# A New Iterative Algorithm for Pseudomonotone Equilibrium Problem and a Finite Family of Demicontractive Mappings

Authors: F. U. Ogbuisi; F. O. Isiogugu
Journal: Abstract and Applied Analysis
 (2020)

---

## Abstract

In this paper, we introduce a new iterative method in a real Hilbert space for approximating a point in the solution set of a pseudomonotone equilibrium problem which is a common fixed point of a finite family of demicontractive mappings. Our result does not require that we impose the condition that the sum of the control sequences used in the finite convex combination is equal to 1. Furthermore, we state and prove a strong convergence result and give some numerical experiments to demonstrate the efficiency and applicability of our iterative method.


---

## Body

## 1. Introduction

In this paper, we will always takeC to be a nonempty closed and convex subset of a real Hilbert space H endowed with inner product ⋅,⋅ and induced norm ⋅, and FT denotes the set of fixed points of a mapping T:C⟶C, that is, FT≔x∈C:x=Tx.Definition 1.
A mappingT:C⟶C is said to be(1)
Nonspreading [1] if(1)Tx−Ty2≤Tx−y2+Ty−x2,∀x,y∈C,or equivalently(2)Tx−Ty2≤x−y2+2x−Tx,y−Ty,∀x,y∈C.(2)
κ-Strictly pseudononspreading [1] if there exists κ∈0,1 such that(3)Tx−Ty2≤x−y2+κx−Tx−y−Ty2+2x−Tx,y−Ty,∀x,y∈C.(3)
β-Strictly pseudocontractive [2] if there exists β∈0,1 such that(4)Tx−Ty2≤x−y2+βx−Tx−y−Ty2∀x,y∈C.(4)
ρ-Demicontractive [3] if there exists ρ∈0,1 such that(5)Tx−y2≤x−y2+ρx−Tx2∀x∈C,y∈FT.
Obviously, the class of demicontractive mappings is more general than the class of quasi-nonexpansive mappings. Moreover, IfT is κ-strictly pseudononspreading (or κ-strictly pseudocontractive) and FT≠Ø, then T is κ-demicontractive.Definition 2.
A bifunctionf:C×C⟶ℝ is(i)
Strongly monotone onC with modulus β>0 (β-strongly monotone on C) if and only if(6)fx,y+fy,x≤−βy−x2,∀x,y∈C;(ii)
monotone onC if and only if(7)fx,y+fy,x≤0,∀x,y∈C;(iii)
pseudomonotone onC if and if(8)fx,y≥0⟹fy,x≤0,∀x,y∈C.
Letf:C×C⟶ℝ be a bifunction such that fx,x=0, for all x∈C. We consider the equilibrium problem (EP) in the sense of Blum and Oettli [4], which is to find(9)x∗∈Csuchthatfx∗,y≥0,∀y∈C.
LetEPC,f denote the set of solutions of EP (9). If fx,y=Ax,y−x∀x,y∈C, where A:C⟶H, then EP (9) reduces to the variational inequality problem:(10)findx∗∈Csuch  thatAx,y−x≥0,∀y∈C.
EPs form a very important area of research and have recently been considered in many research papers. EP (9) is applied in solving problems from optimization, variational inequality, Kakutani fixed point, Nash equilibria in noncooperative game theory, and minimax problems [4, 5].
A popular method that has been applied to solve EP (9) is the subgradient projection method which is developed from the steepest descent projection method in smooth optimization. If bifunction f is convex, subdifferentiable with respect to the second argument, Lipschitz, and strongly monotone on C, then regularization parameters can be chosen such that the subgradient projection method is linearly convergent [6]. However, when f is only monotone, the subgradient projection method may not be convergent. In order to get a method that guarantees convergence for pseudomonotone equilibrium problems (that is, equilibrium problems for pseudomonotone bifunctions) the extragradient (or double projection) method developed by Korpelevich [7] was extended to equilibrium problems. However, the extragradient algorithms involve two projections on the admissible set C, which may be costly to compute if the nature of the admissible set C is complicated. In the light of the need to obtain a more efficient algorithm, the inexact subgradient algorithms using only one projection [8, 9] has been proposed for solving equilibrium problems with paramonotone equilibrium bifunctions. Some other methods that have been utilized to solve equilibrium problems include the auxiliary problem principle method [10], gap function method [11], and the Tikhonov and proximal point regularization methods [12–15].
