Convergence of a General Composite Iterative Method for a Countable Family of Nonexpansive Mappings

Abstract

We propose a general composite iterative method for computing common fixed points of a countable family of nonexpansive mappings in the framework of Hilbert spaces. Our results improve and complement the corresponding ones announced by many others.

1. Introduction and Main Result

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖. A mapping 𝑇∶𝐻→𝐻 is said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐻. The set of fixed points, Fix(𝑇)={𝑥∈𝐻∶𝑇𝑥=𝑥}, of a nonexpansive mapping is always a closed and convex subset of 𝐻.

In addition to nonexpansive mappings, we are going to use contractions and 𝑘-Lipschitzian and 𝜂-strongly monotone operators. A self-mapping 𝑓∶𝐻→𝐻 is a contraction on 𝐻, if there exists a constant 𝛼∈(0,1) such that ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝛼‖𝑥−𝑦‖, for all 𝑥,𝑦∈𝐻. We use Π_𝐻 to denote the collection of mappings 𝑓 verifying the above inequality. That is, Π_𝐻={𝑓∶𝐻→𝐻∣𝑓 is a contraction with constant 𝛼}. A mapping 𝐹∶𝐻→𝐻 is called 𝑘-Lipschitzian if there exists a positive constant 𝑘 such that ‖𝐹𝑥−𝐹𝑦‖≤𝑘‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐻.(1.1)𝐹 is said to be 𝜂-strongly monotone if there exists a positive constant 𝜂 such that ⟨𝐹𝑥−𝐹𝑦,𝑥−𝑦⟩≥𝜂‖𝑥−𝑦‖²,∀𝑥,𝑦∈𝐻.(1.2)

Recently, Jung [1] introduced the following composite iterative scheme for the solution of a specific minimization problem, which involves a closed convex subset 𝐶⊆𝐻, a nonexpansive mapping 𝑇∶𝐶→𝐶, and a contraction 𝑓∶𝐶→𝐶, 𝑥𝑦₁=𝑥∈𝐶arbitrarilychosen,_𝑛𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,=1−𝛽_𝑛𝑦_𝑛+𝛽_𝑛𝑇𝑦_𝑛,𝑛≥1.(1.3) Therein the control sequences {𝛼_𝑛} and {𝛽_𝑛} satisfy certain conditions. He proved that the sequence {𝑥_𝑛} defined by (1.3) converges strongly to 𝑥̃=𝑄(𝑓)∈Fix(𝑇), which is the unique solution of the variational inequality ⟨𝑥̃−𝑓(𝑥̃),𝑥̃−𝑧⟩≤0,𝑧∈Fix(𝑇). The results of Jung [1] are even stronger than stated here. In fact, they hold for a finite family of nonexpansive mappings and in the setting of Banach spaces.

Very recently, Tian [2] considered the following iterative method: for nonexpansive mapping 𝑇∶𝐻→𝐻 with Fix(𝑇)≠∅, 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐹𝑇𝑥_𝑛,𝑛≥1,(1.4) where 𝐹 is a 𝑘-Lipschitzian and 𝜂-strongly monotone operator. He obtained that the sequence {𝑥_𝑛} generated by (1.4) converges to a point 𝑞 in Fix(𝑇), which is the unique solution of the variational inequality ⟨(𝛾𝑓−𝜇𝐹)𝑞,𝑝−𝑞⟩≤0,𝑝∈Fix(𝑇).

In this connection, notice that Aoyama et al. [3] proposed a Halpern approximation method for finding a common fixed point of a countable family of nonexpansive mappings. As their main result, they established the following strong convergence theorem.

