Strong Convergence on Iterative Methods of Cesàro Means for Nonexpansive Mapping in Banach Space

Abstract

Two new iterations with Cesàro's means for nonexpansive mappings are proposed and the strong convergence is obtained as 𝑛→∞. Our main results extend and improve the corresponding results of Xu (2004), Song and Chen (2007), and Yao et al. (2009).

1. Introduction

Let 𝐶 be a nonempty closed convex subset of a real Banach space 𝐸 and let 𝑇 be nonexpansive mapping from 𝐶 into itself (recall that a mapping 𝑇∶𝐶→𝐶 is nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖, ∀𝑥,𝑦∈𝐶). We denote fixed points of 𝑇 as 𝐹(𝑇); that is, 𝐹(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥}.

Recall that a mapping 𝑓∶𝐶→𝐶 is contractive if there exists a constant 𝛽∈(0,1) such that ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝛽‖𝑥−𝑦‖, ∀𝑥,𝑦∈𝐶.

In 1975, Baillon [1] proved the first nonlinear ergodic theorem.

Theorem 1. Suppose that 𝐶 is a nonempty closed convex subset of Hilbert space 𝐸 and 𝑇∶𝐶→𝐶 mapping such that 𝐹(𝑇)≠𝜙; then ∀𝑥∈𝐶, and the Cesàro means 𝑇_𝑛1𝑥=𝑛+1^𝑛_𝑖=0𝑇^𝑖𝑥 (1) weakly converges to a fixed point of 𝑇.

In 1979, Bruck [2] showed the nonlinear ergodic theorem for nonexpansive mapping in uniformly convex Banach space with Fréchet differentiable norms.

In 2004, Xu [3] introduced the following viscosity iterative scheme {𝑥_𝑛} given by 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛, (2) where parameter {𝛼_𝑛}⊂[0,1], satisfying(X1)lim_𝑛→∞𝛼_𝑛=0;(X2)∑^∞_𝑛=0𝛼_𝑛=∞;(X3)either ∑^∞_𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞, or lim_𝑛→∞(𝛼_𝑛+1/𝛼_𝑛)=1.He proved that the explicit iterative scheme {𝑥_𝑛} converges strongly to a fixed point 𝑝 of 𝑇 in uniformly smooth Banach space.

In 2007, Song and Chen [4] defined the following viscosity iteration {𝑥_𝑛} of Cesàro means for nonexpansive mapping 𝑇: 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛1𝑛+1^𝑛_𝑖=0𝑇^𝑖𝑥_𝑛, (3) and they proved that the sequence {𝑥_𝑛} converges strongly to some point in 𝐹(𝑇) in a uniformly convex Banach space with weakly sequentially continuous duality mapping.

In 2009, Yao et al. [5] introduced the following process {𝑥_𝑛}: 𝑥_𝑛+1=𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑇𝑥_𝑛,𝑛≥0. (4) They proved that the sequence {𝑥_𝑛} converges strongly to a fixed point of 𝑇 under the following control conditions of parameters:(YLZ1)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1,  for all 𝑛≥0;(YLZ2)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞;(YLZ3)lim_𝑛→∞𝛾_𝑛=0.

Motivated by the above results, we propose the following new iterations with Cesàro’s means for nonexpansive mappings: 𝑥_𝑛+1=𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+𝛾_𝑛1𝑛+1^𝑛_𝑖=0𝑇^𝑖𝑥_𝑛,𝑛≥0, (5) and viscosity iteration: 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛1𝑛+1^𝑛_𝑖=0𝑇^𝑖𝑥_𝑛,𝑛≥0. (6)

Some examples are given to show the generation of our new iterations with Cesàro’s means as follows.

Example 2. If 𝑛=1, iteration (5) with Cesàro’s means for nonexpansive mappings is 𝑥₂=𝛼₁𝑢+𝛽₁𝑥₁+𝛾₁⋅(1/2)(𝑥₁+𝑇𝑥₁), which is reduced as the same iteration of Yao et al. [5]. If 𝑛≥2, iteration (5) can be written as follows: 𝑥_𝑛+1=𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+𝛾_𝑛⋅1𝑥𝑛+1^𝑛+𝑇𝑥_𝑛+𝑇²𝑥_𝑛+⋯+𝑇^𝑛𝑥_𝑛, (7) which is a generation of Yao et al. [5].