Recently, the problem of finding a common point inEPC,f and the set of fixed points of mappings has become an attractive and interesting subject [16–22]. This interest is because of the possible application of these problems to mathematical models whose constraints can be present as fixed points of mappings and/or (EP). Such a problem occurs, in particular, in the practical problems as signal processing, network resource allocation, image recovery (see [23, 24]).
In 2007, Tada and Takahashi [22] proposed the following iterative algorithm for approximating a common element of the set of solutions of equilibrium problem for monotone bifunctions and the set of fixed points of a nonexpansive mapping T.Algorithm 1.
(11)
z
k
∈
C
such that
f
z
k
,
y
+
1
λ
k
y
−
z
k
,
z
k
−
x
k
≥
0
,
∀
y
∈
C
,
w
k
=
α
k
x
k
+
1
−
α
k
T
z
k
,
C
k
=
z
∈
H
:
w
k
−
z
≤
x
k
−
z
,
D
k
=
z
∈
H
:
x
k
−
z
,
x
0
−
x
k
≥
0
,
x
k
+
1
=
P
C
k
∩
D
k
x
0
,where λk>0 is the regularization parameter at iteration k,x0∈C and PC is the metric projection onto C. They assume that f is a monotone bifunction and obtained a strong convergence result.
Recently, Anh and Muu [25] proposed a new type of algorithm which uses only one projection and does not require any Lipschitz condition for the bifunctions for finding a common point in the solution set of the class of pseudomonotone equilibrium problems and the set of fixed points of nonexpansive mappings. More precisely, they gave an iteration scheme generated as follows.Algorithm 2.
Pickx1∈C. At each iteration k=1,2,…, do the following:(12)computewk=∂εkfxk,⋅xk,takeγk≔maxλk,wk,αk≔βkγk,yk=PCxk−αkwk,xk+1=δkxk+1−δkTyk.
Inspired by Anh and Muu [25], Wangkeeree et al. [26] presented an iterative method for finding hierarchically an element in FT∩EPC,f with respect to a nonexpansive mapping. Precisely, they considered the following problems:(13)findx¯∈Γ≔FT∩EPC,fsuch thatx¯−Sx¯,x¯−x≤0,∀x∈Γ,where T and S are nonexpansive mappings.
Other authors have also considered different algorithms which involve either projection mapping or projection mapping and the construction of sequences of setsCn and Qn for approximating a common solution of pseudomonotone equilibrium problems and fixed point problems of nonexpansive mappings ( see, for example, [27–29]). Those methods are tasking and difficult to compute.
In 2018, Thong and Hieu [30] proposed the following iterative algorithm for the approximation of a common fixed point of a finite family of demicontractive operators. Let xn be a sequence in H defined by(14)x0∈H,xn+1=1−αn−βnxn+βn∑i=1NwiTixn.
Among other standard assumptions, they assumed thatwii=1N is a finite sequence of positive numbers such that(15)∑i=1Nwi=1.
LetK be a nonempty closed and convex subset of a real Hilbert space H. Suppose that Tii=1N, N≥2 is a countable finite family of mappings Ti:K⟶K. In [31], the authors consider the horizontal iteration process generated from an arbitrary x1 for the finite family of mappings Tii=1N, using a finite family of thTii=1N control sequences αnin=1∞i=1N as follows.
ForN = 2,(16)xn+1=αn1xn+1−αn1αn2T1xn+1−αn2T2xn.
ForN = 3,(17)xn+1=αn1xn+1−αn1αn2T1xn+1−αn2αn3T2xn+1−αn3T3xn.
For an arbitrary but finiteN≥2,(18)xn+1=αn1xn+1−αn1αn2T1xn+1−αn2αn3T2xn+1−αn3…αNTN−1xn+1−αNTNxn…=αn1xn+∑i=2Nαni∏j=1i−11−αnjTi−1xn+∏j=1N1−αnjTNxn,n≥1.