Theorem 1.1 (see [3, Theorem 3.4]). Let 𝐸 be a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable, and let 𝐶 be a nonempty closed convex subset of 𝐸. Let {𝑇_𝑛} be a sequence of nonexpansive mappings 𝑇_𝑛∶𝐶→𝐶 such that ^∞_𝑛=1sup‖‖𝑇_𝑛+1𝑧−𝑇_𝑛𝑧‖‖∶𝑧∈𝐵<∞(1.5) for each bounded subset 𝐵 of 𝐶. Suppose in addition that Fix(𝑇)=^∞_𝑛=1Fix𝑇_𝑛≠∅,(1.6) where 𝑇∶𝐶→𝐶 is the nonexpansive mapping defined by 𝑇𝑧=lim_𝑛→∞𝑇_𝑛𝑧. Let 𝛼_𝑛∈(0,1] be a sequence satisfying the conditions (C1) lim_𝑛→∞𝛼_𝑛=0, (C2) ∑^∞𝑛=1𝛼_𝑛=∞, and (C3) ∑^∞𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞ or lim_𝑛→∞(𝛼_𝑛+1/𝛼_𝑛)=1. Given 𝑥∈𝐶, define {𝑥_𝑛}^∞𝑛=1 by 𝑥₁=𝑥,𝑥_𝑛+1=𝛼_𝑛𝑥+1−𝛼_𝑛𝑇_𝑛𝑥_𝑛.(1.7) Then {𝑥_𝑛} converges strongly to 𝑄𝑥, where 𝑄 is a sunny nonexpansive retraction of 𝐸 onto Fix(𝑇).

Inspired by Jung [1], Tian [2] and Aoyama et al. [3], the goal of this paper is to combine these three results into a single method. In this connection, observe that the iteration methods (1.3) and (1.4) just can compute a fixed point of one nonexpansive mapping (or, perhaps, finitely many), while the iteration methods (1.3) and (1.7) do not contain the 𝑘-Lipschitzian and 𝜂-strongly monotone operator 𝐹. On the other hand, in contrast to the result of Jung [1] and Aoyama et al. [3], our result will be restricted to the setting of Hilbert spaces, which on the other hand is natural when dealing with a 𝑘-Lipschitzian and 𝜂-strongly monotone operator 𝐹.

First of all, let us remark that condition (1.5) implies that sequence of mappings {𝑇_𝑛} are uniformly Cauchy on each bounded subset 𝐵⊂𝐶. Hence the limiting map 𝑇 is well defined and is in fact the uniform limit (on 𝐵) of the maps 𝑇_𝑛. In other words, lim_𝑛→∞sup‖‖𝑇𝑧−𝑇_𝑛𝑧‖‖∶𝑧∈𝐵=0.(1.8)

Notice that condition (1.5) is quite strong so that, in general, we cannot apply the result directly to an arbitrary countable family of nonexpansive mappings. Furthermore, we have to assume the nontrivial condition (1.6) that the fixed point set of 𝑇 coincides with the set of common fixed points of the family {𝑇_𝑛}.

Fortunately, as pointed out in [3, Section 4], given an arbitrary countable family of nonexpansive mappings 𝑆_𝑛∶𝐶→𝐶, which have at least one common fixed point, one can find nonnegative numbers 𝛽^𝑘𝑛 (1≤𝑘≤𝑛,𝑛∈ℕ) such that the mappings 𝑇_𝑛𝑥=^𝑛_𝑘=1𝛽^𝑘𝑛𝑆_𝑘𝑥(1.9) are nonexpansive self-maps of 𝐶 satisfying (1.5) and (1.6). More specifically, the 𝑇_𝑛 are constructed as certain convex combinations of the 𝑆₁,…,𝑆_𝑛. (For details, see the reference.) Another possibility of a construction will be mentioned below. Furthermore, we establish the following strong convergence theorem.

Theorem 1.2. Let 𝐻 be a real Hilbert space. Let 𝐹 be a 𝑘-Lipschitzian and 𝜂-strongly monotone operator on 𝐻 with 0<𝜇<2𝜂/𝑘² and 𝑓∈Π_𝐻 with 0<𝛾<𝜇(𝜂−𝜇𝑘²/2)/𝛼=𝜏/𝛼 and 𝜏<1. Assume that {𝑇_𝑛} is a sequence of nonexpansive mappings from 𝐻 into itself such that the condition (1.5). Suppose that 𝑇∶𝐻→𝐻 is defined by 𝑇𝑧=lim_𝑛→∞𝑇_𝑛𝑧 such that the condition (1.6). Let {𝛼_𝑛},{𝛽_𝑛} be sequences in (0,1) satisfying the following conditions:(B1)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞𝑛=1𝛼_𝑛=∞; (B2)𝑏∶=limsup_𝑛→∞𝛽_𝑛<1 and ∑^∞𝑛=1|𝛽_𝑛+1−𝛽_𝑛|<∞; (B3)|𝛼_𝑛+1−𝛼_𝑛|≤𝑜(𝛼_𝑛+1)+𝜎_𝑛 with 𝜎_𝑛≥0 and ∑^∞𝑛=1𝜎_𝑛<∞. Then, for arbitrary 𝑥₁=𝑥∈𝐻, the sequence {𝑥_𝑛}^∞𝑛=1, defined by 𝑥𝑦_𝑛𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛,=1−𝛽_𝑛𝑦_𝑛+𝛽_𝑛𝑇_𝑛𝑦_𝑛,𝑛≥1,(1.10) converges strongly to some 𝑞∈Fix(𝑇), which satisfies the variational inequality ⟨(𝜇𝐹−𝛾𝑓)𝑞,𝑞−𝑧⟩≤0, for all 𝑧∈Fix(𝑇).

