1.Introduction Lu-ChuanCeng, QamrulHasanAnsari, andChing-FengWen ImplicitRelaxedandHybridMethodswithRegularizationforMinimizationProblemsandAsymptoticallyStrictPseudocontractiveMappingsintheIntermediateSense ResearchArticle

Volume 2013, Article ID 854297,14pages http://dx.doi.org/10.1155/2013/854297

Research Article

Implicit Relaxed and Hybrid Methods with Regularization for Minimization Problems and Asymptotically Strict

Pseudocontractive Mappings in the Intermediate Sense

Lu-Chuan Ceng,

Qamrul Hasan Ansari,

and Ching-Feng Wen

1Department of Mathematics, Shanghai Normal University and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China

2Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India

3Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia

4Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 80708, Taiwan

Correspondence should be addressed to Ching-Feng Wen; [email protected] Received 1 February 2013; Accepted 11 March 2013

Academic Editor: Jen-Chih Yao

Copyright © 2013 Lu-Chuan Ceng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We first introduce an implicit relaxed method with regularization for finding a common element of the set of fixed points of an asymptotically strict pseudocontractive mapping𝑆in the intermediate sense and the set of solutions of the minimization problem (MP) for a convex and continuously Frechet differentiable functional in the setting of Hilbert spaces. The implicit relaxed method with regularization is based on three well-known methods: the extragradient method, viscosity approximation method, and gradient projection algorithm with regularization. We derive a weak convergence theorem for two sequences generated by this method. On the other hand, we also prove a new strong convergence theorem by an implicit hybrid method with regularization for the MP and the mapping𝑆. The implicit hybrid method with regularization is based on four well-known methods: the CQ method, extragradient method, viscosity approximation method, and gradient projection algorithm with regularization.

1. Introduction

In 1972, Goebel and Kirk [1] established that every asymptot- ically nonexpansive mapping𝑆 : 𝐶 → 𝐶defined on a non- empty closed convex bounded subset of a uniformly convex Banach space, that is, there exists a sequence{𝑘_𝑛}such that lim_{𝑛 → ∞}𝑘_𝑛= 1and

󵄩󵄩󵄩󵄩𝑆^𝑛𝑥 − 𝑆^𝑛𝑦󵄩󵄩󵄩󵄩 ≤ 𝑘^𝑛󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩, ∀𝑛 ≥ 1, ∀𝑥,𝑦 ∈ 𝐶 (1) has a fixed point in𝐶. It can be easily seen that every nonex- pansive mapping is asymptotically nonexpansive, and every asymptotically nonexpansive mapping is uniformly Lips- chitzian; that is, there exists a constantL> 0such that

󵄩󵄩󵄩󵄩𝑆^𝑛𝑥 − 𝑆^𝑛𝑦󵄩󵄩󵄩󵄩 ≤L󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩, ∀𝑛 ≥ 1, ∀𝑥,𝑦 ∈ 𝐶. (2) Several researchers have weaken the assumption on the map- ping𝑆. Bruck et al. [2] introduced the following concept of

an asymptotically nonexpansive mapping in the intermediate sense.

Definition 1. Let𝐶be a nonempty subset of a normed space 𝑋. A mapping𝑆 : 𝐶 → 𝐶is said to be asymptotically non- expansive in the intermediate sense provided𝑆is uniformly continuous and

lim sup

𝑛 → ∞ sup

𝑥,𝑦∈𝐶(󵄩󵄩󵄩󵄩𝑆^𝑛𝑥 − 𝑆^𝑛𝑦󵄩󵄩󵄩󵄩 −󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩) ≤ 0. (3) They also studied iterative methods for the approximation of fixed points of such mappings.

Recently, Kim and Xu [3] introduced the following con- cept of asymptotically𝜅-strict pseudocontractive mappings in setting of Hilbert spaces.

Definition 2. Let𝐶be a nonempty subset of a Hilbert space 𝐻. A mapping𝑆 : 𝐶 → 𝐶is said to be an asymptotically𝜅- strict pseudocontractive mapping with sequence{𝛾_𝑛}if there

exists a constant𝜅 ∈ [0, 1)and a sequence{𝛾_𝑛}in[0, ∞)with lim_{𝑛 → ∞}𝛾_𝑛= 0such that

󵄩󵄩󵄩󵄩𝑆^𝑛𝑥 − 𝑆^𝑛𝑦󵄩󵄩󵄩󵄩²

≤ (1 + 𝛾_𝑛) 󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩²+ 𝜅󵄩󵄩󵄩󵄩𝑥 − 𝑆^𝑛𝑥 − (𝑦 − 𝑆^𝑛𝑦)󵄩󵄩󵄩󵄩²,

∀𝑛 ≥ 1, ∀𝑥, 𝑦 ∈ 𝐶.

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They studied weak and strong convergence theorems for this class of mappings. It is important to note that every asymptotically𝜅-strict pseudocontractive mapping with se- quence{𝛾_𝑛}is a uniformlyL-Lipschitzian mapping withL= sup{(𝜅 + √1 + (1 − 𝜅)𝛾_𝑛)/(1 + 𝜅) : 𝑛 ≥ 1}.

Very recently, Sahu et al. [4] considered the following concept of asymptotically 𝜅-strict pseudocontractive map- pings in the intermediate sense, which are not necessarily Lipschitzian.

Definition 3. Let𝐶be a nonempty subset of a Hilbert space 𝐻. A mapping𝑆 : 𝐶 → 𝐶is said to be an asymptotically𝜅- strict pseudocontractive mapping in the intermediate sense with sequence{𝛾_𝑛}if there exist a constant𝜅 ∈ [0, 1)and a sequence{𝛾_𝑛}in[0, ∞)with lim_{𝑛 → ∞}𝛾_𝑛= 0such that

lim sup

𝑛 → ∞ sup

𝑥,𝑦∈𝐶(󵄩󵄩󵄩󵄩𝑆^𝑛𝑥 − 𝑆^𝑛𝑦󵄩󵄩󵄩󵄩²− (1 + 𝛾_𝑛) 󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩²

− 𝜅󵄩󵄩󵄩󵄩𝑥 − 𝑆^𝑛𝑥 − (𝑦 − 𝑆^𝑛𝑦)󵄩󵄩󵄩󵄩²) ≤ 0.

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Let

𝑐_𝑛:=max{0, sup

𝑥,𝑦∈𝐶(󵄩󵄩󵄩󵄩𝑆^𝑛𝑥 − 𝑆^𝑛𝑦󵄩󵄩󵄩󵄩²− (1 + 𝛾_𝑛) 󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩²

−𝜅󵄩󵄩󵄩󵄩𝑥 − 𝑆^𝑛𝑥 − (𝑦 − 𝑆^𝑛𝑦)󵄩󵄩󵄩󵄩²) } . (6) Then,𝑐_𝑛 ≥ 0(for all 𝑛 ≥ 1),𝑐_𝑛 → 0(𝑛 → ∞), and (5) reduces to the relation

󵄩󵄩󵄩󵄩𝑆^𝑛𝑥 − 𝑆^𝑛𝑦󵄩󵄩󵄩󵄩²≤ (1 + 𝛾_𝑛) 󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩²

+ 𝜅󵄩󵄩󵄩󵄩𝑥 − 𝑆^𝑛𝑥 − (𝑦 − 𝑆^𝑛𝑦)󵄩󵄩󵄩󵄩²+ 𝑐_𝑛,

∀𝑛 ≥ 1, ∀𝑥, 𝑦 ∈ 𝐶.