Example 3. Let 𝐸=𝑅 with the usual metric, nonexpansive mapping defined by 𝑇𝑥=sin𝑥, the fixed contractive mapping 𝑓(𝑥)=(1/2)𝑥, and the parameters are defined as 𝛼_𝑛=1/2𝑛, 𝛾_𝑛=1/4𝑛, and 𝛽_𝑛=(4𝑛−3)/4𝑛. The new iterations with Cesàro’s means which is related to iterative step 𝑛 can be written as follows: 𝑥_𝑛+1=12𝑛𝑢+4𝑛−3𝑥4𝑛^𝑛+1()𝑥4𝑛𝑛+1^𝑛+𝑇𝑥_𝑛+𝑇²𝑥_𝑛+⋯+𝑇^𝑛𝑥_𝑛,𝑥_𝑛+1=1⋅12𝑛2𝑥_𝑛+4𝑛−3𝑥4𝑛^𝑛+1()𝑥4𝑛𝑛+1^𝑛+𝑇𝑥_𝑛+𝑇²𝑥_𝑛+⋯+𝑇^𝑛𝑥_𝑛. (8)

2. Preliminaries

Throughout the paper, let 𝐸 be a real Banach space with norm ‖⋅‖. The normalized duality mapping 𝐽∶𝐸→2^𝐸∗ is defined by 𝐽(𝑥)=𝑓∈𝐸^∗,𝑥,𝑓=‖𝑥‖𝑓,‖𝑥‖=𝑓,∀𝑥∈𝐸, (9) where 𝐸^∗ denotes the dual space of 𝐸 and ⟨⋅,⋅⟩ denotes the generalized duality pairing. We will denote the single-valued normalized duality mapping by 𝑗.

Let 𝑆∶={𝑥∈𝐸∶‖𝑥‖=1} be the unit sphere of a Banach space. The space is said to have a Gâteaux differentiable norm (or 𝐸 is said to be smooth), if the limit lim_𝑡→0𝑥+𝑡𝑦−‖𝑥‖𝑡 (10) exists for every 𝑥,𝑦∈𝑆, and 𝐸 is said to have a uniformly Gâteaux differentiable norm if for each 𝑦∈𝑆 the limit (10) is attained uniformly for 𝑥∈𝑆. Further, 𝐸 is said to be uniformly smooth if the limit (10) exists uniformly for (𝑥,𝑦)∈𝑆×𝑆.

The following two results can be found in [6].

If 𝐸 is smooth the duality mapping 𝐽 is single-valued and strong-weak^∗ continuous.

If 𝐸 is Banach space with uniformly Gâteaux differentiable norm, then duality mapping 𝐽∶𝐸→𝐸^∗ is single-valued and norm to weak star uniformly continuous on bounded sets of 𝐸.

In order to prove our main results, the following lemmas will be used.

Lemma 4 (see [7]). Let 𝐶 be a nonempty closed convex subset of a uniformly smooth Banach space 𝑋. Let 𝑇∶𝐶→𝐶 be a nonexpansive mapping with 𝐹(𝑇)≠∅. For each fixed 𝑢∈𝐶 and every 𝑡∈(0,1), the unique fixed point 𝑧_𝑡∈𝐶 of the contraction 𝐶∋𝑥⊢𝑡𝑢+(1−𝑡)𝑇𝑥 as 𝑡→0 converges strongly to 𝑥^∗∈𝐹(𝑇) which is the nearest to 𝑢.

Lemma 5 (see [1]). Let 𝑋 be a uniformly smooth Banach space, 𝐶 a closed convex subset of 𝑋, 𝑇∶𝐶→𝐶 a nonexpansive mapping with 𝐹(𝑇)≠𝜙, and 𝑓∶𝐶→𝐶 a fixed contraction. Then 𝑥_𝑡 defined by 𝑥_𝑡=𝑡𝑓(𝑥_𝑡)+(1−𝑡)𝑇𝑥_𝑡 converges strongly to a unique fixed point in 𝐹(𝑇) as 𝑡→0.