The iterative scheme (1.10) is a direct generalization of the three iteration methods considered before. Thus Theorem 1.2 complements or improves Theorem 3.2 of Jung [1], Theorem 3.2 of Tian [2], Theorem 3.4 of Aoyama et al. [3], and Theorem 3.1 of Jung [4].

An important special case is obtained for 𝐹=𝐼, the identity mapping. Then 𝜂=𝑘=1 and we can choose 𝛾=𝜇=1 (compare with iteration scheme (1.3) above).

Besides the basic conditions (B1) and (B2) on the sequences 𝛼_𝑛 and 𝛽_𝑛, we have the “control condition” (B3). It can obviously be replaced by one of the following ones:(B3-1)∑^∞𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞;(B3-2)𝛼_𝑛∈(0,1] for all 𝑛∈ℕ and lim_𝑛→∞(𝛼_𝑛+1/𝛼_𝑛)=1. Indeed, (B3-1) implies (B3) by choosing 𝜎_𝑛=|𝛼_𝑛+1−𝛼_𝑛|, and (B3-2) implies (B3) by choosing 𝜎_𝑛=0. In this sense, (B3) is a weaker condition than the previous condition (C3).

As has already noticed in Theorem 1.1, the assumptions (1.5) and (1.6) do not apply to arbitrary families of nonexpansive mappings. Besides the construction above, we mention another construction, which has appeared in the literature. See [5–8], the references therein, and also Remark 3.1 of Peng and Yao [9]. Proofs are given there.

Let {𝑆_𝑖∶𝐻→𝐻} be a countable family of nonexpansive mappings, and let {𝜉_𝑖} be a sequence of real numbers such that 0≤𝜉_𝑖≤𝑐<1, for all 𝑖≥1. For any 𝑛≥1, define a mapping 𝑇_𝑛∶𝐻→𝐻 as follows: 𝑈𝑈_𝑛,𝑛+1=𝐼,_𝑛,𝑛𝑈=𝜉_𝑛𝑆_𝑛𝑈_𝑛,𝑛+1+1−𝜉_𝑛𝐼,_{𝑛,𝑛−1}𝑈⋮=𝜉_𝑛−1𝑆_𝑛−1𝑈_𝑛,𝑛+1−𝜉_𝑛−1𝐼,_𝑛,𝑘𝑈⋮=𝜉_𝑘𝑆_𝑘𝑈_𝑛,𝑘+1+1−𝜉_𝑘𝐼,_𝑛,2𝑈=𝜉₂𝑆₂𝑈_𝑛,3+1−𝜉₂𝐼,_𝑛,1𝑇=𝜉₁𝑆₁𝑈_𝑛,2+1−𝜉₁𝐼,_𝑛=𝑈_𝑛,1.(1.11)

Proposition 1.3. Let 𝐻 be a real Hilbert space, {𝑆_𝑖∶𝐻→𝐻} a sequence nonexpansive mappings with ⋂^∞𝑖=1Fix(𝑆_𝑖)≠∅, and {𝜉_𝑖} a real sequence such that 0≤𝜉_𝑖≤𝑐<1, for all 𝑖≥1. Define {𝑇_𝑛} as above. Then(1)𝑇_𝑛 is a nonexpansive and Fix(𝑇_𝑛)=⋂^𝑛𝑖=1Fix(𝑆_𝑖) for each 𝑛≥1;(2) the mappings {𝑇_𝑛} satisfy the condition (1.5);(3) defining 𝑇∶𝐻→𝐻 by 𝑇𝑥=lim_𝑛→∞𝑇_𝑛𝑥, we have Fix(𝑇)=^∞_𝑛=1Fix𝑇_𝑛=^∞_𝑛=1Fix𝑆_𝑛.(1.12)

2. Proof of the Main Results

Ahead of the proof, we start with recalling some known auxiliary results.

