Weak and Strong Convergence Theorems for Strictly Pseudononspreading Mappings and Equilibrium Problem in Hilbert Spaces

Abstract

The purpose of this paper is to propose an iterative algorithm for equilibrium problem and a class of strictly pseudononspreading mappings which is more general than the class of nonspreading mappings studied recently in Kurokawa and Takahashi (2010). We explored an auxiliary mapping in our theorems and proofs and under suitable conditions, some weak and strong convergence theorems are proved. The results presented in the paper extend and improve some recent results announced by some authors.

1. Introduction and Preliminaries

Throughout this paper, we assume that 𝐻 is a real Hilbert space and 𝐶 is a nonempty and closed convex subset of 𝐻. In the sequel, we denote by “𝑥_𝑛→𝑥” and “𝑥_𝑛⇀𝑥” the strong and weak convergence of {𝑥_𝑛}, respectively. Denote by 𝐹(𝑇) the set of fixed points of a mapping 𝑇.

Definition 1. Let 𝑇∶𝐶→𝐶 be a mapping.(1)𝑇 is said to be nonexpansive, if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖,∀𝑥, 𝑦∈𝐶.(2)𝑇 is said to be quasinonexpansive, if 𝐹(𝑇) is nonempty and 𝑇𝑥−𝑝≤𝑥−𝑝,∀𝑥∈𝐶,𝑝∈𝐹(𝑇). (1)(3)𝑇 is said to be nonspreading [1, 2], if 2𝑇𝑥−𝑇𝑦²≤𝑇𝑥−𝑦²+𝑇𝑦−𝑥²,∀𝑥,𝑦∈𝐶. (2)It is easy to prove that 𝑇∶𝐶→𝐶 is nonspreading if and only if 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+2𝑥−𝑇𝑥,𝑦−𝑇𝑦∀𝑥,𝑦∈𝐶. (3)(4)𝑇∶𝐶→𝐻 is said to be 𝑘-strictly pseudononspreading in the terminology of Browder-Petryshyn [3], if there exists 𝑘∈[0,1) such that 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+𝑘𝑥−𝑇𝑥−𝑦−𝑇𝑦²+2𝑥−𝑇𝑥,𝑦−𝑇𝑦∀𝑥,𝑦∈𝐶. (4)

Remark 2. (1) If 𝑇∶𝐶→𝐶 is a nonspreading mapping with 𝐹(𝑇)≠∅, then 𝑇 is quasinonexpansive and 𝐹(𝑇) is closed and convex.
(2) Clearly every nonspreading mapping is 𝑘-strictly pseudononspreading with 𝑘=0, but the inverse is not true. This can be seen from the following example.

Example 3. Let ℛ denote the set of all real numbers. Let 𝑇∶ℛ→ℛ be a mapping defined by (),[).𝑇𝑥=𝑥,𝑥∈−∞,0−2𝑥,𝑥∈0,∞ (5) It is easy to see that 𝑇 is a 𝑘-strictly pseudononspreading mapping with 𝑘∈[0,1), but it is not nonspreading (see, [4]).

Definition 4. (1) Let 𝑇∶𝐻→𝐻 be a mapping. 𝐼−𝑇 is said to be demiclosed at 0, if for any sequence {𝑥_𝑛}⊂𝐻 with 𝑥_𝑛⇀𝑥^∗ and ‖(𝐼−𝑇)𝑥_𝑛‖→0, we have 𝑥^∗=𝑇𝑥^∗.
(2) A Banach space 𝐸 is said to have Opial's property, if for any sequence {𝑥_𝑛}⊂𝐸 with 𝑥_𝑛⇀𝑥^∗, we have liminf_𝑛→∞𝑥_𝑛−𝑥^∗<liminf_𝑛→∞𝑥_𝑛−𝑦,∀𝑦∈𝐸with𝑦≠𝑥^∗. (6)
It is well known that each Hilbert space processes opial property.
(3) A mapping 𝑆∶𝐶→𝐶 is said to be semicompact, if for any bounded sequence {𝑥_𝑛}⊂𝐶 with lim_𝑛→∞‖𝑥_𝑛−𝑆𝑥_𝑛‖=0, then there exists a subsequence {𝑥_𝑛𝑖}⊂{𝑥_𝑛} such that {𝑥_𝑛𝑖} converges strongly to some point 𝑥^∗∈𝐶.

