Abstract

We introduce an implicit and explicit iterative schemes for a finite family of nonexpansive semigroups with the Meir-Keeler-type contraction in a Banach space. Then we prove the strong convergence for the implicit and explicit iterative schemes. Our results extend and improve some recent ones in literatures.

1. Introduction

Let 𝐶 be a nonempty subset of a Banach space 𝐸 and 𝑇∶𝐶→𝐶 be a mapping. We call 𝑇 nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐸. The set of all fixed points of 𝑇 is denoted by Fix(𝑇), that is, Fix(𝑇)={𝑥∈𝐶∶𝑥=𝑇𝑥}.

One parameter family 𝒯={𝑇(𝑡)∶𝑡≥0} is said to a semigroup of nonexpansive mappings or nonexpansive semigroup on 𝐶 if the following conditions are satisfied: (1)𝑇(0)𝑥=𝑥 for all 𝑥∈𝐶; (2)𝑇(𝑠+𝑡)=𝑇(𝑠)𝑇(𝑡) for all 𝑠,𝑡≥0; (3)for each 𝑡≥0, ‖𝑇(𝑡)𝑥−𝑇(𝑡)𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶; (4)for each 𝑥∈𝐶, the mapping 𝑇(⋅)𝑥 from ℝ+, where ℝ+ denotes the set of all nonnegative reals, into 𝐶 is continuous.

We denote by Fix(𝒯) the set of all common fixed points of semigroup 𝒯, that is, Fix(𝒯)={𝑥∈𝐶∶𝑇(𝑡)𝑥=𝑥,0≤𝑡<∞} and ℕ by the set of natural numbers.

Now, we recall some recent work on nonexpansive semigroup in literatures. In [1], Shioji and Takahashi introduced the following implicit iteration for a nonexpansive semigroup in a Hilbert space:𝑥𝑛=𝛼𝑛𝑥+1−𝛼𝑛1𝑡𝑛𝑡𝑛0𝑇(𝑠)𝑥𝑛𝑑𝑠,∀𝑛∈ℕ,(1.1) where {𝛼𝑛}⊂(0,1) and {𝑡𝑛}⊂(0,∞). Under the certain conditions on {𝛼𝑛} and {𝑡𝑛}, they proved that the sequence {𝑥𝑛} defined by (1.1) converges strongly to an element in Fix(𝒯).

In [2], Suzuki introduced the following implicit iteration for a nonexpansive semigroup in a Hilbert space:𝑥𝑛=𝛼𝑛𝑢+1−𝛼𝑛𝑇𝑡𝑛𝑥𝑛,∀𝑛∈ℕ,(1.2) where {𝛼𝑛}⊂(0,1) and {𝑡𝑛}⊂(0,∞). Under the conditions that lim𝑛→∞𝑡𝑛=lim𝑛→∞𝛼𝑛/𝑡𝑛=0, he proved that {𝑥𝑛} defined by (1.2) converges strongly to an element of Fix(𝒯). Later on, Xu [3] extended the iteration (1.2) to a uniformly convex Banach space that admits a weakly sequentially continuous duality mapping. Song and Xu [4] also extended the iteration (1.2) to a reflexive and strictly convex Banach space.

In 2007, Chen and He [5] studied the following implicit and explicit viscosity approximation processes for a nonexpansive semigroup in a reflexive Banach space admitting a weakly sequentially continuous duality mapping:𝑥𝑛=𝛼𝑛𝑓𝑥𝑛+1−𝛼𝑛𝑇𝑡𝑛𝑥𝑛,𝑦𝑛+1=𝛽𝑛𝑓𝑦𝑛+1−𝛽𝑛𝑇𝑡𝑛𝑦𝑛,∀𝑛∈ℕ,(1.3) where 𝑓 is a contraction, {𝛼𝑛}⊂(0,1) and {𝑡𝑛}⊂(0,∞). They proved the strong convergence for the above iterations under some certain conditions on the control sequences.

Recently, Chen et al. [6] introduced the following implicit and explicit iterations for nonexpansive semigroups in a reflexive Banach space admitting a weakly sequentially continuous duality mapping:𝑦𝑛=𝛼𝑛𝑥𝑛+1−𝛼𝑛𝑇𝑡𝑛𝑥𝑛,𝑥𝑛=𝛽𝑛𝑓𝑥𝑛+1−𝛽𝑛𝑦𝑛𝑦,∀𝑛∈ℕ,(1.4)𝑛=𝛼𝑛𝑥𝑛+1−𝛼𝑛𝑇𝑡𝑛𝑥𝑛,𝑥𝑛+1=𝛽𝑛𝑓𝑥𝑛+1−𝛽𝑛𝑦𝑛,∀𝑛∈ℕ,(1.5) where 𝑓 is a contraction, {𝛼𝑛}⊂(0,1) and {𝑡𝑛}⊂(0,∞). They proved that {𝑥𝑛} defined by (1.4) and (1.5) converges strongly to an element 𝑞 of Fix(𝒯), which is the unique solution of the following variation inequality problem:⟨(𝑓−𝐼),𝑗(𝑥−𝑞)⟩≤0,∀𝑥∈Fix(𝒯).(1.6)

For more convergence theorems on implicit and explicit iterations for nonexpansive semigroups, refer to [7–13].

In this paper, we introduce an implicit and explicit iterative process by a generalized contraction for a finite family of nonexpansive semigroups in a Banach space. Then we prove the strong convergence for the iterations and our results extend the corresponding ones of Suzuki [2], Xu [3], Chen and He [5], and Chen et al. [6].