Lemma 2.1 (see [10]). Let 𝐹 be a 𝑘-Lipschitzian and 𝜂-strongly monotone operator on a Hilbert space 𝐻 with 𝑘>0,𝜂>0,0<𝜇<2𝜂/𝑘², and 0<𝑡<1. Then 𝑆=(𝐼−𝑡𝜇𝐹)∶𝐻→𝐻 is a contraction with contractive coefficient 1−𝑡𝜏 and 𝜏=(1/2)𝜇(2𝜂−𝜇𝑘²).

Lemma 2.2 (see [2]). Let 𝐻 be a Hilbert space. Assume that {𝑥_𝑡} is defined by 𝑥_𝑡=𝑡𝛾𝑓𝑥_𝑡+(𝐼−𝑡𝜇𝐹)𝑇𝑥_𝑡,(2.1) where 𝐹 is a 𝑘-Lipschitzian and 𝜂-strongly monotone operator on a Hilbert space 𝐻 with 𝑘>0,𝜂>0,0<𝜇<2𝜂/𝑘² and 0<𝑡<1,𝑓∈Π_𝐻, and 𝑇∶𝐻→𝐻 is a nonexpansive mapping. Then 𝑥_𝑡 converges strongly as 𝑡→0 to a fixed point 𝑞 of 𝑇, which solves the variational inequality ⟨(𝜇𝐹−𝛾𝑓)𝑞,𝑞−𝑝⟩≤0,𝑝∈𝐹(𝑇).

Some more auxiliary results are given next.

Lemma 2.3 (see [11]). Let 𝐻 be a Hilbert space, 𝐶 a closed convex subset of 𝐻, and 𝑇∶𝐶→𝐶 a nonexpansive mapping with Fix(𝑇)≠∅. If {𝑥_𝑛} is a sequence in 𝐶 which converges weakly to 𝑥 and if {(𝐼−𝑇)𝑥_𝑛} converges strongly to 𝑦, then (𝐼−𝑇)𝑥=𝑦.

Lemma 2.4 (see [3, Lemma 2.3]). Let {𝑠_𝑛} be a sequence of nonnegative real numbers, {𝛿_𝑛} a sequence of numbers in [0,1] such that ∑^∞𝑛=1𝛿_𝑛=∞, {𝜇_𝑛} a sequence of nonnegative real numbers with ∑^∞𝑛=1𝜇_𝑛<∞, and {𝜃_𝑛} a sequence of real numbers with limsup_𝑛→∞𝜃_𝑛≤0. Suppose that 𝑠_𝑛+1≤1−𝛿_𝑛𝑠_𝑛+𝛿_𝑛𝜃_𝑛+𝜇_𝑛(2.2) for all 𝑛∈ℕ. Then lim_𝑛→∞𝑠_𝑛=0.

Now we are prepared to prove the main result.