Lemma 5 (see [5]). Let 𝐸 be a uniformly convex Banach space and let 𝐵_𝑟(0)∶={𝑥∈𝐸∶‖𝑥‖≤𝑟} be a closed ball with center 0 and radius 𝑟>0. For any given sequence {𝑥₁,𝑥₂,…,𝑥_𝑛,…}⊂𝐵_𝑟(0) and any given number sequence {𝜆₁,𝜆₂,…,𝜆_𝑛,…} with 𝜆_𝑖≥0, ∑^∞_𝑖=1𝜆_𝑖=1, there exists a continuous strictly increasing and convex function 𝑔∶[0,2𝑟)→[0,∞) with 𝑔(0)=0 such that for any 𝑖,𝑗∈𝒩, 𝑖<𝑗 the following holds: ^∞_𝑛=1𝜆_𝑛𝑥_𝑛²≤^∞_𝑛=1𝜆_𝑛𝑥_𝑛²−𝜆_𝑖𝜆_𝑗𝑔𝑥_𝑖−𝑥_𝑗. (7)

Lemma 6. Let 𝐻 be a real Hilbert space, 𝐶 be a nonempty and closed convex subset of 𝐻, and let 𝑇∶𝐶→𝐶 be a 𝑘-strictly pseudononspreading mapping.(i) If 𝐹(𝑇)≠∅, then it is closed and convex.(ii)(𝐼−𝑇) is demiclosed at origin.

Lemma 7. Let 𝑇∶𝐶→𝐶 be a 𝑘-strictly pseudononspreading mapping with 𝑘∈[0,1). Denote by 𝑇_𝛽∶=𝛽𝐼+(1−𝛽)𝑇, where 𝛽∈[𝑘,1), then(i)𝐹(𝑇)=𝐹(𝑇_𝛽); (ii) the following inequality holds: 𝑇_𝛽𝑥−𝑇_𝛽𝑦²≤𝑥−𝑦²+21−𝛽𝑥−𝑇_𝛽𝑥,𝑦−𝑇_𝛽𝑦,∀𝑥,𝑦∈𝐶; (8)(iii)𝑇_𝛽 is a quasinonexpansive mapping, that is, 𝑇_𝛽𝑥−𝑝²≤𝑥−𝑝²,∀𝑥∈𝐶,𝑝∈𝐹(𝑇). (9)

Proof. The conclusion (i) is obvious. Now we prove the conclusion (ii). Since 𝑇 is 𝑘-strictly pseudononspreading, for any 𝑥,𝑦∈𝐶 we have 𝑇_𝛽𝑥−𝑇_𝛽𝑦²=𝛽𝑥−𝑦+1−𝛽𝑇𝑥−𝑇𝑦²=𝛽𝑥−𝑦²+1−𝛽𝑇𝑥−𝑇𝑦²−𝛽1−𝛽𝑥−𝑇𝑥−(𝑦−𝑇𝑦)²≤𝛽𝑥−𝑦²+×1−𝛽𝑥−𝑦²+𝑘𝑥−𝑇𝑥−𝑦−𝑇𝑦²+2𝑥−𝑇𝑥,𝑦−𝑇𝑦−𝛽1−𝛽𝑥−𝑇𝑥−𝑦−𝑇𝑦²=𝑥−𝑦²−+21−𝛽𝑥−𝑇𝑥,𝑦−𝑇𝑦1−𝛽𝛽−𝑘𝑥−𝑇𝑥−𝑦−𝑇𝑦²≤𝑥−𝑦²=+21−𝛽𝑥−𝑇𝑥,𝑦−𝑇𝑦𝑥−𝑦²+21−𝛽𝑥−𝑇_𝛽𝑥,𝑦−𝑇_𝛽𝑦. (10)
Take 𝑦∈𝐹(𝑇) in (8), then 𝑦∈𝐹(𝑇_𝛽). Hence, conclusion (iii) is proved.
This completes the proof.