### 1.1. Question

Is it possible to give an iterative algorithm and obtain a strong convergence result for finding a common element in the set of fixed points of a finite family of demicontractive mappings which also solves equilibrium problems for pseudomonotone bifunctions without imposing the type of condition in (15) on the control sequences?In this paper, motivated by the works of Anh and Muu [25] and Wangkeeree et al. [26], we propose an iterative algorithm for finding a common element in the set of fixed points of a finite family of demicontractive mappings, which also solves equilibrium problems for pseudomonotone bifunctions and prove a strong convergence result which does not require such condition as in (15) on the control sequences. We further give a numerical experiment to demonstrate the performance of our iterative algorithm.


## 1.1. Question

Is it possible to give an iterative algorithm and obtain a strong convergence result for finding a common element in the set of fixed points of a finite family of demicontractive mappings which also solves equilibrium problems for pseudomonotone bifunctions without imposing the type of condition in (15) on the control sequences?In this paper, motivated by the works of Anh and Muu [25] and Wangkeeree et al. [26], we propose an iterative algorithm for finding a common element in the set of fixed points of a finite family of demicontractive mappings, which also solves equilibrium problems for pseudomonotone bifunctions and prove a strong convergence result which does not require such condition as in (15) on the control sequences. We further give a numerical experiment to demonstrate the performance of our iterative algorithm.


