Implicit Relaxed and Hybrid Methods with Regularization for Minimization Problems and Asymptotically Strict Pseudocontractive Mappings in the Intermediate Sense

Abstract

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 mapping 𝑆 in 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 mapping 𝑆. 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 asymptotically nonexpansive mapping 𝑆∶𝐶→𝐶 defined on a nonempty closed convex bounded subset of a uniformly convex Banach space, that is, there exists a sequence {𝑘_𝑛} such that lim_𝑛→∞𝑘_𝑛=1 and 𝑆^𝑛𝑥−𝑆^𝑛𝑦≤𝑘_𝑛𝑥−𝑦,∀𝑛≥1,∀𝑥,𝑦∈𝐶 (1) has a fixed point in 𝐶. It can be easily seen that every nonexpansive mapping is asymptotically nonexpansive, and every asymptotically nonexpansive mapping is uniformly Lipschitzian; that is, there exists a constant ℒ>0 such that 𝑆^𝑛𝑥−𝑆^𝑛𝑦≤ℒ𝑥−𝑦,∀𝑛≥1,∀𝑥,𝑦∈𝐶. (2) Several researchers have weaken the assumption on the mapping 𝑆. Bruck et al. [2] introduced the following concept of an asymptotically nonexpansive mapping in the intermediate sense.

Definition 1. Let 𝐶 be a nonempty subset of a normed space 𝑋. A mapping 𝑆∶𝐶→𝐶 is said to be asymptotically nonexpansive in the intermediate sense provided 𝑆 is uniformly continuous and limsup_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 concept of asymptotically 𝜅-strict pseudocontractive mappings in setting of Hilbert spaces.

Definition 2. Let 𝐶 be a nonempty subset of a Hilbert space 𝐻. A mapping 𝑆∶𝐶→𝐶 is said to be an asymptotically 𝜅-strict pseudocontractive mapping with sequence {𝛾_𝑛} if there exists a constant 𝜅∈[0,1) and a sequence {𝛾_𝑛} in [0,∞) with lim_𝑛→∞𝛾_𝑛=0 such that 𝑆^𝑛𝑥−𝑆^𝑛𝑦²≤1+𝛾_𝑛𝑥−𝑦²+𝜅𝑥−𝑆^𝑛𝑥−𝑦−𝑆^𝑛𝑦²,∀𝑛≥1,∀𝑥,𝑦∈𝐶. (4)

They studied weak and strong convergence theorems for this class of mappings. It is important to note that every asymptotically 𝜅-strict pseudocontractive mapping with sequence {𝛾_𝑛} is a uniformly ℒ-Lipschitzian mapping with ℒ=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 mappings in the intermediate sense, which are not necessarily Lipschitzian.

Definition 3. Let 𝐶 be a nonempty subset of a Hilbert space 𝐻. A mapping 𝑆∶𝐶→𝐶 is said to be an asymptotically 𝜅-strict pseudocontractive mapping in the intermediate sense with sequence {𝛾_𝑛} if there exist a constant 𝜅∈[0,1) and a sequence {𝛾_𝑛} in [0,∞) with lim_𝑛→∞𝛾_𝑛=0 such that limsup_𝑛→∞sup_{𝑥,𝑦∈𝐶}𝑆^𝑛𝑥−𝑆^𝑛𝑦²−1+𝛾_𝑛𝑥−𝑦²−𝜅𝑥−𝑆^𝑛𝑥−𝑦−𝑆^𝑛𝑦²≤0. (5)

Let 𝑐_𝑛∶=max0,sup_{𝑥,𝑦∈𝐶}𝑆^𝑛𝑥−𝑆^𝑛𝑦²−1+𝛾_𝑛𝑥−𝑦²−𝜅𝑥−𝑆^𝑛𝑥−𝑦−𝑆^𝑛𝑦². (6) Then, 𝑐_𝑛≥0 (for all 𝑛≥1), 𝑐_𝑛→0 (𝑛→∞), and (5) reduces to the relation 𝑆^𝑛𝑥−𝑆^𝑛𝑦²≤1+𝛾_𝑛𝑥−𝑦²+𝜅𝑥−𝑆^𝑛𝑥−𝑦−𝑆^𝑛𝑦²+𝑐_𝑛,∀𝑛≥1,∀𝑥,𝑦∈𝐶. (7) Whenever 𝑐_𝑛=0 for all 𝑛≥1 in (7), then 𝑆 is an asymptotically 𝜅-strict pseudocontractive mapping with sequence {𝛾_𝑛}.

Let 𝑓∶𝐶→𝐑 be a convex and continuously Fréchet differentiable functional. We consider the following minimization problem: min_𝑥∈𝐶𝑓(𝑥). (8) We assume that the minimization problem (8) has a solution, and the solution set of this problem is denoted by Γ.

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 approximation 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 regularization for finding a common element of the fixed point set Fix(𝑆) of an asymptotically 𝜅-strict pseudocontractive mapping 𝑆 and the solution set Γ of 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 approximation 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 that 𝐻 is a real Hilbert space whose inner product and norm are denoted by ⟨⋅,⋅⟩ and ‖⋅‖, respectively, and 𝐶 is a nonempty closed convex subset of 𝐻. We write 𝑥_𝑛⇀𝑥 to indicate that the sequence {𝑥_𝑛} converges weakly to 𝑥 and 𝑥_𝑛→𝑥 to indicate that the sequence {𝑥_𝑛} converges strongly to 𝑥. Moreover, we use 𝜔_𝑤(𝑥_𝑛) to denote the weak 𝜔-limit set of the sequence {𝑥_𝑛}, that is, 𝜔_𝑤𝑥_𝑛∶=𝑥∈𝐻∶𝑥_𝑛𝑖𝑥⇀𝑥forsomesubsequence_𝑛𝑖of𝑥_𝑛. (9)

The metric (or nearest point) projection from 𝐻 onto 𝐶 is the mapping 𝑃_𝐶∶𝐻→𝐶 which assigns to each point 𝑥∈𝐻 the unique point 𝑃_𝐶𝑥∈𝐶 satisfying the following property: 𝑥−𝑃_𝐶𝑥=inf_𝑦∈𝐶𝑥−𝑦=∶𝑑(𝑥,𝐶). (10)

We mention some important properties of projections in the following proposition.

Proposition 4. For given 𝑥∈𝐻 and 𝑧∈𝐶, (i)𝑧=𝑃_𝐶𝑥⇔⟨𝑥−𝑧,𝑦−𝑧⟩≤0,  for all  𝑦∈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 all  𝑦∈𝐻.

Consequently, 𝑃_𝐶 is nonexpansive.

Definition 5. A mapping 𝑆∶𝐶→𝐻 is said to be(a)monotone if 𝑆𝑥−𝑆𝑦,𝑥−𝑦≥0,∀𝑥,𝑦∈𝐶; (11)(b)𝜂-strongly monotone if there exists a constant 𝜂>0 such that 𝑆𝑥−𝑆𝑦,𝑥−𝑦≥𝜂𝑥−𝑦²,∀𝑥,𝑦∈𝐶; (12)(c)𝛼-inverse-strongly monotone (𝛼-ism) if there exists a constant 𝛼>0 such that 𝑆𝑥−𝑆𝑦,𝑥−𝑦≥𝛼𝑆𝑥−𝑆𝑦²,∀𝑥,𝑦∈𝐶. (13)

Obviously, if 𝑆 is 𝛼-inverse-strongly monotone, then it is monotone and (1/𝛼)-Lipschitz continuous. It can be easily seen that if 𝑆∶𝐶→𝐶 is nonexpansive, then 𝐼−𝑆 is monotone. It is also easy to see that a projection mapping 𝑃_𝐶 is 1-ism. The inverse strongly monotone (also known as cocoercive) 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. Let 𝑋 be a real inner product space. Then, 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑥+𝑦,∀𝑥,𝑦∈𝑋. (14)

Lemma 7 (see [7, Proposition 2.4]). Let {𝑥_𝑛} be a bounded sequence in a reflexive Banach space 𝑋. If 𝜔_𝑤({𝑥_𝑛})={𝑥}, then 𝑥_𝑛⇀𝑥.

Let 𝐴∶𝐶→𝐻 be a nonlinear mapping. The classical variational inequality problem (VIP) is to find 𝑥∈𝐶 such 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 inequality can be characterized be a fixed point of a projection mapping. Therefore, by using Proposition 4(i), we have the following result.

Lemma 8. Let 𝐴∶𝐶→𝐻 be a monotone mapping. Then, 𝑢∈VI(𝐶,𝐴)⟺𝑢=𝑃_𝐶(𝑢−𝜆𝐴𝑢),𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝜆>0. (16)

Lemma 9. The following assertions hold: (a) ‖𝑥−𝑦‖²=‖𝑥‖²−‖𝑦‖²−2⟨𝑥−𝑦,𝑦⟩ for all 𝑥,𝑦∈𝐻;(b) ‖𝜆𝑥+𝜇𝑦+𝜈𝑧‖²=𝜆‖𝑥‖²+𝜇‖𝑦‖²+𝜈‖𝑧‖²−𝜆𝜇‖𝑥−𝑦‖²−𝜇𝜈‖𝑦−𝑧‖²−𝜆‖𝑥−𝑧‖² for all 𝑥,𝑦,𝑧∈𝐻 and 𝜆,𝜇,𝜈∈[0,1] with 𝜆+𝜇+𝜈=1 [14];(c) if {𝑥_𝑛} is a sequence in 𝐻 such that 𝑥_𝑛⇀𝑥, then limsup_𝑛→∞𝑥_𝑛−𝑦²=limsup_𝑛→∞𝑥_𝑛−𝑥²+𝑥−𝑦²,∀𝑦∈𝐻. (17)

Lemma 10 (see [4, Lemma 2.5]). For given points 𝑥,𝑦,𝑧∈𝐻 and given also a real number 𝑎∈𝐑, the set 𝑣∈𝐶∶𝑦−𝑣²≤‖𝑥−𝑣‖²+⟨𝑧,𝑣⟩+𝑎 (18) is convex (and closed).