In the sequel, we assume that 𝜙∶𝐶×𝐶→ℛ is a bifunction satisfying the following conditions:(A1)𝜙(𝑥,𝑥)=0,∀𝑥∈𝐶;(A2)𝜙 is monotone, that is, 𝜙(𝑥,𝑦)+𝜙(𝑦,𝑥)≤0,∀𝑥,𝑦∈𝐶;(A3)lim_𝑡↓0𝜙(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝜙(𝑥,𝑦),∀𝑥,𝑦,𝑧∈𝐶;(A4) for each 𝑥∈𝐶, 𝑦↦𝜙(𝑥,𝑦) is convex and lower semicontinuous.

Recalled that the “so-called” equilibrium problem for a bifunction function  𝜙 is to find a point 𝑥^∗∈𝐶, such that 𝜙𝑥^∗,𝑦≥0,∀𝑦∈𝐶. (11)

Lemma 8 (see [6, 7]). Let 𝐶 be a nonempty and closed convex subset of a Hilbert space 𝐻 and let 𝜙∶𝐶×𝐶→ℛ be a bi-function satisfying conditions: (A1), (A2), (A3), and (A4). Then, for any 𝑟>0 and 𝑥∈𝐶, there exists 𝑧∈𝐶 such that 𝜙+1𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶. (12) Furthermore, if for given 𝑟>0, we define a mapping 𝑇_𝑟∶𝐶→𝐶 by 𝑇_𝑟+1𝑥∶=𝑧∈𝐶∶𝜙𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶, (13) then the following hold:(1)𝑇_𝑟 is single-valued;(2)𝑇_𝑟 is firmly nonexpansive, that is, ‖𝑇_𝑟𝑥−𝑇_𝑟𝑦‖²≤⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝑥−𝑦⟩;(3)𝐹(𝑇_𝑟)=Ω, where Ω is the set of solutions of the equilibrium problem (11);(4)Ω is a closed and convex subset of 𝐶.

Concerning the weak and strong convergence problem for some kinds of iterative algorithms for nonspreading mappings, 𝑘-strictly pseudononspreading mappings and other kind of nonlinear mappings have been considered in Osilike and Isiogugu [4], Igarashi et al. [8], Iemoto and Takahashi [9], Kurokawa and Takahashi [10], and Kim [11–28]. The purpose of this paper is to propose an iterative algorithm for an infinite family of strictly pseudononspreading mappings and equilibrium problem. Under suitable conditions, some weak and strong convergence theorems are proved. The results presented in the paper extend and improve the corresponding results in [4, 8–11].

2. Main Results

Throughout this section, we assume that the following conditions are satisfied. (1)𝐻 is a real Hilbert spaces, 𝐶 is a nonempty and close convex subset of 𝐻. (2) For each 𝑆_𝑖∶𝐶→𝐶, 𝑖=1,2,… is a 𝑘_𝑖-strictly pseudononspreading mapping with 𝑘∶=sup_𝑖≥1𝑘_𝑖∈(0,1). For given 𝛽∈[𝑘,1), denoted by 𝑆_𝑖,𝛽∶=𝛽𝐼+(1−𝛽)𝑆_𝑖, for each 𝑖=1,2,…, it follows from (8) that 𝑆_𝑖,𝛽𝑥−𝑆_𝑖,𝛽𝑦²≤𝑥−𝑦²+21−𝛽𝑥−𝑆_𝑖,𝛽𝑥,𝑦−𝑆_𝑖,𝛽𝑦,∀𝑥,𝑦∈𝐶. (14)(3)𝜙∶𝐶×𝐶→ℛ is a bifunction satisfying the conditions (A1)–(A4). Then it follows from Lemma 8 that the mapping 𝑇_𝑟 defined by (13) is single valued, 𝑧=𝑇_𝑟𝑥, 𝐹(𝑇_𝑟)=Ω (where Ω is the solution set of the equilibrium problem (11)), and Ω is a closed and convex subset of 𝐶.

