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

Authors: Lu-Chuan Ceng; Qamrul Hasan Ansari; Ching-Feng Wen
Journal: Abstract and Applied Analysis
 (2013)

---

## 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 mappingS 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.


---

## Body

## 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 {kn} such that limn→∞kn=1 and
(1)∥Snx-Sny∥≤kn∥x-y∥,∀n≥1,∀x,y∈C
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
(2)∥Snx-Sny∥≤ℒ∥x-y∥,∀n≥1,∀x,y∈C.
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.
LetC 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
(3)limsupn→∞supx,y∈C(∥Snx-Sny∥-∥x-y∥)≤0.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.
LetC 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 {γn} if there exists a constant κ∈[0,1) and a sequence {γn} in [0,∞) with limn→∞γn=0 such that
(4)∥Snx-Sny∥2≤(1+γn)∥x-y∥2+κ∥x-Snx-(y-Sny)∥2,∀n≥1,∀x,y∈C.They studied weak and strong convergence theorems for this class of mappings. It is important to note that every asymptoticallyκ-strict pseudocontractive mapping with sequence {γn} is a uniformly ℒ-Lipschitzian mapping with ℒ=sup{(κ+1+(1-κ)γn)/(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.
LetC 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 {γn} if there exist a constant κ∈[0,1) and a sequence {γn} in [0,∞) with limn→∞γn=0 such that
(5)limsupn→∞supx,y∈C(∥Snx-Sny∥2-(1+γn)∥x-y∥2-κ∥x-Snx-(y-Sny)∥2)≤0.Let(6)cn:=max{0,supx,y∈C(∥Snx-Sny∥2-(1+γn)∥x-y∥2cn:=max{0,supx,y∈C(-κ∥x-Snx-(y-Sny)∥2){supx,y∈C(∥Snx-Sny∥2-(1+γn)∥x-y∥2}}.
Then, cn≥0 (for all n≥1), cn→0 (n→∞), and (5) reduces to the relation
(7)∥Snx-Sny∥2≤(1+γn)∥x-y∥2+κ∥x-Snx-(y-Sny)∥2+cn,∀n≥1,∀x,y∈C.
Whenever cn=0 for all n≥1 in (7), then S is an asymptotically κ-strict pseudocontractive mapping with sequence {γn}.Letf:C→R be a convex and continuously Fréchet differentiable functional. We consider the following minimization problem:
(8)minx∈Cf(x).
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 asymptoticallyκ-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 thatH 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 xn⇀x to indicate that the sequence {xn} converges weakly to x and xn→x to indicate that the sequence {xn} converges strongly to x. Moreover, we use ωw(xn) to denote the weak ω-limit set of the sequence {xn}, that is,
(9)ωw(xn):={x∈H:xni⇀xforsomesubsequence{xni}of{xn}}.The metric (or nearest point) projection fromH onto C is the mapping PC:H→C which assigns to each point x∈H the unique point PCx∈C satisfying the following property:
(10)∥x-PCx∥=infy∈C∥x-y∥=:d(x,C).We mention some important properties of projections in the following proposition.Proposition 4.
For givenx∈H and z∈C, (i)
z=PCx⇔〈x-z,y-z〉≤0,  for all  y∈C;(ii)
z=PCx⇔∥x-z∥2≤∥x-y∥2-∥y-z∥2,  for all  y∈C;(iii)
〈PCx-PCy,x-y〉≥∥PCx-PCy∥2,  for all  y∈H.Consequently,PC is nonexpansive.Definition 5.
A mappingS:C→H is said to be(a)
monotone if(11)〈Sx-Sy,x-y〉≥0,∀x,y∈C;(b)
η-strongly monotone if there exists a constant η>0 such that
(12)〈Sx-Sy,x-y〉≥η∥x-y∥2,∀x,y∈C;(c)
α-inverse-strongly monotone (α-ism) if there exists a constant α>0 such that
(13)〈Sx-Sy,x-y〉≥α∥Sx-Sy∥2,∀x,y∈C.Obviously, ifS 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 PC 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.
LetX be a real inner product space. Then,
(14)∥x+y∥2≤∥x∥2+2〈y,x+y〉,∀x,y∈X.Lemma 7 (see [7, Proposition 2.4]).
Let{xn} be a bounded sequence in a reflexive Banach space X. If ωw({xn})={x}, then xn⇀x.LetA:C→H be a nonlinear mapping. The classical variational inequality problem (VIP) is to find x-∈C such that
(15)〈Ax-,y-x-〉≥0,∀y∈C.
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 Proposition4(i), we have the following result.Lemma 8.
LetA:C→H be a monotone mapping. Then,
(16)u∈VI(C,A)⟺u=PC(u-λAu),forsomeλ>0.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{xn} is a sequence in H such that xn⇀x, then
(17)limsupn→∞∥xn-y∥2=limsupn→∞∥xn-x∥2+∥x-y∥2,∀y∈H.Lemma 10 (see [4, Lemma 2.5]).
For given pointsx,y,z∈H and given also a real number a∈R, the set
(18){v∈C:∥y-v∥2≤∥x-v∥2+〈z,v〉+a}
is convex (and closed).Lemma 11 (see [4, Lemma 2.6]).
LetC 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 {γn}. Then,
(19)∥Snx-Sny∥≤11-κ({(1+(1-κ)γn)∥x-y∥2+(1-κ)cn}κ∥x-y∥+(1+(1-κ)γn)∥x-y∥2+(1-κ)cn)
for all x,y∈C and n≥1.Lemma 12 (see [4, Lemma 2.7]).
LetC 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 {γn}. Let {xn} be a sequence in C such that ∥xn-xn+1∥→0 and ∥xn-Snxn∥→0 as n→∞. Then, ∥xn-Sxn∥→0 as n→∞.Lemma 13 (Demiclosedness Principle, [see [4, Proposition 3.1]]).
LetC 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 {γn}. Then I-S is demiclosed at zero in the sense that if {xn} is a sequence in C such that xn⇀x∈C and limsupm→∞limsupn→∞∥xn-Smxn∥=0, then (I-S)x=0.Lemma 14 (see [4, Proposition 3.2]).
LetC 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 {γn} 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{an}n=1∞, {bn}n=1∞, and {δn}n=1∞ be sequences of nonnegative real numbers satisfying the inequality
(20)an+1≤(1+δn)an+bn,∀n≥1.
