a strong convergence and numerical illustration for the

A STRONG CONVERGENCE AND NUMERICAL ILLUSTRATION FOR THE ITERATIVE METHODS TO SOLVE A SPLIT COMMON NULL POINT PROBLEM AND A VARIATIONAL INEQUALITY IN HILBERT SPACES

Nguyen Thi Dinh¹, Pham Thanh Hieu^2*

1Hanoi University of Science and Technology

2TNU -University of Agriculture and Forestry

ARTICLE INFO Received:

Revised:

Published:

KEYWORDS Split feasibility problem Null point problem Variational inequality Hillbert spaces

Nonexpansive mapping

ABSTRACT

In this paper, we introduce two iterative methods to approximate the so- lution of a split common null point problem and a variational inequality problem in Hilbert spaces. These problems have many important applica- tions in the fields of signal processing, image processing, optimal control and many other mathematical problems as well as real word situations. The considered methods are generated based on the Halpern method and the viscocity one which have been applied for many other problems such as the fixed point problem and the variational inequality. The strong conver- gence of the method is proven with some certain conditions imposed on the parameters. Finally, a numerical example for solving an optimization problem in Euclidean spaces is given to illustrate the strong convergence of the proposed methods.

SỰ HỘI TỤ MẠNH VÀ VÍ DỤ SỐ CHO PHƯƠNG PHÁP LẶP GIẢI BÀI TOÁN KHÔNG ĐIỂM CHUNG TÁCH VÀ BÀI TOÁN BẤT ĐẲNG THỨC BIẾN PHÂN TRONG KHÔNG GIAN HILBERT

Nguyễn Thị Dinh¹,Phạm Thanh Hiếu^2*

1Trường Đại học Bách khoa Hà Nội

2Trường Đại học Nông Lâm - ĐH Thái Nguyên

THÔNG TIN BÀI BÁO Ngày nhận bài:

Ngày hoàn thiện:

Ngày đăng:

TỪ KHÓA Bài toán chấp nhận tách Bài toán không điểm Bất đẳng thức biến phân Không gian Hilbert Ánh xạ không giãn

TÓM TẮT

Trong bài báo này, chúng tôi giới thiệu hai thuật toán lặp để giải bài toán không điểm chung tách và bài toán bất đẳng thức biến phân đơn điệu trong không gian Hilbert. Các bài toán này có nhiều ứng dụng quan trọng trong những lĩnh vực như xử lý tín hiệu, xử lý ảnh, điều khiển tối ưu và nhiều lĩnh vực khác của toán học cũng như trong đời sống. Các phương pháp mà chúng tôi đề xuất dựa trên phương pháp lặp Halpern và phương pháp xấp xỉ mềm đã được áp dụng để giải các bài toán điểm bất động và bất đẳng thức biến phân. Sự hội tụ mạnh của thuật toán đã được chứng minh cùng với một số điều kiện nhất định đặt lên các dãy tham số. Cuối cùng chúng tôi đưa ra một ví dụ số giải bài toán tối ưu trong không gian hữu hạn chiều để minh họa cho sự hội tụ của thuật toán.

*Corresponding author:Email:phamthanhhieu@tuaf.edu.vn

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05/11/2021 05/11/2021

05/11/2021 05/11/2021 09/6/2021 09/6/2021

DOI: https://doi.org/10.34238/tnu-jst.4619

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TNU Journal of Science and Technology

1. Introduction

LetCandQbe nonempty closed and convex subsets of real Hilbert spacesH₁andH₂, respectively. Let T :H1→H2be a bounded linear operator and letT^∗:H2→H1be the adjoint ofT. The split feasibility problem (SFP) is formulated as follows:

find an elementx^∗∈S=C∩T⁻¹(Q). (1)

Problem (1) first introduced by Censor and Elfving [1] for modeling inverse problems plays an important role in many disciplines, such as, medical image reconstruction and signal processing. Recently, (1) has been attracted by many mathematicians (see for example, [2] - [4] and references there in).

Problem (1) also consists of so called the convex constrained linear inverse problem as a special case, which is formulated as finding an elementx^∗∈H₁such that

x^∗∈CandT x^∗=b∈Q.