2. Preliminaries

Let 𝐸 be a Banach space and 𝐸∗ the duality space of 𝐸. We denote the normalized mapping from 𝐸 to 2𝐸∗ by 𝐽 defined by𝐽(𝑥)=𝑗∈𝐸∗∶⟨𝑥,𝑗𝑥⟩=‖𝑥‖2=‖𝑗‖,∀𝑥∈𝐸,(2.1) where ⟨⋅,⋅⟩ denotes the generalized duality pairing. For any 𝑥,𝑦∈𝐸 with 𝑗(𝑥)∈𝐽(𝑥) and 𝑗(𝑥+𝑦)∈𝐽(𝑥+𝑦), it is well known that the following inequality holds: ‖𝑥‖2+2⟨𝑦,𝑗(𝑥)⟩≤‖𝑥+𝑦‖2≤‖𝑥‖2+2⟨𝑦,𝑗(𝑥+𝑦)⟩.(2.2)

The dual mapping 𝐽 is called weakly sequentially continuous if 𝐽 is single valued, and {𝑥𝑛}⇀𝑥∈𝐸, where ⇀ denotes the weak convergence, then 𝐽(𝑥𝑛) weakly star converges to 𝐽(𝑥) [14–16]. A Banach space 𝐸 is called to satisfy Opial’s condition [17] if for any sequence {𝑥𝑛} in 𝐸, 𝑥𝑛⇀𝑥, limsup𝑛→∞‖‖𝑥𝑛‖‖−𝑥<limsup𝑛→∞‖‖𝑥𝑛‖‖−𝑦,∀𝑦∈𝐸with𝑥≠𝑦.(2.3) It is known that if 𝐸 admits a weakly sequentially continuous duality mapping 𝐽, then 𝐸 is smooth and satisfies Opial’s condition [14].

A function 𝜓∶ℝ+→ℝ+ is said to be an 𝐿-function if 𝜓(0)=0, 𝜓(𝑡)>0 for any 𝑡>0, and for every 𝑡>0 and 𝑠>0, there exists 𝑢>𝑠 such that 𝜓(𝑡)≤𝑠, for all 𝑡∈[𝑠,𝑢]. This implies that 𝜓(𝑡)<𝑡 for all 𝑡>0.

Let 𝑓∶𝐶→𝐶 be a mapping. 𝑓 is said to be a (𝜓,𝐿)-contraction if there exists a 𝐿-function 𝜓∶ℝ+→ℝ+ such that ‖𝑓(𝑥)−𝑓(𝑦)‖<𝜓(‖𝑥−𝑦‖) for all 𝑥,𝑦∈𝐶 with 𝑥≠𝑦. Obviously, if 𝜓(𝑡)=𝑘𝑡 for all 𝑡>0, where 𝑘∈(0,1), then 𝑓 is a contraction. 𝑓 is called a Meir-Keeler-type mapping if for each 𝜖>0, there exists 𝛿(𝜖)>0 such that for all 𝑥,𝑦∈𝐶, if 𝜖<‖𝑥−𝑦‖<𝜖+𝛿, then ‖𝑓(𝑥)−𝑓(𝑦)‖<𝜖.

In this paper, we always assume that 𝜓(𝑡) is continuous, strictly increasing and lim𝑡→∞𝜂(𝑡)=∞, where 𝜂(𝑡)=𝑡−𝜓(𝑡), is strictly increasing and onto.

The following lemmas will be used in next section.

Lemma 2.1 (see [18]). Let (𝑋,𝑑) be a metric space and 𝑓∶𝑋→𝑋 be a mapping. The following assertions are equivalent: (i)𝑓 is a Meir-Keeler-type mapping;(ii)there exists an 𝐿-function 𝜓∶ℝ+→ℝ+ such that 𝑓 is a (𝜓,𝐿)-contraction.

Lemma 2.2 (see [19]). Let 𝐸 be a Banach space and 𝐶 be a convex subset of 𝐸. Let 𝑇∶𝐶→𝐶 be a nonexpansive mapping and 𝑓 be a (𝜓,𝐿)-contraction. Then the following assertions hold: (i)𝑇∘𝑓 is a (𝜓,𝐿)-contraction on 𝐶 and has a unique fixed point in 𝐶;(ii)for each 𝛼∈(0,1), the mapping 𝑥↦𝛼𝑓(𝑥)+(1−𝛼)𝑇𝑥 is of Meir-Keeler-type and it has a unique fixed point in 𝐶.

Lemma 2.3 (see [20]). Let 𝐸 be a Banach space and 𝐶 be a convex subset of 𝐸. Let 𝑓∶𝐶→𝐶 be a Meir-Keeler-type contraction. Then for each 𝜖>0 there exists 𝑟∈(0,1) such that, for each 𝑥,𝑦∈𝐶 with ‖𝑥−𝑦‖≥𝜖,‖𝑓(𝑥)−𝑓(𝑦)‖≤𝑟‖𝑥−𝑦‖.

Lemma 2.4 (see [21]). Let C be a closed convex subset of a strictly convex Banach space E. Let 𝑇𝑚∶𝐶→𝐶 be a nonexpansive mapping for each 1≤𝑚≤𝑟, where 𝑟 is some integer. Suppose that ∩𝑟𝑚=1Fix(𝑇𝑚) is nonempty. Let {𝜆𝑛} be a sequence of positive numbers with ∑𝑟𝑛=1𝜆𝑛=1. Then the mapping 𝑆∶𝐶→𝐶 defined by 𝑆𝑥=𝑟𝑚=1𝜆𝑚𝑇𝑚𝑥,∀𝑥∈𝐶,(2.4) is well defined, nonexpansive and Fix(𝑆)=∩𝑟𝑚=1Fix(𝑇𝑚) holds.

Lemma 2.5 (see [22]). Assume that {𝛼𝑛} is a sequence of nonnegative real numbers such that 𝛼𝑛+1≤1−𝛾𝑛𝛼𝑛+𝛿𝑛,𝑛∈ℕ,(2.5) where {𝛾𝑛} is a sequence in (0,1) and {𝛿𝑛} is a sequence in ℝ such that(i)lim𝑛→∞𝛾𝑛=0;(ii)∑∞𝑛=1𝛾𝑛=∞;(iii)limsup𝑛→∞𝛿𝑛/𝛾𝑛≤0 or ∑∞𝑛=1|𝛿𝑛|<∞.Then lim𝑛→∞𝛼𝑛=0.