We are now in a position to give the following result.

Theorem 9. Let 𝐻,𝐶, {𝑆_𝑖}, 𝑘,𝛽, {𝑆_𝑖,𝛽}, 𝜙, 𝑇_𝑟, and Ω be the same as above. Let {𝑥_𝑛} and {𝑢_𝑛} be the sequences defined by 𝑥₁𝜙𝑢∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝛼_0,𝑛𝑢_𝑛+^∞_𝑖=1𝛼_𝑖,𝑛𝑆_𝑖,𝛽𝑢_𝑛, (15) where {𝛼_𝑖,𝑛}⊂(0,1) and {𝑟_𝑛} satisfy the following conditions:(a)∑^∞_𝑖=0𝛼_𝑖,𝑛=1, for each 𝑛≥1; (b) for each 𝑖≥1,  liminf_𝑛→∞𝛼_0,𝑛𝛼_𝑖,𝑛>0; (c){𝑟_𝑛}⊂(0,∞) and liminf_𝑛→∞𝑟_𝑛>0. (I) If ℱ∶=(⋂^∞_𝑖=1𝐹(𝑆_𝑖))⋂Ω≠∅, then both {𝑥_𝑛} and {𝑢_𝑛} converge weakly to some point 𝑥^∗∈ℱ; (II) in addition, if there exists some positive integer 𝑚 such that 𝑆_𝑚 is semicompact, then both {𝑥_𝑛} and {𝑢_𝑛} converge strongly to 𝑥^∗∈ℱ.