If ∑n=1∞‍δn<∞ and ∑n=1∞‍bn<∞, then limn→∞an exists. If, in addition, {an}n=1∞ has a subsequence which converges to zero, then limn→∞an=0.Corollary 16 (see [16, page 303]).
Let{an}n=0∞ and {bn}n=0∞ be two sequences of nonnegative real numbers satisfying the inequality
(21)an+1≤an+bn,∀n≥0.
If ∑n=0∞‍bn converges, then limn→∞an exists.Lemma 17 (see [17]).
Every Hilbert spaceH has the Kadec-Klee property; that is, given a sequence {xn}⊂H and a point x∈H, we have
(22)∥xn∥⟶∥x∥,xn⇀x⇒xn⟶x.It is well known that every Hilbert spaceH satisfies Opial’s condition [18]; that is, for any sequence {xn} with xn⇀x, we have
(23)liminfn→∞∥xn-x∥<liminfn→∞∥xn-y∥,∀y∈Hwithy≠x.A set-valued mappingT:H→2H 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→2H 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 NCv be the normal cone to C at v∈C, that is, NCv={w∈H:〈v-u,w〉≥0,∀u∈C}. Define
(24)Tv={Av+NCv,ifv∈C,∅,ifv∉C.
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 asymptoticallyκ-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.
LetC 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 {γn} such that Fix(S)∩Γ is nonempty. Let {γn} and {cn} be defined as in Definition 3. Let {xn} and {yn} be the sequences generated by
(25)x1=x∈Cchosenarbitrary,yn=PC(xn-λnμn∇fαn(xn)-λn(1-μn)∇fαn(yn)),tn=PC(xn-λn∇fαn(yn)-λn(1-μn)∇fαn(tn)),xn+1=(1-σn-βn)tn+σnQtn+βnSntn,∀n≥1,
where {αn} is a sequence (0,∞), {λn}, {μn} are sequences in (0,1], and {σn}, {βn} are sequences in [0,1] satisfying the following conditions: (i)
δ≤βn and σn+βn≤1-κ-τ, for all n≥1 for some δ,τ∈(0,1);(ii)
∑n=1∞‍αn<∞, ∑n=1∞‍γn<∞, ∑n=1∞‍σn<∞ and ∑n=1∞‍cn<∞;(iii)
limn→∞μn=1;(iv)
λn(αn+L)<1, for all n≥1 and {λn}⊂[a,b] for some a,b∈(0,(1/L)).
Then, the sequences{xn}, {yn} converge weakly to some point u∈Fix(S)∩Γ.Remark 19.
Observe that for allx,y∈C and all n≥1(26)∥PC(xn-λnμn∇fαn(xn)-λn(1-μn)∇fαn(x))-PC(xn-λnμn∇fαn(xn)-λn(1-μn)∇fαn(y))∥≤∥(xn-λnμn∇fαn(xn)-λn(1-μn)∇fαn(x))-(xn-λnμn∇fαn(xn)-λn(1-μn)∇fαn(y))∥=λn(1-μn)∥∇fαn(x)-∇fαn(y)∥≤λn(αn+L)∥x-y∥.
By Banach contraction principle, we know that for each n≥1, there exists a unique yn∈C such that
(27)yn=PC(xn-λnμn∇fαn(xn)-λn(1-μn)∇fαn(yn)).
Also, observe that for allx,y∈C and all n≥1(28)∥PC(xn-λn∇fαn(yn)-λn(1-μn)∇fαn(x))-PC(xn-λn∇fαn(yn)-λn(1-μn)∇fαn(y))∥≤∥(xn-λn∇fαn(yn)-λn(1-μn)∇fαn(x))-(xn-λn∇fαn(yn)-λn(1-μn)∇fαn(y))∥=λn(1-μn)∥∇fαn(x)-∇fαn(y)∥≤λn(αn+L)∥x-y∥.
Utilizing Banach contraction principle, for each n≥1, there exists a unique tn∈C such that
(29)tn=PC(xn-λn∇fαn(yn)-λn(1-μn)∇fαn(tn)).Proof of Theorem18.
Note that theL-Lipschitz continuity of the gradient ∇f implies that ∇f is (1/L)-ism [20], that is,
(30)〈∇f(x)-∇f(y),x-y〉≥1L∥∇f(x)-∇f(y)∥2,∀x,y∈C.
Thus, ∇f is monotone and L-Lipschitz continuous.
We divide the rest of the proof into several steps.
Step
1. limn→∞∥xn-u∥ exists for each u∈Fix(S)∩Γ.
Indeed, note thattn=PC(xn-λn∇fαn(yn)-λn(1-μn)∇fαn(tn)) for all n≥1. Take u∈Fix(S)∩Γ arbitrarily. From Proposition 4(ii), monotonicity of ∇f, and u∈Γ, we have
(31)∥tn-u∥2≤∥(xn-λn∇fαn(yn)-λn(1-μn)∇fαn(tn))-u∥2-∥(xn-λn∇fαn(yn)-λn(1-μn)∇fαn(tn))-tn∥2=∥xn-λn(1-μn)∇fαn(tn)-u∥2-∥xn-λn(1-μn)∇fαn(tn)-tn∥2+2λn〈∇fαn(yn),u-tn〉=∥xn-λn(1-μn)∇fαn(tn)-u∥2-∥xn-λn(1-μn)∇fαn(tn)-tn∥2+2λn(〈∇fαn(yn),u-yn〉+〈∇fαn(yn),yn-tn〉)=∥xn-λn(1-μn)∇fαn(tn)-u∥2-∥xn-λn(1-μn)∇fαn(tn)-tn∥2+2λn(〈∇fαn(yn)-∇fαn(u),u-yn〉+〈∇fαn(u),u-yn〉+〈∇fαn(yn),yn-tn〉)≤∥xn-λn(1-μn)∇fαn(tn)-u∥2-∥xn-λn(1-μn)∇fαn(tn)-tn∥2+2λn(αn〈u,u-yn〉+〈∇fαn(yn),yn-tn〉)=∥xn-u∥2-∥xn-tn∥2-2λn(1-μn)〈∇fαn(tn),tn-u〉+2λn(αn〈u,u-yn〉+〈∇fαn(yn),yn-tn〉)=∥xn-u∥2-∥xn-yn∥2-2〈xn-yn,yn-tn〉-∥yn-tn∥2+2λn(αn〈u,u-yn〉+〈∇fαn(yn),yn-tn〉)-2λn(1-μn)×(〈∇fαn(tn)-∇fαn(u),tn-u〉+〈∇fαn(u),tn-u〉)≤∥xn-u∥2-∥xn-yn∥2-∥yn-tn∥2+2〈xn-λn∇fαn(yn)-yn,tn-yn〉+2λnαn(〈u,u-yn〉+(1-μn)〈u,u-tn〉)≤∥xn-u∥2-∥xn-yn∥2-∥yn-tn∥2+2〈xn-λn∇fαn(yn)-yn,tn-yn〉+2λnαn∥u∥(∥yn-u∥+∥tn-u∥).