3. Main Results

In this section, by a generalized contraction mapping we mean a Meir-Keeler-type mapping or (𝜓,𝐿)- contraction. In the rest of the paper we suppose that 𝜓 from the definition of the (𝜓,𝐿)-contraction is continuous, strictly increasing and 𝜂(𝑡) is strictly increasing and onto, where 𝜂(𝑡)=𝑡−𝜓(𝑡), for all 𝑡∈ℝ+. As a consequence, we have the 𝜂(𝑡) is a bijection on ℝ+.

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a reflexive Banach space 𝐸 which admits a weakly sequentially continuous duality mapping 𝐽 from 𝐸 into 𝐸∗. For every 𝑖=1,…,𝑁(𝑁≥1), let 𝒯𝑖={𝑇𝑖(𝑡)∶𝑡≥0} be a semigroup of nonexpansive mappings on 𝐶 such that ℱ=∩𝑁𝑖=1Fix(𝒯𝑖)≠∅ and 𝑓∶𝐶→𝐶 be a generalized contraction on 𝐶. Let {𝛼𝑛},{𝛽𝑛}⊂[0,1) and {𝑡𝑛}⊂(0,∞) be the sequences satisfying lim𝑛→∞𝑡𝑛=lim𝑛→∞(𝛼𝑛/𝑡𝑛)=0 and limsup𝑛→∞𝛽𝑛<1. Let {𝑥𝑛} be a sequence generated by 𝑥𝑛=𝛼𝑛𝑓𝑥𝑛+1−𝛼𝑛𝑁𝑁𝑖=1𝑦𝑖𝑛,𝑦𝑖𝑛=𝛽𝑛𝑥𝑛+1−𝛽𝑛𝑇𝑖𝑡𝑛𝑥𝑛,𝑖=1,…,𝑁.(3.1) Then {𝑥𝑛} converges strongly to a point 𝑥∗∈ℱ, which is the unique solution to the following variational inequality: (𝑓−𝐼)𝑥∗,𝑗𝑥−𝑥∗≤0,∀𝑥∈ℱ.(3.2)