Proof. First, we prove the conclusion (I). The proof is divided into three steps.
Step  1. We prove that the sequences {𝑥_𝑛}, {𝑢_𝑛}, {𝑆_𝑖,𝛽𝑢_𝑛}, and {𝑆_𝑖,𝛽𝑥_𝑛},𝑖≥1 all are bounded, and for each 𝑝∈ℱ the limits lim_𝑛→∞‖𝑥_𝑛−𝑝‖, lim_𝑛→∞‖𝑢_𝑛−𝑝‖ exist and lim_𝑛→∞𝑥_𝑛−𝑝=lim_𝑛→∞𝑢_𝑛−𝑝. (16)
In fact, it follows from Lemma 8 that 𝑢_𝑛=𝑇_𝑟𝑛𝑥_𝑛, 𝑝=𝑇_𝑟𝑛𝑝, and 𝑢_𝑛−𝑝=𝑇_𝑟𝑛𝑥_𝑛−𝑇_𝑟𝑛𝑝≤𝑥_𝑛−𝑝,∀𝑛≥1. (17) Since 𝑝∈ℱ, by Lemma 7(i), 𝑝∈⋂^∞_𝑖=1𝐹(𝑆_𝑖,𝛽). Hence, it follows from (17) and (9) that 𝑥_𝑛+1−𝑝=𝛼_0,𝑛𝑢_𝑛+^∞_𝑖=1𝛼_𝑖,𝑛𝑆_𝑖,𝛽𝑢_𝑛−𝑝≤𝛼_0,𝑛𝑢_𝑛−𝑝+^∞_𝑖=1𝛼_𝑖,𝑛𝑆_𝑖,𝛽𝑢_𝑛−𝑝≤𝛼_0,𝑛𝑢_𝑛−𝑝+^∞_𝑖=1𝛼_𝑖,𝑛𝑢_𝑛=𝑢−𝑝_𝑛−𝑝≤𝑥_𝑛−𝑝,∀𝑛≥1. (18) This implies that for each 𝑝∈ℱ, the limits lim_𝑛→∞‖𝑥_𝑛−𝑝‖ and lim_𝑛→∞‖𝑢_𝑛−𝑝‖ exist. And so {𝑥_𝑛} and {𝑢_𝑛} are bounded and (16) holds.
Furthermore, by (9), it is easy to see that for each 𝑖≥1, {𝑆_𝑖,𝛽𝑢_𝑛} and {𝑆_𝑖,𝛽𝑥_𝑛} are also bounded.
Step  2. Next we prove that for each 𝑖≥1 the following holds: lim_𝑛→∞𝑥_𝑛−𝑆_𝑖𝑥_𝑛=lim_𝑛→∞𝑢_𝑛−𝑆_𝑖𝑢_𝑛=0. (19) In fact, by Lemma 5 for any positive integer 𝑖≥1 and 𝑝∈ℱ, we have 𝑥_𝑛+1−𝑝²=𝛼_0,𝑛𝑢_𝑛−𝑝+^∞_𝑖=1𝛼_𝑖,𝑛𝑆_𝑖,𝛽𝑢_𝑛−𝑝²≤𝛼_0,𝑛𝑢_𝑛−𝑝²+^∞_𝑖=1𝛼_𝑖,𝑛𝑆_𝑖,𝛽𝑢_𝑛−𝑝²−𝛼_0,𝑛𝛼_𝑖,𝑛𝑔𝑢_𝑛−𝑆_𝑖,𝛽𝑢_𝑛≤𝛼_0,𝑛𝑢_𝑛−𝑝²+^∞_𝑖=1𝛼_𝑖,𝑛𝑢_𝑛−𝑝²−𝛼_0,𝑛𝛼_𝑖,𝑛𝑔𝑢_𝑛−𝑆_𝑖,𝛽𝑢_𝑛≤𝑢_𝑛−𝑝²−𝛼_0,𝑛𝛼_𝑖,𝑛𝑔𝑢_𝑛−𝑆_𝑖,𝛽𝑢_𝑛≤𝑥_𝑛−𝑝²−𝛼_0,𝑛𝛼_𝑖,𝑛𝑔𝑢_𝑛−𝑆_𝑖,𝛽𝑢_𝑛. (20) This shows that 𝛼_0,𝑛𝛼_𝑖,𝑛𝑔𝑢_𝑛−𝑆_𝑖,𝛽𝑢_𝑛≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²⟶0(as𝑛⟶∞). (21) Since 𝑔 is a continuous and strictly increasing function with 𝑔(0)=0. By condition (b), it yields that lim_𝑛→∞𝑢_𝑛−𝑆_𝑖,𝛽𝑢_𝑛=0. (22) Therefore, we have lim_𝑛→∞𝑢_𝑛−𝑆_𝑖𝑢_𝑛=lim_𝑛→∞1𝑢1−𝛽^𝑛−𝑆_𝑖,𝛽𝑢_𝑛=0. (23) On the other hand, it follows from Lemma 8 that 𝑢_𝑛=𝑇_𝑟𝑛𝑥_𝑛 and for each 𝑝∈ℱ𝑢_𝑛−𝑝²=𝑇_𝑟𝑛𝑥_𝑛−𝑇_𝑟𝑛𝑝²≤𝑇_𝑟𝑛𝑥_𝑛−𝑇_𝑟𝑛𝑝,𝑥_𝑛=𝑢−𝑝_𝑛−𝑝,𝑥_𝑛=1−𝑝2𝑢_𝑛−𝑝²+𝑥_𝑛−𝑝²−𝑥_𝑛−𝑢_𝑛². (24) This shows that 𝑢_𝑛−𝑝²≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝑢_𝑛². (25) In view of (20) and (25) 𝑥_𝑛+1−𝑝²≤𝑢_𝑛−𝑝²≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝑢_𝑛², (26) that is, 𝑥_𝑛−𝑢_𝑛²≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²⟶0(as𝑛⟶∞). (27) In view of (27), (22), (14), and noting that {𝑥_𝑛−𝑆_𝑖,𝛽𝑥_𝑛} is bounded, we have 𝑥_𝑛−𝑆_𝑖,𝛽𝑥_𝑛≤𝑥_𝑛−𝑢_𝑛+𝑢_𝑛−𝑆_𝑖,𝛽𝑢_𝑛+𝑆_𝑖,𝛽𝑢_𝑛−𝑆_𝑖,𝛽𝑥_𝑛≤𝑥_𝑛−𝑢_𝑛+𝑢_𝑛−𝑆_𝑖,𝛽𝑢_𝑛+𝑥_𝑛−𝑢_𝑛²+2𝑢1−𝛽^𝑛−𝑆_𝑖,𝛽𝑢_𝑛,𝑥_𝑛−𝑆_𝑖,𝛽𝑥_𝑛^1/2⟶0(as𝑛⟶∞). (28) Therefore, we have lim_𝑛→∞𝑥_𝑛−𝑆_𝑖𝑥_𝑛=lim_𝑛→∞1𝑥1−𝛽^𝑛−𝑆_𝑖,𝛽𝑥_𝑛=0. (29)