Since yn=PC(xn-λnμn∇fαn(xn)-λn(1-μn)∇fαn(yn)) and ∇fαn is (αn+L)-Lipschitz continuous, by Proposition 4(i), we have
(32)〈xn-λn∇fαn(yn)-yn,tn-yn〉=〈xn-λnμn∇fαn(xn)-λn(1-μn)∇fαn(yn)-yn,tn-yn{xn-λnμn∇fαn(xn)-λn(1-μn)∇fαn(yn)}〉+λnμn〈∇fαn(xn)-∇fαn(yn),tn-yn〉≤λnμn〈∇fαn(xn)-∇fαn(yn),tn-yn〉≤λnμn∥∇fαn(xn)-∇fαn(yn)∥∥tn-yn∥≤λn(αn+L)∥xn-yn∥∥tn-yn∥.
So, we have
(33)∥tn-u∥2≤∥xn-u∥2-∥xn-yn∥2-∥yn-tn∥2∥tn-u∥2≤+2λn(αn+L)∥xn-yn∥∥tn-yn∥∥tn-u∥2≤+2λnαn∥u∥(∥yn-u∥+∥tn-u∥)∥tn-u∥2≤∥xn-u∥2-∥xn-yn∥2-∥yn-tn∥2∥tn-u∥2≤+λn2(αn+L)2∥xn-yn∥2+∥tn-yn∥2∥tn-u∥2≤+2λnαn∥u∥(∥yn-u∥+∥tn-u∥)∥tn-u∥2=∥xn-u∥2+(λn2(αn+L)2-1)∥xn-yn∥2∥tn-u∥2≤+2λnαn∥u∥(∥yn-u∥+∥tn-u∥)∥tn-u∥2≤∥xn-u∥2+2λnαn∥u∥(∥yn-u∥+∥tn-u∥).
Therefore, from (33), xn+1=(1-σn-βn)tn+σnf(tn)+βnSntn and u=Su. By Lemma 9(b), we have
(34)∥xn+1-u∥2=∥(1-σn-βn)(tn-u)+σn(Qtn-u)+βn(Sntn-u)∥2≤(1-σn-βn)∥tn-u∥2+σn∥Qtn-u∥2+βn∥Sntn-u∥2-(1-σn-βn)βn∥tn-Sntn∥2≤(1-σn-βn)∥tn-u∥2+σn∥Qtn-u∥2+βn[(1+γn)∥tn-u∥2+κ∥tn-Sntn∥2+cn]-(1-σn-βn)βn∥tn-Sntn∥2=[1-σn-βn+βn(1+γn)]∥tn-u∥2+σn∥Qtn-u∥2+βn(κ-(1-σn-βn))∥tn-Sntn∥2+βncn≤(1+γn)∥tn-u∥2+σn∥Qtn-u∥2+βn(κ-(1-σn-βn))∥tn-Sntn∥2+cn≤(1+γn)∥tn-u∥2+σn∥Qtn-u∥2+cn≤(1+γn)[∥xn-u∥2+2λnαn∥u∥(∥yn-u∥+∥tn-u∥)]+σn∥Qtn-u∥2+cn=(1+γn)∥xn-u∥2+2αnλn(1+γn)∥u∥×(∥yn-u∥+∥tn-u∥)+σn∥Qtn-u∥2+cn.
Since ∑n=1∞‍αn<∞, ∑n=1∞‍γn<∞, ∑n=1∞‍σn<∞, and ∑n=1∞‍cn<∞, from the boundedness of C, it follows that
(35)∑n=1∞{{+σn∥Qtn-u∥2+cn}2αnλn(1+γn)∥u∥(∥yn-u∥+∥tn-u∥)‍+σn∥Qtn-u∥2+cn{2αnλn(1+γn)∥u∥(∥yn-u∥+∥tn-u∥)}}<∞.
By Lemma 15, we have
(36)limn→∞∥xn-u∥existsforallu∈Fix(S)∩Γ.
Step
2. limn→∞∥xn-yn∥=0 and limn→∞∥xn-tn∥=0.
Indeed, substituting (33) in (34), we get
(37)∥xn+1-u∥2≤(1+γn)∥tn-u∥2+σn∥Qtn-u∥2+cn≤(1+γn)[∥xn-u∥2+(λn2(αn+L)2-1)∥xn-yn∥2+2λnαn∥u∥(∥yn-u∥+∥tn-u∥){∥xn-u∥2+(λn2(αn+L)2-1)∥xn-yn∥2}]+σn∥Qtn-u∥2+cn=∥xn-u∥2+γn∥xn-u∥2+(1+γn)(λn2(αn+L)2-1)∥xn-yn∥2+2αnλn(1+γn)∥u∥(∥yn-u∥+∥tn-u∥)+σn∥Qtn-u∥2+cn,
which hence implies that
(38)(1+γn)(b2(αn+L)2-1)∥xn-yn∥2≤(1+γn)(λn2(αn+L)2-1)∥xn-yn∥2≤∥xn-u∥2-∥xn+1-u∥2+γn∥xn-u∥2+2αnλn(1+γn)∥u∥(∥yn-u∥+∥tn-u∥)+σn∥Qtn-u∥2+cn.
Since limn→∞∥xn-u∥ exists, αn→0, γn→0, σn→0, and cn→0, from the boundedness of C, it follows that
(39)limn→∞∥xn-yn∥=0.