Lemma 6 (see [3]). Let 𝐸 be a real Banach space and let 𝐽 be the normalized duality mapping. Then for any given 𝑥,𝑦∈𝐸, one has 𝑥+𝑦²≤‖𝑥‖²,.+2𝑦,𝑗𝑥+𝑦∀𝑗𝑥+𝑦∈𝐽𝑥+𝑦 (11)

Lemma 7 (see [8]). Assume {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛾_𝑛𝛿_𝑛,𝑛≥0, (12) where {𝛾_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence such that(1)∑^∞_𝑛=0𝛾_𝑛=∞;(2)limsup_𝑛→∞𝛿_𝑛≤0. Then lim_𝑛→∞𝑎_𝑛=0.

3. Main Results

Let 𝐶 be a nonempty closed convex subset of a uniformly smooth Banach space. Let 𝑇∶𝐶→𝐶 be a nonexpansive mapping such that 𝐹(𝑇)≠𝜙. Let {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} be three real sequences in (0,1) satisfying(i)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1, for all 𝑛≥0;(ii)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞;(iii)lim_𝑛→∞𝛾_𝑛=0.

In the following, we will present the first main result. For each 𝑛≥0 and 𝑡∈(0,1), let 𝑥_𝑡,𝑛 be the unique fixed point of the contractive mapping 𝑇_𝑡,𝑛 given by 𝑇_𝑡,𝑛𝑥=1−𝛼_𝑛𝑡𝛾_𝑛+𝑡𝛽_𝑛()𝛾𝑢+1−𝑡_𝑛𝛾_𝑛+𝑡𝛽_𝑛1𝑛+1^𝑛_𝑖=0𝑇^𝑖𝑥. (13) That is, 𝑥_𝑡,𝑛=1−𝛼_𝑛𝑡𝛾_𝑛+𝑡𝛽_𝑛()𝛾𝑢+1−𝑡_𝑛𝛾_𝑛+𝑡𝛽_𝑛1𝑛+1^𝑛_𝑖=0𝑇^𝑖𝑥_𝑡,𝑛. (14) From Lemma 4, for fixed 𝑛, we have lim_𝑡→0𝑥_𝑡,𝑛=𝑝∈𝐹(𝑇), (15) which is the unique fixed point.

Theorem 8. Let 𝐶 be a nonempty closed convex subset of a uniformly smooth Banach space 𝑋. Let 𝑇∶𝐶→𝐶 be a nonexpansive mapping such that 𝐹(𝑇)≠𝜙. Let {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} be three real sequences in (0,1) satisfying conditions (i)–(iii). Then, for given 𝑥₀∈𝐶 arbitrarily, let the sequence {𝑥_𝑛} be generated iteratively by (5). Then the sequence {𝑥_𝑛} defined by (5) converges strongly to a fixed point of 𝑇.