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Whenever𝑐_𝑛 = 0for all𝑛 ≥ 1in (7), then𝑆is an asymptoti- cally𝜅-strict pseudocontractive mapping with sequence{𝛾_𝑛}.

Let𝑓 : 𝐶 → R be a convex and continuously Fr´echet differentiable functional. We consider the following mini- mization problem:

min𝑥∈𝐶𝑓 (𝑥) . (8)

We assume that the minimization problem (8) has a solution, and the solution set of this problem is denoted byΓ.

To develop some new iterative methods for computing the approximate solutions of the minimization problem is one of the main areas of research in optimization and approxi- mation theory. In the recent past, some study has also been done in the direction to suggest some iterative algorithms to

compute the fixed point of a mapping which is also a solution of some minimization problem; for further detail, we refer to [5] and the references therein.

The main aim of this paper is to propose some iterative schemes for finding a common solution of fixed point set of an asymptotically 𝜅-strict pseudocontractive mapping and the solution set of the minimization problem. In particular, we introduce an implicit relaxed algorithm with regular- ization for finding a common element of the fixed point set Fix(𝑆) of an asymptotically 𝜅-strict pseudocontractive mapping𝑆and the solution setΓof minimization problem (8). This implicit relaxed method with regularization is based on three well-known methods, namely, the extragradient method [6], viscosity approximation method, and gradient projection algorithm with regularization. We also propose an implicit hybrid algorithm with regularization for finding an element of Fix(𝑆) ∩ Γ. The implicit hybrid method with regularization is based on four well-known methods, namely, the CQ method, extragradient method, viscosity approx- imation method, and gradient projection algorithm with regularization. The weak and strong convergence results of these two algorithms are established, respectively.

2. Preliminaries

Throughout the paper, unless otherwise specified, we use the following assumptions, notations, and terminologies.

We assume that𝐻is a real Hilbert space whose inner product and norm are denoted by⟨⋅, ⋅⟩and‖ ⋅ ‖, respectively, and𝐶 is a nonempty closed convex subset of𝐻. We write 𝑥_𝑛 ⇀ 𝑥to indicate that the sequence{𝑥_𝑛}converges weakly to𝑥and𝑥_𝑛 → 𝑥to indicate that the sequence{𝑥_𝑛}converges strongly to𝑥. Moreover, we use𝜔_𝑤(𝑥_𝑛)to denote the weak𝜔- limit set of the sequence{𝑥_𝑛}, that is,

𝜔_𝑤(𝑥_𝑛) := {𝑥 ∈ 𝐻 : 𝑥_𝑛𝑖⇀ 𝑥

for some subsequence {𝑥_𝑛𝑖}of{𝑥_𝑛}} . (9) The metric (or nearest point) projection from𝐻onto𝐶is the mapping𝑃_𝐶: 𝐻 → 𝐶which assigns to each point𝑥 ∈ 𝐻 the unique point𝑃_𝐶𝑥 ∈ 𝐶satisfying the following property:

󵄩󵄩󵄩󵄩𝑥 − 𝑃^𝐶𝑥󵄩󵄩󵄩󵄩 =inf

𝑦∈𝐶󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩 =: 𝑑(𝑥,𝐶). (10) We mention some important properties of projections in the following proposition.

Proposition 4. For given𝑥 ∈ 𝐻and𝑧 ∈ 𝐶, (i)𝑧 = 𝑃_𝐶𝑥 ⇔ ⟨𝑥 − 𝑧, 𝑦 − 𝑧⟩ ≤ 0, for all𝑦 ∈ 𝐶;

(ii)𝑧 = 𝑃_𝐶𝑥 ⇔ ‖𝑥 − 𝑧‖² ≤ ‖𝑥 − 𝑦‖²− ‖𝑦 − 𝑧‖², for all 𝑦 ∈ 𝐶;

(iii)⟨𝑃_𝐶𝑥 − 𝑃_𝐶𝑦, 𝑥 − 𝑦⟩≥ ‖𝑃_𝐶𝑥 − 𝑃_𝐶𝑦‖², for all𝑦 ∈ 𝐻.

Consequently,𝑃_𝐶is nonexpansive.

Definition 5. A mapping𝑆 : 𝐶 → 𝐻is said to be (a) monotone if

⟨𝑆𝑥 − 𝑆𝑦, 𝑥 − 𝑦⟩ ≥ 0, ∀𝑥, 𝑦 ∈ 𝐶; (11)

(b)𝜂-strongly monotone if there exists a constant𝜂 > 0 such that

⟨𝑆𝑥 − 𝑆𝑦, 𝑥 − 𝑦⟩ ≥ 𝜂󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩², ∀𝑥, 𝑦 ∈ 𝐶; (12) (c)𝛼-inverse-strongly monotone (𝛼-ism) if there exists a

constant𝛼 > 0such that

⟨𝑆𝑥 − 𝑆𝑦, 𝑥 − 𝑦⟩ ≥ 𝛼󵄩󵄩󵄩󵄩𝑆𝑥 − 𝑆𝑦󵄩󵄩󵄩󵄩², ∀𝑥, 𝑦 ∈ 𝐶. (13) Obviously, if𝑆is𝛼-inverse-strongly monotone, then it is monotone and(1/𝛼)-Lipschitz continuous. It can be easily seen that if 𝑆 : 𝐶 → 𝐶 is nonexpansive, then𝐼 − 𝑆 is monotone. It is also easy to see that a projection mapping𝑃_𝐶is 1-ism. The inverse strongly monotone (also known as cocoer- cive) operators have been applied widely in solving practical problems in various fields.

We need some facts and tools which are listed in the form of the following lemmas.

Lemma 6. Let𝑋be a real inner product space. Then,

󵄩󵄩󵄩󵄩𝑥 + 𝑦󵄩󵄩󵄩󵄩²≤ ‖𝑥‖²+ 2 ⟨𝑦, 𝑥 + 𝑦⟩ , ∀𝑥, 𝑦 ∈ 𝑋. (14) Lemma 7 (see [7, Proposition 2.4]). Let{𝑥_𝑛}be a bounded sequence in a reflexive Banach space𝑋. If𝜔_𝑤({𝑥_𝑛}) = {𝑥}, then 𝑥_𝑛⇀ 𝑥.

Let𝐴 : 𝐶 → 𝐻be a nonlinear mapping. The classical variational inequality problem (VIP) is to find𝑥 ∈ 𝐶such that

⟨𝐴𝑥, 𝑦 − 𝑥⟩ ≥ 0, ∀𝑦 ∈ 𝐶. (15)

The solution set of VIP is denoted by VI(𝐶, 𝐴).

The theory of variational inequalities is a well-established subject in applied mathematics, nonlinear analysis, and optimization. For further details on variational inequalities, we refer to [8–13] and the references therein.

It is well known that the solution of a variational inequal- ity can be characterized be a fixed point of a projection map- ping. Therefore, by using Proposition4(i), we have the fol- lowing result.

Lemma 8. Let𝐴 : 𝐶 → 𝐻be a monotone mapping. Then, 𝑢 ∈VI(𝐶, 𝐴) ⇐⇒ 𝑢 = 𝑃_𝐶(𝑢 − 𝜆𝐴𝑢) , 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝜆 > 0.

(16) Lemma 9. The following assertions hold:

(a)‖𝑥 − 𝑦‖²= ‖𝑥‖²− ‖𝑦‖²− 2⟨𝑥 − 𝑦, 𝑦⟩for all𝑥, 𝑦 ∈ 𝐻;

(b) ‖𝜆𝑥 + 𝜇𝑦 +]𝑧‖² = 𝜆‖𝑥‖² + 𝜇‖𝑦‖² + ]‖𝑧‖² − 𝜆𝜇‖𝑥 − 𝑦‖²−𝜇]‖𝑦 − 𝑧‖²−𝜆‖𝑥 − 𝑧‖² for all𝑥, 𝑦, 𝑧 ∈ 𝐻 and𝜆, 𝜇,]∈ [0, 1]with𝜆 + 𝜇 +]= 1[14];

(c)if{𝑥_𝑛}is a sequence in𝐻such that𝑥_𝑛⇀ 𝑥, then lim sup

𝑛 → ∞ 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑦󵄩󵄩󵄩󵄩²=lim sup

𝑛 → ∞ 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑥󵄩󵄩󵄩󵄩²+ 󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩², ∀𝑦 ∈ 𝐻.

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Lemma 10 (see [4, Lemma 2.5]). For given points𝑥, 𝑦, 𝑧 ∈ 𝐻 and given also a real number𝑎 ∈R, the set

{V∈ 𝐶 : 󵄩󵄩󵄩󵄩𝑦 −V󵄩󵄩󵄩󵄩²≤ ‖𝑥 −V‖²+ ⟨𝑧,V⟩ + 𝑎} (18) is convex (and closed).

Lemma 11 (see [4, Lemma 2.6]). Let𝐶be a nonempty subset of a Hilbert space 𝐻and𝑆 : 𝐶 → 𝐶 an asymptotically 𝜅-strict pseudocontractive mapping in the intermediate sense with sequence{𝛾_𝑛}. Then,

󵄩󵄩󵄩󵄩𝑆^𝑛𝑥 − 𝑆^𝑛𝑦󵄩󵄩󵄩󵄩

≤ 1

1 − 𝜅(𝜅 󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩

+√(1 + (1 − 𝜅) 𝛾_𝑛) 󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩²+ (1 − 𝜅) 𝑐_𝑛) (19)

for all𝑥, 𝑦 ∈ 𝐶and𝑛 ≥ 1.

Lemma 12 (see [4, Lemma 2.7]). Let𝐶be a nonempty subset of a Hilbert space𝐻and𝑆 : 𝐶 → 𝐶a uniformly continuous asymptotically𝜅-strict pseudocontractive mapping in the inter- mediate sense with sequence{𝛾_𝑛}. Let{𝑥_𝑛}be a sequence in𝐶 such that‖𝑥_𝑛− 𝑥_𝑛+1‖ → 0and‖𝑥_𝑛− 𝑆^𝑛𝑥_𝑛‖ → 0as𝑛 → ∞.

Then,‖𝑥_𝑛− 𝑆𝑥_𝑛‖ → 0as𝑛 → ∞.

Lemma 13 (Demiclosedness Principle, [see [4, Proposition 3.1]]). Let𝐶be a nonempty closed convex subset of a Hilbert space𝐻and 𝑆 : 𝐶 → 𝐶be a continuous asymptotically 𝜅-strict pseudocontractive mapping in the intermediate sense with sequence{𝛾_𝑛}. Then𝐼−𝑆is demiclosed at zero in the sense that if{𝑥_𝑛}is a sequence in𝐶such that𝑥_𝑛 ⇀ 𝑥 ∈ 𝐶and lim sup_{𝑚 → ∞}lim sup_{𝑛 → ∞}‖𝑥_𝑛− 𝑆^𝑚𝑥_𝑛‖ = 0, then(𝐼 − 𝑆)𝑥 = 0.

Lemma 14 (see [4, Proposition 3.2]). Let 𝐶be a nonempty closed convex subset of a Hilbert space𝐻and𝑆 : 𝐶 → 𝐶a con- tinuous asymptotically𝜅-strict pseudocontractive mapping in the intermediate sense with sequence{𝛾_𝑛}such thatFix(𝑆) ̸= 0.

Then,Fix(𝑆)is closed and convex.

To prove a weak convergence theorem by an implicit relaxed method with regularization for the minimization problem (8) and the fixed point problem of an asymptoti- cally𝜅-strict pseudocontractive mapping in the intermediate sense, we need the following lemma due to Osilike et al. [15].

Lemma 15 (see [15, page 80]). Let{𝑎_𝑛}^∞_𝑛=1,{𝑏_𝑛}^∞_𝑛=1, and{𝛿_𝑛}^∞_𝑛=1 be sequences of nonnegative real numbers satisfying the ine- quality

𝑎_𝑛+1≤ (1 + 𝛿_𝑛) 𝑎_𝑛+ 𝑏_𝑛, ∀𝑛 ≥ 1. (20) If∑^∞_𝑛=1𝛿_𝑛 < ∞and∑^∞_𝑛=1𝑏_𝑛 < ∞, thenlim_{𝑛 → ∞}𝑎_𝑛exists. If, in addition,{𝑎_𝑛}^∞_𝑛=1has a subsequence which converges to zero, thenlim_{𝑛 → ∞}𝑎_𝑛= 0.

Corollary 16 (see [16, page 303]). Let {𝑎_𝑛}^∞_𝑛=0 and {𝑏_𝑛}^∞_𝑛=0 be two sequences of nonnegative real numbers satisfying the inequality

𝑎_𝑛+1≤ 𝑎_𝑛+ 𝑏_𝑛, ∀𝑛 ≥ 0. (21) If∑^∞_𝑛=0𝑏_𝑛converges, thenlim_{𝑛 → ∞}𝑎_𝑛exists.

Lemma 17 (see [17]). Every Hilbert space𝐻has the Kadec- Klee property; that is, given a sequence{𝑥_𝑛} ⊂ 𝐻and a point 𝑥 ∈ 𝐻, we have

󵄩󵄩󵄩󵄩𝑥^𝑛󵄩󵄩󵄩󵄩 󳨀→ ‖𝑥‖, 𝑥_𝑛⇀ 𝑥 󳨐⇒ 𝑥_𝑛 󳨀→ 𝑥. (22) It is well known that every Hilbert space𝐻satisfies Opial’s condition [18]; that is, for any sequence{𝑥_𝑛}with𝑥_𝑛⇀ 𝑥, we have

lim inf_{𝑛 → ∞} 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑥󵄩󵄩󵄩󵄩 <lim inf_{𝑛 → ∞} 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑦󵄩󵄩󵄩󵄩 ,

∀𝑦 ∈ 𝐻with𝑦 ̸= 𝑥. (23) A set-valued mapping𝑇 : 𝐻 → 2^𝐻is called monotone if for all𝑥, 𝑦 ∈ 𝐻, 𝑓 ∈ 𝑇𝑥and𝑔 ∈ 𝑇𝑦 imply⟨𝑥−𝑦, 𝑓−𝑔⟩≥ 0. A monotone mapping𝑇 : 𝐻 → 2^𝐻is maximal if its graph𝐺(𝑇) is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping𝑇is maximal if and only if for(𝑥, 𝑓) ∈ 𝐻 × 𝐻,⟨𝑥 − 𝑦, 𝑓 − 𝑔⟩≥ 0for all(𝑦, 𝑔) ∈ 𝐺(𝑇)implies𝑓 ∈ 𝑇𝑥. Let𝐴 : 𝐶 → 𝐻be a monotone and𝐿-Lipschitz continuous mapping, and let𝑁_𝐶V be the normal cone to𝐶atV ∈ 𝐶, that is,𝑁_𝐶V = {𝑤 ∈ 𝐻 :

⟨V− 𝑢, 𝑤⟩ ≥ 0, ∀ 𝑢 ∈ 𝐶}. Define

𝑇V= {𝐴V+ 𝑁_𝐶V, ifV∈ 𝐶,

0, ifV∉ 𝐶. (24)

It is known that in this case𝑇is maximal monotone, and0 ∈ 𝑇Vif and only ifV∈ Ω; see [19].