Meantime, utilizing the arguments similar to those in (33), we have
(40)∥tn-u∥2≤∥xn-u∥2-∥xn-yn∥2-∥yn-tn∥2∥tn-u∥2≤+2λn(αn+L)∥xn-yn∥∥tn-yn∥∥tn-u∥2≤+2λnαn∥u∥(∥yn-u∥+∥tn-u∥)∥tn-u∥2≤∥xn-u∥2-∥xn-yn∥2-∥yn-tn∥2∥tn-u∥2≤+λn2(αn+L)2∥tn-yn∥2+∥xn-yn∥2∥tn-u∥2≤+2λnαn∥u∥(∥yn-u∥+∥tn-u∥)∥tn-u∥2=∥xn-u∥2+(λn2(αn+L)2-1)∥tn-yn∥2∥tn-u∥2≤+2λnαn∥u∥(∥yn-u∥+∥tn-u∥).
Substituting (40) in (34), we get
(41)∥xn+1-u∥2≤(1+γn)∥tn-u∥2+σn∥Qtn-u∥2+cn≤(1+γn)[∥xn-u∥2+(λn2(αn+L)2-1)∥tn-yn∥2+2λnαn∥u∥(∥yn-u∥+∥tn-u∥)]+σn∥Qtn-u∥2+cn=∥xn-u∥2+γn∥xn-u∥2+(1+γn)(λn2(αn+L)2-1)∥tn-yn∥2+2αnλn(1+γn)∥u∥(∥yn-u∥+∥tn-u∥)+σn∥Qtn-u∥2+cn,
which hence implies that
(42)(1+γn)(b2(αn+L)2-1)∥tn-yn∥2≤(1+γn)(λn2(αn+L)2-1)∥tn-yn∥2≤∥xn-u∥2-∥xn+1-u∥2+γn∥xn-u∥2+2αnλn(1+γn)∥u∥(∥yn-u∥+∥tn-u∥)+σn∥Qtn-u∥2+cn.
Since limn→∞∥xn-u∥ exists, αn→0, γn→0, σn→0, and cn→0, from the boundedness of C it follows that
(43)limn→∞∥tn-yn∥=0.
This together with ∥xn-yn∥→0 implies that
(44)limn→∞∥xn-tn∥=0.
Step
3.  limn→∞∥xn+1-xn∥=0 and limn→∞∥xn-Sxn∥=0.
Indeed, from (33) and (34), we conclude that
(45)∥xn+1-u∥2≤(1+γn)∥tn-u∥2+σn∥Qtn-u∥2+βn(κ-(1-σn-βn))∥tn-Sntn∥2+cn≤(1+γn)[∥xn-u∥2+2λnαn∥u∥(∥yn-u∥+∥tn-u∥)]+σn∥Qtn-u∥2+βn(κ-(1-σn-βn))∥tn-Sntn∥2+cn=∥xn-u∥2+γn∥xn-u∥2+2αnλn(1+γn)∥u∥(∥yn-u∥+∥tn-u∥)+σn∥Qtn-u∥2-βn((1-σn-βn)-κ)∥tn-Sntn∥2+cn,
which together with condition (i) yields
(46)δτ∥tn-Sntn∥2≤βn((1-σn-βn)-κ)∥tn-Sntn∥2≤∥xn-u∥2-∥xn+1-u∥2+γn∥xn-u∥2+2αnλn(1+γn)∥u∥(∥yn-u∥+∥tn-u∥)+σn∥Qtn-u∥2+cn.
Since limn→∞∥xn-u∥ exists, αn→0, γn→0, σn→0, and cn→0, from the boundedness of C it follows that
(47)limn→∞∥tn-Sntn∥=0.
Note that
(48)∥xn+1-xn∥≤∥xn+1-tn∥+∥tn-xn∥=∥σn(Qtn-tn)+βn(Sntn-tn)∥+∥tn-xn∥≤σn∥Qtn-tn∥+∥Sntn-tn∥+∥tn-xn∥.
From the boundedness of C, σn→0, ∥tn-xn∥→0, and ∥Sntn-tn∥→0, we deduce that
(49)limn→∞∥xn+1-xn∥=0.
Furthermore, observe that
(50)∥xn-Snxn∥≤∥xn-tn∥+∥tn-Sntn∥+∥Sntn-Snxn∥.
Utilizing Lemma 11, we have
(51)∥Sntn-Snxn∥≤11-κ×({+(1+(1-κ)γn)∥tn-xn∥2+(1-κ)cn}κ∥tn-xn∥+(1+(1-κ)γn)∥tn-xn∥2+(1-κ)cn)
for every n=1,2,…. Hence, it follows from ∥xn-tn∥→0 that ∥Sntn-Snxn∥→0. Thus, from (50) and ∥tn-Sntn∥→0, we get ∥xn-Snxn∥→0. Since ∥xn+1-xn∥→0, ∥xn-Snxn∥→0 as n→∞, and S is uniformly continuous, we obtain from Lemma 12 that ∥xn-Sxn∥→0 as n→∞.
Step
4. ωw(xn)⊂Fix(S)∩Γ.
Indeed, from the boundedness of{xn}, we know that ωw({xn})≠∅. Take u^∈ωw({xn}) arbitrarily. Then, there exists a subsequence {xni} of {xn} such that {xni} converges weakly to u^. Note that S is uniformly continuous and ∥xn-Sxn∥→0. Hence, it is easy to see that ∥xn-Smxn∥→0 for all m≥1. By Lemma 13, we obtain u^∈Fix(S). Furthermore, we show u^∈Γ. Since xn-tn→0 and xn-yn→0, we have tni⇀u^ and yni⇀u^. Let
(52)Tv={∇f(v)+NCv,ifv∈C,∅,ifv∉C,
where NCv 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)+NCv and hence w-∇f(v)∈NCv. So, we have 〈v-t,w-∇f(v)〉≥0 for all t∈C. On the other hand, from tn=PC(xn-λn∇fαn(yn)-λn(1-μn)∇fαn(tn)) and v∈C, we have
(53)〈xn-λn∇fαn(yn)-λn(1-μn)∇fαn(tn)-tn,tn-v〉≥0,
and hence
(54)〈v-tn,tn-xnλn+∇fαn(yn)+(1-μn)∇fαn(tn)〉≥0.