The conclusion is proved.
Step  3. Next we prove that the weak-accumulation point set 𝑊_𝜔(𝑥_𝑛) of the sequence {𝑥_𝑛} is a singleton and 𝑊_𝜔(𝑥_𝑛)⊂ℱ.
In fact, for any 𝑤∈𝑊_𝜔(𝑥_𝑛), their exists a subsequence {𝑥_𝑛𝑖}⊂{𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑤. It follows from (27) that 𝑢_𝑛𝑖⇀𝑤. Since 𝑢_𝑛=𝑇_𝑟𝑛𝑥_𝑛, from (15) and condition (A2) we have 𝑦−𝑢_𝑛𝑖,1𝑟_𝑛𝑖𝑢_𝑛𝑖−𝑥_𝑛𝑖≥𝜙𝑦,𝑢_𝑛𝑖,∀𝑦∈𝐶. (30) Since (1/𝑟_𝑛𝑖)(𝑢_𝑛𝑖−𝑥_𝑛𝑖)→0(as𝑛_𝑖→∞) and 𝑢_𝑛𝑖⇀𝑤, it follows from condition (A4) that 𝜙𝑦,𝑤≤0,∀𝑦∈𝐶. (31) For any 𝑡∈(0,1),𝑦∈𝐶, letting 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑤, then 𝑦_𝑡∈𝐶. By condition (A1) and (A4), we have 0=𝜙𝑦_𝑡,𝑦_𝑡≤𝑡𝜙𝑦_𝑡,𝑦+(1−𝑡)𝜙𝑦_𝑡,𝑤≤𝑡𝜙𝑦_𝑡,𝑦. (32) This implies that 𝜙(𝑦_𝑡,𝑦)≥0. Letting 𝑡→0, by condition (A3) we have 𝜙𝑤,𝑦≥0,∀𝑦∈𝐶. (33) This shows that 𝑤∈𝐶 is a solution to the equilibrium (11), that is, 𝑤∈Ω.
On the other hand, by Lemma 6, for each 𝑖≥1, 𝐼−𝑆_𝑖 is demiclosed at 0. In view of (19), we know that 𝑤∈ℱ. Due to the arbitrariness of 𝑤∈𝑊_𝜔(𝑥_𝑛), we have 𝑊_𝜔(𝑥_𝑛)⊂ℱ.
Now we prove that 𝑊_𝜔(𝑥_𝑛) is a singleton. Suppose to the contrary that there exist 𝑥^∗,𝑦^∗∈𝑊_𝜔(𝑥_𝑛) with 𝑥^∗≠𝑦^∗. Therefore, there exist subsequences {𝑥_𝑛𝑖} and {𝑥_𝑛𝑗} in {𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑥^∗ and 𝑥_𝑛𝑗⇀𝑦^∗. Since 𝑥^∗,𝑦^∗∈ℱ, by (16), the limits lim_𝑛→∞‖𝑥_𝑛−𝑥^∗‖ and lim_𝑛→∞‖𝑥_𝑛−𝑦^∗‖ exist. By using the opial property of 𝐻, we have liminf_{𝑛𝑖→∞}𝑥_𝑛𝑖−𝑥^∗<liminf_{𝑛𝑖→∞}𝑥_𝑛𝑖−𝑦^∗=lim_𝑛→∞𝑥_𝑛−𝑦^∗=lim_{𝑛𝑗→∞}𝑥_𝑛𝑗−𝑦^∗<liminf_{𝑛𝑗→∞}𝑥_𝑛𝑗−𝑥^∗=lim_𝑛→∞𝑥_𝑛−𝑥^∗=liminf_{𝑛𝑖→∞}𝑥_𝑛𝑖−𝑥^∗. (34) This is a contradiction. Therefore, 𝑊_𝜔(𝑥_𝑛) is a singleton. Without loss of generality, we can assume that 𝑊_𝜔(𝑥_𝑛)={𝑥^∗} and 𝑥_𝑛⇀𝑥^∗. By using (15) and (19), we have 𝑢_𝑛⇀𝑥^∗.
This completes the proof of the conclusion (I).
Next we prove the conclusion (II).
Without loss of generality, we can assume that 𝑆₁ is semicompact. From (19) we have that 𝑥_𝑛−𝑆₁𝑥_𝑛⟶0(as𝑛⟶∞). (35) Therefore, there exists a subsequence of {𝑥_𝑛𝑖}⊂{𝑥_𝑛} such that 𝑥_𝑛𝑖→𝑢^∗∈𝐶. Since 𝑥_𝑛𝑖⇀𝑥^∗, we have 𝑥^∗=𝑢^∗ and so 𝑥_𝑛𝑖→𝑥^∗∈ℱ. By virtue of (16), we have lim_𝑛→∞𝑢_𝑛−𝑥^∗=0,lim_𝑛→∞𝑥_𝑛−𝑥^∗=0, (36)
This completes the proof of Theorem 9.