Lemma 11 (see [4, Lemma 2.6]). Let 𝐶 be a nonempty subset of a Hilbert space 𝐻 and 𝑆∶𝐶→𝐶 an asymptotically 𝜅-strict pseudocontractive mapping in the intermediate sense with sequence {𝛾_𝑛}. Then, 𝑆^𝑛𝑥−𝑆^𝑛𝑦≤1𝜅+1−𝜅𝑥−𝑦1+(1−𝜅)𝛾_𝑛𝑥−𝑦²+(1−𝜅)𝑐_𝑛 (19) for all 𝑥,𝑦∈𝐶 and 𝑛≥1.

Lemma 12 (see [4, Lemma 2.7]). Let 𝐶 be a nonempty subset of a Hilbert space 𝐻 and 𝑆∶𝐶→𝐶 a uniformly continuous asymptotically 𝜅-strict pseudocontractive mapping in the intermediate sense with sequence {𝛾_𝑛}. Let {𝑥_𝑛} be a sequence in 𝐶 such that ‖𝑥_𝑛−𝑥_𝑛+1‖→0 and ‖𝑥_𝑛−𝑆^𝑛𝑥_𝑛‖→0 as 𝑛→∞. Then, ‖𝑥_𝑛−𝑆𝑥_𝑛‖→0 as 𝑛→∞.

Lemma 13 (Demiclosedness Principle, [see [4, Proposition 3.1]]). Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻 and 𝑆∶𝐶→𝐶 be a continuous asymptotically 𝜅-strict pseudocontractive mapping in the intermediate sense with sequence {𝛾_𝑛}. Then 𝐼−𝑆 is demiclosed at zero in the sense that if {𝑥_𝑛} is a sequence in 𝐶 such that 𝑥_𝑛⇀𝑥∈𝐶 and limsup_𝑚→∞limsup_𝑛→∞‖𝑥_𝑛−𝑆^𝑚𝑥_𝑛‖=0, then (𝐼−𝑆)𝑥=0.

Lemma 14 (see [4, Proposition 3.2]). Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻 and 𝑆∶𝐶→𝐶 a continuous asymptotically 𝜅-strict pseudocontractive mapping in the intermediate sense with sequence {𝛾_𝑛} such that Fix(𝑆)≠∅. 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 asymptotically 𝜅-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 inequality 𝑎_𝑛+1≤1+𝛿_𝑛𝑎_𝑛+𝑏_𝑛,∀𝑛≥1. (20) If ∑^∞_𝑛=1𝛿_𝑛<∞ and ∑^∞_𝑛=1𝑏_𝑛<∞, then lim_𝑛→∞𝑎_𝑛 exists. If, in addition, {𝑎_𝑛}^∞_𝑛=1 has a subsequence which converges to zero, then lim_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_𝑛 converges, then lim_𝑛→∞𝑎_𝑛 exists.

Lemma 17 (see [17]). Every Hilbert space 𝐻 has the Kadec-Klee property; that is, given a sequence {𝑥_𝑛}⊂𝐻 and a point 𝑥∈𝐻, we have 𝑥_𝑛⟶‖𝑥‖,𝑥_𝑛⇀𝑥⇒𝑥_𝑛⟶𝑥. (22)

It is well known that every Hilbert space 𝐻 satisfies Opial’s condition [18]; that is, for any sequence {𝑥_𝑛} with 𝑥_𝑛⇀𝑥, we have liminf_𝑛→∞𝑥_𝑛−𝑥<liminf_𝑛→∞𝑥_𝑛−𝑦,∀𝑦∈𝐻with𝑦≠𝑥. (23)

A set-valued mapping 𝑇∶𝐻→2^𝐻 is called monotone if for all 𝑥,𝑦∈𝐻,𝑓∈𝑇𝑥 and 𝑔∈𝑇𝑦 imply ⟨𝑥−𝑦,𝑓−𝑔⟩≥0. A monotone mapping 𝑇∶𝐻→2^𝐻 is maximal if its graph 𝐺(𝑇) is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping 𝑇 is maximal if and only if for (𝑥,𝑓)∈𝐻×𝐻, ⟨𝑥−𝑦,𝑓−𝑔⟩≥0 for all (𝑦,𝑔)∈𝐺(𝑇) implies 𝑓∈𝑇𝑥. Let 𝐴∶𝐶→𝐻 be a monotone and 𝐿-Lipschitz continuous mapping, and let 𝑁_𝐶𝑣 be the normal cone to 𝐶 at 𝑣∈𝐶, that is, 𝑁_𝐶𝑣={𝑤∈𝐻∶⟨𝑣−𝑢,𝑤⟩≥0,∀𝑢∈𝐶}. Define 𝑇𝑣=𝐴𝑣+𝑁_𝐶𝑣,if𝑣∈𝐶,∅,if𝑣∉𝐶. (24) It is known that in this case 𝑇 is maximal monotone, and 0∈𝑇𝑣 if and only if 𝑣∈Ω; see [19].

3. Weak Convergence Theorem

In this section, we will prove a weak convergence theorem 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→𝐶 in the intermediate sense and the set of solutions of the minimization problem (8) for a convex functional 𝑓∶𝐶→𝐑 with 𝐿-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→𝐶 a 𝜌-contraction with coefficient 𝜌∈[0,1), and 𝑆∶𝐶→𝐶 a uniformly continuous asymptotically 𝜅-strict pseudocontractive mapping in the intermediate sense with sequence {𝛾_𝑛} such that Fix(𝑆)∩Γ is nonempty. Let {𝛾_𝑛} and {𝑐_𝑛} be defined as in Definition 3. Let {𝑥_𝑛} and {𝑦_𝑛} be the sequences generated by 𝑥₁𝑦=𝑥∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦,_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛𝜇_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑦_𝑛,𝑡_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑡_𝑛,𝑥_𝑛+1=1−𝜎_𝑛−𝛽_𝑛𝑡_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_𝑛 and 𝜎_𝑛+𝛽_𝑛≤1−𝜅−𝜏, for all 𝑛≥1 for 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 all 𝑛≥1 and {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 all 𝑥,𝑦∈𝐶 and all 𝑛≥1𝑃_𝐶𝑥_𝑛−𝜆_𝑛𝜇_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛(𝑥)−𝑃_𝐶𝑥_𝑛−𝜆_𝑛𝜇_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑦≤𝑥_𝑛−𝜆_𝑛𝜇_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛(𝑥)−𝑥_𝑛−𝜆_𝑛𝜇_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑦=𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛(𝑥)−∇𝑓_𝛼𝑛𝑦≤𝜆_𝑛𝛼_𝑛+𝐿𝑥−𝑦. (26) By Banach contraction principle, we know that for each 𝑛≥1, there exists a unique 𝑦_𝑛∈𝐶 such that 𝑦_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛𝜇_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑦_𝑛. (27)
Also, observe that for all 𝑥,𝑦∈𝐶 and all 𝑛≥1𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛(𝑥)−𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑦≤𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛(𝑥)−𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑦=𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛(𝑥)−∇𝑓_𝛼𝑛𝑦≤𝜆_𝑛𝛼_𝑛+𝐿𝑥−𝑦. (28) Utilizing Banach contraction principle, for each 𝑛≥1, there exists a unique 𝑡_𝑛∈𝐶 such that 𝑡_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑡_𝑛. (29)