3. Weak Convergence Theorem

In this section, we will prove a weak convergence theo- rem for an implicit relaxed method with regularization for finding a common element of the set of fixed points of an asymptotically 𝜅-strict pseudocontractive mapping 𝑆 : 𝐶 → 𝐶in the intermediate sense and the set of solutions of the minimization problem (8) for a convex functional 𝑓 : 𝐶 → R with𝐿-Lipschitz continuous gradient∇𝑓. This implicit relaxed method with regularization is based on the extragradient method, viscosity approximation method, and gradient projection algorithm (GPA) with regularization.

Theorem 18. Let 𝐶 be a nonempty bounded closed convex subset of a real Hilbert space 𝐻. Let 𝑓 : 𝐶 → R be a convex functional with 𝐿-Lipschitz continuous gradient ∇𝑓, 𝑄 : 𝐶 → 𝐶a 𝜌-contraction with coefficient 𝜌 ∈ [0, 1), and𝑆 : 𝐶 → 𝐶 a uniformly continuous asymptotically𝜅- strict pseudocontractive mapping in the intermediate sense with sequence{𝛾_𝑛}such thatFix(𝑆) ∩ Γis nonempty. Let{𝛾_𝑛}

and{𝑐_𝑛}be defined as in Definition3. Let{𝑥_𝑛}and{𝑦_𝑛}be the sequences generated by

𝑥₁= 𝑥 ∈ 𝐶 𝑐ℎ𝑜𝑠𝑒𝑛 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦,

𝑦_𝑛 = 𝑃_𝐶(𝑥_𝑛− 𝜆_𝑛𝜇_𝑛∇𝑓_𝛼𝑛(𝑥_𝑛) − 𝜆_𝑛(1 − 𝜇_𝑛) ∇𝑓_𝛼𝑛(𝑦_𝑛)) , 𝑡_𝑛= 𝑃_𝐶(𝑥_𝑛− 𝜆_𝑛∇𝑓_𝛼𝑛(𝑦_𝑛) − 𝜆_𝑛(1 − 𝜇_𝑛) ∇𝑓_𝛼𝑛(𝑡_𝑛)) , 𝑥_𝑛+1= (1 − 𝜎_𝑛− 𝛽_𝑛) 𝑡_𝑛+ 𝜎_𝑛𝑄𝑡_𝑛+ 𝛽_𝑛𝑆^𝑛𝑡_𝑛, ∀𝑛 ≥ 1,

(25) where{𝛼_𝑛}is a sequence (0, ∞),{𝜆_𝑛},{𝜇_𝑛} are sequences in (0, 1], and {𝜎_𝑛}, {𝛽_𝑛} are sequences in [0, 1] satisfying the following conditions:

(i)𝛿 ≤ 𝛽_𝑛and𝜎_𝑛+ 𝛽_𝑛 ≤ 1 − 𝜅 − 𝜏, for all𝑛 ≥ 1for some 𝛿, 𝜏 ∈ (0, 1);

(ii)∑^∞_𝑛=1𝛼_𝑛 < ∞,∑^∞_𝑛=1𝛾_𝑛 < ∞,∑^∞_𝑛=1𝜎_𝑛 < ∞ and

∑^∞_𝑛=1𝑐_𝑛< ∞;

(iii) lim_{𝑛 → ∞}𝜇_𝑛= 1;

(iv)𝜆_𝑛(𝛼_𝑛+ 𝐿) < 1, for all𝑛 ≥ 1and{𝜆_𝑛} ⊂ [𝑎, 𝑏]for some 𝑎, 𝑏 ∈ (0, (1/𝐿)).

Then, the sequences{𝑥_𝑛},{𝑦_𝑛}converge weakly to some point 𝑢 ∈Fix(𝑆) ∩ Γ.

Remark 19. Observe that for all𝑥, 𝑦 ∈ 𝐶and all𝑛 ≥ 1

󵄩󵄩󵄩󵄩󵄩𝑃^𝐶(𝑥_𝑛− 𝜆_𝑛𝜇_𝑛∇𝑓_𝛼𝑛(𝑥_𝑛) − 𝜆_𝑛(1 − 𝜇_𝑛) ∇𝑓_𝛼𝑛(𝑥))

−𝑃_𝐶(𝑥_𝑛− 𝜆_𝑛𝜇_𝑛∇𝑓_𝛼𝑛(𝑥_𝑛) − 𝜆_𝑛(1 − 𝜇_𝑛) ∇𝑓_𝛼𝑛(𝑦))󵄩󵄩󵄩󵄩󵄩

≤ 󵄩󵄩󵄩󵄩󵄩(𝑥^𝑛− 𝜆_𝑛𝜇_𝑛∇𝑓_𝛼𝑛(𝑥_𝑛) − 𝜆_𝑛(1 − 𝜇_𝑛) ∇𝑓_𝛼𝑛(𝑥))

− (𝑥_𝑛− 𝜆_𝑛𝜇_𝑛∇𝑓_𝛼𝑛(𝑥_𝑛) − 𝜆_𝑛(1 − 𝜇_𝑛) ∇𝑓_𝛼𝑛(𝑦))󵄩󵄩󵄩󵄩󵄩

= 𝜆_𝑛(1 − 𝜇_𝑛) 󵄩󵄩󵄩󵄩󵄩∇𝑓^𝛼𝑛(𝑥) − ∇𝑓_𝛼𝑛(𝑦)󵄩󵄩󵄩󵄩󵄩

≤ 𝜆_𝑛(𝛼_𝑛+ 𝐿) 󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩.

(26) By Banach contraction principle, we know that for each𝑛 ≥ 1, there exists a unique𝑦_𝑛 ∈ 𝐶such that

𝑦_𝑛 = 𝑃_𝐶(𝑥_𝑛− 𝜆_𝑛𝜇_𝑛∇𝑓_𝛼𝑛(𝑥_𝑛) − 𝜆_𝑛(1 − 𝜇_𝑛) ∇𝑓_𝛼𝑛(𝑦_𝑛)) . (27) Also, observe that for all𝑥, 𝑦 ∈ 𝐶and all𝑛 ≥ 1

󵄩󵄩󵄩󵄩󵄩𝑃^𝐶(𝑥_𝑛− 𝜆_𝑛∇𝑓_𝛼𝑛(𝑦_𝑛) − 𝜆_𝑛(1 − 𝜇_𝑛) ∇𝑓_𝛼𝑛(𝑥))

−𝑃_𝐶(𝑥_𝑛− 𝜆_𝑛∇𝑓_𝛼𝑛(𝑦_𝑛) − 𝜆_𝑛(1 − 𝜇_𝑛) ∇𝑓_𝛼𝑛(𝑦))󵄩󵄩󵄩󵄩󵄩

≤ 󵄩󵄩󵄩󵄩󵄩(𝑥^𝑛− 𝜆_𝑛∇𝑓_𝛼𝑛(𝑦_𝑛) − 𝜆_𝑛(1 − 𝜇_𝑛) ∇𝑓_𝛼𝑛(𝑥))

− (𝑥_𝑛− 𝜆_𝑛∇𝑓_𝛼𝑛(𝑦_𝑛) − 𝜆_𝑛(1 − 𝜇_𝑛) ∇𝑓_𝛼𝑛(𝑦))󵄩󵄩󵄩󵄩󵄩

= 𝜆_𝑛(1 − 𝜇_𝑛) 󵄩󵄩󵄩󵄩󵄩∇𝑓^𝛼𝑛(𝑥) − ∇𝑓_𝛼𝑛(𝑦)󵄩󵄩󵄩󵄩󵄩

≤ 𝜆_𝑛(𝛼_𝑛+ 𝐿) 󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩.