Proof. Taking a fixed point 𝑝∈𝐹(𝑇), we have 𝑥_𝑛+1−𝑝=𝛼_𝑛𝑢−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛1𝑛+1^𝑛_𝑖=0𝑇^𝑖𝑥_𝑛−𝑇^𝑖𝑝≤𝛼_𝑛𝑢−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑥_𝑛−𝑝=𝛼_𝑛𝑢−𝑝+1−𝛼_𝑛𝑥_𝑛,𝑥−𝑝≤max𝑢−𝑝_𝑛−𝑝. (16) By induction, we get that {𝑥_𝑛} is bounded. We observe that (14) can be rewritten as follows: 𝑥_𝑡,𝑛(𝛽=𝑡𝑢+1−𝑡)^𝑛1−𝛼_𝑛𝑥_𝑡,𝑛+𝛾_𝑛1−𝛼_𝑛11+𝑛^𝑛_𝑖=0𝑇^𝑖𝑥_𝑡,𝑛,lim_𝑡→0𝑥_𝑡,𝑛=𝑝∈𝐹(𝑇),∀𝑛≥0. (17) By (14), we have 𝑥_𝑡,𝑛−𝑥_𝑛=𝑡𝑢−𝑥_𝑛+(𝛽1−𝑡)^𝑛1−𝛼_𝑛𝑥_𝑡,𝑛−𝑥_𝑛+𝛾_𝑛1−𝛼_𝑛11+𝑛^𝑛_𝑖=0𝑇^𝑖𝑥_𝑡,𝑛−𝑥_𝑛. (18) Applying Lemma 6 to (18), we have 𝑥_𝑡,𝑛−𝑥_𝑛≤(1−𝑡)²𝛽_𝑛1−𝛼_𝑛𝑥_𝑡,𝑛−𝑥_𝑛+𝛾_𝑛1−𝛼_𝑛11+𝑛^𝑛_𝑖=0𝑇^𝑖𝑥_𝑡,𝑛−𝑥_𝑛²+2𝑡𝑢−𝑥_𝑛,𝑗𝑥_𝑡,𝑛−𝑥_𝑛≤(1−𝑡)²𝛽_𝑛1−𝛼_𝑛𝑥_𝑡,𝑛−𝑥_𝑛+𝛾_𝑛1−𝛼_𝑛11+𝑛^𝑛_𝑖=0𝑇^𝑖𝑥_𝑡,𝑛−𝑇^𝑖𝑥_𝑛+𝛾_𝑛1−𝛼_𝑛11+𝑛^𝑛_𝑖=0𝑇^𝑖𝑥_𝑛−𝑥_𝑛²+2𝑡𝑥_𝑡,𝑛−𝑥_𝑛²+2𝑡𝑢−𝑥_𝑡,𝑛,𝑗𝑥_𝑡,𝑛−𝑥_𝑛≤(1−𝑡)²𝑥_𝑡,𝑛−𝑥_𝑛+𝛾_𝑛1−𝛼_𝑛11+𝑛^𝑛_𝑖=0𝑇^𝑖𝑥_𝑛−𝑥_𝑛²+2𝑡𝑥_𝑡,𝑛−𝑥_𝑛²+2𝑡𝑢−𝑥_𝑡,𝑛,𝑗𝑥_𝑡,𝑛−𝑥_𝑛=1+𝑡²𝑥_𝑡,𝑛−𝑥_𝑛²+(1−𝑡)²𝛾_𝑛1−𝛼_𝑛11+𝑛^𝑛_𝑖=0𝑇^𝑖𝑥_𝑛−𝑥_𝑛²+2(1−𝑡)²𝛾_𝑛1−𝛼_𝑛𝑥_𝑡,𝑛−𝑥_𝑛11+𝑛^𝑛_𝑖=0𝑇^𝑖𝑥_𝑛−𝑥_𝑛+2𝑡𝑢−𝑥_𝑡,𝑛,𝑗𝑥_𝑡,𝑛−𝑥_𝑛≤1+𝑡²𝑥_𝑡,𝑛−𝑥_𝑛²+𝛾_𝑛1−𝛼_𝑛𝑄+2𝑡𝑢−𝑥_𝑡,𝑛,𝑗𝑥_𝑡,𝑛−𝑥_𝑛, (19) where 𝑄 is some constant such that 1sup1+𝑛^𝑛_𝑖=0𝑇^𝑖𝑥_𝑛−𝑥_𝑛+2𝑥_𝑡,𝑛−𝑥_𝑛11+𝑛^𝑛_𝑖=0𝑇^𝑖𝑥_𝑛−𝑥_𝑛,𝑛≥0≤𝑄. (20) Hence, we get 𝑢−𝑥_𝑡,𝑛,𝑗𝑥_𝑛−𝑥_𝑡,𝑛≤𝑡2𝑄₁+1𝛾2𝑡1−^𝑛𝛼_𝑛𝑄, (21) where 𝑄₁ is a constant such that sup𝑥_𝑡,𝑛−𝑥_𝑛,𝑛≥0,0<𝑡<1≤𝑄₁. (22) It follows that limsup_𝑡→0limsup_𝑛→∞𝑢−𝑥_𝑡,𝑛,𝑗𝑥_𝑛−𝑥_𝑡,𝑛≤0. (23) By the fact that the order of limsup_𝑡→0 and limsup_𝑛→∞ is changeable, we have limsup_𝑛→∞𝑢−𝑝,𝑗𝑥_𝑛−𝑝≤0. (24) Finally, we prove 𝑥_𝑛→𝑝. Indeed, applying Lemma 6 to (5), we obtain 𝑥_𝑛+1−𝑝²=𝛼_𝑛(𝑢−𝑝)+𝛽_𝑛(𝑥_𝑛−𝑝)+𝛾_𝑛1𝑛+1^𝑛_𝑖=0𝑇^𝑖𝑥_𝑛−𝑝²≤𝛽_𝑛(𝑥_𝑛−𝑝)+𝛾_𝑛1𝑛+1^𝑛_𝑖=0𝑇^𝑖𝑥_𝑛−𝑇^𝑖𝑝²+2𝛼_𝑛𝑢−𝑝,𝑗𝑥_𝑛+1≤𝛽−𝑝_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑥_𝑛−𝑝²+2𝛼_𝑛𝑢−𝑝,𝑗𝑥_𝑛+1≤−𝑝1−𝛼_𝑛𝑥_𝑛−𝑝²+2𝛼_𝑛𝑢−𝑝,𝑗𝑥_𝑛+1−𝑝. (25) Hence, by Lemma 7, we have that 𝑥_𝑛→𝑝 as 𝑛→∞.
The proof is complete.