Proof of Theorem 1.2. We divide the proof into seven steps.
Step 1. We claim that {𝑥_𝑛} is bounded. Taking any point 𝑞∈⋂^∞𝑛=1Fix(𝑇_𝑛) and using Lemma 2.1, we obtain ‖‖𝑥_𝑛+1−𝑞‖‖=≤‖‖‖‖𝑦1−𝛽_𝑛𝑦_𝑛−𝑞+𝛽_𝑛𝑇_𝑛𝑦_𝑛−𝑞_𝑛‖‖=‖‖𝛼−𝑞_𝑛‖‖≤1−𝛾𝛼𝑓_𝑛𝑥_𝑛−𝜇𝐹𝑞+𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−𝐼−𝛼_𝑛𝜇𝐹𝑞‖‖≤1−𝛼_𝑛𝜏‖‖𝑥_𝑛−𝑞‖‖+𝛼_𝑛‖‖𝛾𝑓𝑥_𝑛−𝜇𝐹𝑞‖‖≤1−𝛼_𝑛𝜏‖‖𝑥_𝑛−𝑞‖‖+𝛼_𝑛‖‖𝛾𝑓𝑥_𝑛−𝛾𝑓(𝑞)‖‖+𝛼_𝑛‖𝛾𝑓(𝑞)−𝜇𝐹𝑞‖≤1−𝛼_𝑛𝜏‖‖𝑥_𝑛−𝑞‖‖+𝛼_𝑛𝛾𝛼‖‖𝑥_𝑛−𝑞‖‖+𝛼_𝑛‖𝛾𝑓(𝑞)−𝜇𝐹𝑞‖(𝜏−𝛾𝛼)‖‖𝑥_𝑛−𝑞‖‖+𝛼_𝑛(𝜏−𝛾𝛼)‖𝛾𝑓(𝑞)−𝜇𝐹𝑞‖≤max‖‖𝑥_𝑛𝜏−𝛾𝛼−𝑞‖‖,‖𝛾𝑓(𝑞)−𝜇𝐹𝑞‖𝜏−𝛾𝛼.(2.3) By induction, it follows ‖‖𝑥_𝑛−𝑞‖‖≤max‖‖𝑥₁−𝑞‖‖,‖𝛾𝑓(𝑞)−𝜇𝐹𝑞‖𝜏−𝛾𝛼,𝑛≥1,(2.4) and hence {𝑥_𝑛} is bounded. From this, we also obtain that {𝑦_𝑛},{𝑇_𝑛𝑦_𝑛},{𝐹𝑇_𝑛𝑥_𝑛}, and {𝑓(𝑥_𝑛)} are all bounded. In what follows, let 𝐵 stand for some bounded set of 𝐻, which contains all of {𝑥_𝑛},{𝑦_𝑛},{𝑇_𝑛𝑦_𝑛},{𝐹𝑇_𝑛𝑥_𝑛}, {𝑓(𝑥_𝑛)}.
Step 2. We show that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0.
Let 𝑀=sup{‖𝑇_𝑛𝑦_𝑛−𝑦_𝑛‖,‖𝑓(𝑥_𝑛)‖,‖𝐹𝑇_𝑛𝑥_𝑛‖∶𝑛∈ℕ}<∞. From the definition of {𝑥_𝑛}, we obtain ‖‖𝑥_𝑛+2−𝑥_𝑛+1‖‖==‖‖‖‖11−−𝛽𝛽_𝑛+1𝑛+1𝑦_𝑛+1+𝛽_𝑛+1𝑇_𝑛+1𝑦_𝑛+1−1−𝛽_𝑛𝑦_𝑛−𝛽_𝑛𝑇_𝑛𝑦_𝑛‖‖𝑦_𝑛+1+𝛽_𝑛+1𝑇_𝑛+1𝑦_𝑛+1−1−𝛽_𝑛+1𝑦_𝑛−𝛽_𝑛+1−𝛽_𝑛𝑦_𝑛=‖‖−1𝛽−_𝑛+1𝛽𝑇_𝑛𝑦_𝑛𝑛+1+𝛽_𝑛+1−𝛽_𝑛𝑇_𝑛𝑦_𝑛‖‖≤1−𝛽_𝑛+1𝑦_𝑛+1−𝑦_𝑛+𝛽_𝑛+1𝑇_𝑛+1𝑦_𝑛+1−𝑇_𝑛𝑦_𝑛+𝛽_𝑛+1−𝛽_𝑛𝑇_𝑛𝑦_𝑛−𝑦_𝑛‖‖≤1−𝛽_𝑛+1‖‖𝑦_𝑛+1−𝑦_𝑛‖‖+𝛽_𝑛+1‖‖𝑇_𝑛+1𝑦_𝑛+1−𝑇_𝑛𝑦_𝑛‖‖+||𝛽_𝑛+1−𝛽_𝑛||𝑀+||𝛽_𝑛+1‖‖𝑦_𝑛+1−𝑦_𝑛‖‖+𝛽_𝑛+1‖‖𝑇_𝑛+1𝑦_𝑛+1−𝑇_𝑛𝑦_𝑛+1‖‖+𝛽_𝑛+1‖‖𝑦_𝑛+1−𝑦_𝑛‖‖≤‖‖𝑦_𝑛+1−𝛽_𝑛||𝑀−𝑦_𝑛‖‖+‖‖𝑇_𝑛+1𝑦_𝑛+1−𝑇_𝑛𝑦_𝑛+1‖‖+||𝛽_𝑛+1−𝛽_𝑛||𝑀(2.5) for all 𝑛∈ℕ. From (1.10), we have 𝑦𝑦_𝑛𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛,=𝛼_𝑛+1𝛾𝑓𝑥_𝑛+1+𝐼−𝛼_𝑛+1𝜇𝐹𝑇_𝑛+1𝑥_𝑛+1.