Thus the result follows from Theorem 2.

Corollary 6. Let 𝐸 be a strictly convex and reflexive Banach space having a weakly sequentially continuous duality mapping 𝐽_𝜑 with gauge function 𝜑. Let 𝐶 be a nonempty closed convex subset of 𝐸 and 𝑇∶𝐶→𝒦𝒞(𝐸) a multivalued ∗-nonexpansive nonself-mapping such that 𝐹(𝑇)≠∅. Let 𝑓∶𝐶→𝐶 be a contraction with constant 𝑘. Suppose that, for each 𝑣∈𝐶 and 𝑡∈(0,1), the contraction 𝑆_𝑡 defined by 𝑆_𝑡𝑥=𝑡𝑃_𝑇𝑥+(1−𝑡)𝑣 has a fixed point 𝑥_𝑡∈𝐶. Let {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} be three sequences in (0,1) satisfying the following conditions:(i)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1;(ii)lim_𝑛→∞𝛼_𝑛=0 and lim_𝑛→∞(𝛽_𝑛/𝛼_𝑛)=0. For arbitrary initial value 𝑥₀∈𝐶, let the sequence {𝑥_𝑛} be generated by (19). Then {𝑥_𝑛} converges strongly to a fixed point of 𝑇.

Corollary 7. Let 𝐸 be a uniformly convex Banach space having a weakly sequentially continuous duality mapping 𝐽_𝜑 with gauge function 𝜑. Let 𝐶 be a nonempty closed convex subset of 𝐸 and 𝑇∶𝐶→𝒦(𝐸) a multivalued nonself-mapping satisfying the weak inwardness condition such that 𝑃_𝑇 is nonexpansive. Let 𝑓∶𝐶→𝐶 be a contraction with constant 𝑘. Let {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} be three sequences in (0,1) satisfying the following conditions:(i)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1;(ii)lim_𝑛→∞𝛼_𝑛=0 and lim_𝑛→∞(𝛽_𝑛/𝛼_𝑛)=0. For arbitrary initial value 𝑥₀∈𝐶, let the sequence {𝑥_𝑛} be generated by (19). Then {𝑥_𝑛} converges strongly to a fixed point of 𝑇.

Proof. Define, for each 𝑣∈𝐶 and 𝑡∈(0,1), the contraction 𝑆_𝑡∶𝐶→𝒦(𝐸) by 𝑆_𝑡𝑥∶=𝑡𝑃_𝑇𝑥+(1−𝑡)𝑣,𝑥∈𝐶. (44) As it is easily seen that 𝑆_𝑡 also satisfies the weak inwardness condition: 𝑆_𝑡𝑥⊂𝐼_𝐶(𝑥) for all 𝑥∈𝐶, it follows from Lemma 1 that 𝑆_𝑡 has a fixed point denoted by 𝑥_𝑡. Thus the result follows from Theorem 2.

Remark 8. (1) As in [31], Shahzad and Zegeye [15] gave the following example of a multivalued 𝑇 such that 𝑃_𝑇 is nonexpansive. Let 𝐶=[0,∞), and let 𝑇 be defined by 𝑇𝑥=[𝑥,2𝑥] for 𝑥∈𝐶. Then 𝑃_𝑇𝑥={𝑥} for 𝑥∈𝐶. Also 𝑇 is ∗-nonexpansive but not nonexpansive (see [31]).
(2) Corollaries 4–7 develop Corollaries 3.3–3.6 of Ceng and Yao [22] to the viscosity iterative method in different Banach spaces.
(3) By replacing the iterative scheme (11) in Theorem CY with the iterative scheme (19) in Theorem 2 and using the same proof lines as Theorem CY together with our method, we can also establish the viscosity iteration version of Theorem CY.

Acknowledgments

The author thanks the anonymous referees for their valuable comments and suggests, which improved the presentation of this paper and for providing some recent related papers. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012000895).