Now we will give the second main result. In order to prove the strong convergence of viscosity iterative (6), we assume that 𝑧_𝑡,𝑛 is the unique fixed point of the following contractive mapping 𝑇_𝑡,𝑛 given by 𝑇_𝑡,𝑛𝑥=1−𝛼_𝑛𝑡𝛾_𝑛+𝑡𝛽_𝑛𝑓(𝑥)+(1−𝑡)𝛾_𝑛𝛾_𝑛+𝑡𝛽_𝑛1𝑛+1^𝑛_𝑖=0𝑇^𝑖𝑥. (26) That is, 𝑧_𝑡,𝑛=1−𝛼_𝑛𝑡𝛾_𝑛+𝑡𝛽_𝑛𝑓𝑧_𝑡,𝑛+(1−𝑡)𝛾_𝑛𝛾_𝑛+𝑡𝛽_𝑛1𝑛+1^𝑛_𝑖=0𝑇^𝑖𝑧_𝑡,𝑛. (27) From Lemma 5, for fixed 𝑛, we have lim_𝑡→0𝑧_𝑡,𝑛=𝑝∈𝐹(𝑇), (28) which is the unique fixed point.

Theorem 9. Let 𝐶 be a nonempty closed convex subset of a uniformly smooth Banach space 𝑋. Let 𝑇∶𝐶→𝐶 be a nonexpansive mapping such that 𝐹(𝑇)≠𝜙. Let {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} be three real sequences in (0,1) satisfying the following control conditions (i–iii). Then, for given 𝑥₀∈𝐶 arbitrarily, the sequence {𝑥_𝑛} defined by (6) converges strongly to a fixed point of 𝑇.