(2.6) With 𝑀₁=(𝜇+𝛾)𝑀, we thus have ‖‖𝑦_𝑛+1−𝑦_𝑛‖‖==‖‖‖‖𝛼𝐼−_𝑛+1𝛾𝛼𝑓𝑥_𝑛+1𝑛+1+𝐼−𝛼_𝑛+1𝜇𝐹𝑇_𝑛+1𝑥_𝑛+1−𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛‖‖𝜇𝐹𝑇_𝑛+1𝑥_𝑛+1−𝐼−𝛼_𝑛+1𝜇𝐹𝑇_𝑛𝑥_𝑛−𝛼_𝑛+1−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛≤1+−𝛼𝛼_𝑛+1𝛾_𝑛+1𝑓𝑥_𝑛+1−𝑓𝑥_𝑛+𝛼_𝑛+1−𝛼_𝑛𝛾𝑓𝑥_𝑛‖‖+𝛾||𝛼_𝑛+1𝜏‖‖𝑇_𝑛+1𝑥_𝑛+1−𝑇_𝑛𝑥_𝑛‖‖+||𝛼_𝑛+1−𝛼_𝑛||𝜇‖‖𝐹𝑇_𝑛𝑥_𝑛‖‖+𝛼_𝑛+1𝛾𝛼‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤1−𝛼_𝑛+1−𝛼_𝑛||‖‖𝑓𝑥_𝑛‖‖𝜏‖‖𝑇_𝑛+1𝑥_𝑛+1−𝑇_𝑛𝑥_𝑛‖‖+𝛼_𝑛+1𝛾𝛼‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+||𝛼_𝑛+1−𝛼_𝑛||𝑀₁≤+1𝑀−₁𝛼_𝑛+1𝜏‖‖𝑇_𝑛+1𝑥_𝑛+1−𝑇_𝑛+1𝑥_𝑛‖‖+‖‖𝑇_𝑛+1𝑥_𝑛−𝑇_𝑛𝑥_𝑛‖‖+𝛼_𝑛+1𝛾𝛼‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤1−𝛼||𝛼_𝑛+1𝑛+1−𝛼_𝑛||(𝜏−𝛾𝛼)‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+‖‖𝑇_𝑛+1𝑥_𝑛−𝑇_𝑛𝑥_𝑛‖‖+𝑀₁||𝛼_𝑛+1−𝛼_𝑛||.(2.7)
Combining (2.5) and (2.7), we conclude that ‖‖𝑥_𝑛+2−𝑥_𝑛+1‖‖≤+12−s𝛼up_𝑛+1(‖‖𝜏𝑇−_𝑛+1𝛾𝛼)‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+𝑀₂||𝛼_𝑛+1−𝛼_𝑛||+||𝛽_𝑛+1−𝛽_𝑛||𝑧−𝑇_𝑛𝑧‖‖∶𝑧∈𝐵,(2.8) where 𝑀₂=max{𝑀,𝑀₁}. Using the assumption (B3), we obtain ‖‖𝑥_𝑛+2−𝑥_𝑛+1‖‖≤1−𝛼_𝑛+1(𝜏−𝛾𝛼)‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+𝑀₂𝑜𝛼_𝑛+1+𝜇_𝑛(2.9) with 𝜇_𝑛=𝑀₂(𝜎_𝑛+|𝛽_𝑛+1−𝛽_𝑛|)+2sup{‖𝑇_𝑛+1𝑧−𝑇_𝑛𝑧‖∶𝑧∈𝐵}. The various assumptions imply that ∑^∞𝑛=1𝜇_𝑛<∞. Now it follows from Lemma 2.4 that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0.
Step 3. Next we show that lim_𝑛→∞‖‖𝑇_𝑛𝑥_𝑛−𝑦_𝑛‖‖=lim_𝑛→∞‖‖𝑥_𝑛−𝑦_𝑛‖‖=0.(2.10) In fact, from Step 1, {𝛾𝑓(𝑥_𝑛)−𝜇𝐹𝑇_𝑛𝑥_𝑛} is bounded. Moreover, it follows from lim_𝑛→∞𝛼_𝑛=0 that ‖‖𝑇_𝑛𝑥_𝑛−𝑦_𝑛‖‖=‖‖𝛼_𝑛𝛾𝑓𝑥_𝑛−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛‖‖=𝛼_𝑛‖‖𝛾𝑓𝑥_𝑛−𝜇𝐹𝑇_𝑛𝑥_𝑛‖‖⟶0(𝑛⟶∞).