Proof. First, we show that the sequence {𝑥𝑛} generated by (3.1) is well defined. For every 𝑛∈ℕ and 𝑖=1,…,𝑁, let 𝑈𝑖𝑛=𝛽𝑛𝐼+(1−𝛽𝑛)𝑇𝑖(𝑡𝑛) and define 𝑊𝑛∶𝐶→𝐶 by 𝑊𝑛𝑥=𝛼𝑛𝑓(𝑥)+1−𝛼𝑛𝐺𝑛𝑥,∀𝑥∈𝐶,(3.3) where 𝐺𝑛∑𝑥=(1/𝑁)𝑁𝑖=1𝑈𝑖𝑛𝑥. Since 𝑈𝑖𝑛 is nonexpansive, 𝐺𝑛 is nonexpansive. By Lemma 2.2 we see that 𝑊𝑛 is a Meir-Keeler-type contraction for each 𝑛∈ℕ. Hence, each 𝑊𝑛 has a unique fixed point, denoted as 𝑥𝑛, which uniquely solves the fixed point equation (3.3). Hence {𝑥𝑛} generated by (3.1) is well defined.
Now we prove that {𝑥𝑛} generated by (3.1) is bounded. For any 𝑝∈ℱ, we have ‖‖𝑦𝑖𝑛‖‖−𝑝≤𝛽𝑛‖‖𝑥𝑛‖‖+−𝑝1−𝛽𝑛‖‖𝑇𝑖𝑡𝑛𝑥𝑛‖‖≤‖‖𝑥−𝑝𝑛‖‖−𝑝.(3.4) Using (3.4), we get ‖𝑥𝑛−𝑝‖2=𝛼𝑛𝑓𝑥𝑛+1−𝛼𝑛𝑁𝑁𝑖=1𝑦𝑖𝑛𝑥−𝑝,𝑗𝑛−𝑝=𝛼𝑛𝑓𝑥𝑛𝑥−𝑓(𝑝),𝑗𝑛−𝑝+𝛼𝑛𝑥𝑓(𝑝)−𝑝,𝑗𝑛+−𝑝1−𝛼𝑛𝑁𝑁𝑖=1𝑦𝑖𝑛𝑥−𝑝,𝑗𝑛−𝑝≤𝛼𝑛𝜓‖‖𝑥𝑛‖‖‖‖𝑥−𝑝𝑛‖‖−𝑝+𝛼𝑛‖‖𝑥‖𝑓(𝑝)−𝑝‖𝑛‖‖+−𝑝1−𝛼𝑛𝑁𝑁𝑖=1‖‖𝑦𝑖𝑛‖‖‖‖𝑥−𝑝𝑛‖‖−𝑝=𝛼𝑛𝜓‖‖𝑥𝑛‖‖‖‖𝑥−𝑝𝑛‖‖−𝑝+𝛼𝑛‖‖‖𝑥𝑓(𝑝)−𝑝‖𝑛‖‖+−𝑝1−𝛼𝑛‖‖𝑥𝑛‖‖−𝑝2(3.5) and hence ‖‖𝑥𝑛‖‖‖‖𝑥−𝑝≤𝜓𝑛‖‖−𝑝+‖𝑓(𝑝)−𝑝‖,(3.6) which implies that 𝜂‖‖𝑥𝑛‖‖=‖‖𝑥−𝑝𝑛‖‖‖‖𝑥−𝑝−𝜓𝑛‖‖−𝑝≤‖𝑓(𝑝)−𝑝‖.(3.7) Hence ‖‖𝑥𝑛‖‖−𝑝≤𝜂−1(‖𝑓(𝑝)−𝑝‖).(3.8) This shows that {𝑥𝑛} is bounded, and so are {𝑇𝑖(𝑡𝑛)𝑥𝑛}, {𝑓(𝑥𝑛)} and {𝑦𝑖𝑛}.
Since 𝐸 is reflexivity and {𝑥𝑛} is bounded, there exists a subsequence {𝑥𝑛𝑗}⊂{𝑥𝑛} such that 𝑥𝑛𝑗⇀𝑥∗ for some 𝑥∗∈𝐶 as 𝑗→∞. Now we prove that 𝑥∗∈ℱ. For any fixed 𝑡>0, we have 𝑁𝑖=1‖‖𝑥𝑛𝑗−𝑇𝑖(𝑡)𝑥∗‖‖≤𝑁𝑖=1⎡⎢⎢⎣[𝑡/𝑡𝑛𝑖]−1𝑘=0‖‖𝑇𝑖(𝑘+1)𝑡𝑛𝑗𝑥𝑛𝑗−𝑇𝑖𝑘𝑡𝑛𝑗𝑥𝑛𝑗‖‖+‖‖‖‖𝑇𝑖𝑡𝑡𝑛𝑗𝑡𝑛𝑗𝑥𝑛𝑗−𝑇𝑖𝑡𝑡𝑛𝑗𝑡𝑛𝑗𝑥∗‖‖‖‖+‖‖‖‖𝑇𝑖𝑡𝑡𝑛𝑗𝑡𝑛𝑗𝑥𝑛𝑗−𝑇𝑖(𝑡)𝑥∗‖‖‖‖≤𝑁𝑖=1𝑡𝑡𝑛𝑗‖‖𝑇𝑖𝑡𝑛𝑗𝑥𝑛𝑗−𝑥𝑛𝑗‖‖+‖‖𝑥𝑛𝑗−𝑥∗‖‖+‖‖‖‖𝑇𝑖𝑡𝑡−𝑡𝑛𝑗𝑡𝑛𝑗𝑥𝑛𝑗−𝑥∗‖‖‖‖≤𝑁𝑖=1𝑡𝑡𝑛𝑗‖‖𝑇𝑖𝑡𝑛𝑗𝑥𝑛𝑗−𝑥𝑛𝑗‖‖+‖‖𝑥𝑛𝑗−𝑥∗‖‖‖‖𝑇+max𝑖(𝑠)𝑥∗−𝑥∗‖‖∶0≤𝑠≤𝑡𝑛𝑗≤𝑁𝛼𝑛𝑗𝑡/𝑡𝑛𝑗1−𝛼𝑛𝑗1−𝛽𝑛𝑗‖‖𝑥𝑛𝑗𝑥−𝑓𝑛𝑗‖‖‖‖𝑥+𝑁𝑛𝑗−𝑥∗‖‖+𝑁𝑖=1‖‖𝑇max𝑖(𝑠)𝑥∗−𝑥∗‖‖∶0≤𝑠≤𝑡𝑛𝑗≤𝑁𝑡1−𝛼𝑛𝑗1−𝛽𝑛𝑗𝛼𝑛𝑗𝑡𝑛𝑗‖‖𝑥𝑛𝑗𝑥−𝑓𝑛𝑗‖‖‖‖𝑥+𝑁𝑛𝑗−𝑥∗‖‖+𝑁𝑖=1‖‖𝑇max𝑖(𝑠)𝑥∗−𝑥∗‖‖∶0≤𝑠≤𝑡𝑛𝑗.(3.9) By hypothesis on {𝑡𝑛},{𝛼𝑛}, {𝛽𝑛}, we have lim𝑗→∞𝑁𝑡1−𝛼𝑛𝑗1−𝛽𝑛𝑗𝛼𝑛𝑗𝑡𝑛𝑗=0.(3.10) Further, from (3.9) we get limsup𝑁𝑗→∞𝑖=1‖‖𝑥𝑛𝑗−𝑇𝑖(𝑡)𝑥∗‖‖≤limsup𝑗→∞𝑁‖‖𝑥𝑛𝑗−𝑥∗‖‖.(3.11) Since 𝐸 admits a weakly sequentially duality mapping, we see that 𝐸 satisfies Opial’s condition. Thus if 𝑥∗∉ℱ, we have limsup𝑗→∞𝑁‖‖𝑥𝑛𝑗−𝑥∗‖‖<limsup𝑁𝑗→∞𝑖=1‖‖𝑥𝑛𝑗−𝑇𝑖𝑥∗‖‖.(3.12) This contradicts (3.11). So 𝑥∗∈ℱ.