(28)

Utilizing Banach contraction principle, for each𝑛 ≥ 1, there exists a unique𝑡_𝑛 ∈ 𝐶such that

𝑡_𝑛= 𝑃_𝐶(𝑥_𝑛− 𝜆_𝑛∇𝑓_𝛼𝑛(𝑦_𝑛) − 𝜆_𝑛(1 − 𝜇_𝑛) ∇𝑓_𝛼𝑛(𝑡_𝑛)) . (29) Proof of Theorem18. Note that the𝐿-Lipschitz continuity of the gradient∇𝑓implies that∇𝑓is(1/𝐿)-ism [20], that is,

⟨∇𝑓 (𝑥) − ∇𝑓 (𝑦) , 𝑥 − 𝑦⟩ ≥ 1

𝐿󵄩󵄩󵄩󵄩∇𝑓(𝑥) − ∇𝑓(𝑦)󵄩󵄩󵄩󵄩²,

∀𝑥, 𝑦 ∈ 𝐶.

(30) Thus,∇𝑓is monotone and𝐿-Lipschitz continuous.

We divide the rest of the proof into several steps.

Step 1. lim_{𝑛 → ∞}‖𝑥_𝑛− 𝑢‖exists for each𝑢 ∈Fix(𝑆) ∩ Γ.

Indeed, note that 𝑡_𝑛 = 𝑃_𝐶(𝑥_𝑛 − 𝜆_𝑛∇𝑓_𝛼𝑛(𝑦_𝑛) − 𝜆_𝑛(1 − 𝜇_𝑛)∇𝑓_𝛼𝑛(𝑡_𝑛)) for all𝑛 ≥ 1. Take 𝑢 ∈ Fix(𝑆) ∩ Γ arbitrarily.

From Proposition4(ii), monotonicity of∇𝑓, and𝑢 ∈ Γ, we have

󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²

≤ 󵄩󵄩󵄩󵄩󵄩(𝑥^𝑛− 𝜆_𝑛∇𝑓_𝛼𝑛(𝑦_𝑛) − 𝜆_𝑛(1 − 𝜇_𝑛) ∇𝑓_𝛼𝑛(𝑡_𝑛)) − 𝑢󵄩󵄩󵄩󵄩󵄩²

− 󵄩󵄩󵄩󵄩󵄩(𝑥^𝑛− 𝜆_𝑛∇𝑓_𝛼𝑛(𝑦_𝑛) − 𝜆_𝑛(1 − 𝜇_𝑛) ∇𝑓_𝛼𝑛(𝑡_𝑛)) − 𝑡_𝑛󵄩󵄩󵄩󵄩󵄩²

= 󵄩󵄩󵄩󵄩󵄩𝑥^𝑛− 𝜆_𝑛(1 − 𝜇_𝑛) ∇𝑓_𝛼𝑛(𝑡_𝑛) − 𝑢󵄩󵄩󵄩󵄩󵄩²

− 󵄩󵄩󵄩󵄩󵄩𝑥^𝑛− 𝜆_𝑛(1 − 𝜇_𝑛) ∇𝑓_𝛼𝑛(𝑡_𝑛) − 𝑡_𝑛󵄩󵄩󵄩󵄩󵄩² + 2𝜆_𝑛⟨∇𝑓_𝛼𝑛(𝑦_𝑛) , 𝑢 − 𝑡_𝑛⟩

= 󵄩󵄩󵄩󵄩󵄩𝑥^𝑛− 𝜆_𝑛(1 − 𝜇_𝑛) ∇𝑓_𝛼𝑛(𝑡_𝑛) − 𝑢󵄩󵄩󵄩󵄩󵄩²

− 󵄩󵄩󵄩󵄩󵄩𝑥^𝑛− 𝜆_𝑛(1 − 𝜇_𝑛) ∇𝑓_𝛼𝑛(𝑡_𝑛) − 𝑡_𝑛󵄩󵄩󵄩󵄩󵄩²

+ 2𝜆_𝑛(⟨∇𝑓_𝛼𝑛(𝑦_𝑛) , 𝑢 − 𝑦_𝑛⟩ + ⟨∇𝑓_𝛼𝑛(𝑦_𝑛) , 𝑦_𝑛− 𝑡_𝑛⟩)

= 󵄩󵄩󵄩󵄩󵄩𝑥^𝑛− 𝜆_𝑛(1 − 𝜇_𝑛) ∇𝑓_𝛼𝑛(𝑡_𝑛) − 𝑢󵄩󵄩󵄩󵄩󵄩²

− 󵄩󵄩󵄩󵄩󵄩𝑥^𝑛− 𝜆_𝑛(1 − 𝜇_𝑛) ∇𝑓_𝛼𝑛(𝑡_𝑛) − 𝑡_𝑛󵄩󵄩󵄩󵄩󵄩² + 2𝜆_𝑛(⟨∇𝑓_𝛼𝑛(𝑦_𝑛) − ∇𝑓_𝛼𝑛(𝑢) , 𝑢 − 𝑦_𝑛⟩

+ ⟨∇𝑓_𝛼𝑛(𝑢) , 𝑢 − 𝑦_𝑛⟩ + ⟨∇𝑓_𝛼𝑛(𝑦_𝑛) , 𝑦_𝑛− 𝑡_𝑛⟩)

≤ 󵄩󵄩󵄩󵄩󵄩𝑥^𝑛− 𝜆_𝑛(1 − 𝜇_𝑛) ∇𝑓_𝛼𝑛(𝑡_𝑛) − 𝑢󵄩󵄩󵄩󵄩󵄩²

− 󵄩󵄩󵄩󵄩󵄩𝑥^𝑛− 𝜆_𝑛(1 − 𝜇_𝑛) ∇𝑓_𝛼𝑛(𝑡_𝑛) − 𝑡_𝑛󵄩󵄩󵄩󵄩󵄩²

+ 2𝜆_𝑛(𝛼_𝑛⟨𝑢, 𝑢 − 𝑦_𝑛⟩ + ⟨∇𝑓_𝛼𝑛(𝑦_𝑛) , 𝑦_𝑛− 𝑡_𝑛⟩)

= 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩²− 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑡_𝑛󵄩󵄩󵄩󵄩²