Therefore, from 〈v-t,w-∇f(v)〉≥0 for all t∈C and tni∈C, we have
(55)〈v-tni,w〉≥〈v-tni,∇f(v)〉≥〈v-tni,∇f(v)〉-〈v-tni,tni-xniλni+∇fαni(yni)+(1-μni)∇fαni(tni){v-tni,tni-xniλni+∇fαni(yni)}〉=〈v-tni,∇f(v)〉-〈v-tni,tni-xniλni+∇f(yni)〉-αni〈v-tni,yni〉-(1-μni)〈v-tni,∇fαni(tni)〉=〈v-tni,∇f(v)-∇f(tni)〉+〈v-tni,∇f(tni)-∇f(yni)〉-〈v-tni,tni-xniλni〉-αni〈v-tni,yni〉-(1-μni)〈v-tni,∇fαni(tni)〉≥〈v-tni,∇f(tni)-∇f(yni)〉-〈v-tni,tni-xniλni〉-αni〈v-tni,yni〉-(1-μni)〈v-tni,∇fαni(tni)〉.
Since ∥∇f(tn)-∇f(yn)∥→0 (due to the Lipschitz continuity of ∇f), (tn-xn)/(λn)→0 (due to {λn}⊂[a,b]), αn→0, and μn→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 ωw({xn})⊂Fix(S)∩Γ.
Step
5. {xn} and {yn} converge weakly to the same point u∈Fix(S)∩Γ.
Indeed, it is sufficient to show thatωw(xn) is a single-point set because xn-yn→0 as n→∞. Since ωw(xn)≠∅, let us take two points u,u^∈ωw(xn) arbitrarily. Then, there exist two subsequences {xnj} and {xmk} of {xn} such that xnj⇀u and xmk⇀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 limn→∞∥xn-u∥ and limn→∞∥xn-u^∥. Let us show that u=u^. Assume that u≠u^. From the Opial condition [18], it follows that
(56)limn→∞∥xn-u∥=liminfj→∞∥xnj-u∥<liminfj→∞∥xnj-u^∥limn→∞∥xn-u∥=limn→∞∥xn-u^∥=liminfk→∞∥xmk-u^∥limn→∞∥xn-u∥<liminfk→∞∥xmk-u∥=limn→∞∥xn-u∥.
This leads to a contradiction. Thus, we must have u=u^. This implies that ωw(xn) is a single-point set. Without loss of generality, we may write ωw(xn)={u}. Consequently, by Lemma 7, we obtain that xn⇀u∈Fix(S)∩Γ. Since xn-yn→0 as n→∞, we also have
(57)yn⇀u∈Fix(S)∩Γ.
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 asymptoticallyκ-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.
LetC 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 {γn} such that Fix(S)∩Γ is nonempty and bounded. Let {γn} and {cn} be defined as in Definition 3. Let {xn},{yn}, and {zn} be the sequences generated by
(58)x1=x∈Cchosenarbitrary,yn=PC(xn-λnμn∇fαn(xn)-λn(1-μn)∇fαn(yn)),tn=PC(xn-λn∇fαn(yn)-λn(1-μn)∇fαn(tn)),zn=(1-σn-βn)tn+σnQtn+βnSntn,Cn={z∈C:∥zn-z∥2≤∥xn-z∥2+θn},Qn={z∈C:〈xn-z,x-xn〉≥0},xn+1=PCn∩Qnx,∀n≥1,
where {αn}⊂(0,∞), {λn}, {μn}⊂(0,1], {σn}, {βn}⊂[0,1], θn=cn+(αn+γn+σn)Δn and
(59)Δn=supz∈Fix(S)∩Γ{33μn4∥xn-z∥2+[((1-ρ)-1+γn)2+32μn4αn2]×(∥z∥2+∥Qz-z∥2){33μn4}}<∞.
Assume that the following conditions hold:(i)
δ≤βn and σn+βn≤1-κ, for all n≥1 for some δ∈(0,1);(ii)
limn→∞αn=0 and limn→∞σn=0;(iii)
limn→∞μn=1;(iv)
λn(αn+L)<1, for all n≥1 and {λn}⊂[a,b] for some a,b∈(0,(1/L)).
Then, the sequences{xn}, {yn}, and {zn} converge strongly to the same point PFix(S)∩Γx.Proof.
Utilizing the conditionλn(αn+L)<1 for all n≥1 and repeating the same arguments as in Remark 19, we can see that {yn} and {zn} are defined well. Note that the L-Lipschitz continuity of the gradient ∇f implies that ∇f is (1/L)-ism [20], that is,
(60)〈∇f(x)-∇f(y),x-y〉≥1L∥∇f(x)-∇f(y)∥2,∀x,y∈C.
Observe that
(61)(α+L)〈∇fα(x)-∇fα(y),x-y〉=(α+L)[α∥x-y∥2+〈∇f(x)-∇f(y),x-y〉]=α2∥x-y∥2+α〈∇f(x)-∇f(y),x-y〉+αL∥x-y∥2+L〈∇f(x)-∇f(y),x-y〉≥α2∥x-y∥2+2α〈∇f(x)-∇f(y),x-y〉+∥∇f(x)-∇f(y)∥2=∥α(x-y)+∇f(x)-∇f(y)∥2=∥∇fα(x)-∇fα(y)∥2.
Hence, it follows that ∇fα=αI+∇f is (1/(α+L))-ism. It is clear that Cn is closed and Qn is closed and convex for every n=1,2,…. As the defining inequality in Cn is equivalent to the inequality
(62)〈2(xn-zn),z〉≤∥xn∥2-∥zn∥2+θn.
By Lemma 10, we also have that Cn is convex for every n=1,2,…. As Qn={z∈C:〈xn-z,x-xn〉≥0}, we have 〈xn-z,x-xn〉≥0 for all z∈Qn, and by Proposition 4(i), we get xn=PQnx.
We divide the rest of the proof into several steps.
Step
1. {xn}, {yn}, and {zn} are bounded.