In (3.5), replacing 𝑝 with 𝑥∗ and 𝑛 with 𝑛𝑗, we see that ‖‖𝑥𝑛𝑗−𝑥∗‖‖2=𝛼𝑛𝑗𝑓𝑥𝑛𝑗𝑥−𝑓∗𝑥,𝑗𝑛𝑗−𝑥∗+𝛼𝑛𝑗𝑓𝑥∗−𝑥∗𝑥,𝑗𝑛𝑗−𝑥∗+1−𝛼𝑛𝑗𝑁𝑁𝑖=1𝑦𝑖𝑛𝑗−𝑥∗𝑥,𝑗𝑛𝑗−𝑥∗≤𝛼𝑛𝑗𝜓‖‖𝑥𝑛𝑗−𝑥∗‖‖‖‖𝑥𝑛𝑗−𝑥∗‖‖+𝛼𝑛𝑗𝑓𝑥∗−𝑥∗𝑥,𝑗𝑛𝑗−𝑥∗+1−𝛼𝑛𝑗𝑁𝑁𝑖=1‖‖𝑦𝑖𝑛𝑗−𝑥∗‖‖‖‖𝑥𝑛𝑗−𝑥∗‖‖≤𝛼𝑛𝑗𝜓‖‖𝑥𝑛𝑗−𝑥∗‖‖‖‖𝑥𝑛𝑗−𝑥∗‖‖+𝛼𝑛𝑗𝑓𝑥∗−𝑥∗𝑥,𝑗𝑛𝑗−𝑥∗+1−𝛼𝑛𝑗‖‖𝑥𝑛‖‖−𝑝2,(3.13) which implies that ‖‖𝑥𝑛𝑗−𝑥∗‖‖𝜓‖‖𝑥𝑛𝑗−𝑥∗‖‖−‖‖𝑥𝑛𝑗−𝑥∗‖‖≤𝑓𝑥∗−𝑥∗𝑥,𝑗𝑛𝑗−𝑥∗.(3.14) Now we prove that {𝑥𝑛} is relatively sequentially compact. Since 𝑗 is weakly sequentially continuous, we have lim𝑗→∞‖‖𝑥𝑛𝑗−𝑥∗‖‖𝜓‖‖𝑥𝑛𝑗−𝑥∗‖‖−‖‖𝑥𝑛𝑗−𝑥∗‖‖≤0,(3.15) which implies that lim𝑗→∞‖‖𝑥𝑛𝑗−𝑥∗‖‖=0,orlim𝑗→∞𝜓‖‖𝑥𝑛𝑗−𝑥∗‖‖−‖‖𝑥𝑛𝑗−𝑥∗‖‖=0.(3.16) If lim𝑗→∞‖𝑥𝑛𝑗−𝑥∗‖=0, then {𝑥𝑛} is relatively sequentially compact. If lim𝑗→∞(𝜓(‖𝑥𝑛𝑗−𝑥∗‖)−‖𝑥𝑛𝑗−𝑥∗‖)=0, we have lim𝑗→∞‖𝑥𝑛𝑗−𝑥∗‖=lim𝑗→∞𝜓(‖𝑥𝑛𝑗−𝑥∗‖). Since 𝜓 is continuous, lim𝑗→∞‖𝑥𝑛𝑗−𝑥∗‖=𝜓(lim𝑗→∞‖𝑥𝑛𝑗−𝑥∗‖). By the definition of 𝜓, we conclude that lim𝑗→∞‖𝑥𝑛𝑗−𝑥∗‖=0, which implies that {𝑥𝑛} is relatively sequentially compact.
Next, we prove that 𝑥∗ is the solution to (3.2). Indeed, for any 𝑥∈ℱ, we have ‖‖𝑥𝑛‖‖−𝑥2=𝛼𝑛𝑓𝑥𝑛−𝑥𝑛+𝑥𝑛𝑥−𝑥,𝑗𝑛+−𝑥1−𝛼𝑛𝑁𝑁𝑖=1𝑦𝑖𝑛𝑥−𝑥,𝑗𝑛−𝑥=𝛼𝑛𝑓𝑥𝑛−𝑥𝑛𝑥,𝑗𝑛−𝑥+𝛼𝑛𝑥𝑛𝑥−𝑥,𝑗𝑛+−𝑥1−𝛼𝑛𝑁𝑁𝑖=1𝛽𝑛𝑥𝑛𝑥−𝑥,𝑗𝑛+−𝑥1−𝛽𝑛𝑇𝑖𝑡𝑛𝑥𝑛𝑥−𝑥,𝑗𝑛−𝑥∗≤𝛼𝑛𝑓𝑥𝑛−𝑥𝑛𝑥,𝑗𝑛−𝑥+𝛼𝑛‖‖𝑥𝑛‖‖−𝑥2+1−𝛼𝑛𝑁𝑁𝑖=1𝛽𝑛‖‖𝑥𝑛‖‖−𝑥2+1−𝛽𝑛‖‖𝑇𝑖𝑡𝑛𝑥𝑛‖‖‖‖𝑥−𝑥𝑛‖‖−𝑥≤𝛼𝑛𝑓𝑥𝑛−𝑥𝑛𝑥,𝑗𝑛−𝑥+𝛼𝑛‖‖𝑥𝑛‖‖−𝑥2+1−𝛼𝑛𝑁𝑁𝑖=1𝛽𝑛‖‖𝑥𝑛‖‖−𝑥2+1−𝛽𝑛‖‖𝑥𝑛‖‖−𝑥2=𝛼𝑛𝑓𝑥𝑛−𝑥𝑛𝑥,𝑗𝑛+‖‖𝑥−𝑥𝑛‖‖−𝑥2.(3.17) Therefore, 𝑓𝑥𝑛−𝑥𝑛,𝑗𝑥−𝑥𝑛≤0.(3.18) Since 𝑥𝑛𝑗⇀𝑥∗ and 𝑗 is weakly sequentially continuous, we have 𝑓𝑥∗−𝑥∗,𝑗𝑥−𝑥∗=lim𝑗→∞𝑓𝑥𝑛𝑗−𝑥𝑛𝑗,𝑗𝑥−𝑥𝑛𝑗≤0.(3.19) This shows that 𝑥∗ is the solution of the variational inequality (3.2).
Finally, we prove that 𝑥∗ is the unique solution of the variational inequality (3.2). Assume that ̂𝑥∈ℱ with ̂𝑥≠𝑥∗ is another solution of (3.2). Then there exists 𝜖>0 such that ‖̂𝑥−𝑥∗‖>𝜖. By Lemma 2.3 there exists 𝑟∈(0,1) such that ‖𝑓(̂𝑥)−𝑓(𝑥∗)‖≤𝑟‖̂𝑥−𝑥∗‖. Since both ̂𝑥 and 𝑥∗ are the solution of (3.2), we have 𝑓𝑥∗−𝑥∗,𝑗̂𝑥−𝑥∗𝑓𝑥≤0,(̂𝑥)−̂𝑥,𝑗∗−̂𝑥≤0.(3.20) Adding the above inequalities, we get 0<(1−𝑟)𝜖2<(1−𝑟)‖̂𝑥−𝑥∗‖2≤⟨(𝐼−𝑓)𝑥∗𝑥−(𝐼−𝑓)̂𝑥⟩,𝑗∗−̂𝑥≤0,(3.21) which is a contradiction. Therefore, we must have ̂𝑥=𝑥∗, which implies that 𝑥∗ is the unique solution of (3.2).
In a similar way it can be shown that each cluster point of sequence {𝑥𝑛} is equal to 𝑥∗. Therefore, the entire sequence {𝑥𝑛} converges strongly to 𝑥∗. This completes the proof.