− 2𝜆_𝑛(1 − 𝜇_𝑛) ⟨∇𝑓_𝛼𝑛(𝑡_𝑛) , 𝑡_𝑛− 𝑢⟩

+ 2𝜆_𝑛(𝛼_𝑛⟨𝑢, 𝑢 − 𝑦_𝑛⟩ + ⟨∇𝑓_𝛼𝑛(𝑦_𝑛) , 𝑦_𝑛− 𝑡_𝑛⟩)

= 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩²− 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩²

− 2 ⟨𝑥_𝑛− 𝑦_𝑛, 𝑦_𝑛− 𝑡_𝑛⟩ − 󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑡_𝑛󵄩󵄩󵄩󵄩²

+ 2𝜆_𝑛(𝛼_𝑛⟨𝑢, 𝑢 − 𝑦_𝑛⟩ + ⟨∇𝑓_𝛼𝑛(𝑦_𝑛) , 𝑦_𝑛− 𝑡_𝑛⟩)

− 2𝜆_𝑛(1 − 𝜇_𝑛)

× (⟨∇𝑓_𝛼𝑛(𝑡_𝑛) − ∇𝑓_𝛼𝑛(𝑢) , 𝑡_𝑛− 𝑢⟩ + ⟨∇𝑓_𝛼𝑛(𝑢) , 𝑡_𝑛− 𝑢⟩)

≤ 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩²− 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩²− 󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑡_𝑛󵄩󵄩󵄩󵄩² + 2 ⟨𝑥_𝑛− 𝜆_𝑛∇𝑓_𝛼𝑛(𝑦_𝑛) − 𝑦_𝑛, 𝑡_𝑛− 𝑦_𝑛⟩ + 2𝜆_𝑛𝛼_𝑛(⟨𝑢, 𝑢 − 𝑦_𝑛⟩ + (1 − 𝜇_𝑛) ⟨𝑢, 𝑢 − 𝑡_𝑛⟩)

≤ 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩²− 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩²− 󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑡_𝑛󵄩󵄩󵄩󵄩² + 2 ⟨𝑥_𝑛− 𝜆_𝑛∇𝑓_𝛼𝑛(𝑦_𝑛) − 𝑦_𝑛, 𝑡_𝑛− 𝑦_𝑛⟩ + 2𝜆_𝑛𝛼_𝑛‖𝑢‖ (󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑢󵄩󵄩󵄩󵄩 +󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩) .

(31) Since𝑦_𝑛 = 𝑃_𝐶(𝑥_𝑛− 𝜆_𝑛𝜇_𝑛∇𝑓_𝛼𝑛(𝑥_𝑛) − 𝜆_𝑛(1 − 𝜇_𝑛)∇𝑓_𝛼𝑛(𝑦_𝑛))and

∇𝑓_𝛼𝑛is(𝛼_𝑛+ 𝐿)-Lipschitz continuous, by Proposition4(i), we have

⟨𝑥_𝑛− 𝜆_𝑛∇𝑓_𝛼𝑛(𝑦_𝑛) − 𝑦_𝑛, 𝑡_𝑛− 𝑦_𝑛⟩

= ⟨𝑥_𝑛− 𝜆_𝑛𝜇_𝑛∇𝑓_𝛼𝑛(𝑥_𝑛) − 𝜆_𝑛(1 − 𝜇_𝑛) ∇𝑓_𝛼𝑛(𝑦_𝑛)

−𝑦_𝑛, 𝑡_𝑛− 𝑦_𝑛⟩

+ 𝜆_𝑛𝜇_𝑛⟨∇𝑓_𝛼𝑛(𝑥_𝑛) − ∇𝑓_𝛼𝑛(𝑦_𝑛) , 𝑡_𝑛− 𝑦_𝑛⟩

≤ 𝜆_𝑛𝜇_𝑛⟨∇𝑓_𝛼𝑛(𝑥_𝑛) − ∇𝑓_𝛼𝑛(𝑦_𝑛) , 𝑡_𝑛− 𝑦_𝑛⟩

≤ 𝜆_𝑛𝜇_𝑛󵄩󵄩󵄩󵄩󵄩∇𝑓^𝛼𝑛(𝑥_𝑛) − ∇𝑓_𝛼𝑛(𝑦_𝑛)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩

≤ 𝜆_𝑛(𝛼_𝑛+ 𝐿) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩.

(32) So, we have

󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²≤ 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩²− 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩²− 󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑡_𝑛󵄩󵄩󵄩󵄩² + 2𝜆_𝑛(𝛼_𝑛+ 𝐿) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩

+ 2𝜆_𝑛𝛼_𝑛‖𝑢‖ (󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑢󵄩󵄩󵄩󵄩 +󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩)

≤ 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩²− 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩²− 󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑡_𝑛󵄩󵄩󵄩󵄩² + 𝜆²_𝑛(𝛼_𝑛+ 𝐿)²󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩²+ 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩² + 2𝜆_𝑛𝛼_𝑛‖𝑢‖ (󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑢󵄩󵄩󵄩󵄩 +󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩)

= 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ (𝜆²_𝑛(𝛼_𝑛+ 𝐿)²− 1) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩² + 2𝜆_𝑛𝛼_𝑛‖𝑢‖ (󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑢󵄩󵄩󵄩󵄩 +󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩)

≤ 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 2𝜆_𝑛𝛼_𝑛‖𝑢‖ (󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑢󵄩󵄩󵄩󵄩 +󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩) . (33)

Therefore, from (33),𝑥_𝑛+1= (1−𝜎_𝑛−𝛽_𝑛)𝑡_𝑛+𝜎_𝑛𝑓(𝑡_𝑛)+𝛽_𝑛𝑆^𝑛𝑡_𝑛 and𝑢 = 𝑆𝑢. By Lemma9(b), we have

󵄩󵄩󵄩󵄩𝑥^𝑛+1− 𝑢󵄩󵄩󵄩󵄩²

= 󵄩󵄩󵄩󵄩(1 − 𝜎^𝑛− 𝛽_𝑛) (𝑡_𝑛− 𝑢) + 𝜎_𝑛(𝑄𝑡_𝑛− 𝑢) + 𝛽_𝑛(𝑆^𝑛𝑡_𝑛− 𝑢)󵄩󵄩󵄩󵄩²

≤ (1 − 𝜎_𝑛− 𝛽_𝑛) 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 𝜎_𝑛󵄩󵄩󵄩󵄩𝑄𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 𝛽_𝑛󵄩󵄩󵄩󵄩𝑆^𝑛𝑡_𝑛− 𝑢󵄩󵄩󵄩󵄩²

− (1 − 𝜎_𝑛− 𝛽_𝑛) 𝛽_𝑛󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑆^𝑛𝑡_𝑛󵄩󵄩󵄩󵄩²

≤ (1 − 𝜎_𝑛− 𝛽_𝑛) 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 𝜎_𝑛󵄩󵄩󵄩󵄩𝑄𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩² + 𝛽_𝑛[(1 + 𝛾_𝑛) 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 𝜅󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑆^𝑛𝑡_𝑛󵄩󵄩󵄩󵄩²+ 𝑐_𝑛]

− (1 − 𝜎_𝑛− 𝛽_𝑛) 𝛽_𝑛󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑆^𝑛𝑡_𝑛󵄩󵄩󵄩󵄩²

= [1 − 𝜎_𝑛− 𝛽_𝑛+ 𝛽_𝑛(1 + 𝛾_𝑛)] 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 𝜎_𝑛󵄩󵄩󵄩󵄩𝑄𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩² + 𝛽_𝑛(𝜅 − (1 − 𝜎_𝑛− 𝛽_𝑛)) 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑆^𝑛𝑡_𝑛󵄩󵄩󵄩󵄩²+ 𝛽_𝑛𝑐_𝑛