(2.11) This is the first part. From condition (B2), we obtain, for 𝑛 sufficiently large and 𝑏<𝑏₁<1, ‖‖𝑥_𝑛+1−𝑦_𝑛‖‖≤=𝛽𝛽_𝑛𝑛‖‖𝑇_𝑛𝑦_𝑛−𝑦_𝑛‖‖≤𝑏₁‖‖𝑇_𝑛𝑦_𝑛−𝑇_𝑛𝑥_𝑛‖‖+‖‖𝑇_𝑛𝑥_𝑛−𝑦_𝑛‖‖≤𝑏₁‖‖𝑦_𝑛−𝑥_𝑛‖‖+‖‖𝑇_𝑛𝑥_𝑛−𝑦_𝑛‖‖‖‖𝑦_𝑛−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+‖‖𝑇_𝑛𝑥_𝑛−𝑦_𝑛‖‖,(2.12) which implies ‖‖𝑥_𝑛+1−𝑦_𝑛‖‖≤𝑏¹1−𝑏₁‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+‖‖𝑇_𝑛𝑥_𝑛−𝑦_𝑛‖‖.(2.13) Hence, using Step 2 and the above, we obtain lim_𝑛→∞‖𝑥_𝑛+1−𝑦_𝑛‖=0 and thus lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0.
Step 4. We show that lim_𝑛→∞‖𝑦_𝑛−𝑇_𝑛𝑦_𝑛‖=0.
Indeed, we have ‖‖𝑦_𝑛−𝑇_𝑛𝑦_𝑛‖‖≤≤‖‖‖‖𝑦𝑦_𝑛𝑛−𝑇_𝑛𝑥_𝑛‖‖+‖‖𝑇_𝑛𝑥_𝑛−𝑇_𝑛𝑦_𝑛‖‖−𝑇_𝑛𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝑦_𝑛‖‖,(2.14) and now the statement follows from Step 3.
Step 5. We show that ‖𝑇𝑦_𝑛−𝑦_𝑛‖→0 as 𝑛→∞.
Observe that ‖‖𝑇𝑦_𝑛−𝑦_𝑛‖‖≤≤s‖‖u𝑇p𝑦_𝑛−‖‖𝑇𝑇𝑧_𝑛𝑦−_𝑛‖‖𝑇_𝑛+‖‖𝑇_𝑛𝑦_𝑛−𝑦_𝑛‖‖𝑧‖‖∶𝑧∈𝐵+‖‖𝑇_𝑛𝑦_𝑛−𝑦_𝑛‖‖,(2.15) and now apply Step 4 and the fact that 𝑇_𝑛 converges uniformly on 𝐵 to 𝑇 (which is a consequence of (1.5)).
Step 6. We claim that limsup_𝑛→∞⟨𝛾𝑓(𝑞)−𝜇𝐹𝑞,𝑦_𝑛−𝑞⟩≤0, where 𝑞=lim_𝑡→0𝑥_𝑡 with 𝑥_𝑡 defined by (2.1) and the existence of 𝑞 being guaranteed by Lemma 2.2.
Assume the contrary. Then there exists a subsequence {𝑦_𝑛𝑘} of {𝑦_𝑛} such that the limit 𝑐∶=lim_𝑘→∞𝛾𝑓(𝑞)−𝜇𝐹𝑞,𝑦_𝑛𝑘−𝑞(2.16) exists and is greater than zero. Since {𝑦_𝑛} is bounded, there exists a subsequence of {𝑦_𝑛𝑘}, which converges weakly to some 𝑧∈𝐻. We can assume without loss of generality that {𝑦_𝑛𝑘} is this subsequence. From ‖𝑇𝑦_𝑛−𝑦_𝑛‖→0, we obtain that 𝑇𝑦_𝑛𝑘 converges weakly to 𝑧. Now Lemma 2.3 implies that 𝑧∈Fix(𝑇). Hence from weak convergence and Lemma 2.2, lim_𝑘→∞𝛾𝑓(𝑞)−𝜇𝐹𝑞,𝑦_𝑛𝑘−𝑞=⟨(𝛾𝑓−𝜇𝐹)𝑞,𝑧−𝑞⟩≤0.(2.17) This is a contradiction.