Proof of Theorem 18. Note that the 𝐿-Lipschitz continuity of the gradient ∇𝑓 implies that ∇𝑓 is (1/𝐿)-ism [20], that is, (𝑥)𝑦≥1∇𝑓−∇𝑓,𝑥−𝑦𝐿∇𝑓(𝑥)−∇𝑓𝑦²,∀𝑥,𝑦∈𝐶. (30) Thus, ∇𝑓 is monotone and 𝐿-Lipschitz continuous.
We divide the rest of the proof into several steps.
Step 1. lim_𝑛→∞‖𝑥_𝑛−𝑢‖ exists for each 𝑢∈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 all 𝑛≥1. Take 𝑢∈Fix(𝑆)∩Γ arbitrarily. From Proposition 4(ii), monotonicity of ∇𝑓, and 𝑢∈Γ, we have 𝑡_𝑛−𝑢²≤𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑡_𝑛−𝑢²−𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑡_𝑛−𝑡_𝑛²=𝑥_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑡_𝑛−𝑢²−𝑥_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑡_𝑛−𝑡_𝑛²+2𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛,𝑢−𝑡_𝑛=𝑥_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑡_𝑛−𝑢²−𝑥_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑡_𝑛−𝑡_𝑛²+2𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛,𝑢−𝑦_𝑛+∇𝑓_𝛼𝑛𝑦_𝑛,𝑦_𝑛−𝑡_𝑛=𝑥_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑡_𝑛−𝑢²−𝑥_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑡_𝑛−𝑡_𝑛²+2𝜆_𝑛⟨∇𝑓_𝛼𝑛𝑦_𝑛−∇𝑓_𝛼𝑛(𝑢),𝑢−𝑦_𝑛⟩+∇𝑓_𝛼𝑛(𝑢),𝑢−𝑦_𝑛+∇𝑓_𝛼𝑛𝑦_𝑛,𝑦_𝑛−𝑡_𝑛≤𝑥_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑡_𝑛−𝑢²−𝑥_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑡_𝑛−𝑡_𝑛²+2𝜆_𝑛𝛼_𝑛𝑢,𝑢−𝑦_𝑛+∇𝑓_𝛼𝑛𝑦_𝑛,𝑦_𝑛−𝑡_𝑛=𝑥_𝑛−𝑢²−𝑥_𝑛−𝑡_𝑛²−2𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑡_𝑛,𝑡_𝑛−𝑢+2𝜆_𝑛𝛼_𝑛𝑢,𝑢−𝑦_𝑛+∇𝑓_𝛼𝑛𝑦_𝑛,𝑦_𝑛−𝑡_𝑛=𝑥_𝑛−𝑢²−𝑥_𝑛−𝑦_𝑛²−2𝑥_𝑛−𝑦_𝑛,𝑦_𝑛−𝑡_𝑛−𝑦_𝑛−𝑡_𝑛²+2𝜆_𝑛𝛼_𝑛𝑢,𝑢−𝑦_𝑛+∇𝑓_𝛼𝑛𝑦_𝑛,𝑦_𝑛−𝑡_𝑛−2𝜆_𝑛1−𝜇_𝑛×∇𝑓_𝛼𝑛𝑡_𝑛−∇𝑓_𝛼𝑛(𝑢),𝑡_𝑛−𝑢+∇𝑓_𝛼𝑛(𝑢),𝑡_𝑛≤𝑥−𝑢_𝑛−𝑢²−𝑥_𝑛−𝑦_𝑛²−𝑦_𝑛−𝑡_𝑛²+2𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝑦_𝑛,𝑡_𝑛−𝑦_𝑛+2𝜆_𝑛𝛼_𝑛𝑢,𝑢−𝑦_𝑛+1−𝜇_𝑛𝑢,𝑢−𝑡_𝑛≤𝑥_𝑛−𝑢²−𝑥_𝑛−𝑦_𝑛²−𝑦_𝑛−𝑡_𝑛²+2𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝑦_𝑛,𝑡_𝑛−𝑦_𝑛+2𝜆_𝑛𝛼_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛−𝑢. (31) Since 𝑦_𝑛=𝑃_𝐶(𝑥_𝑛−𝜆_𝑛𝜇_𝑛∇𝑓_𝛼𝑛(𝑥_𝑛)−𝜆_𝑛(1−𝜇_𝑛)∇𝑓_𝛼𝑛(𝑦_𝑛)) and ∇𝑓_𝛼𝑛 is (𝛼_𝑛+𝐿)-Lipschitz continuous, by Proposition 4(i), we have 𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝑦_𝑛,𝑡_𝑛−𝑦_𝑛=𝑥_𝑛−𝜆_𝑛𝜇_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝑦_𝑛,𝑡_𝑛−𝑦_𝑛+𝜆_𝑛𝜇_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−∇𝑓_𝛼𝑛𝑦_𝑛,𝑡_𝑛−𝑦_𝑛≤𝜆_𝑛𝜇_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−∇𝑓_𝛼𝑛𝑦_𝑛,𝑡_𝑛−𝑦_𝑛≤𝜆_𝑛𝜇_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−∇𝑓_𝛼𝑛𝑦_𝑛𝑡_𝑛−𝑦_𝑛≤𝜆_𝑛𝛼_𝑛+𝐿𝑥_𝑛−𝑦_𝑛𝑡_𝑛−𝑦_𝑛. (32) So, we have 𝑡_𝑛−𝑢²≤𝑥_𝑛−𝑢²−𝑥_𝑛−𝑦_𝑛²−𝑦_𝑛−𝑡_𝑛²+2𝜆_𝑛𝛼_𝑛+𝐿𝑥_𝑛−𝑦_𝑛𝑡_𝑛−𝑦_𝑛+2𝜆_𝑛𝛼_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛≤𝑥−𝑢_𝑛−𝑢²−𝑥_𝑛−𝑦_𝑛²−𝑦_𝑛−𝑡_𝑛²+𝜆²_𝑛𝛼_𝑛+𝐿²𝑥_𝑛−𝑦_𝑛²+𝑡_𝑛−𝑦_𝑛²+2𝜆_𝑛𝛼_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛=𝑥−𝑢_𝑛−𝑢²+𝜆²_𝑛𝛼_𝑛+𝐿²−1𝑥_𝑛−𝑦_𝑛²+2𝜆_𝑛𝛼_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛≤𝑥−𝑢_𝑛−𝑢²+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 and 𝑢=𝑆𝑢. By Lemma 9(b), we have 𝑥_𝑛+1−𝑢²=1−𝜎_𝑛−𝛽_𝑛𝑡_𝑛−𝑢+𝜎_𝑛𝑄𝑡_𝑛−𝑢+𝛽_𝑛𝑆^𝑛𝑡_𝑛−𝑢²≤1−𝜎_𝑛−𝛽_𝑛𝑡_𝑛−𝑢²+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝛽_𝑛𝑆^𝑛𝑡_𝑛−𝑢²−1−𝜎_𝑛−𝛽_𝑛𝛽_𝑛𝑡_𝑛−𝑆^𝑛𝑡_𝑛²≤1−𝜎_𝑛−𝛽_𝑛𝑡_𝑛−𝑢²+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝛽_𝑛1+𝛾_𝑛𝑡_𝑛−𝑢²+𝜅𝑡_𝑛−𝑆^𝑛𝑡_𝑛²+𝑐_𝑛−1−𝜎_𝑛−𝛽_𝑛𝛽_𝑛𝑡_𝑛−𝑆^𝑛𝑡_𝑛²=1−𝜎_𝑛−𝛽_𝑛+𝛽_𝑛1+𝛾_𝑛𝑡_𝑛−𝑢²+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝛽_𝑛𝜅−1−𝜎_𝑛−𝛽_𝑛𝑡_𝑛−𝑆^𝑛𝑡_𝑛²+𝛽_𝑛𝑐_𝑛≤1+𝛾_𝑛𝑡_𝑛−𝑢²+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝛽_𝑛𝜅−1−𝜎_𝑛−𝛽_𝑛𝑡_𝑛−𝑆^𝑛𝑡_𝑛²+𝑐_𝑛≤1+𝛾_𝑛𝑡_𝑛−𝑢²+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝑐_𝑛≤1+𝛾_𝑛𝑥_𝑛−𝑢²+2𝜆_𝑛𝛼_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛−𝑢+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝑐_𝑛=1+𝛾_𝑛𝑥_𝑛−𝑢²+2𝛼_𝑛𝜆_𝑛1+𝛾_𝑛‖𝑢‖×𝑦_𝑛−𝑢+𝑡_𝑛−𝑢+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝑐_𝑛. (34) Since ∑^∞_𝑛=1𝛼_𝑛<∞, ∑^∞_𝑛=1𝛾_𝑛<∞, ∑^∞_𝑛=1𝜎_𝑛<∞, and ∑^∞_𝑛=1𝑐_𝑛<∞, from the boundedness of 𝐶, it follows that ^∞_𝑛=12𝛼_𝑛𝜆_𝑛1+𝛾_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛−𝑢+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝑐_𝑛<∞. (35) By Lemma 15, we have lim_𝑛→∞𝑥_𝑛−𝑢existsforall𝑢∈Fix(𝑆)∩Γ. (36)
Step 2. lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0 and lim_𝑛→∞‖𝑥_𝑛−𝑡_𝑛‖=0.
Indeed, substituting (33) in (34), we get 𝑥_𝑛+1−𝑢²≤1+𝛾_𝑛𝑡_𝑛−𝑢²+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝑐_𝑛≤1+𝛾_𝑛𝑥_𝑛−𝑢²+𝜆²_𝑛𝛼_𝑛+𝐿²−1𝑥_𝑛−𝑦_𝑛²+2𝜆_𝑛𝛼_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛−𝑢+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝑐_𝑛=𝑥_𝑛−𝑢²+𝛾_𝑛𝑥_𝑛−𝑢²+1+𝛾_𝑛𝜆²_𝑛𝛼_𝑛+𝐿²−1𝑥_𝑛−𝑦_𝑛²+2𝛼_𝑛𝜆_𝑛1+𝛾_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛−𝑢+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝑐_𝑛, (37) which hence implies that 1+𝛾_𝑛𝑏²𝛼_𝑛+𝐿²−1𝑥_𝑛−𝑦_𝑛²≤1+𝛾_𝑛𝜆²_𝑛𝛼_𝑛+𝐿²−1𝑥_𝑛−𝑦_𝑛²≤𝑥_𝑛−𝑢²−𝑥_𝑛+1−𝑢²+𝛾_𝑛𝑥_𝑛−𝑢²+2𝛼_𝑛𝜆_𝑛1+𝛾_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛−𝑢+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+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 of 𝐶, it follows that lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0. (39) Meantime, utilizing the arguments similar to those in (33), we have 𝑡_𝑛−𝑢²≤𝑥_𝑛−𝑢²−𝑥_𝑛−𝑦_𝑛²−𝑦_𝑛−𝑡_𝑛²+2𝜆_𝑛𝛼_𝑛+𝐿𝑥_𝑛−𝑦_𝑛𝑡_𝑛−𝑦_𝑛+2𝜆_𝑛𝛼_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛≤𝑥−𝑢_𝑛−𝑢²−𝑥_𝑛−𝑦_𝑛²−𝑦_𝑛−𝑡_𝑛²+𝜆²_𝑛𝛼_𝑛+𝐿²𝑡_𝑛−𝑦_𝑛²+𝑥_𝑛−𝑦_𝑛²+2𝜆_𝑛𝛼_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛=𝑥−𝑢_𝑛−𝑢²+𝜆²_𝑛𝛼_𝑛+𝐿²−1𝑡_𝑛−𝑦_𝑛²+2𝜆_𝑛𝛼_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛−𝑢. (40) Substituting (40) in (34), we get 𝑥_𝑛+1−𝑢²≤1+𝛾_𝑛𝑡_𝑛−𝑢²+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝑐_𝑛≤1+𝛾_𝑛𝑥_𝑛−𝑢²+𝜆²_𝑛𝛼_𝑛+𝐿²−1𝑡_𝑛−𝑦_𝑛²+2𝜆_𝑛𝛼_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛−𝑢+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝑐_𝑛=𝑥_𝑛−𝑢²+𝛾_𝑛𝑥_𝑛−𝑢²+1+𝛾_𝑛𝜆²_𝑛𝛼_𝑛+𝐿²−1𝑡_𝑛−𝑦_𝑛²+2𝛼_𝑛𝜆_𝑛1+𝛾_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛−𝑢+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝑐_𝑛, (41) which hence implies that 1+𝛾_𝑛𝑏²𝛼_𝑛+𝐿²−1𝑡_𝑛−𝑦_𝑛²≤1+𝛾_𝑛𝜆²_𝑛𝛼_𝑛+𝐿²−1𝑡_𝑛−𝑦_𝑛²≤𝑥_𝑛−𝑢²−𝑥_𝑛+1−𝑢²+𝛾_𝑛𝑥_𝑛−𝑢²+2𝛼_𝑛𝜆_𝑛1+𝛾_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛−𝑢+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+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 of 𝐶 it follows that lim_𝑛→∞𝑡_𝑛−𝑦_𝑛=0. (43) This together with ‖𝑥_𝑛−𝑦_𝑛‖→0 implies that lim_𝑛→∞𝑥_𝑛−𝑡_𝑛=0. (44)
Step 3.  lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0 and lim_𝑛→∞‖𝑥_𝑛−𝑆𝑥_𝑛‖=0.
Indeed, from (33) and (34), we conclude that 𝑥_𝑛+1−𝑢²≤1+𝛾_𝑛𝑡_𝑛−𝑢²+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝛽_𝑛𝜅−1−𝜎_𝑛−𝛽_𝑛𝑡_𝑛−𝑆^𝑛𝑡_𝑛²+𝑐_𝑛≤1+𝛾_𝑛𝑥_𝑛−𝑢²+2𝜆_𝑛𝛼_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛−𝑢+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝛽_𝑛𝜅−1−𝜎_𝑛−𝛽_𝑛𝑡_𝑛−𝑆^𝑛𝑡_𝑛²+𝑐_𝑛=𝑥_𝑛−𝑢²+𝛾_𝑛𝑥_𝑛−𝑢²+2𝛼_𝑛𝜆_𝑛1+𝛾_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛−𝑢+𝜎_𝑛𝑄𝑡_𝑛−𝑢²−𝛽_𝑛1−𝜎_𝑛−𝛽_𝑛−𝜅𝑡_𝑛−𝑆^𝑛𝑡_𝑛²+𝑐_𝑛, (45) which together with condition (i) yields 𝛿𝜏𝑡_𝑛−𝑆^𝑛𝑡_𝑛²≤𝛽_𝑛1−𝜎_𝑛−𝛽_𝑛−𝜅𝑡_𝑛−𝑆^𝑛𝑡_𝑛²≤𝑥_𝑛−𝑢²−𝑥_𝑛+1−𝑢²+𝛾_𝑛𝑥_𝑛−𝑢²+2𝛼_𝑛𝜆_𝑛1+𝛾_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛−𝑢+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+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 of 𝐶 it follows that lim_𝑛→∞𝑡_𝑛−𝑆^𝑛𝑡_𝑛=0. (47) Note that 𝑥_𝑛+1−𝑥_𝑛≤𝑥_𝑛+1−𝑡_𝑛+𝑡_𝑛−𝑥_𝑛=𝜎_𝑛𝑄𝑡_𝑛−𝑡_𝑛+𝛽_𝑛𝑆^𝑛𝑡_𝑛−𝑡_𝑛+𝑡_𝑛−𝑥_𝑛≤𝜎_𝑛𝑄𝑡_𝑛−𝑡_𝑛+𝑆^𝑛𝑡_𝑛−𝑡_𝑛+𝑡_𝑛−𝑥_𝑛. (48) From the boundedness of 𝐶, 𝜎_𝑛→0, ‖𝑡_𝑛−𝑥_𝑛‖→0, and ‖𝑆^𝑛𝑡_𝑛−𝑡_𝑛‖→0, we deduce that lim_𝑛→∞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 Lemma 11, we have 𝑆^𝑛𝑡_𝑛−𝑆^𝑛𝑥_𝑛≤1×𝜅𝑡1−𝜅_𝑛−𝑥_𝑛+1+(1−𝜅)𝛾_𝑛𝑡_𝑛−𝑥_𝑛²+(1−𝜅)𝑐_𝑛 (51) for every 𝑛=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‖→0 as 𝑛→∞, and 𝑆 is uniformly continuous, we obtain from Lemma 12 that ‖𝑥_𝑛−𝑆𝑥_𝑛‖→0 as 𝑛→∞.
Step 4. 𝜔_𝑤(𝑥_𝑛)⊂Fix(𝑆)∩Γ.
Indeed, from the boundedness of {𝑥_𝑛}, we know that 𝜔_𝑤({𝑥_𝑛})≠∅. Take 𝑢∈𝜔_𝑤({𝑥_𝑛}) arbitrarily. Then, there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that {𝑥_𝑛𝑖} converges weakly to 𝑢. Note that 𝑆 is uniformly continuous and ‖𝑥_𝑛−𝑆𝑥_𝑛‖→0. Hence, it is easy to see that ‖𝑥_𝑛−𝑆^𝑚𝑥_𝑛‖→0 for all 𝑚≥1. By Lemma 13, we obtain 𝑢∈Fix(𝑆). Furthermore, we show 𝑢∈Γ. Since 𝑥_𝑛−𝑡_𝑛→0 and 𝑥_𝑛−𝑦_𝑛→0, we have 𝑡_𝑛𝑖⇀𝑢 and 𝑦_𝑛𝑖⇀𝑢. Let (𝑣)𝑇𝑣=∇𝑓+𝑁_𝐶𝑣,if𝑣∈𝐶,∅,if𝑣∉𝐶, (52) where 𝑁_𝐶𝑣 is the normal cone to 𝐶 at 𝑣∈𝐶. We have already mentioned that in this case the mapping 𝑇 is maximal monotone, and 0∈𝑇𝑣 if and only if 𝑣∈VI(𝐶,∇𝑓); see [19] for more details. Let 𝐺(𝑇) be the graph of 𝑇 and let (𝑣,𝑤)∈𝐺(𝑇). Then, we have 𝑤∈𝑇𝑣=∇𝑓(𝑣)+𝑁_𝐶𝑣 and hence 𝑤−∇𝑓(𝑣)∈𝑁_𝐶𝑣. So, we have ⟨𝑣−𝑡,𝑤−∇𝑓(𝑣)⟩≥0 for all 𝑡∈𝐶. On the other hand, from 𝑡_𝑛=𝑃_𝐶(𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛(𝑦_𝑛)−𝜆_𝑛(1−𝜇_𝑛)∇𝑓_𝛼𝑛(𝑡_𝑛)) and 𝑣∈𝐶, we have 𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑡_𝑛−𝑡_𝑛,𝑡_𝑛−𝑣≥0, (53) and hence 𝑣−𝑡_𝑛,𝑡_𝑛−𝑥_𝑛𝜆_𝑛+∇𝑓_𝛼𝑛𝑦_𝑛+1−𝜇_𝑛∇𝑓_𝛼𝑛𝑡_𝑛≥0. (54) Therefore, from ⟨𝑣−𝑡,𝑤−∇𝑓(𝑣)⟩≥0 for all 𝑡∈𝐶 and 𝑡_𝑛𝑖∈𝐶, we have 𝑣−𝑡_𝑛𝑖≥,𝑤𝑣−𝑡_𝑛𝑖(𝑣≥,∇𝑓)𝑣−𝑡_𝑛𝑖(𝑣−,∇𝑓)𝑣−𝑡_𝑛𝑖,𝑡_𝑛𝑖−𝑥_𝑛𝑖𝜆_𝑛𝑖+∇𝑓_𝛼𝑛𝑖𝑦_𝑛𝑖+1−𝜇_𝑛𝑖∇𝑓_𝛼𝑛𝑖𝑡_𝑛𝑖=𝑣−𝑡_𝑛𝑖,∇𝑓(𝑣)−𝑣−𝑡_𝑛𝑖,𝑡_𝑛𝑖−𝑥_𝑛𝑖𝜆_𝑛𝑖+∇𝑓𝑦_𝑛𝑖−𝛼_𝑛𝑖𝑣−𝑡_𝑛𝑖,𝑦_𝑛𝑖−1−𝜇_𝑛𝑖𝑣−𝑡_𝑛𝑖,∇𝑓_𝛼𝑛𝑖𝑡_𝑛𝑖=𝑣−𝑡_𝑛𝑖,∇𝑓(𝑣)−∇𝑓𝑡_𝑛𝑖+𝑣−𝑡_𝑛𝑖,∇𝑓𝑡_𝑛𝑖−∇𝑓𝑦_𝑛𝑖−𝑣−𝑡_𝑛𝑖,𝑡_𝑛𝑖−𝑥_𝑛𝑖𝜆_𝑛𝑖−𝛼_𝑛𝑖𝑣−𝑡_𝑛𝑖,𝑦_𝑛𝑖−1−𝜇_𝑛𝑖𝑣−𝑡_𝑛𝑖,∇𝑓_𝛼𝑛𝑖𝑡_𝑛𝑖≥𝑣−𝑡_𝑛𝑖,∇𝑓𝑡_𝑛𝑖−∇𝑓𝑦_𝑛𝑖−𝑣−𝑡_𝑛𝑖,𝑡_𝑛𝑖−𝑥_𝑛𝑖𝜆_𝑛𝑖−𝛼_𝑛𝑖𝑣−𝑡_𝑛𝑖,𝑦_𝑛𝑖−1−𝜇_𝑛𝑖𝑣−𝑡_𝑛𝑖,∇𝑓_𝛼𝑛𝑖𝑡_𝑛𝑖. (55) Since ‖∇𝑓(𝑡_𝑛)−∇𝑓(𝑦_𝑛)‖→0 (due to the Lipschitz continuity of ∇𝑓), (𝑡_𝑛−𝑥_𝑛)/(𝜆_𝑛)→0 (due to {𝜆_𝑛}⊂[𝑎,𝑏]), 𝛼_𝑛→0, and 𝜇_𝑛→1, we obtain ⟨𝑣−𝑢,𝑤⟩≥0 as 𝑖→∞. Since 𝑇 is maximal monotone, we have 𝑢∈𝑇⁻¹0 and hence 𝑢∈VI(𝐶,∇𝑓). Clearly, 𝑢∈Γ. Consequently, 𝑢∈Fix(𝑆)∩Γ. This implies that 𝜔_𝑤({𝑥_𝑛})⊂Fix(𝑆)∩Γ.
Step 5. {𝑥_𝑛} and {𝑦_𝑛} converge weakly to the same point 𝑢∈Fix(𝑆)∩Γ.
Indeed, it is sufficient to show that 𝜔_𝑤(𝑥_𝑛) is a single-point set because 𝑥_𝑛−𝑦_𝑛→0 as 𝑛→∞. Since 𝜔_𝑤(𝑥_𝑛)≠∅, let us take two points 𝑢,𝑢∈𝜔_𝑤(𝑥_𝑛) arbitrarily. Then, there exist two subsequences {𝑥_𝑛𝑗} and {𝑥_𝑚𝑘} of {𝑥_𝑛} such that 𝑥_𝑛𝑗⇀𝑢 and 𝑥_𝑚𝑘⇀𝑢, respectively. In terms of Step 4, we know that 𝑢,𝑢∈Fix(𝑆)∩Γ. Meantime, according to Step 1, we also know that there exist both lim_𝑛→∞‖𝑥_𝑛−𝑢‖ and lim_𝑛→∞‖𝑥_𝑛−𝑢‖. Let us show that 𝑢=𝑢. Assume that 𝑢≠𝑢. From the Opial condition [18], it follows that lim_𝑛→∞𝑥_𝑛−𝑢=liminf_𝑗→∞𝑥_𝑛𝑗−𝑢<liminf_𝑗→∞𝑥_𝑛𝑗−𝑢=lim_𝑛→∞𝑥_𝑛−𝑢=liminf_𝑘→∞𝑥_𝑚𝑘−𝑢<liminf_𝑘→∞𝑥_𝑚𝑘−𝑢=lim_𝑛→∞𝑥_𝑛−𝑢. (56) This leads to a contradiction. Thus, we must have 𝑢=𝑢. This implies that 𝜔_𝑤(𝑥_𝑛) is a single-point set. Without loss of generality, we may write 𝜔_𝑤(𝑥_𝑛)={𝑢}. Consequently, by Lemma 7, we obtain that 𝑥_𝑛⇀𝑢∈Fix(𝑆)∩Γ. Since 𝑥_𝑛−𝑦_𝑛→0 as 𝑛→∞, we also have 𝑦_𝑛⇀𝑢∈Fix(𝑆)∩Γ. (57) This completes the proof.