Indeed, takeu∈Fix(S)∩Γ arbitrarily. Taking into account λn(αn+L)<1, for all n≥1, we deduce that
(63)∥xn-u-λnμn(∇fαn(xn)-∇fαn(u))∥2≤∥xn-u∥2+λnμn(λnμn-2αn+L)∥∇fαn(xn)-∇fαn(u)∥2≤∥xn-u∥2+λn(λn-2αn+L)∥∇fαn(xn)-∇fαn(u)∥2≤∥xn-u∥2,∥yn-u∥=∥PC(xn-λnμn∇fαn(xn)-λn(1-μn)∇fαn(yn))-PC(u-λnμn∇f(u)-λn(1-μn)∇f(u)){(xn-λnμn∇fαn(xn)-λn(1-μn)∇fαn(yn))}{(xn-λnμn∇fαn(xn)-λn(1-μn)∇fαn(yn))}∥≤∥(xn-λnμn∇fαn(xn)-λn(1-μn)∇fαn(yn))-(u-λnμn∇f(u)-λn(1-μn)∇f(u)){(xn-λnμn∇fαn(xn)-λn(1-μn)∇fαn(yn))}{(xn-λnμn∇fαn(xn)-λn(1-μn)∇fαn(yn))}∥≤∥xn-u-λnμn(∇fαn(xn)-∇f(u))∥+λn(1-μn)∥∇fαn(yn)-∇f(u)∥≤∥xn-u-λnμn(∇fαn(xn)-∇fαn(u))∥+λnμn∥∇fαn(u)-∇f(u)∥+λn(1-μn)×[∥∇fαn(yn)-∇fαn(u)∥+∥∇fαn(u)-∇f(u)∥]≤∥xn-u∥+λnμnαn∥u∥+λn(1-μn)[(αn+L)∥yn-u∥+αn∥u∥]=∥xn-u∥+λn(1-μn)(αn+L)∥yn-u∥+λnαn∥u∥≤∥xn-u∥+(1-μn)∥yn-u∥+λnαn∥u∥,
which implies that
(64)∥yn-u∥≤1μn(∥xn-u∥+λnαn∥u∥).
Meantime, we also have
(65)∥tn-u∥=∥PC(xn-λn∇fαn(yn)-λn(1-μn)∇fαn(tn))-PC(u-λn∇f(u)-λn(1-μn)∇f(u)){PC(xn-λn∇fαn(yn)-λn(1-μn)∇fαn(tn))}∥≤∥(xn-λn∇fαn(yn)-λn(1-μn)∇fαn(tn))-(u-λn∇f(u)-λn(1-μn)∇f(u)){(xn-λn∇fαn(yn)-λn(1-μn)∇fαn(tn))}{(xn-λn∇fαn(yn)-λn(1-μn)∇fαn(tn))}∥≤∥xn-u∥+λn∥∇fαn(yn)-∇f(u)∥+λn(1-μn)∥∇fαn(tn)-∇f(u)∥≤∥xn-u∥+λn[∥∇fαn(yn)-∇fαn(u)∥+∥∇fαn(u)-∇f(u)∥]+λn(1-μn)×[∥∇fαn(tn)-∇fαn(u)∥+∥∇fαn(u)-∇f(u)∥]≤∥xn-u∥+λn[(αn+L)∥yn-u∥+αn∥u∥]+λn(1-μn)[(αn+L)∥tn-u∥+αn∥u∥]≤∥xn-u∥+∥yn-u∥+λnαn∥u∥+(1-μn)∥tn-u∥+λn(1-μn)αn∥u∥≤∥xn-u∥+∥yn-u∥+2λnαn∥u∥+(1-μn)∥tn-u∥,
which hence implies that
(66)∥tn-u∥≤1μn[∥xn-u∥+∥yn-u∥+2λnαn∥u∥]≤1μn{∥xn-u∥+1μn(∥xn-u∥+λnαn∥u∥)+2λnαn∥u∥}=(1μn+1μn2)∥xn-u∥+(1μn2+2μn)λnαn∥u∥≤1+2μnμn2(∥xn-u∥+λnαn∥u∥).
Thus, from (64) and (66), it follows that
(67)∥yn-u∥+∥tn-u∥≤1μn(∥xn-u∥+λnαn∥u∥)∥yn-u∥+∥tn-u∥≤+1+2μnμn2(∥xn-u∥+λnαn∥u∥)∥yn-u∥+∥tn-u∥=1+3μnμn2(∥xn-u∥+λnαn∥u∥)∥yn-u∥+∥tn-u∥≤4μn2(∥xn-u∥+λnαn∥u∥),
which hence implies that
(68)(∥yn-u∥+∥tn-u∥)2≤16μn4(∥xn-u∥+λnαn∥u∥)2(∥yn-u∥+∥tn-u∥)2≤16μn4(2∥xn-u∥2+2λn2αn2∥u∥2)(∥yn-u∥+∥tn-u∥)2≤16μn4(2∥xn-u∥2+2αn2∥u∥2)(∥yn-u∥+∥tn-u∥)2≤32μn4∥xn-u∥2+32μn4αn2∥u∥2.