## 2. Preliminaries

In the sequel, we shall need the following definitions and lemmas. LetH be a real Hilbert space, and C a nonempty, closed, and convex subset of H. By PC, we denote the metric projection operator onto C, that is,(19)PCx∈C:x−PCx≤x−y,∀y∈C.Lemma 1.
Suppose thatC is a nonempty, closed, and convex subset in H. Then, PC has the following properties:(20)x−y2≥x−PCx2+y−PCx2,∀x∈H,y∈C.Lemma 2 (see [32]).
Letak and bk be two nonnegative real sequences satisfying the following conditions:(21)ak+1≤ak+bk∀k≥0,∑k=1∞bk<+∞.Then,limk⟶∞ak exists.Lemma 3 (see [33]).
LetH be a real Hilbert space, C a closed convex subset of H, and let T:C⟶C be a continuous pseudocontractive mapping, then(i)
F(T) is closed convex subset of C(ii)
I−T is demiclosed at zero, i.e., if xn is a sequence in C such that xn⇀x and Txn−xn⟶0, as n⟶∞, then x=TxLemma 4 (see [1]).
LetC be a nonempty, closed, and convex subset of a real Hilbert space H, and let T:C⟶C be a ρ-strictly pseudononspreading mapping. If FT≠Ø, then it is closed and convex.Lemma 5 (see [1]).
LetC be a nonempty, closed, and convex subset of a real Hilbert space H, and let T:C⟶C be a ρ-strictly pseudononspreading mapping. Then, I−T is demiclosed at 0.Definition 3.
LetC be a nonempty closed and convex subset of a Hilbert space E. Let f:C×C⟶ℝ be a bifunction where fx,⋅ is a convex function for each x∈C. Then, the ϵ-subdifferential (ϵ-diagonal subdifferential) of f at x denoted by ∂ϵfx,⋅x is given by (22)∂ϵfx,⋅x=w∈H:fx,y−fx,x+ϵ≥w,y−x,∀y∈C.Lemma 6.
Letαii=1N⊆ℝ be a countable subset of the set of real numbers ℝ, where N≥2 is an arbitrary integer. Then, the following holds:(23)α1+∑i=2Nαi∏j=1i−11−αj+∏j=1N1−αj=1.Proof.
This result has been proved in [31], but for the sake of completeness, we present the proof again here. For N=2,(24)α1+∑i=22αi∏j=1i−11−αj+∏j=121−αj=α1+α21−α1+1−α11−α2=α1+1−α1α2+1−α2=α1+1−α1=1.
We assume it is true forN and prove for N + 1.(25)α1+∑i=2N+1αi∏j=1i−11−αj+∏j=1N+11−αj=α1+∑i=2Nαi∏j=1i−11−αj+αN+1∏j=1N1−αj+∏j=1N+11−αj=α1+∑i=2Nαi∏j=1i−11−αj+∏j=1N1−αjαN+1+1−αN+1=α1+∑i=2Nαi∏j=1i−11−αj+∏j=1N+11−αj=1.Remark 1.
Lemma6 holds if αii=1N is replaced with αii=0N.Lemma 7 (see also [31]).
Lett and u be arbitrary elements of a real Hilbert space H, and let N∈ℕ be such that 2≤N. Let vii=1N⊆H and αii=1N⊆0,1 be a countable finite subset of H and ℝ, respectively. Define(26)y=α1t+∑i=2Nαi∏j=1i−11−αjvi−1+∏j=1N1−αjvN.Then,(27)y−u2≤α1t−u2+∑i=2Nαi∏j=1i−11−αjvi−1−u2+∏j=1N1−αjvN−u2−α1∑i=2Nαi∏j=1i−11−αjt−vi−12+∏j=1N1−αjt−vN2.Proof.
Letwk=∑i=k+1Nαi∏j=ki−11−αjvi−1+∏j=kN1−αjvN, k=1,2,…,N−1 and wN=1−αNvN. Observe that, for k≤N−1, wk=1−αkαk+1vk+wk+1. Using the well-known identity(28)βx+1−βy2=βx2+1−βy2−β1−βx−y2,which holds for all x,y∈H and for all t∈0,1, we have(29)y−u2=α1t+∑i=2Nαi∏j=1i−11−αjvi−1+∏j=1N1−αjvN−u2=αkt+w1−u2=α1t+1−α1α2v1+w2−u2=α1t−u2+1−α1α2v1+w2−u2−α11−α1t−α2v1+w22=α1t−u2+1−α1α2v1−u2+1−α2α3v2+w3−u2−α21−α2v1−α3v2+w32−α11−α1α2t−v12+1−α2t−α3v2+w32−α21−α2v1−α3v2+w32,=α1t−u2+1−α1α2v1−u2+1−α11−α2α3v2+w3−u2−1−α1α21−α2v1−α3v2+w32−α11−α1α2t−v12−α11−α11−α2t−α3v2+w32+α11−α1α21−α2v1−α3v2+w32=α1t−u2+1−α1α2v1−u2−α11−α1α2t−v12+1−α11−α2α3v2+w3−u2−α11−α11−α2t−α3v2+w32−α21−α11−α21−α1v1−α3v2+w32,13=α1t−u2+1−α1α2v1−u2−α11−α1α2t−v12−α21−α11−α21−α1v1−α3v2+w32+1−α11−α2α3v2+w3−u2−α11−α11−α2t−α3v2+w32=α1t−u2+1−α1α2v1−u2−α11−α1α2t−v12−α21−α11−α21−α1v1−α3v2+w32+1−α11−α2α3v2+1−α3α4v3+w4−u2−α11−α11−α2α3v2+1−α3α4v3+w4−t2=α1t−u2+1−α1α2v1−u2−α11−α1α2t−vk2+1−α11−α2α3v2−u2+1−α11−α21−α3α4v3+w4−u2−1−α11−α2α31−α3v2−α4v3+w42−α11−α11−α2α3v2−t2−α11−α11−α21−α3α4v3+w4−t2+α11−α11−α2α31−α3v2−α4v3+w42−α21−α11−α21−α1v1−α3v2+w32,=α1t−u2+1−α1α2v1−u2−α11−α1α2t−v12+1−α11−α2α3v2−u2+1−α11−α21−α3α4v3+w4−u2−α11−α11−α2α3v2−t2−α11−α11−α21−α3α4v3+w4−t2−α21−α11−α21−α1v1−α3v2+w32−1−α11−α2α31−α3v2−α4v3+w42,=α1t−u2+∑i=23αi∏j=1i−11−αjvi−1−u2−α1∑i=23αi∏j=1i−11−αjt−vi−12−1−α1∑i=23αi∏j=1i1−αjvi−1−αi+1vi+wi+12+∏j=131−αjα4v3+w4−u2−α1∏j=131−αjt−α4v3+w42+α3∏j=131−αju−v22⋮=α1t−u2+∑i=2Nαi∏j=1i−11−αjvi−1−u2+∐j=1N1−αjvN−u2−α1∑i=2Nαi∏j=1i−11−αjt−vi−12+∏j=1N1−αjt−vN2−1−α1∑i=2N−1αi∏j=1i1−αjvi−1−αi+1vi+wi+12+αN∏j−1N1−αjv−vN−12.