4. Strong Convergence Theorem

In this section, we establish a strong convergence theorem for an implicit hybrid method with regularization for finding a common element of the set of fixed points of an asymptotically 𝜅-strict pseudocontractive mapping 𝑆∶𝐶→𝐶 in the intermediate sense and the set of solutions of the MP (8) for a convex functional 𝑓∶𝐶→𝐑 with 𝐿-Lipschitz continuous gradient ∇𝑓. This implicit hybrid method with regularization is based on the CQ method, extragradient method, viscosity approximation method, and gradient projection algorithm (GPA) with regularization.

Theorem 20. Let 𝐶 be a nonempty 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→𝐶 a 𝜌-contraction with coefficient 𝜌∈[0,1), and 𝑆∶𝐶→𝐶 a uniformly continuous asymptotically 𝜅-strict pseudocontractive mapping in the intermediate sense with sequence {𝛾_𝑛} such that Fix(𝑆)∩Γ is nonempty and bounded. Let {𝛾_𝑛} and {𝑐_𝑛} be defined as in Definition 3. Let {𝑥_𝑛},{𝑦_𝑛}, and {𝑧_𝑛} be the sequences generated by 𝑥₁𝑦=𝑥∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦,_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛𝜇_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑦_𝑛,𝑡_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑡_𝑛,𝑧_𝑛=1−𝜎_𝑛−𝛽_𝑛𝑡_𝑛+𝜎_𝑛𝑄𝑡_𝑛+𝛽_𝑛𝑆^𝑛𝑡_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑧_𝑛−𝑧²≤𝑥_𝑛−𝑧²+𝜃_𝑛,𝑄_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥,∀𝑛≥1, (58) where {𝛼_𝑛}⊂(0,∞), {𝜆_𝑛}, {𝜇_𝑛}⊂(0,1], {𝜎_𝑛}, {𝛽_𝑛}⊂[0,1], 𝜃_𝑛=𝑐_𝑛+(𝛼_𝑛+𝛾_𝑛+𝜎_𝑛)Δ_𝑛 and Δ_𝑛=sup_{𝑧∈Fix(𝑆)∩Γ}33𝜇⁴_𝑛𝑥_𝑛−𝑧²+1−𝜌⁻¹+𝛾_𝑛²+32𝜇⁴_𝑛𝛼²_𝑛×‖𝑧‖²+‖𝑄𝑧−𝑧‖²<∞. (59)
Assume that the following conditions hold: (i)𝛿≤𝛽_𝑛 and 𝜎_𝑛+𝛽_𝑛≤1−𝜅, for all 𝑛≥1 for some 𝛿∈(0,1);(ii)lim_𝑛→∞𝛼_𝑛=0 and lim_𝑛→∞𝜎_𝑛=0;(iii)lim_𝑛→∞𝜇_𝑛=1;(iv)𝜆_𝑛(𝛼_𝑛+𝐿)<1, for all 𝑛≥1 and {𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,(1/𝐿)).
Then, the sequences {𝑥_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛} converge strongly to the same point 𝑃_{Fix(𝑆)∩Γ}𝑥.