Repeating the same arguments as in (33) and (34), we can obtain that
(69)∥tn-u∥2≤∥xn-u∥2+(λn2(αn+L)2-1)∥xn-yn∥2+2λnαn∥u∥(∥yn-u∥+∥tn-u∥)≤∥xn-u∥2+2λnαn∥u∥(∥yn-u∥+∥tn-u∥),(70)∥zn-u∥2≤(1+γn)∥tn-u∥2+σn∥Qtn-u∥2+βn(κ-(1-σn-βn))∥tn-Sntn∥2+cn≤(1+γn)∥tn-u∥2+σn∥Qtn-u∥2+cn≤(1+γn)[∥xn-u∥2+2λnαn∥u∥(∥yn-u∥+∥tn-u∥)]+σn∥Qtn-u∥2+cn=(1+γn)∥xn-u∥2+2αnλn(1+γn)∥u∥×(∥yn-u∥+∥tn-u∥)+σn∥Qtn-u∥2+cn≤(1+γn)∥xn-u∥2+αn[λn2(1+γn)2∥u∥2+(∥yn-u∥+∥tn-u∥)2]+σn(∥Qtn-Qu∥+∥Qu-u∥)2+cn≤(1+γn)∥xn-u∥2+αn[(1+γn)2∥u∥2+(∥yn-u∥+∥tn-u∥)2]+σn[ρ∥tn-u∥+(1-ρ)11-ρ∥Qu-u∥]2+cn≤(1+γn)∥xn-u∥2+αn[(1+γn)2∥u∥2+(∥yn-u∥+∥tn-u∥)2]+σn[ρ∥tn-u∥2+(1-ρ)1(1-ρ)2∥Qu-u∥2]+cn≤∥xn-u∥2+γn∥xn-u∥2+αn(1+γn)2∥u∥2+(αn+γn)(∥yn-u∥+∥tn-u∥)2+σn∥tn-u∥2+σn(1-ρ)-1∥Qu-u∥2+cn≤∥xn-u∥2+(αn+γn)∥xn-u∥2+αn(1+γn)2∥u∥2+(αn+γn)(∥yn-u∥+∥tn-u∥)2+σn(∥yn-u∥+∥tn-u∥)2+σn((1-ρ)-1+γn)∥Qu-u∥2+cn≤∥xn-u∥2+(αn+γn)∥xn-u∥2+αn((1-ρ)-1+γn)2∥u∥2+(αn+γn+σn)(∥yn-u∥+∥tn-u∥)2+σn((1-ρ)-1+γn)∥Qu-u∥2+cn≤∥xn-u∥2+(αn+γn+σn)[∥xn-u∥2+(∥yn-u∥+∥tn-u∥)2]+αn((1-ρ)-1+γn)2∥u∥2+σn((1-ρ)-1+γn)∥Qu-u∥2+cn≤∥xn-u∥2+(αn+γn+σn)×[∥xn-u∥2+(∥yn-u∥+∥tn-u∥)2]+(αn+γn+σn)((1-ρ)-1+γn)2(∥u∥2+∥Qu-u∥2)+cn=∥xn-u∥2+(αn+γn+σn)×{{+((1-ρ)-1+γn)2(∥u∥2+∥Qu-u∥2)}∥xn-u∥2+(∥yn-u∥+∥tn-u∥)2+((1-ρ)-1+γn)2(∥u∥2+∥Qu-u∥2)}+cn≤∥xn-u∥2+(αn+γn+σn)×{∥xn-u∥2+32μn4∥xn-u∥2+32μn4αn2∥u∥2+((1-ρ)-1+γn)2(∥u∥2+∥Qu-u∥2){+32μn4∥xn-u∥2+32μn4αn2∥u∥2}}+cn=∥xn-u∥2+(αn+γn+σn)×{32+μn4μn4∥xn-u∥2+[((1-ρ)-1+γn)2+32μn4αn2]×∥u∥2+((1-ρ)-1+γn)2∥Qu-u∥2{32+μn4μn4}}+cn≤∥xn-u∥2+(αn+γn+σn)×{33μn4∥xn-u∥2+[((1-ρ)-1+γn)2+32μn4αn2]×(∥u∥2+∥Qu-u∥2){33μn4∥xn-u∥2+[((1-ρ)-1+γn)2+32μn4αn2]}}+cn≤∥xn-u∥2+cn+(αn+γn+σn)Δn,
for every n=1,2,… and hence u∈Cn. So, Fix(S)∩Γ⊂Cn for every n=1,2,…. Next, let us show by mathematical induction that {xn} is well-defined and Fix(S)∩Γ⊂Cn∩Qn for every n=1,2,…. For n=1, we have Q1=C. Hence, we obtain Fix(S)∩Γ⊂C1∩Q1. Suppose that xk is given and Fix(S)∩Γ⊂Ck∩Qk for some integer k≥1. Since Fix(S)∩Γ is nonempty, Ck∩Qk is a nonempty closed convex subset of C. So, there exists a unique element xk+1∈Ck∩Qk such that xk+1=PCk∩Qkx. It is also obvious that there holds 〈xk+1-z,x-xk+1〉≥0 for every z∈Ck∩Qk. Since Fix(S)∩Γ⊂Ck∩Qk, we have 〈xk+1-z,x-xk+1〉≥0 for every z∈Fix(S)∩Γ and hence Fix(S)∩Γ⊂Qk+1. Therefore, we obtain Fix(S)∩Γ⊂Ck+1∩Qk+1.
Step
2. limn→∞∥xn-xn+1∥=0 and limn→∞∥xn-zn∥=0.
Indeed, letq=PFix(S)∩Γx. From xn+1=PCn∩Qnx and q∈Fix(S)∩Γ⊂Cn∩Qn, we have
(71)∥xn+1-x∥≤∥q-x∥
for every n=1,2,…. Therefore, {xn} is bounded. From (64), (66), and (70), we also obtain that {yn}, {tn}, and {zn} are bounded. Since xn+1∈Cn∩Qn⊂Qn and xn=PQnx, we have
(72)∥xn-x∥≤∥xn+1-x∥
for every n=1,2,…. Therefore, there exists limn→∞∥xn-x∥. Since xn=PQnx and xn+1∈Qn, using Proposition 4(ii), we have
(73)∥xn+1-xn∥2≤∥xn+1-x∥2-∥xn-x∥2
for every n=1,2,…. This implies that
(74)limn→∞∥xn+1-xn∥=0.
Since xn+1∈Cn, we have
(75)∥zn-xn+1∥2≤∥xn-xn+1∥2+θn,
which implies that
(76)∥zn-xn+1∥≤∥xn-xn+1∥+θn.
Hence, we get
(77)∥xn-zn∥≤∥xn-xn+1∥+∥xn+1-zn∥≤2∥xn+1-xn∥+θn
for every n=1,2,…. From ∥xn+1-xn∥→0 and θn→0, we conclude that ∥xn-zn∥→0 as n→∞.
Step
3. limn→∞∥xn-yn∥=0, limn→∞∥xn-tn∥=0 and limn→∞∥xn-Sxn∥=0.
Indeed, substituting (69) in (70), we get
(78)∥zn-u∥2≤(1+γn)∥tn-u∥2+σn∥Qtn-u∥2+cn≤(1+γn)[∥xn-u∥2+(λn2(αn+L)2-1)×∥xn-yn∥2+2λnαn∥u∥(∥yn-u∥+∥tn-u∥){∥xn-u∥2+(λn2(αn+L)2-1)}]+σn∥Qtn-u∥2+cn=∥xn-u∥2+γn∥xn-u∥2+(1+γn)(λn2(αn+L)2-1)×∥xn-yn∥2+2αnλn(1+γn)∥u∥(∥yn-u∥+∥tn-u∥)+σn∥Qtn-u∥2+cn,
which hence implies that
(79)(1+γn)(b2(αn+L)2-1)∥xn-yn∥2≤(1+γn)(λn2(αn+L)2-1)∥xn-yn∥2≤∥xn-u∥2-∥zn-u∥2+γn∥xn-u∥2+2αnλn(1+γn)∥u∥(∥yn-u∥+∥tn-u∥)+σn∥Qtn-u∥2+cn≤(∥xn-u∥+∥zn-u∥)∥xn-zn∥+γn∥xn-u∥2+2αnλn(1+γn)∥u∥(∥yn-u∥+∥tn-u∥)+σn∥Qtn-u∥2+cn.