Step 7. We show that {𝑥_𝑛} converges strongly to 𝑞. From (1.10), we have ‖‖𝑥_𝑛+1−𝑞‖‖²=‖‖1−𝛽_𝑛𝑦_𝑛−𝑞+𝛽_𝑛𝑇_𝑛𝑦_𝑛−𝑞‖‖²≤‖‖𝑦_𝑛−𝑞‖‖²=‖‖𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−𝑞‖‖²=‖‖𝛼_𝑛𝛾𝑓𝑥_𝑛−𝜇𝐹𝑞+𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−𝐼−𝛼_𝑛𝜇𝐹𝑞‖‖².(2.18) Using the general inequality ‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑥+𝑦⟩, we can further estimate the above by ≤≤‖‖1𝐼−−𝛼_𝑛𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−𝐼−𝛼_𝑛𝜇𝐹𝑞‖‖²+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝜇𝐹𝑞,𝑦_𝑛−𝑞≤1−𝛼_𝑛𝜏²‖‖𝑥_𝑛−𝑞‖‖²+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝛾𝑓(𝑞),𝑦_𝑛−𝑞+2𝛼_𝑛⟨𝛾𝑓(𝑞)−𝜇𝐹𝑞,𝑦_𝑛−𝑞⟩≤1−2𝛼𝜏_𝑛²‖‖𝑥_𝑛−𝑞‖‖²+2𝛼_𝑛𝛾𝛼‖‖𝑥_𝑛−𝑞‖‖‖‖𝑦_𝑛−𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝑞‖‖+2𝛼_𝑛⟨𝛾𝑓(𝑞)−𝜇𝐹𝑞,𝑦_𝑛−𝑞⟩+2𝛼_𝑛(𝜏−𝛾𝛼)‖‖𝑥_𝑛−𝑞‖‖²+𝛼_𝑛𝜏²‖‖𝑥_𝑛−𝑞‖‖²+2𝛼_𝑛𝛾𝛼‖‖𝑥_𝑛−𝑞‖‖‖‖𝑦_𝑛−𝑥_𝑛‖‖⟨𝛾𝑓(𝑞)−𝜇𝐹𝑞,𝑦_𝑛−𝑞⟩.(2.19) Introducing 𝑀₃=sup{‖𝑥_𝑛−𝑞‖∶𝑛≥1}, 𝛿_𝑛=2𝛼_𝑛(𝜏−𝛾𝛼)>0, and 𝜃_𝑛=𝛼^𝑛𝜏²𝑀²³2(𝜏−𝛾𝛼)+𝜏𝛾𝛼−𝑀𝛾𝛼³‖‖𝑦^𝑛−𝑥_𝑛‖‖+1𝜏−𝛾𝛼⟨𝛾𝑓(𝑞)−𝜇𝐹𝑞,𝑦^𝑛−𝑞⟩,(2.20) we obtain ‖‖𝑥_𝑛+1−𝑞‖‖²≤1−𝛿_𝑛‖‖𝑥_𝑛−𝑞‖‖²+𝛿_𝑛𝜃_𝑛.(2.21) Because 𝛿_𝑛→0, ∑^∞𝑛=1𝛿_𝑛=∞, and limsup_𝑛→∞𝜃_𝑛≤0, it follows from Lemma 2.4 that the sequence {𝑥_𝑛} converges strongly to 𝑞. This finishes the proof of the main part of the theorem.
From the definition of 𝑞 as 𝑞=lim_𝑡→0𝑥_𝑡 (see Step 6) and from Lemma 2.2, we obtain ⟨(𝜇𝐹−𝛾𝑓)𝑞,𝑞−𝑧⟩≤0, for all 𝑧∈Fix(𝑇)=⋂^∞𝑛=1Fix(𝑇_𝑛). This completes the proof.

Acknowledgment

This author is supported by the Natural Science Foundation of Yancheng Teachers University under Grant (11YCKL009).