Taking 𝜙≡0 and 𝑟_𝑛=1,forall𝑛≥1 in Theorem 9, we have 𝑥_𝑛=𝑢_𝑛,forall𝑛≥1, Therefore, the following theorem can be obtained from Theorem 9 immediately.

Theorem 10. Let 𝐻,𝐶, {𝑆_𝑖}, 𝑘,𝛽 and {𝑆_𝑖,𝛽} be the same as in Theorem 9. Let {𝑥_𝑛} be the sequences defined by 𝑥₁𝑥∈𝐶,𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛+1=𝛼_0,𝑛𝑥_𝑛+^∞_𝑖=1𝛼_𝑖,𝑛𝑆_𝑖,𝛽𝑥_𝑛, (37) where {𝛼_𝑖,𝑛}⊂(0,1) satisfies the following conditions:(a)∑^∞_𝑖=0𝛼_𝑖,𝑛=1, for each 𝑛≥1;(b) for each 𝑖≥1,  liminf_𝑛→∞𝛼_0,𝑛𝛼_𝑖,𝑛>0.(I) If ℱ∶=(⋂^∞_𝑖=1𝐹(𝑆_𝑖))≠∅, then both {𝑥_𝑛} converge weakly to some point 𝑥^∗∈ℱ;(II) in addition, if there exists some positive integer 𝑚 such that 𝑆_𝑚 is semicompact, then {𝑥_𝑛} converge strongly to 𝑥^∗∈ℱ.

Remark 11. Theorems 9 and 10 improve and extend the corresponding recent results of [4, 8–11].

Acknowledgments

The authors would like to express their thanks to the referees for their helpful comments and advices. This work was supported by the Natural Science Foundation of Yunnan Province (Grant no. 2011FB074).