≤ (1 + 𝛾_𝑛) 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 𝜎_𝑛󵄩󵄩󵄩󵄩𝑄𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩² + 𝛽_𝑛(𝜅 − (1 − 𝜎_𝑛− 𝛽_𝑛)) 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑆^𝑛𝑡_𝑛󵄩󵄩󵄩󵄩²+ 𝑐_𝑛

≤ (1 + 𝛾_𝑛) 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 𝜎_𝑛󵄩󵄩󵄩󵄩𝑄𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 𝑐_𝑛

≤ (1 + 𝛾_𝑛) [󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 2𝜆_𝑛𝛼_𝑛‖𝑢‖ (󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑢󵄩󵄩󵄩󵄩 +󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩)]

+ 𝜎_𝑛󵄩󵄩󵄩󵄩𝑄𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 𝑐_𝑛

= (1 + 𝛾_𝑛) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 2𝛼_𝑛𝜆_𝑛(1 + 𝛾_𝑛) ‖𝑢‖

× (󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑢󵄩󵄩󵄩󵄩 +󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩) + 𝜎^𝑛󵄩󵄩󵄩󵄩𝑄𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 𝑐_𝑛.

(34) Since ∑^∞_𝑛=1𝛼_𝑛 < ∞, ∑^∞_𝑛=1𝛾_𝑛 < ∞, ∑^∞_𝑛=1𝜎_𝑛 < ∞, and

∑^∞_𝑛=1𝑐_𝑛< ∞, from the boundedness of𝐶, it follows that

∑∞

𝑛=1{2𝛼_𝑛𝜆_𝑛(1 + 𝛾_𝑛) ‖𝑢‖ (󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑢󵄩󵄩󵄩󵄩 +󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩) +𝜎_𝑛󵄩󵄩󵄩󵄩𝑄𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 𝑐_𝑛} < ∞.

(35)

By Lemma15, we have

𝑛 → ∞lim 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩exists for all𝑢 ∈Fix(𝑆) ∩ Γ. (36)

Step 2. lim_{𝑛 → ∞}‖𝑥_𝑛− 𝑦_𝑛‖ = 0and lim_{𝑛 → ∞}‖𝑥_𝑛− 𝑡_𝑛‖ = 0.

Indeed, substituting (33) in (34), we get

󵄩󵄩󵄩󵄩𝑥^𝑛+1− 𝑢󵄩󵄩󵄩󵄩²

≤ (1 + 𝛾_𝑛) 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 𝜎_𝑛󵄩󵄩󵄩󵄩𝑄𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 𝑐_𝑛

≤ (1 + 𝛾_𝑛) [󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ (𝜆²_𝑛(𝛼_𝑛+ 𝐿)²− 1) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩² +2𝜆_𝑛𝛼_𝑛‖𝑢‖ (󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑢󵄩󵄩󵄩󵄩 +󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩)]

+ 𝜎_𝑛󵄩󵄩󵄩󵄩𝑄𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 𝑐_𝑛

= 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 𝛾_𝑛󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩²

+ (1 + 𝛾_𝑛) (𝜆²_𝑛(𝛼_𝑛+ 𝐿)²− 1) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩² + 2𝛼_𝑛𝜆_𝑛(1 + 𝛾_𝑛) ‖𝑢‖ (󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑢󵄩󵄩󵄩󵄩 +󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩) + 𝜎_𝑛󵄩󵄩󵄩󵄩𝑄𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 𝑐_𝑛,

(37) which hence implies that

(1 + 𝛾_𝑛) (𝑏²(𝛼_𝑛+ 𝐿)²− 1) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩²

≤ (1 + 𝛾_𝑛) (𝜆²_𝑛(𝛼_𝑛+ 𝐿)²− 1) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩²

≤ 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩²− 󵄩󵄩󵄩󵄩𝑥^𝑛+1− 𝑢󵄩󵄩󵄩󵄩²+ 𝛾_𝑛󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩² + 2𝛼_𝑛𝜆_𝑛(1 + 𝛾_𝑛) ‖𝑢‖ (󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑢󵄩󵄩󵄩󵄩 +󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩) + 𝜎_𝑛󵄩󵄩󵄩󵄩𝑄𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 𝑐_𝑛.

(38)

Since lim_{𝑛 → ∞}‖𝑥_𝑛−𝑢‖exists,𝛼_𝑛 → 0,𝛾_𝑛 → 0,𝜎_𝑛 → 0, and 𝑐_𝑛 → 0, from the boundedness of𝐶, it follows that

𝑛 → ∞lim 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩 = 0. (39) Meantime, utilizing the arguments similar to those in (33), we have

󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²≤ 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩²− 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩²− 󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑡_𝑛󵄩󵄩󵄩󵄩² + 2𝜆_𝑛(𝛼_𝑛+ 𝐿) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩

+ 2𝜆_𝑛𝛼_𝑛‖𝑢‖ (󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑢󵄩󵄩󵄩󵄩 +󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩)

≤ 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩²− 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩²− 󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑡_𝑛󵄩󵄩󵄩󵄩² + 𝜆²_𝑛(𝛼_𝑛+ 𝐿)²󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩²+ 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩² + 2𝜆_𝑛𝛼_𝑛‖𝑢‖ (󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑢󵄩󵄩󵄩󵄩 +󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩)

= 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ (𝜆²_𝑛(𝛼_𝑛+ 𝐿)²− 1) 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩² + 2𝜆_𝑛𝛼_𝑛‖𝑢‖ (󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑢󵄩󵄩󵄩󵄩 +󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩) .

(40)

Substituting (40) in (34), we get

󵄩󵄩󵄩󵄩𝑥^𝑛+1− 𝑢󵄩󵄩󵄩󵄩²

≤ (1 + 𝛾_𝑛) 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 𝜎_𝑛󵄩󵄩󵄩󵄩𝑄𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 𝑐_𝑛

≤ (1 + 𝛾_𝑛) [󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ (𝜆²_𝑛(𝛼_𝑛+ 𝐿)²− 1) 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩² +2𝜆_𝑛𝛼_𝑛‖𝑢‖ (󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑢󵄩󵄩󵄩󵄩 +󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩)] + 𝜎^𝑛󵄩󵄩󵄩󵄩𝑄𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 𝑐_𝑛

= 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 𝛾_𝑛󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩²

+ (1 + 𝛾_𝑛) (𝜆²_𝑛(𝛼_𝑛+ 𝐿)²− 1) 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩² + 2𝛼_𝑛𝜆_𝑛(1 + 𝛾_𝑛) ‖𝑢‖ (󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑢󵄩󵄩󵄩󵄩 +󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩) + 𝜎_𝑛󵄩󵄩󵄩󵄩𝑄𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 𝑐_𝑛,

(41) which hence implies that

(1 + 𝛾_𝑛) (𝑏²(𝛼_𝑛+ 𝐿)²− 1) 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩²

≤ (1 + 𝛾_𝑛) (𝜆²_𝑛(𝛼_𝑛+ 𝐿)²− 1) 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩²

≤ 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩²− 󵄩󵄩󵄩󵄩𝑥^𝑛+1− 𝑢󵄩󵄩󵄩󵄩²+ 𝛾_𝑛󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩² + 2𝛼_𝑛𝜆_𝑛(1 + 𝛾_𝑛) ‖𝑢‖ (󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑢󵄩󵄩󵄩󵄩 +󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩) + 𝜎_𝑛󵄩󵄩󵄩󵄩𝑄𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 𝑐_𝑛.