Proof. Taking a fixed point 𝑞∈𝐹(𝑇), we have 𝑥_𝑛+1−𝑞≤𝛼_𝑛𝑓𝑥_𝑛−𝑞+𝛽_𝑛𝑥_𝑛−𝑞+𝛾_𝑛11+𝑛^𝑛_𝑖=0𝑇^𝑖𝑥_𝑛−𝑇^𝑖𝑞≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑞+𝑓𝑞−𝑞+𝛽_𝑛𝑥_𝑛−𝑞+𝛾_𝑛𝑥_𝑛−𝑞≤𝛼_𝑛𝛽𝑥_𝑛−𝑞+𝛼_𝑛𝑓𝑞+−𝑞1−𝛼_𝑛𝑥_𝑛−𝑞=𝛼_𝑛𝑓𝑞−𝑞+1−1−𝛽𝛼_𝑛𝑥_𝑛1−𝑞≤max𝑓𝑞,𝑥1−𝛽−𝑞_𝑛−𝑞. (29) By induction, 𝑥_𝑛1−𝑞≤max𝑓𝑞,𝑥1−𝛽−𝑞₀−𝑞,𝑛≥0 (30) and {𝑥_𝑛} is bounded so are {𝑇^𝑖𝑥_𝑛} and {𝑓(𝑥_𝑛)}.
We observe that (27) can be rewritten as 𝑧_𝑡,𝑛=𝑡𝑓𝑧_𝑡,𝑛+(𝛽1−𝑡)^𝑛1−𝛼_𝑛𝑧_𝑡,𝑛+𝛾_𝑛1−𝛼_𝑛11+𝑛^𝑛_𝑖=0𝑇^𝑖𝑧_𝑡,𝑛, (31) and lim_𝑡→0𝑧_𝑡,𝑛=𝑞∈𝐹(𝑇), for all 𝑛≥0.
From (31), we have 𝑧_𝑡,𝑛−𝑥_𝑛=𝑡𝑓𝑧_𝑡,𝑛−𝑥_𝑛+(𝛽1−𝑡)^𝑛1−𝛼_𝑛𝑧_𝑡,𝑛−𝑥_𝑛+𝛾_𝑛1−𝛼_𝑛11+𝑛^𝑛_𝑖=0𝑇^𝑖𝑧_𝑡,𝑛−𝑥_𝑛. (32) Applying Lemma 6 to (32), we get 𝑧_𝑡,𝑛−𝑥_𝑛²≤(1−𝑡)²𝛽_𝑛1−𝛼_𝑛𝑧_𝑡,𝑛−𝑥_𝑛+𝛾_𝑛1−𝛼_𝑛11+𝑛^𝑛_𝑖=0𝑇^𝑖𝑧_𝑡,𝑛−𝑥_𝑛²+2𝑡𝑓𝑧_𝑡,𝑛−𝑥_𝑛,𝑗𝑧_𝑡,𝑛−𝑥_𝑛≤(1−𝑡)²𝛽_𝑛1−𝛼_𝑛𝑧_𝑡,𝑛−𝑥_𝑛+𝛾_𝑛1−𝛼_𝑛11+𝑛^𝑛_𝑖=0𝑇^𝑖𝑧_𝑡,𝑛−𝑇^𝑖𝑥_𝑛+𝛾_𝑛1−𝛼_𝑛11+𝑛^𝑛_𝑖=0𝑇^𝑖𝑥_𝑛−𝑥_𝑛²+2𝑡𝑧_𝑡,𝑛−𝑥_𝑛²+2𝑡𝑓𝑧_𝑡,𝑛−𝑧_𝑡,𝑛,𝑗𝑧_𝑡,𝑛−𝑥_𝑛≤(1−𝑡)²𝑧_𝑡,𝑛−𝑥_𝑛+𝛾_𝑛1−𝛼_𝑛11+𝑛^𝑛_𝑖=0𝑇^𝑖𝑥_𝑛−𝑥_𝑛²+2𝑡𝑧_𝑡,𝑛−𝑥_𝑛²+2𝑡𝑓𝑧_𝑡,𝑛−𝑧_𝑡,𝑛,𝑗𝑧_𝑡,𝑛−𝑥_𝑛=1+𝑡²𝑧_𝑡,𝑛−𝑥_𝑛²+(1−𝑡)²𝛾_𝑛1−𝛼_𝑛11+𝑛^𝑛_𝑖=0𝑇^𝑖𝑥_𝑛−𝑥_𝑛²+2(1−𝑡)²𝛾_𝑛1−𝛼_𝑛11+𝑛^𝑛_𝑖=0𝑇^𝑖𝑥_𝑛−𝑥_𝑛𝑧_𝑡,𝑛−𝑥_𝑛+2𝑡𝑓𝑧_𝑡,𝑛−𝑧_𝑡,𝑛,𝑗𝑧_𝑡,𝑛−𝑥_𝑛≤1+𝑡²𝑧_𝑡,𝑛−𝑥_𝑛²+𝛾_𝑛1−𝛼_𝑛𝑀+2𝑡𝑓𝑧_𝑡,𝑛−𝑧_𝑡,𝑛,𝑗𝑧_𝑡,𝑛−𝑥_𝑛, (33) where 𝑀 is some constant such that 1sup1+𝑛^𝑛_𝑖=0𝑇^𝑖𝑥_𝑛−𝑥_𝑛²+2𝑇^𝑖𝑥_𝑛−𝑥_𝑛𝑧_𝑡,𝑛−𝑥_𝑛,𝑛≥0≤𝑀. (34) Hence, we have 𝑓𝑧_𝑡,𝑛−𝑧_𝑡,𝑛,𝑗𝑥_𝑛−𝑧_𝑡,𝑛≤𝑡2𝑀₁+1𝛾2𝑡1−^𝑛𝛼_𝑛𝑀, (35) where 𝑀₁ is also a constant such that 𝑀₁≥sup𝑧_𝑡,𝑛−𝑥_𝑛²,𝑛≥0,0<𝑡<1. (36) It follows that limsup_𝑡→0limsup_𝑛→∞𝑓𝑧_𝑡,𝑛−𝑧_𝑡,𝑛,𝑗𝑥_𝑛−𝑧_𝑡,𝑛≤0. (37) Since the order of limsup_𝑡→0 and limsup_𝑛→∞ is exchangeable, hence limsup_𝑛→∞𝑓𝑞−𝑞,𝑗𝑥_𝑛−𝑞≤0. (38)