If letting 𝛽𝑛=0 for all 𝑛∈ℕ in Theorem 3.1, then we get the following.

Corollary 3.2. Let 𝐶 be a nonempty closed convex subset of a reflexive Banach space 𝐸 which admits a weakly sequentially continuous duality mapping 𝐽 from 𝐸 into 𝐸∗. For every 𝑖=1,…,𝑁 ( 𝑁≥1), let 𝒯𝑖={𝑇𝑖(𝑡)∶𝑡≥0} be a semigroup of nonexpansive mappings on 𝐶 such that ℱ=∩𝑁𝑖=1Fix(𝒯𝑖)≠∅ and 𝑓∶𝐶→𝐶 be a generalized contraction on 𝐶. Let {𝛼𝑛}⊂[0,1) and {𝑡𝑛}⊂(0,∞) be sequences satisfying lim𝑛→∞𝑡𝑛=lim𝑛→∞(𝛼𝑛/𝑡𝑛)=0. Let {𝑥𝑛} be a sequence generated by 𝑥𝑛=𝛼𝑛𝑓𝑥𝑛+1−𝛼𝑛𝑁𝑁𝑖=1𝑇𝑖𝑡𝑛𝑥𝑛.(3.22) Then {𝑥𝑛} converges strongly to a point 𝑥∗∈ℱ, which is the unique solution to the following variational inequality: (𝑓−𝐼)𝑥∗,𝑗𝑥−𝑥∗≤0,∀𝑥∈ℱ.(3.23)

Theorem 3.3. Let 𝐶 be a nonempty closed convex subset of a reflexive and strictly convex Banach space 𝐸 which admits a weakly sequentially continuous duality mapping 𝐽 from 𝐸 into 𝐸∗. For every 𝑖=1,⋯,𝑁(𝑁≥1), let 𝒯𝑖={𝑇𝑖(𝑡)∶𝑡≥0} be a semigroup of nonexpansive mappings on 𝐶 such that ℱ=∩𝑁𝑖=1Fix(𝒯𝑖)≠∅ and 𝑓∶𝐶→𝐶 be a generalized contraction on 𝐶. Let {𝛼𝑛},{𝛽𝑛}⊂[0,1) and {𝑡𝑛}⊂(0,∞) be the sequences satisfying lim𝑛→∞𝑡𝑛=lim𝑛→∞(𝛽𝑛/𝑡𝑛)=0. Let {𝑥𝑛} be a sequence generated 𝑦𝑖𝑛=𝛼𝑛𝑥𝑛+1−𝛼𝑛𝑇𝑖𝑡𝑛𝑥𝑛𝑥,𝑖=1,…,𝑁,𝑛+1=𝛽𝑛𝑓𝑥𝑛+1−𝛽𝑛𝑁𝑁𝑖=1𝑦𝑖𝑛,∀𝑛∈ℕ.(3.24) Then {𝑥𝑛} converges strongly to a point 𝑥∗∈ℱ, which is the unique solution of variational inequality (3.2).