## 3. Main Results

We now give a strong convergence result for the approximation of a solution of a pseudomonotone equilibrium problem which is also a common fixed point of a finite family of demicontractive mappings.Letf:C×C⟶ℝ be a bifunction that satisfies the following conditions: (B1) f is pseudomonotone on C, fx,x=0, and fx,⋅ is lower semicontinuous for all x∈C; (B2) f⋅,y is sequentially weakly upper semicontinuous on C for each fixed point y∈C, i.e., if xn⊂C is a sequence converging weakly to x∈C, then limsupn⟶∞fxn,y≤fx,y; (B3) fx,⋅ is convex and subdifferentiable on E for every fixed x∈C; (B4) If xk⊆C is bounded ϵk↓0 as k⟶∞, then the sequence yk with yk∈∂ϵkfxk,⋅xk is bounded; (B5) f is pseudomonotone on C with respect to every x∈EPC,f and satisfies the strict paramonotonicity property, i.e.,(30)x∈EPC,f,y∈C,fy,x=0⟹y∈EPC,f.It has been proved that under the conditions (B1)–(B3), the solution setEPf,C of EP (9) is closed and convex [34].Algorithm 3.
Step 1. Let λk,βk and δk be sequences of nonnegative real numbers and let Ti,i=1,2,…,N, be ρi-demicontractive mappings on C. Suppose that the following conditions are satisfied:(1)
0<η≤λk<λ,ρ=max1≤i≤Nρiand0≤ρ<a<δk,i<b<1,limk⟶∞δk,1=1/2.(2)
βk>0,∑k=1∞βk=+∞and∑k=1∞βk2<+∞.(3)
∑k=1∞βkεk<+∞.
Step 2.
(31)
x
1
∈
C
;
y
k
∈
∂
ε
k
f
x
k
,
⋅
x
k
;
γ
k
≔
max
λ
k
,
y
k
and
α
k
≔
β
k
γ
k
;
w
k
=
P
C
x
k
−
α
k
y
k
;
x
k
+
1
=
δ
k
,
1
w
k
+
∑
i
=
2
N
δ
k
,
i
∏
j
=
1
i
−
1
1
−
δ
k
,
j
T
i
−
1
w
k
+
∏
j
=
1
N
1
−
δ
k
,
j
T
N
w
k
.Theorem 1.
LetC be a closed and convex subset of a real Hilbert space H, and let f:C×C⟶ℝ be a bifunction that satisfies conditions (B1)–(B5). Let Ti:C⟶C,i=1,2,…,N be a finite family of ρi-demicontractive mappings such that I−Ti is demiclosed at 0 for each i and Ω≔EPC,f∩∩i=1NFTi≠Ø. Then, the sequences xk and wk generated by Algorithm 3 converges strongly to a point p∈Ω, where p=limk⟶∞PΩxk.Proof.
Letx∗∈Ω. From wk∈PCxk−αkyk and x∗∈C, we have(32)xk−αkyk−wk,wk−x∗≥0.
Therefore,(33)x∗−wk,xk−wk≤αkyk,x∗−wk=αkyk,x∗−xk+αkyk,xk−wk≤αkyk,x∗−xk+αkykxk−wk.
Sincexk∈C, we also have(34)xk−αkyk−wk,wk−xk≥0.
From (33) and (34), we have(35)xk−wk,xk−wk=xk−wk2≤αkyk,xk−wk≤αkykxk−wk.
That is,(36)xk−wk≤αkyk.
Therefore,(37)αkykxk−wk≤αkyk2=βkykγk2=βk2ykmaxλk,yk2≤βk2.
Moreover, sincexk∈C and yk∈∂ϵkfxk,⋅xk, we have(38)fxk,x∗+ϵk=fxk,x∗−fxk,xk+ϵk≥yk,x∗−xk.