Proof. Utilizing the condition 𝜆_𝑛(𝛼_𝑛+𝐿)<1 for all 𝑛≥1 and repeating the same arguments as in Remark 19, we can see that {𝑦_𝑛} and {𝑧_𝑛} are defined well. Note that the 𝐿-Lipschitz continuity of the gradient ∇𝑓 implies that ∇𝑓 is (1/𝐿)-ism [20], that is, (𝑥)𝑦≥1∇𝑓−∇𝑓,𝑥−𝑦𝐿∇𝑓(𝑥)−∇𝑓𝑦²,∀𝑥,𝑦∈𝐶. (60) Observe that (𝛼+𝐿)∇𝑓_𝛼(𝑥)−∇𝑓_𝛼𝑦=(𝛼,𝑥−𝑦𝛼+𝐿)𝑥−𝑦²+∇𝑓(𝑥)−∇𝑓𝑦,𝑥−𝑦=𝛼²𝑥−𝑦²+𝛼∇𝑓(𝑥)−∇𝑓𝑦,𝑥−𝑦+𝛼𝐿𝑥−𝑦²+𝐿∇𝑓(𝑥)−∇𝑓𝑦,𝑥−𝑦≥𝛼²𝑥−𝑦²(𝑥)𝑦+(𝑥)𝑦+2𝛼∇𝑓−∇𝑓,𝑥−𝑦∇𝑓−∇𝑓²=𝛼𝑥−𝑦+∇𝑓(𝑥)−∇𝑓𝑦²=∇𝑓_𝛼(𝑥)−∇𝑓_𝛼𝑦². (61) Hence, it follows that ∇𝑓_𝛼=𝛼𝐼+∇𝑓 is (1/(𝛼+𝐿))-ism. It is clear that 𝐶_𝑛 is closed and 𝑄_𝑛 is closed and convex for every 𝑛=1,2,…. As the defining inequality in 𝐶_𝑛 is equivalent to the inequality 2𝑥_𝑛−𝑧_𝑛,𝑧≤𝑥_𝑛²−𝑧_𝑛²+𝜃_𝑛. (62) By Lemma 10, we also have that 𝐶_𝑛 is convex for every 𝑛=1,2,…. As 𝑄_𝑛={𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝑥−𝑥_𝑛⟩≥0}, we have ⟨𝑥_𝑛−𝑧,𝑥−𝑥_𝑛⟩≥0 for all 𝑧∈𝑄_𝑛, and by Proposition 4(i), we get 𝑥_𝑛=𝑃_𝑄𝑛𝑥.
We divide the rest of the proof into several steps.
Step 1. {𝑥_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛} are bounded.
Indeed, take 𝑢∈Fix(𝑆)∩Γ arbitrarily. Taking into account 𝜆_𝑛(𝛼_𝑛+𝐿)<1, for all 𝑛≥1, we deduce that 𝑥_𝑛−𝑢−𝜆_𝑛𝜇_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−∇𝑓_𝛼𝑛(𝑢)²≤𝑥_𝑛−𝑢²+𝜆_𝑛𝜇_𝑛𝜆_𝑛𝜇_𝑛−2𝛼_𝑛+𝐿∇𝑓_𝛼𝑛𝑥_𝑛−∇𝑓_𝛼𝑛(𝑢)²≤𝑥_𝑛−𝑢²+𝜆_𝑛𝜆_𝑛−2𝛼_𝑛+𝐿∇𝑓_𝛼𝑛𝑥_𝑛−∇𝑓_𝛼𝑛(𝑢)²≤𝑥_𝑛−𝑢²,𝑦_𝑛=𝑃−𝑢_𝐶𝑥_𝑛−𝜆_𝑛𝜇_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝑃_𝐶𝑢−𝜆_𝑛𝜇_𝑛∇𝑓(𝑢)−𝜆_𝑛1−𝜇_𝑛(𝑢)≤𝑥∇𝑓_𝑛−𝜆_𝑛𝜇_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝑢−𝜆_𝑛𝜇_𝑛∇𝑓(𝑢)−𝜆_𝑛1−𝜇_𝑛(𝑢)≤𝑥∇𝑓_𝑛−𝑢−𝜆_𝑛𝜇_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−∇𝑓(𝑢)+𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑦_𝑛(𝑢≤𝑥−∇𝑓)_𝑛−𝑢−𝜆_𝑛𝜇_𝑛∇𝑓_𝛼𝑛𝑥_𝑛−∇𝑓_𝛼𝑛(𝑢)+𝜆_𝑛𝜇_𝑛∇𝑓_𝛼𝑛(𝑢)−∇𝑓(𝑢)+𝜆_𝑛1−𝜇_𝑛×∇𝑓_𝛼𝑛𝑦_𝑛−∇𝑓_𝛼𝑛(𝑢)+∇𝑓_𝛼𝑛(𝑢)(𝑢≤𝑥−∇𝑓)_𝑛−𝑢+𝜆_𝑛𝜇_𝑛𝛼_𝑛‖𝑢‖+𝜆_𝑛1−𝜇_𝑛𝛼_𝑛+𝐿𝑦_𝑛−𝑢+𝛼_𝑛‖𝑢=𝑥_𝑛‖−𝑢+𝜆_𝑛1−𝜇_𝑛𝛼_𝑛+𝐿𝑦_𝑛−𝑢+𝜆_𝑛𝛼_𝑛‖𝑢‖≤𝑥_𝑛−𝑢+1−𝜇_𝑛𝑦_𝑛−𝑢+𝜆_𝑛𝛼_𝑛‖𝑢‖, (63) which implies that 𝑦_𝑛≤1−𝑢𝜇_𝑛𝑥_𝑛−𝑢+𝜆_𝑛𝛼_𝑛‖𝑢‖. (64) Meantime, we also have 𝑡_𝑛=𝑃−𝑢_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑡_𝑛−𝑃_𝐶𝑢−𝜆_𝑛∇𝑓(𝑢)−𝜆_𝑛1−𝜇_𝑛(𝑢)≤𝑥∇𝑓_𝑛−𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑡_𝑛−𝑢−𝜆_𝑛∇𝑓(𝑢)−𝜆_𝑛1−𝜇_𝑛(𝑢)≤𝑥∇𝑓_𝑛−𝑢+𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−∇𝑓(𝑢)+𝜆_𝑛1−𝜇_𝑛∇𝑓_𝛼𝑛𝑡_𝑛(𝑢≤𝑥−∇𝑓)_𝑛−𝑢+𝜆_𝑛∇𝑓_𝛼𝑛𝑦_𝑛−∇𝑓_𝛼𝑛(𝑢)+∇𝑓_𝛼𝑛(𝑢)−∇𝑓(𝑢)+𝜆_𝑛1−𝜇_𝑛×∇𝑓_𝛼𝑛𝑡_𝑛−∇𝑓_𝛼𝑛(𝑢)+∇𝑓_𝛼𝑛(𝑢)(𝑢≤𝑥−∇𝑓)_𝑛−𝑢+𝜆_𝑛𝛼_𝑛+𝐿𝑦_𝑛−𝑢+𝛼_𝑛‖𝑢‖+𝜆_𝑛1−𝜇_𝑛𝛼_𝑛+𝐿𝑡_𝑛−𝑢+𝛼_𝑛‖𝑢≤𝑥_𝑛‖−𝑢+𝑦_𝑛−𝑢+𝜆_𝑛𝛼_𝑛‖𝑢‖+1−𝜇_𝑛𝑡_𝑛−𝑢+𝜆_𝑛1−𝜇_𝑛𝛼_𝑛‖𝑢‖≤𝑥_𝑛−𝑢+𝑦_𝑛−𝑢+2𝜆_𝑛𝛼_𝑛‖𝑢‖+1−𝜇_𝑛𝑡_𝑛−𝑢, (65) which hence implies that 𝑡_𝑛≤1−𝑢𝜇_𝑛𝑥_𝑛−𝑢+𝑦_𝑛−𝑢+2𝜆_𝑛𝛼_𝑛‖𝑢≤1‖𝜇_𝑛𝑥_𝑛+1−𝑢𝜇_𝑛𝑥_𝑛−𝑢+𝜆_𝑛𝛼_𝑛‖𝑢‖+2𝜆_𝑛𝛼_𝑛‖𝑢=1‖𝜇_𝑛+1𝜇²_𝑛𝑥_𝑛+1−𝑢𝜇²_𝑛+2𝜇_𝑛𝜆_𝑛𝛼_𝑛‖𝑢‖≤1+2𝜇_𝑛𝜇²_𝑛𝑥_𝑛−𝑢+𝜆_𝑛𝛼_𝑛‖𝑢‖. (66) Thus, from (64) and (66), it follows that 𝑦_𝑛−𝑢+𝑡_𝑛≤1−𝑢𝜇_𝑛𝑥_𝑛−𝑢+𝜆_𝑛𝛼_𝑛‖𝑢+‖1+2𝜇_𝑛𝜇²_𝑛𝑥_𝑛−𝑢+𝜆_𝑛𝛼_𝑛‖𝑢‖=1+3𝜇_𝑛𝜇²_𝑛𝑥_𝑛−𝑢+𝜆_𝑛𝛼_𝑛‖𝑢‖≤4𝜇²_𝑛𝑥_𝑛−𝑢+𝜆_𝑛𝛼_𝑛‖𝑢‖, (67) which hence implies that 𝑦_𝑛−𝑢+𝑡_𝑛−𝑢²≤16𝜇⁴_𝑛𝑥_𝑛−𝑢+𝜆_𝑛𝛼_𝑛‖𝑢‖²≤16𝜇⁴_𝑛2𝑥_𝑛−𝑢²+2𝜆²_𝑛𝛼²_𝑛‖𝑢‖²≤16𝜇⁴_𝑛2𝑥_𝑛−𝑢²+2𝛼²_𝑛‖𝑢‖²≤32𝜇⁴_𝑛𝑥_𝑛−𝑢²+32𝜇⁴_𝑛𝛼²_𝑛‖𝑢‖². (68) Repeating the same arguments as in (33) and (34), we can obtain that 𝑡_𝑛−𝑢²≤𝑥_𝑛−𝑢²+𝜆²_𝑛𝛼_𝑛+𝐿²−1𝑥_𝑛−𝑦_𝑛²+2𝜆_𝑛𝛼_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛≤𝑥−𝑢_𝑛−𝑢²+2𝜆_𝑛𝛼_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛,𝑧−𝑢_𝑛−𝑢²≤1+𝛾_𝑛𝑡_𝑛−𝑢²+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝛽_𝑛𝜅−1−𝜎_𝑛−𝛽_𝑛𝑡_𝑛−𝑆^𝑛𝑡_𝑛²+𝑐_𝑛≤1+𝛾_𝑛𝑡_𝑛−𝑢²+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝑐_𝑛≤1+𝛾_𝑛𝑥_𝑛−𝑢²+2𝜆_𝑛𝛼_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛−𝑢+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝑐_𝑛=1+𝛾_𝑛𝑥_𝑛−𝑢²+2𝛼_𝑛𝜆_𝑛1+𝛾_𝑛‖𝑢‖×𝑦_𝑛−𝑢+𝑡_𝑛−𝑢+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝑐_𝑛≤1+𝛾_𝑛𝑥_𝑛−𝑢²+𝛼_𝑛𝜆²_𝑛1+𝛾_𝑛²‖𝑢‖²+𝑦_𝑛−𝑢+𝑡_𝑛−𝑢²+𝜎_𝑛𝑄𝑡_𝑛−𝑄𝑢+‖𝑄𝑢−𝑢‖²+𝑐_𝑛≤1+𝛾_𝑛𝑥_𝑛−𝑢²+𝛼_𝑛1+𝛾_𝑛²‖𝑢‖²+𝑦_𝑛−𝑢+𝑡_𝑛−𝑢²+𝜎_𝑛𝜌𝑡_𝑛+1−𝑢1−𝜌‖1−𝜌𝑄𝑢−𝑢‖²+𝑐_𝑛≤1+𝛾_𝑛𝑥_𝑛−𝑢²+𝛼_𝑛1+𝛾_𝑛²‖𝑢‖²+𝑦_𝑛−𝑢+𝑡_𝑛−𝑢²+𝜎_𝑛𝜌𝑡_𝑛−𝑢²+11−𝜌1−𝜌²‖𝑄𝑢−𝑢‖²+𝑐_𝑛≤𝑥_𝑛−𝑢²+𝛾_𝑛𝑥_𝑛−𝑢²+𝛼_𝑛1+𝛾_𝑛²‖𝑢‖²+𝛼_𝑛+𝛾_𝑛𝑦_𝑛−𝑢+𝑡_𝑛−𝑢²+𝜎_𝑛𝑡_𝑛−𝑢²+𝜎_𝑛1−𝜌⁻¹‖𝑄𝑢−𝑢‖²+𝑐_𝑛≤𝑥_𝑛−𝑢²+𝛼_𝑛+𝛾_𝑛𝑥_𝑛−𝑢²+𝛼_𝑛1+𝛾_𝑛²‖𝑢‖²+𝛼_𝑛+𝛾_𝑛𝑦_𝑛−𝑢+𝑡_𝑛−𝑢²+𝜎_𝑛𝑦_𝑛−𝑢+𝑡_𝑛−𝑢²+𝜎_𝑛1−𝜌⁻¹+𝛾_𝑛‖𝑄𝑢−𝑢‖²+𝑐_𝑛≤𝑥_𝑛−𝑢²+𝛼_𝑛+𝛾_𝑛𝑥_𝑛−𝑢²+𝛼_𝑛1−𝜌⁻¹+𝛾_𝑛²‖𝑢‖²+𝛼_𝑛+𝛾_𝑛+𝜎_𝑛𝑦_𝑛−𝑢+𝑡_𝑛−𝑢²+𝜎_𝑛1−𝜌⁻¹+𝛾_𝑛‖𝑄𝑢−𝑢‖²+𝑐_𝑛≤𝑥_𝑛−𝑢²+𝛼_𝑛+𝛾_𝑛+𝜎_𝑛𝑥_𝑛−𝑢²+𝑦_𝑛−𝑢+𝑡_𝑛−𝑢²+𝛼_𝑛1−𝜌⁻¹+𝛾_𝑛²‖𝑢‖²+𝜎_𝑛1−𝜌⁻¹+𝛾_𝑛‖𝑄𝑢−𝑢‖²+𝑐_𝑛≤𝑥_𝑛−𝑢²+𝛼_𝑛+𝛾_𝑛+𝜎_𝑛×𝑥_𝑛−𝑢²+𝑦_𝑛−𝑢+𝑡_𝑛−𝑢²+𝛼_𝑛+𝛾_𝑛+𝜎_𝑛1−𝜌⁻¹+𝛾_𝑛²‖𝑢‖²+‖𝑄𝑢−𝑢‖²+𝑐_𝑛=𝑥_𝑛−𝑢²+𝛼_𝑛+𝛾_𝑛+𝜎_𝑛×𝑥_𝑛−𝑢²+𝑦_𝑛−𝑢+𝑡_𝑛−𝑢²+1−𝜌⁻¹+𝛾_𝑛²‖𝑢‖²+‖𝑄𝑢−𝑢‖²+𝑐_𝑛≤𝑥_𝑛−𝑢²+𝛼_𝑛+𝛾_𝑛+𝜎_𝑛×𝑥_𝑛−𝑢²+32𝜇⁴_𝑛𝑥_𝑛−𝑢²+32𝜇⁴_𝑛𝛼²_𝑛‖𝑢‖²+1−𝜌⁻¹+𝛾_𝑛²‖𝑢‖²+‖𝑄𝑢−𝑢‖²+𝑐_𝑛=𝑥_𝑛−𝑢²+𝛼_𝑛+𝛾_𝑛+𝜎_𝑛×32+𝜇⁴_𝑛𝜇⁴_𝑛𝑥_𝑛−𝑢²+1−𝜌⁻¹+𝛾_𝑛²+32𝜇⁴_𝑛𝛼²_𝑛×‖𝑢‖²+1−𝜌⁻¹+𝛾_𝑛²‖𝑄𝑢−𝑢‖²+𝑐_𝑛≤𝑥_𝑛−𝑢²+𝛼_𝑛+𝛾_𝑛+𝜎_𝑛×33𝜇⁴_𝑛𝑥_𝑛−𝑢²+1−𝜌⁻¹+𝛾_𝑛²+32𝜇⁴_𝑛𝛼²_𝑛×‖𝑢‖²+‖𝑄𝑢−𝑢‖²+𝑐_𝑛≤𝑥_𝑛−𝑢²+𝑐_𝑛+𝛼_𝑛+𝛾_𝑛+𝜎_𝑛Δ_𝑛, (69)(70) for every 𝑛=1,2,… and hence 𝑢∈𝐶_𝑛. So, Fix(𝑆)∩Γ⊂𝐶_𝑛 for every 𝑛=1,2,…. Next, let us show by mathematical induction that {𝑥_𝑛} is well-defined and Fix(𝑆)∩Γ⊂𝐶_𝑛∩𝑄_𝑛 for every 𝑛=1,2,…. For 𝑛=1, we have 𝑄₁=𝐶. Hence, we obtain Fix(𝑆)∩Γ⊂𝐶₁∩𝑄₁. Suppose that 𝑥_𝑘 is given and Fix(𝑆)∩Γ⊂𝐶_𝑘∩𝑄_𝑘 for some integer 𝑘≥1. Since Fix(𝑆)∩Γ is nonempty, 𝐶_𝑘∩𝑄_𝑘 is a nonempty closed convex subset of 𝐶. So, there exists a unique element 𝑥_𝑘+1∈𝐶_𝑘∩𝑄_𝑘 such that 𝑥_𝑘+1=𝑃_{𝐶𝑘∩𝑄𝑘}𝑥. It is also obvious that there holds ⟨𝑥_𝑘+1−𝑧,𝑥−𝑥_𝑘+1⟩≥0 for every 𝑧∈𝐶_𝑘∩𝑄_𝑘. Since Fix(𝑆)∩Γ⊂𝐶_𝑘∩𝑄_𝑘, we have ⟨𝑥_𝑘+1−𝑧,𝑥−𝑥_𝑘+1⟩≥0 for every 𝑧∈Fix(𝑆)∩Γ and hence Fix(𝑆)∩Γ⊂𝑄_𝑘+1. Therefore, we obtain Fix(𝑆)∩Γ⊂𝐶_𝑘+1∩𝑄_𝑘+1.
Step 2. lim_𝑛→∞‖𝑥_𝑛−𝑥_𝑛+1‖=0 and lim_𝑛→∞‖𝑥_𝑛−𝑧_𝑛‖=0.
Indeed, let 𝑞=𝑃_{Fix(𝑆)∩Γ}𝑥. From 𝑥_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥 and 𝑞∈Fix(𝑆)∩Γ⊂𝐶_𝑛∩𝑄_𝑛, we have 𝑥_𝑛+1−𝑥≤𝑞−𝑥 (71) for every 𝑛=1,2,…. Therefore, {𝑥_𝑛} is bounded. From (64), (66), and (70), we also obtain that {𝑦_𝑛}, {𝑡_𝑛}, and {𝑧_𝑛} are bounded. Since 𝑥_𝑛+1∈𝐶_𝑛∩𝑄_𝑛⊂𝑄_𝑛 and 𝑥_𝑛=𝑃_𝑄𝑛𝑥, we have 𝑥_𝑛−𝑥≤𝑥_𝑛+1−𝑥 (72) for every 𝑛=1,2,…. Therefore, there exists lim_𝑛→∞‖𝑥_𝑛−𝑥‖. Since 𝑥_𝑛=𝑃_𝑄𝑛𝑥 and 𝑥_𝑛+1∈𝑄_𝑛, using Proposition 4(ii), we have 𝑥_𝑛+1−𝑥_𝑛²≤𝑥_𝑛+1−𝑥²−𝑥_𝑛−𝑥² (73) for every 𝑛=1,2,…. This implies that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (74) Since 𝑥_𝑛+1∈𝐶_𝑛, we have 𝑧_𝑛−𝑥_𝑛+1²≤𝑥_𝑛−𝑥_𝑛+1²+𝜃_𝑛, (75) which implies that 𝑧_𝑛−𝑥_𝑛+1≤𝑥_𝑛−𝑥_𝑛+1+𝜃_𝑛. (76) Hence, we get 𝑥_𝑛−𝑧_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑧_𝑛≤2𝑥_𝑛+1−𝑥_𝑛+𝜃_𝑛 (77) for every 𝑛=1,2,…. From ‖𝑥_𝑛+1−𝑥_𝑛‖→0 and 𝜃_𝑛→0, we conclude that ‖𝑥_𝑛−𝑧_𝑛‖→0 as 𝑛→∞.
Step 3. lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0, lim_𝑛→∞‖𝑥_𝑛−𝑡_𝑛‖=0 and lim_𝑛→∞‖𝑥_𝑛−𝑆𝑥_𝑛‖=0.
Indeed, substituting (69) in (70), we get 𝑧_𝑛−𝑢²≤1+𝛾_𝑛𝑡_𝑛−𝑢²+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝑐_𝑛≤1+𝛾_𝑛𝑥_𝑛−𝑢²+𝜆²_𝑛𝛼_𝑛+𝐿²×𝑥−1_𝑛−𝑦_𝑛²+2𝜆_𝑛𝛼_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛−𝑢+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝑐_𝑛=𝑥_𝑛−𝑢²+𝛾_𝑛𝑥_𝑛−𝑢²+1+𝛾_𝑛𝜆²_𝑛𝛼_𝑛+𝐿²×𝑥−1_𝑛−𝑦_𝑛²+2𝛼_𝑛𝜆_𝑛1+𝛾_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛−𝑢+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝑐_𝑛, (78) which hence implies that 1+𝛾_𝑛𝑏²𝛼_𝑛+𝐿²−1𝑥_𝑛−𝑦_𝑛²≤1+𝛾_𝑛𝜆²_𝑛𝛼_𝑛+𝐿²−1𝑥_𝑛−𝑦_𝑛²≤𝑥_𝑛−𝑢²−𝑧_𝑛−𝑢²+𝛾_𝑛𝑥_𝑛−𝑢²+2𝛼_𝑛𝜆_𝑛1+𝛾_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛−𝑢+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝑐_𝑛≤𝑥_𝑛−𝑢+𝑧_𝑛−𝑢𝑥_𝑛−𝑧_𝑛+𝛾_𝑛𝑥_𝑛−𝑢²+2𝛼_𝑛𝜆_𝑛1+𝛾_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛−𝑢+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝑐_𝑛. (79) Since ‖𝑥_𝑛−𝑧_𝑛‖→0, 𝛼_𝑛→0, 𝛾_𝑛→0, 𝜎_𝑛→0, and 𝑐_𝑛→0, from the boundedness of {𝑥_𝑛}, {𝑦_𝑛}, {𝑡_𝑛}, and {𝑧_𝑛}, it follows that lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0. (80) Meantime, utilizing the arguments similar to those in (69), we have 𝑡_𝑛−𝑢²≤𝑥_𝑛−𝑢²−𝑥_𝑛−𝑦_𝑛²−𝑦_𝑛−𝑡_𝑛²+2𝜆_𝑛𝛼_𝑛+𝐿𝑥_𝑛−𝑦_𝑛𝑡_𝑛−𝑦_𝑛+2𝜆_𝑛𝛼_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛≤𝑥−𝑢_𝑛−𝑢²−𝑥_𝑛−𝑦_𝑛²−𝑦_𝑛−𝑡_𝑛²+𝜆²_𝑛𝛼_𝑛+𝐿²𝑡_𝑛−𝑦_𝑛²+𝑥_𝑛−𝑦_𝑛²+2𝜆_𝑛𝛼_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛=𝑥−𝑢_𝑛−𝑢²+𝜆²_𝑛𝛼_𝑛+𝐿²−1𝑡_𝑛−𝑦_𝑛²+2𝜆_𝑛𝛼_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛−𝑢. (81) Substituting (81) in (70), we get 𝑧_𝑛−𝑢²≤1+𝛾_𝑛𝑡_𝑛−𝑢²+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝑐_𝑛≤1+𝛾_𝑛𝑥_𝑛−𝑢²+𝜆²_𝑛𝛼_𝑛+𝐿²−1𝑡_𝑛−𝑦_𝑛²+2𝜆_𝑛𝛼_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛−𝑢+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝑐_𝑛=𝑥_𝑛−𝑢²+𝛾_𝑛𝑥_𝑛−𝑢²+1+𝛾_𝑛𝜆²_𝑛𝛼_𝑛+𝐿²×𝑡−1_𝑛−𝑦_𝑛²+2𝛼_𝑛𝜆_𝑛1+𝛾_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛−𝑢+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝑐_𝑛, (82) which hence implies that 1+𝛾_𝑛𝑏²𝛼_𝑛+𝐿²−1𝑡_𝑛−𝑦_𝑛²≤1+𝛾_𝑛𝜆²_𝑛𝛼_𝑛+𝐿²−1𝑡_𝑛−𝑦_𝑛²≤𝑥_𝑛−𝑢²−𝑧_𝑛−𝑢²+𝛾_𝑛𝑥_𝑛−𝑢²+2𝛼_𝑛𝜆_𝑛1+𝛾_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛−𝑢+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝑐_𝑛≤𝑥_𝑛−𝑢+𝑧_𝑛−𝑢𝑥_𝑛−𝑧_𝑛+𝛾_𝑛𝑥_𝑛−𝑢²+2𝛼_𝑛𝜆_𝑛1+𝛾_𝑛‖𝑢‖𝑦_𝑛−𝑢+𝑡_𝑛−𝑢+𝜎_𝑛𝑄𝑡_𝑛−𝑢²+𝑐_𝑛. (83) Since ‖𝑥_𝑛−𝑧_𝑛‖→0, 𝛼_𝑛→0, 𝛾_𝑛→0, 𝜎_𝑛→0, and 𝑐_𝑛→0, from the boundedness of {𝑥_𝑛}, {𝑦_𝑛}, {𝑡_𝑛}, and {𝑧_𝑛}, it follows that lim_𝑛→∞𝑡_𝑛−𝑦_𝑛=0. (84) This together with ‖𝑥_𝑛−𝑦_𝑛‖→0 implies that lim_𝑛→∞𝑥_𝑛−𝑡_𝑛=0. (85)
Since 𝑧_𝑛=(1−𝜎_𝑛−𝛽_𝑛)𝑡_𝑛+𝜎_𝑛𝑄𝑡_𝑛+𝛽_𝑛𝑆^𝑛𝑡_𝑛, we have 𝛽_𝑛(𝑆^𝑛𝑡_𝑛−𝑡_𝑛)=(𝑧_𝑛−𝑡_𝑛)−𝜎_𝑛(𝑄𝑡_𝑛−𝑡_𝑛). Then, 𝛿𝑆^𝑛𝑡_𝑛−𝑡_𝑛≤𝛽_𝑛𝑆^𝑛𝑡_𝑛−𝑡_𝑛≤𝑧_𝑛−𝑡_𝑛+𝜎_𝑛𝑄𝑡_𝑛−𝑡_𝑛≤𝑧_𝑛−𝑥_𝑛+𝑥_𝑛−𝑡_𝑛+𝜎_𝑛𝑄𝑡_𝑛−𝑡_𝑛 (86) and hence ‖𝑡_𝑛−𝑆^𝑛𝑡_𝑛‖→0. Furthermore, observe that 𝑥_𝑛−𝑆^𝑛𝑥_𝑛≤𝑥_𝑛−𝑡_𝑛+𝑡_𝑛−𝑆^𝑛𝑡_𝑛+𝑆^𝑛𝑡_𝑛−𝑆^𝑛𝑥_𝑛. (87) Utilizing Lemma 11, we have 𝑆^𝑛𝑡_𝑛−𝑆^𝑛𝑥_𝑛≤1𝜅𝑡1−𝜅^𝑛−𝑥_𝑛+1+(1−𝜅)𝛾_𝑛𝑡_𝑛−𝑥_𝑛²+(1−𝜅)𝑐_𝑛 (88) for every 𝑛=1,2,…. Hence, it follows from ‖𝑥_𝑛−𝑡_𝑛‖→0 that ‖𝑆^𝑛𝑡_𝑛−𝑆^𝑛𝑥_𝑛‖→0. Thus, from (87) 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‖→0 as 𝑛→∞, and 𝑆 is uniformly continuous, we obtain from Lemma 12 that ‖𝑥_𝑛−𝑆𝑥_𝑛‖→0 as 𝑛→∞.
Step 4. 𝜔_𝑤(𝑥_𝑛)⊂Fix(𝑆)∩Γ.
Indeed, repeating the same arguments as in Step 4 of the proof of Theorem 18, we can derive the desired conclusion.
Step 5. {𝑥_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛} converge strongly to 𝑞=𝑃_{Fix(𝑆)∩Γ}𝑥.
Indeed, take 𝑢∈𝜔_𝑤(𝑥_𝑛) arbitrarily. Then, 𝑢∈Fix(𝑆)∩Γ according to Step 4. Moreover, there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑢. Hence, from 𝑞=𝑃_{Fix(𝑆)∩Γ}𝑥, 𝑢∈Fix(𝑆)∩Γ, and (71), we have 𝑞−𝑥≤‖𝑢−𝑥‖≤liminf_𝑖→∞𝑥_𝑛𝑖−𝑥≤limsup_𝑖→∞𝑥_𝑛𝑖−𝑥≤𝑞−𝑥. (89) So, we obtain lim_𝑖→∞𝑥_𝑛𝑖−𝑥=‖𝑢−𝑥‖. (90) From 𝑥_𝑛𝑖−𝑥⇀𝑢−𝑥, we have 𝑥_𝑛𝑖−𝑥→𝑢−𝑥 due to the Kadec-Klee property of Hilbert spaces [17]. So, it is clear that 𝑥_𝑛𝑖→𝑢. Since 𝑥_𝑛=𝑃_𝑄𝑛𝑥 and 𝑞∈Fix(𝑆)∩Γ⊂𝐶_𝑛∩𝑄_𝑛⊂𝑄_𝑛, we have −𝑞−𝑥_𝑛𝑖²=𝑞−𝑥_𝑛𝑖,𝑥_𝑛𝑖−𝑥+𝑞−𝑥_𝑛𝑖≥,𝑥−𝑞𝑞−𝑥_𝑛𝑖,𝑥−𝑞. (91) As 𝑖→∞, we obtain −‖𝑞−𝑢‖²≥⟨𝑞−𝑢,𝑥−𝑞⟩≥0 by 𝑞=𝑃_{Fix(𝑆)∩Γ}𝑥 and 𝑢∈Fix(𝑆)∩Γ. Hence, we have 𝑢=𝑞. This implies that 𝑥_𝑛→𝑞. It is easy to see that 𝑦_𝑛→𝑞 and 𝑧_𝑛→𝑞. This completes the proof.

Acknowledgments

In this research, first author was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and Ph.D. Profram Foundation of Ministry of Education of China (20123127110002). The research part of the second author was done during his visit to KFUPM, Dhahran, Saudi Arabia. The third author was partially supported by a Grant from the NSC 101-2115-M-037-001.