Proof. Let 𝑝∈ℱ and 𝑀=max{‖𝑥1−𝑝‖,𝜂−1(‖𝑓(𝑝)−𝑝‖}. Now we show by induction that ‖‖𝑥𝑛‖‖−𝑝≤𝑀,∀𝑛∈ℕ.(3.25) It is obvious that (3.25) holds for 𝑛=1. Suppose that (3.25) holds for some 𝑛=𝑘, where 𝑘>1. Observe that ‖‖𝑦𝑖𝑘‖‖=‖‖𝛼−𝑝𝑘𝑥𝑘+−𝑝1−𝛼𝑘𝑇𝑖𝑡𝑘𝑥𝑘‖‖−𝑝≤𝛼𝑘‖‖𝑥𝑘‖‖+−𝑝1−𝛼𝑘‖‖𝑇𝑖𝑡𝑘𝑥𝑘‖‖≤‖‖𝑥−𝑝𝑘‖‖.−𝑝(3.26) Now, by using (3.24) and (3.26), we have ‖𝑥𝑘+1‖‖‖‖𝛽−𝑝‖=𝑘𝑓𝑥𝑘+−𝑝1−𝛽𝑘𝑁𝑁𝑖=1𝑦𝑖𝑘‖‖‖‖−𝑝≤𝛽𝑘‖‖𝑓𝑥𝑘‖‖−𝑓(𝑝)+𝛽𝑘‖𝑓(𝑝)−𝑝‖+1−𝛽𝑘𝑁𝑁𝑖=1‖‖𝑦𝑖𝑘‖‖−𝑝≤𝛽𝑘𝜓‖‖𝑥𝑘‖‖−𝑝+𝛽𝑘‖𝑓(𝑝)−𝑝‖+1−𝛽𝑘𝑁𝑁𝑖=1‖‖𝑥𝑘‖‖−𝑝=𝛽𝑘𝜓‖‖𝑥𝑘‖‖−𝑝+𝛽𝑘‖𝑓(𝑝)−𝑝‖+1−𝛽𝑘‖‖𝑥𝑘‖‖−𝑝=𝛽𝑘𝜓‖‖𝑥𝑘‖‖−𝑝+𝛽𝑘𝜂𝜂−1+‖𝑓(𝑝)−𝑝‖1−𝛽𝑘‖‖𝑥𝑘‖‖−𝑝≤𝛽𝑘𝜓(𝑀)+𝛽𝑘𝜂(𝑀)+1−𝛽𝑘𝑀=𝛽𝑘𝜓(𝑀)+𝛽𝑘(𝑀−𝜓(𝑀))+1−𝛽𝑘𝑀=𝑀.(3.27) By induction we conclude that (3.25) holds for all 𝑛∈ℕ. Therefore, {𝑥𝑛} is bounded and so are {𝑓(𝑥𝑛)}, {𝑦𝑖𝑛}, {𝑇𝑖(𝑡𝑛)𝑥𝑛}.
For each 𝑖=1,…,𝑁 and 𝑛∈ℕ, define the mapping 𝑈(𝑡𝑛∑)=(1/𝑁)𝑁𝑖=1𝑆𝑖(𝑡𝑛), where 𝑆𝑖(𝑡𝑛)=𝛼𝑛𝐼+(1−𝛼𝑛)𝑇𝑖(𝑡𝑛). Then we rewrite the sequence (3.24) to 𝑥𝑛+1=𝛽𝑛𝑓𝑥𝑛+1−𝛽𝑛𝑈𝑡𝑛𝑥𝑛.(3.28) Obviously, each 𝑈(𝑡𝑛) is nonexpansive. Since {𝑥𝑛} is bounded and 𝐸 is reflexive, we may assume that some subsequence {𝑥𝑛𝑗} of {𝑥𝑛} converges weakly to 𝑝. Next we show that 𝑝∈ℱ. Put 𝑥𝑗=𝑥𝑛𝑗, 𝛽𝑗=𝛽𝑛𝑗, and 𝑡𝑗=𝑡𝑛𝑗 for each 𝑗∈ℕ. Fix 𝑡>0. By (3.28) we have ‖‖𝑥𝑗‖‖=−𝑈(𝑡)𝑝[𝑡/𝑡𝑗]−1𝑘=0‖‖𝑈(𝑘+1)𝑡𝑗𝑥𝑗−𝑈𝑘𝑡𝑗𝑥𝑗‖‖+‖‖‖𝑈𝑡𝑡𝑗𝑡𝑗𝑥𝑗𝑡−𝑈𝑡𝑗𝑡𝑗𝑝‖‖‖+‖‖‖𝑈𝑡𝑡𝑗𝑡𝑗‖‖‖≤𝑡𝑝−𝑈(𝑡)𝑝𝑡𝑗‖‖𝑈𝑡𝑗𝑥𝑗−𝑥𝑗+1‖‖+‖‖𝑥𝑗+1‖‖+‖‖‖𝑈𝑡−𝑝𝑡−𝑡𝑗𝑡𝑗‖‖‖=𝑡𝑝−𝑝𝑡𝑗𝛽𝑗‖‖𝑈𝑡𝑗𝑥𝑗𝑥−𝑓𝑗‖‖+‖‖𝑥𝑗+1‖‖+‖‖‖𝑈𝑡−𝑝𝑡−𝑡𝑗𝑡𝑗‖‖‖≤𝑝−𝑝𝑡𝛽𝑗𝑡𝑗‖‖𝑈𝑡𝑗𝑥𝑗𝑥−𝑓𝑗‖‖+‖‖𝑥𝑗+1‖‖−𝑝+max‖𝑈(𝑠)𝑝−𝑝‖∶0≤𝑠≤𝑡𝑗.(3.29) So, for all 𝑗∈ℕ, we have limsup𝑗→∞‖‖𝑥𝑗‖‖−𝑈(𝑡)𝑝≤limsup𝑗→∞‖‖𝑥𝑗+1‖‖−𝑝=limsup𝑗→∞‖‖𝑥𝑗‖‖−𝑝.(3.30)
Since 𝐸 has a weakly sequentially continuous duality mapping satisfying Opials’ condition, this implies 𝑝=𝑈(𝑡)𝑝. By Lemma 2.4, we have Fix(𝑈(𝑡))=∩𝑁𝑖=1Fix(𝑇𝑖(𝑡)) for each 𝑡>0. Therefore, 𝑝∈ℱ. In view of the variational inequality (3.2) and the assumption that duality mapping 𝐽 is weakly sequentially continuous, we conclude that limsup𝑛→∞𝑥(𝑓−𝐼)𝑞,𝑗𝑛+1−𝑞=lim𝑗→∞𝑥(𝑓−𝐼)𝑞,𝑗𝑛𝑗+1−𝑞=⟨(𝐼−𝑓)𝑞,𝑗(𝑝−𝑞)⟩≤0.(3.31)
Finally, we prove that 𝑥𝑛→𝑞 as 𝑛→∞. Suppose that ‖𝑥𝑛−𝑞‖↛0. Then there exists 𝜖>0 and subsequence {𝑥𝑛𝑗} of {𝑥𝑛} such that ‖𝑥𝑛𝑗−𝑞‖≥𝜖 for all 𝑗∈ℕ. Put 𝑥𝑗=𝑥𝑛𝑗, 𝛽𝑗=𝛽𝑛𝑗 and 𝑡𝑗=𝑡𝑛𝑗. By Lemma 2.3 one has ‖𝑓(𝑥𝑗)−𝑓(𝑞)‖≤𝑟‖𝑥𝑗−𝑞‖ for all 𝑗∈ℕ. Now, from (2.2) and (3.28) we have ‖𝑥𝑗+1−𝑞‖2=‖‖(1−𝛽𝑛)(𝑈(𝑡𝑗)𝑥𝑗−𝑞)+𝛽𝑛(𝑓(𝑥𝑗‖‖)−𝑞)2≤1−𝛽𝑗2‖‖𝑈𝑡𝑗𝑥𝑗‖‖−𝑞2+2𝛽𝑗𝑓𝑥𝑗𝑥−𝑞,𝑗𝑗+1≤−𝑞1−𝛽𝑗2‖‖𝑥𝑗‖‖−𝑞2+2𝛽𝑛𝑓𝑥𝑗𝑥−𝑓(𝑞),𝑗𝑗+1−𝑞+2𝛽𝑗𝑥𝑓(𝑞)−𝑞,𝑗𝑗+1≤−𝑞1−𝛽𝑗2‖‖𝑥𝑗‖‖−𝑞2+2𝛽𝑗𝑟‖‖𝑥𝑗‖‖‖‖𝑥−𝑞𝑗+1‖‖−𝑞+2𝛽𝑛𝑥𝑓(𝑞)−𝑞,𝑗𝑗+1≤−𝑞1−𝛽𝑗2‖‖𝑥𝑗‖‖−𝑞2+𝛽𝑗𝑟‖‖𝑥𝑗‖‖−𝑞2+‖‖𝑥𝑗+1‖‖−𝑞2+2𝛽𝑗𝑓𝑥(𝑞)−𝑞,𝑗𝑗+1=−𝑞1−𝛽𝑗2+𝛽𝑗𝑟‖‖𝑥𝑗‖‖−𝑞2+𝛽𝑗𝑟‖‖𝑥𝑗+1‖‖−𝑞2+2𝛽𝑗𝑥𝑓(𝑞)−𝑞,𝑗𝑗+1.−𝑞(3.32) It follows that ‖‖𝑥𝑗+1‖‖≤1−(2−𝑟)𝛽𝑗+𝛽2𝑗1−𝛽𝑗𝑟‖‖𝑥𝑗‖‖−𝑞2+2𝛽𝑗1−𝛽𝑗𝑟𝑥𝑓(𝑞)−𝑞,𝑗𝑗+1≤−𝑞1−𝛽𝑗𝑟−2(1−𝑟)𝛽𝑗1−𝛽𝑗𝑟‖‖𝑥𝑗‖‖−𝑞2+2𝛽𝑗1−𝛽𝑗𝑟𝑓𝑥(𝑞)−𝑞,𝑗𝑗+1−𝑞+𝛽2𝑗𝑀=1−2(1−𝑟)𝛽𝑗1−𝛽𝑗𝑟‖‖𝑥𝑗‖‖−𝑞2+2𝛽𝑗1−𝛽𝑗𝑟𝑥𝑓(𝑞)−𝑞,𝑗𝑗+1−𝑞+𝛽2𝑗𝑀≤1−2(1−𝑟)𝛽𝑗‖‖𝑥𝑗‖‖−𝑞2+𝛽𝑗2𝑓𝑥1−𝑟(𝑞)−𝑞,𝑗𝑗+1−𝑞+𝛽𝑗𝑀,(3.33) where 𝑀 is a constant.
Let 𝛾𝑗=2(1−𝑟)𝛽𝑗 and 𝛿𝑗=𝛽𝑗((2/(1−𝑟))⟨𝑓(𝑞)−𝑞,𝑗(𝑥𝑗+1−𝑞)⟩+𝛽𝑗𝑀). It follows from (3.33) that ‖‖𝑥𝑗+1‖‖≤−𝑞1−𝛾𝑗‖‖𝑥𝑗‖‖−𝑞+𝛿𝑗.(3.34) It is easy to see that 𝛾𝑗→0, ∑∞𝑗=1𝛾𝑗=∞ and (noting (3.28)) limsup𝑗→∞𝛿𝑗𝛾𝑗1=limsup(1−𝑟)2𝑥𝑓(𝑞)−𝑞,𝑗𝑗+1+𝑀−𝑞𝛽2(1−𝑟)𝑗,limsup𝑛→∞1(1−𝑟)2𝑓𝑥(𝑞)−𝑞,𝑗𝑗+1−𝑞≤0.(3.35) Using Lemma 2.5, we conclude that ‖𝑥𝑗−𝑞‖→0 as 𝑗→∞. It is a contradiction. Therefore, 𝑥𝑛→𝑞 as 𝑛→∞. This completes the proof.