It then follows from the definitions ofαk and γk that(39)αk=βkγk=βkmaxλk,yk≤βkλk.
Thus, from (33)–(39), we obtain(40)x∗−wk,xk−wk≤αkfxk,x∗+βkϵkλk+βk2.
But(41)2x∗−wk,xk−wk=wk−x∗2+xk−wk2−xk−x∗2.
Therefore, from (40) and (41), we get(42)wk−x∗2≤xk−x∗2−xk−yk2+2αkfxk,x∗+2βkϵkλk+2βk2.
Now, from Proposition 2.9,x∗∈FTi, and Ti is ρi-demicontractive, for i=1,2,…,N, we have(43)xk+1−x∗2≤δk,1wk−x∗2+∑i=2Nδk,i∏j=1i−11−δk,jTi−1wk−x∗2+∏j=1N1−δk,jTNwk−x∗2−δk,1∑i=2Nδk,i∏j=1i−11−δk,jwk−Ti−1wk2+∏j=1N1−δk,jwk−TNwk2≤δk,1wk−x∗2+∑i=2Nδk,i∏j=1i−11−δk,jwk−x∗2+ρi−1wk−Ti−1wk2+∏j=1N1−δk,jwk−x∗+ρNTNwk−wk2−δk,1∑i=2Nδk,i∏j=1i−11−δk,jwk−Ti−1wk2+∏j=1N1−δk,jwk−TNwk2≤wk−x∗2−Dk,where(44)Dk=δk,1−ρ∑i=2Nδk,i∏j=1i−11−δk,jwk−Ti−1wk2+∏j=1N1−δk,jwk−TNwk2.
From (42) and (43), we have(45)xk+1−x∗2≤xk−x∗2−xk−wk2+2αkfxk,x∗+2βnϵkλk+2βk2−Dk.
Furthermore, sincex∗∈Ω, then fx∗,x≥0 for all x∈C. Therefore, by the pseudomonotonicity of f with respect to x∗, we have fx,x∗≤0, for all x∈C. It then follows from xk∈C that fxk,x∗≤0. Hence, we have from (45) that(46)xk+1−x∗2≤xk−x∗2+2βkϵkλk+2βk2.
Applying Lemma2 to (46), we obtain the existence of(47)l≔limk⟶∞xk−x∗.
Sincef is pseudomonotone on C and fx∗,xk≥0, we have −fxk,x∗≥0. Thus, from (45), we obtain(48)0≤Dk+2αk−fxk,x∗≤xk−x∗2−xk+1−x∗2+2βkϵkλk+2βk2.
Summing up the inequality (48), for every k, we have(49)0≤∑k=1∞Dk+∑k=1∞2αk−fxk,x∗≤x1−x∗2+2∑k=1∞βkϵkλk+2∑βk2.
Since the sequencesyk and λk are bounded, then there exists a constant M>0 such that maxλk,yk≤M. Thus, we have(50)αk=βkγk=βkmaxλk,yk≥βkM.
Thus, from (49), we obtain(51)0≤∑k=1∞Dk+2M∑k=1∞βk−fxk,x∗≤∑k=1∞Dk+2∑k=1∞αk−fxk,x∗≤+∞.
Thus,∑k=1∞Dk<+∞ and ∑k=1∞βk−fxk,x∗<+∞.
Since∑k=1∞βk=∞ and −fxk,x∗≥0, we have that(52)limsupk⟶∞fxk,x∗=0.
For anyx∗∈Ω, suppose that xkτ is the subsequence of xk such that(53)limsupk⟶∞fxk,x∗=limτ⟶∞fxkτ,x∗,and without loss of generality, we may assume that xkτ⇀x¯ as τ⟶∞ for some x¯∈C.
Next, we show thatx¯ is in EPC,f. Observe that since f⋅,x∗ is weakly upper semicontinuous, we have(54)fx¯,x∗≥limsupτ⟶∞fxkτ,x∗=limτ⟶∞fxkτ,x∗=limsupkfxk,x∗=0.
But sincef is pseudomonotone with respect to x∗ and fx∗,x¯≥0, we have(55)fx¯,x∗≤0.
From (54) and (55), we conclude that fx¯,x∗=0. Thus, by condition B6, we obtain that x¯ is in EPC,f.
Since∑k=1∞Dk<+∞, we have that(56)wk−Tiwk⟶0,k⟶∞,i=1,2,…,N.