(42)

Since lim_{𝑛 → ∞}‖𝑥_𝑛−𝑢‖exists,𝛼_𝑛 → 0,𝛾_𝑛 → 0,𝜎_𝑛 → 0, and 𝑐_𝑛 → 0, from the boundedness of𝐶it follows that

𝑛 → ∞lim 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑦_𝑛󵄩󵄩󵄩󵄩 = 0. (43) This together with‖𝑥_𝑛− 𝑦_𝑛‖ → 0implies that

𝑛 → ∞lim 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑡_𝑛󵄩󵄩󵄩󵄩 = 0. (44) Step 3. lim_{𝑛 → ∞}‖𝑥_𝑛+1− 𝑥_𝑛‖ = 0and lim_{𝑛 → ∞}‖𝑥_𝑛− 𝑆𝑥_𝑛‖ = 0.

Indeed, from (33) and (34), we conclude that

󵄩󵄩󵄩󵄩𝑥^𝑛+1− 𝑢󵄩󵄩󵄩󵄩²

≤ (1 + 𝛾_𝑛) 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 𝜎_𝑛󵄩󵄩󵄩󵄩𝑄𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩² + 𝛽_𝑛(𝜅 − (1 − 𝜎_𝑛− 𝛽_𝑛)) 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑆^𝑛𝑡_𝑛󵄩󵄩󵄩󵄩²+ 𝑐_𝑛

≤ (1 + 𝛾_𝑛) [󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 2𝜆_𝑛𝛼_𝑛‖𝑢‖ (󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑢󵄩󵄩󵄩󵄩 +󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩)]

+ 𝜎_𝑛󵄩󵄩󵄩󵄩𝑄𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²

+ 𝛽_𝑛(𝜅 − (1 − 𝜎_𝑛− 𝛽_𝑛)) 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑆^𝑛𝑡_𝑛󵄩󵄩󵄩󵄩²+ 𝑐_𝑛

= 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 𝛾_𝑛󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩²

+ 2𝛼_𝑛𝜆_𝑛(1 + 𝛾_𝑛) ‖𝑢‖ (󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑢󵄩󵄩󵄩󵄩 +󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩)

+ 𝜎_𝑛󵄩󵄩󵄩󵄩𝑄𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²− 𝛽_𝑛((1 − 𝜎_𝑛− 𝛽_𝑛) − 𝜅) 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑆^𝑛𝑡_𝑛󵄩󵄩󵄩󵄩² + 𝑐_𝑛,

(45)

which together with condition (i) yields 𝛿𝜏󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑆^𝑛𝑡_𝑛󵄩󵄩󵄩󵄩²

≤ 𝛽_𝑛((1 − 𝜎_𝑛− 𝛽_𝑛) − 𝜅) 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑆^𝑛𝑡_𝑛󵄩󵄩󵄩󵄩²

≤ 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩²− 󵄩󵄩󵄩󵄩𝑥^𝑛+1− 𝑢󵄩󵄩󵄩󵄩²+ 𝛾_𝑛󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑢󵄩󵄩󵄩󵄩² + 2𝛼_𝑛𝜆_𝑛(1 + 𝛾_𝑛) ‖𝑢‖ (󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑢󵄩󵄩󵄩󵄩 +󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩) + 𝜎_𝑛󵄩󵄩󵄩󵄩𝑄𝑡^𝑛− 𝑢󵄩󵄩󵄩󵄩²+ 𝑐_𝑛.

(46)

Since lim_{𝑛 → ∞}‖𝑥_𝑛−𝑢‖exists,𝛼_𝑛 → 0,𝛾_𝑛 → 0,𝜎_𝑛 → 0, and 𝑐_𝑛 → 0, from the boundedness of𝐶it follows that

𝑛 → ∞lim 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑆^𝑛𝑡_𝑛󵄩󵄩󵄩󵄩 = 0. (47) Note that

󵄩󵄩󵄩󵄩𝑥^𝑛+1− 𝑥_𝑛󵄩󵄩󵄩󵄩

≤ 󵄩󵄩󵄩󵄩𝑥^𝑛+1− 𝑡_𝑛󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩

= 󵄩󵄩󵄩󵄩𝜎^𝑛(𝑄𝑡_𝑛− 𝑡_𝑛) + 𝛽_𝑛(𝑆^𝑛𝑡_𝑛− 𝑡_𝑛)󵄩󵄩󵄩󵄩 +󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩

≤ 𝜎_𝑛󵄩󵄩󵄩󵄩𝑄𝑡^𝑛− 𝑡_𝑛󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑆^𝑛𝑡_𝑛− 𝑡_𝑛󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩.

(48)

From the boundedness of𝐶,𝜎_𝑛 → 0,‖𝑡_𝑛− 𝑥_𝑛‖ → 0, and

‖𝑆^𝑛𝑡_𝑛− 𝑡_𝑛‖ → 0, we deduce that

𝑛 → ∞lim 󵄩󵄩󵄩󵄩𝑥^𝑛+1− 𝑥_𝑛󵄩󵄩󵄩󵄩 = 0. (49) Furthermore, observe that

󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑆^𝑛𝑥_𝑛󵄩󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑡_𝑛󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑆^𝑛𝑡_𝑛󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑆^𝑛𝑡_𝑛− 𝑆^𝑛𝑥_𝑛󵄩󵄩󵄩󵄩.

(50) Utilizing Lemma11, we have

󵄩󵄩󵄩󵄩𝑆^𝑛𝑡_𝑛− 𝑆^𝑛𝑥_𝑛󵄩󵄩󵄩󵄩

≤ 1

1 − 𝜅

× (𝜅 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩

+√(1 + (1 − 𝜅) 𝛾𝑛) 󵄩󵄩󵄩󵄩𝑡^𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩²+ (1 − 𝜅) 𝑐𝑛) (51) for every𝑛 = 1, 2, . . . .Hence, it follows from‖𝑥_𝑛− 𝑡_𝑛‖ → 0 that‖𝑆^𝑛𝑡_𝑛−𝑆^𝑛𝑥_𝑛‖ → 0. Thus, from (50) and‖𝑡_𝑛−𝑆^𝑛𝑡_𝑛‖ → 0, we get‖𝑥_𝑛 − 𝑆^𝑛𝑥_𝑛‖ → 0. Since‖𝑥_𝑛+1− 𝑥_𝑛‖ → 0,‖𝑥_𝑛 − 𝑆^𝑛𝑥_𝑛‖ → 0as𝑛 → ∞, and𝑆is uniformly continuous, we obtain from Lemma12that‖𝑥_𝑛− 𝑆𝑥_𝑛‖ → 0as𝑛 → ∞.

Step 4.𝜔_𝑤(𝑥_𝑛) ⊂Fix(𝑆) ∩ Γ.

Indeed, from the boundedness of {𝑥_𝑛}, we know that 𝜔_𝑤({𝑥_𝑛}) ̸= 0. Takê𝑢 ∈ 𝜔_𝑤({𝑥_𝑛})arbitrarily. Then, there exists a subsequence{𝑥_𝑛𝑖}of{𝑥_𝑛}such that{𝑥_𝑛𝑖}converges weakly tô𝑢. Note that𝑆is uniformly continuous and‖𝑥_𝑛−𝑆𝑥_𝑛‖ → 0.