If letting 𝛼𝑛=0 for all 𝑛∈ℕ in Theorem 3.3, then we get the following.

Corollary 3.4. Let 𝐶 be a nonempty closed convex subset of a reflexive and strictly convex Banach space 𝐸 which admits a weakly sequentially continuous duality mapping 𝐽 from 𝐸 into 𝐸∗. For every 𝑖=1,…,𝑁(𝑁≥1), let 𝒯𝑖={𝑇𝑖(𝑡)∶𝑡≥0} be a semigroup of nonexpansive mappings on 𝐶 such that ℱ=∩𝑁𝑖=1Fix(𝒯𝑖)≠∅ and 𝑓∶𝐶→𝐶 be a generalized contraction on 𝐶. Let {𝛽𝑛}⊂[0,1) and {𝑡𝑛}⊂(0,∞) be sequences satisfying lim𝑛→∞𝑡𝑛=lim𝑛→∞(𝛽𝑛/𝑡𝑛)=0. Let {𝑥𝑛} be a sequence generated 𝑥𝑛+1=𝛽𝑛𝑓𝑥𝑛+1−𝛽𝑛𝑁𝑁𝑖=1𝑇𝑖𝑡𝑛𝑥𝑛,∀𝑛∈ℕ.(3.36) Then {𝑥𝑛} converges strongly to a point 𝑥∗∈ℱ, which is the unique solution of variational inequality (3.2).

Remark 3.5. Theorem 3.1 and Corollary 3.2 extend the corresponding ones of Suzuki [2], Xu [3], and Chen and He [5] from one nonexpansive semigroup to a finite family of nonexpansive semigroups. But Theorem 3.3 and Corollary 3.4 are not the extension of Theorem 3.2 of Chen and He [5] since Banach space in Theorem 3.3 and Corollary 3.4 is required to be strictly convex. But if letting 𝑁=1 in Theorem 3.3 and Corollary 3.4, we can remove the restriction on strict convexity and hence they extend Theorem 3.2 of Chen and He [5] from a contraction to a generalized contraction.

Remark 3.6. Our Theorem 3.1 extends and improves Theorems 3.2 and 4.2 of Song and Xu [4] from a nonexpansive semigroup to a finite family of nonexpansive semigroups and a contraction to a generalized contraction. Our conditions on the control sequences are different with ones of Song and Xu [4].

Acknowledgment

This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (Grant Number: 2011-0021821).