Furthermore, withwk=PCxk−αkyk and xk∈C, we have(57)xk−wk2≤αkyk,x∗−wk≤αkykxk−wk=βkmaxλk,ykykxk−wk≤βkxk−wk,which implies that, and therefore(58)xk−wk⟶0,k⟶∞.
It then follows fromxkτ⇀x¯ that wkτ⇀x¯. Since I−Ti,i=1,2,…,N are demiclosed at 0, and Tiwk−wk⟶0,k⟶∞,i=1,2,…,N. We have that x¯∈∩i=1NFTi.
Now, observe that(59)xk+1−wk2≤δk,1wk−wk2+∑i=2Nδk,i∏j=1i−11−δk,jTi−1wk−wk2+∏j=1N1−δk,jTNwk−wk2⟶0,k⟶∞.
Consequently,(60)xk+1−xk≤xk+1−wk+wk−xk⟶0,k⟶∞.
We now show that(61)limk⟶∞xk=limk⟶∞wk=limk⟶∞PΩxk=x¯.
It follows from (46) that, for all x∗∈Ω,(62)xk+1−x∗2≤xk−x∗2+ξk,where ξk≔2βkϵk/λk+2βk2>0, for all k≥0, and ∑k=1∞ξk<+∞.
Now, using Lemma1, we have(63)xk+1−PΩxk+12≤xk+1−PΩwk2≤δk,1wk−PΩwk2+∑i=2Nδk,i∏j=1i−11−δk,jTi−1wk−PΩwk2+∏j=1N1−δk,jTNwk−PΩwk2≤δk,1wk−PΩwk2+∑i=2Nδk,i∏j=1i−11−δk,jTi−1wk−wk2−∑i=2Nδk,i∏j=1i−11−δk,jwk−PΩwk2+∏j=1N1−δk,jTNwk−wk2−∏j=1N1−δk,jwk−PΩwk2.
That is,(64)xk+1−PΩxk+12≤δk,1wk−PΩwk2+∑i=2Nδk,i∏j=1i−11−δk,jTi−1wk−wk2+∏j=1N1−δk,jTNwk−wk2−1−δk,1wk−PΩwk2≤2δk,1−1wk−PΩwk2+∑i=2Nδk,i∏j=1i−11−δk,jTi−1wk−wk2+∏j=1N1−δk,jTNwk−wk2⟶0,k⟶∞.
For allm>k, since Ω is convex, we have 1/2PΩxm+PΩxk∈Ω. Therefore,(65)PΩxm−PΩxk2=2xm−PΩxm2+2xm−PΩxk2−4xm−12PΩxm+PΩxk2≤2xm−PΩxm2+2xm−PΩxk2−4xm−PΩxm2=2xm−PΩxk2−2xm−PΩxm2.
Replacingx∗ with PΩxk in (62), we have that(66)xm−PΩxk2≤xm−1−PΩxk2+ξm−1≤xm−2−PΩxk2+ξm−1+ξm−2⋮≤xk−PΩxk2+∑j=km−1ξj.
From (65) and (66), we have(67)PΩxm−PΩxk2≤xk−PΩxk2−2xm−PΩxm2+∑j=km−1ξj.
Hence,(68)limm⟶∞,k⟶∞PΩxm−PΩxk2=0,which implies that PΩxk is a Cauchy sequence. Hence, PΩxk strongly converges to some point z¯∈Ω. However, τ⟶∞, we obtain in the limit that(69)z¯=limτ⟶∞PΩxkτ=PΩx¯=x¯∈Ω.
Thus, from (64), we conclude that xk⟶∞, and consequently wk⟶x¯.Corollary 1.
LetC be a closed and convex subset of a real Hilbert space H, and let f:C×C⟶ℝ be a bifunction that satisfies conditions (B1)–(B5). Let Ti:C⟶C,i=1,2,…,N be a finite family of ρi-strictly pseudononspreading mappings, such that Ω≔EPC,f∩∩i=1NFTi≠Ø. Then, the sequences xk and wk generated by Algorithm 3 converge strongly to a point p∈Ω, where p=limk⟶∞PΩxk.Corollary 2.
Let C be a closed and convex subset of a real Hilbert spaceH and let f:C×C⟶ℝ be a bifunction that satisfies conditions (B1)–(B5). Let Ti:C⟶C,i=1,2,…,N be a finite family of ρi-strictly pseudocontractive mappings, such that Ω≔EPC,f∩∩i=1NFTi≠Ø. Then, the sequences xk and wk generated by Algorithm 3 converge strongly to a point p∈Ω, where p=limk⟶∞PΩxk.