Finally, we prove that 𝑥_𝑛→𝑞. Indeed, applying Lemma 6 to (6), we obtain 𝑥_𝑛+1−𝑞²=𝛼_𝑛𝑓𝑥_𝑛−𝑞+𝛽_𝑛𝑥_𝑛−𝑞+𝛾_𝑛11+𝑛^𝑛_𝑖=0𝑇^𝑖𝑥_𝑛−𝑇^𝑖𝑞²≤𝛽_𝑛𝑥_𝑛−𝑞+𝛾_𝑛11+𝑛^𝑛_𝑖=0𝑇^𝑖𝑥_𝑛−𝑇^𝑖𝑞²+2𝛼_𝑛𝑓𝑥_𝑛−𝑞,𝑗𝑥_𝑛+1≤𝛽−𝑞_𝑛𝑥_𝑛−𝑞+𝛾_𝑛𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑞,𝑗𝑥_𝑛+1−𝑞+2𝛼_𝑛𝑓𝑞−𝑞,𝑗𝑥_𝑛+1≤−𝑞1−𝛼_𝑛²𝑥_𝑛−𝑞²+𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑞²+𝑥_𝑛+1−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝑗𝑥_𝑛+1−𝑞. (39) Therefore, we have 𝑥_𝑛+1−𝑞²≤1−2𝛼_𝑛+𝛽²𝛼_𝑛+𝛼²_𝑛1−𝛼_𝑛𝑥_𝑛−𝑞²+2𝛼_𝑛1−𝛼_𝑛𝑓𝑞−𝑞,𝑗𝑥_𝑛+1≤−𝑞1−1−𝛽²1−𝛼_𝑛𝛼_𝑛𝑥_𝑛−𝑞²+𝛼²_𝑛1−𝛼_𝑛𝑥_𝑛−𝑞²+2𝛼_𝑛1−𝛼_𝑛𝑓𝑞−𝑞,𝑗𝑥_𝑛+1≤𝛼−𝑞1−_𝑛𝑥_𝑛−𝑞²+𝛼_𝑛𝛽_𝑛. (40) Put 𝛼_𝑛=1−𝛽²1−𝛼_𝑛𝛼_𝑛,𝛽_𝑛=𝑀₁1−𝛽²𝛼_𝑛+21−𝛽²𝑓𝑞−𝑞,𝑗𝑥_𝑛+1−𝑞. (41) It follows that 𝑥_𝑛+1−𝑞²≤1−𝛼_𝑛𝑥_𝑛−𝑞²+𝛼_𝑛𝛽_𝑛. (42) It is easily seen from (ii) and (38) that 𝛼_𝑛⟶0,^∞_𝑛=0𝛼_𝑛=∞,limsup_𝑛→∞𝛽_𝑛≤0. (43) Hence, applying Lemma 7 to (42), we have that 𝑥_𝑛→𝑞 as 𝑛→∞.
The proof is complete.

Remark 10. Our main result extends the main result of Yao et al. to Cesàro means and viscosity iteration method. Our results are new and the proofs are simple and different from many others.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.