Since ∥xn-zn∥→0, αn→0, γn→0, σn→0, and cn→0, from the boundedness of {xn}, {yn}, {tn}, and {zn}, it follows that
(80)limn→∞∥xn-yn∥=0.
Meantime, utilizing the arguments similar to those in (69), we have
(81)∥tn-u∥2≤∥xn-u∥2-∥xn-yn∥2-∥yn-tn∥2+2λn(αn+L)∥xn-yn∥∥tn-yn∥+2λnαn∥u∥(∥yn-u∥+∥tn-u∥)≤∥xn-u∥2-∥xn-yn∥2-∥yn-tn∥2+λn2(αn+L)2∥tn-yn∥2+∥xn-yn∥2+2λnαn∥u∥(∥yn-u∥+∥tn-u∥)=∥xn-u∥2+(λn2(αn+L)2-1)∥tn-yn∥2+2λnαn∥u∥(∥yn-u∥+∥tn-u∥).
Substituting (81) in (70), we get
(82)∥zn-u∥2≤(1+γn)∥tn-u∥2+σn∥Qtn-u∥2+cn≤(1+γn)[∥xn-u∥2+(λn2(αn+L)2-1)∥tn-yn∥2+2λnαn∥u∥(∥yn-u∥+∥tn-u∥){∥xn-u∥2+(λn2(αn+L)2-1)∥tn-yn∥2}]+σn∥Qtn-u∥2+cn=∥xn-u∥2+γn∥xn-u∥2+(1+γn)(λn2(αn+L)2-1)×∥tn-yn∥2+2αnλn(1+γn)∥u∥(∥yn-u∥+∥tn-u∥)+σn∥Qtn-u∥2+cn,
which hence implies that
(83)(1+γn)(b2(αn+L)2-1)∥tn-yn∥2≤(1+γn)(λn2(αn+L)2-1)∥tn-yn∥2≤∥xn-u∥2-∥zn-u∥2+γn∥xn-u∥2+2αnλn(1+γn)∥u∥(∥yn-u∥+∥tn-u∥)+σn∥Qtn-u∥2+cn≤(∥xn-u∥+∥zn-u∥)∥xn-zn∥+γn∥xn-u∥2+2αnλn(1+γn)∥u∥(∥yn-u∥+∥tn-u∥)+σn∥Qtn-u∥2+cn.
Since ∥xn-zn∥→0, αn→0, γn→0, σn→0, and cn→0, from the boundedness of {xn}, {yn}, {tn}, and {zn}, it follows that
(84)limn→∞∥tn-yn∥=0.
This together with ∥xn-yn∥→0 implies that
(85)limn→∞∥xn-tn∥=0.
Sincezn=(1-σn-βn)tn+σnQtn+βnSntn, we have βn(Sntn-tn)=(zn-tn)-σn(Qtn-tn). Then,
(86)δ∥Sntn-tn∥≤βn∥Sntn-tn∥δ∥Sntn-tn∥≤∥zn-tn∥+σn∥Qtn-tn∥δ∥Sntn-tn∥≤∥zn-xn∥+∥xn-tn∥+σn∥Qtn-tn∥
and hence ∥tn-Sntn∥→0. Furthermore, observe that
(87)∥xn-Snxn∥≤∥xn-tn∥+∥tn-Sntn∥+∥Sntn-Snxn∥.
Utilizing Lemma 11, we have
(88)∥Sntn-Snxn∥≤11-κ({(1+(1-κ)γn)∥tn-xn∥2+(1-κ)cn}κ∥tn-xn∥+(1+(1-κ)γn)∥tn-xn∥2+(1-κ)cn)
for every n=1,2,…. Hence, it follows from ∥xn-tn∥→0 that ∥Sntn-Snxn∥→0. Thus, from (87) and ∥tn-Sntn∥→0, we get ∥xn-Snxn∥→0. Since ∥xn+1-xn∥→0, ∥xn-Snxn∥→0 as n→∞, and S is uniformly continuous, we obtain from Lemma 12 that ∥xn-Sxn∥→0 as n→∞.
Step
4. ωw(xn)⊂Fix(S)∩Γ.
Indeed, repeating the same arguments as in Step 4 of the proof of Theorem18, we can derive the desired conclusion.
Step
5. {xn}, {yn}, and {zn} converge strongly to q=PFix(S)∩Γx.
Indeed, takeu^∈ωw(xn) arbitrarily. Then, u^∈Fix(S)∩Γ according to Step 4. Moreover, there exists a subsequence {xni} of {xn} such that xni⇀u^. Hence, from q=PFix(S)∩Γx, u^∈Fix(S)∩Γ, and (71), we have
(89)∥q-x∥≤∥u^-x∥≤liminfi→∞∥xni-x∥∥q-x∥≤limsupi→∞∥xni-x∥≤∥q-x∥.
So, we obtain
(90)limi→∞∥xni-x∥=∥u^-x∥.
From xni-x⇀u^-x, we have xni-x→u^-x due to the Kadec-Klee property of Hilbert spaces [17]. So, it is clear that xni→u^. Since xn=PQnx and q∈Fix(S)∩Γ⊂Cn∩Qn⊂Qn, we have
(91)-∥q-xni∥2=〈q-xni,xni-x〉+〈q-xni,x-q〉-∥q-xni∥2≥〈q-xni,x-q〉.
As i→∞, we obtain -∥q-u^∥2≥〈q-u^,x-q〉≥0 by q=PFix(S)∩Γx and u^∈Fix(S)∩Γ. Hence, we have u^=q. This implies that xn→q. It is easy to see that yn→q and zn→q. This completes the proof.



---

*Source: 854297-2013-04-16.xml*