## 4. Numerical Example

We now give a numerical example to demonstrate the performance and efficiency of our proposed iterative algorithm. LetH=ℝ and C=−3,1. Set fx,y=2y−2x and define Ti:C⟶C,i=1,2,…,N by(70)Tix=x,ifx∈−3,0;−3ix1+i,ifx∈0,1.Observe thatΩ≔EPC,f∩∩i=1NFTi=−3. Moreover, ρi=2i−1/1+4i, and thus letting N=5, we have that ρ=max1≤i≤5ρi=3/7. Take λk=1,ϵk=0,βk=1/k and δk,i=k+4i/2k+7i. Therefore, (32) is expressed as an iteration:(71)x1∈C;wk=−3ifxk−βk<−3;xk−βkifxk−βk∈−3,1;1,ifxk−βk>1,xk+1=δk,1wk+∑i=2Nδk,i∏j=1i−11−δk,jTi−1wk+∏j=1N1−δk,jTNwk.We make different choices ofx1 and use xn+1−xn/x2−x1<0.00001 for stopping criterion. Figures 1–4 are the graphs of the numerical computations of Algorithm 4.2 (71) corresponding, respectively, to x1=−1,x1=0,x1=1, and x1=−3.Figure 1
x
1
=
−
1.Figure 2
x
1
=
0
.Figure 3
x
1
=
1
.Figure 4
x
1
=
−
3.



---

*Source: 3183529-2020-03-20.xml*