In caseCis a closed convex subset of a Hilbert spaceH, henceCis the set of null points of the maximal monotone operatorA, which is defined byA=∂i_C, wherei_Cis the indicator function ofCand∂i_C is the subdifferential operator ofi_C.So,(1.1)can be deduced from the split common null point problem (SCNPP), which consists of finding a pointx^∗∈H₁such that

0∈A₁(x^∗)and 0∈A₂(T x^∗), (2)

whereA_i:H_i→2^Hi,i=1,2 are maximal monotone operators.

LetH₁,H₂be real Hilbert spaces,A:H₁→2^H1 andB:H₂→2^H2be maximal monotone operators onH₁, H₂respectively. LetS:H₁→H₁be a nonexpansive mapping andT:H₁→H₂be a bounded linear operator.

The split common null point problem and a fixed point problem are defined by: findx^†∈H₁such as x^†∈Ω:=A⁻¹0^\T⁻¹(B⁻¹0)^\Fix(S). (3) Now consider the variational inequality problem in Hilbert space. The theory of variational inequality problem (in short VIP) was first considered by Stampacchia [5] in the early 1960s. Since then, it has played the important rule in optimization and nonlinear analysis. To be practice, letCbe a nonempty, closed and convex subset of the Hilbert spaceH, andF:H−→Hbe a mapping. Then the VIP is defined as follow:

Findx^∗∈Csuch thathF(x^∗),x−x^∗i ≥0 ∀x∈C. (4) The (VIP) defined forCandF can be denoted by VIP(C,F). The applications of the above problems, (SFP)-(SCNP)-(NPF)-(VIP), have been studied and mentioned in many publications, to name a few see [5] - [8].

In 2020, Tuyen et al. [4] proved the strong convergence of their hybrid method for solving the split common null point problem and the fixed point problem of a nonexpansive mapping. The medthod is based on the Halpern method [9] and the viscocity one which have been used for fixed point problems and variational inequalities (see [4,8] and references therein). In this paper, based on the method introduced in [4], we present two iterative schemes to solve (2)-(4) for finding a common solution of the split common null point problem and the variational inequality in Hilbert spaces.

The remaining part of this paper is organized as follows: the next section are some notations, definitions and lemmas that will be used for the validity and convergence of the algorithm. The third section is devoted to the proof of our strong convergence result. In Section 4, a numerical example is also discussed to illustrate the convergence of the proposed methods.

2. Preliminaries

LetH be a real Hilbert space. In what follows, we writex^k−→xto indicate that the sequence{x^k} converges strongly toxwhilex^k*xto indicate that the sequence{x^k}converges weakly tox.

Definition 2.1. A mappingF:C−→Cis said to beL-Lipschitz continuous if there exists a positive constant Lsuch that

kF(x)−F(y)k ≤Lkx−yk ∀x,y∈C.

If 0<L<1,Fis said to be contraction mapping. IfL=1,Fis said to be nonexpansive mapping.

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Definition 2.2. LetS:C−→Cbe a nonexpansive mapping. A pointx∈Cis said to be a fixed point ofFif F(x) =x.

Throughout this paper, we denote the set of all fixed points ofFby Fix(S), i.e. Fix(S) ={x∈C|S(x) =x}.

Definition 2.3. LetCbe a set in the Hilbert spaceH. For eachx∈H,PC(x)is an element inCsuch that:

kx−P_C(x)k ≤ kx−yk ∀y∈C.

The mappingP_C:H−→Cis called the metric projection fromHontoC.

The metric projectionP_Cis a nonexpansive mapping.

For an operatorA:H→2^H, we define its domain, range, and graph as follows:

D(A) ={x∈H:A(x)6=/0}

R(A) =∪{Az:z∈D(A)}

andG(A) ={(x,y)∈H×H:x∈D(A),y∈A(x)},respectively. The inverseA⁻¹ofAis defined byx∈A⁻¹(y) if and only ify∈A(x). The operatorAis said to be monotone if, for eachx,y∈D(A),hu−v,x−yi ≥0 for allu∈A(x)and v∈A(y). We denote byI^H the identity operator onH. A monotone operatorA is said to be maximal monotone if there is no proper monotone extension ofAorR I^H+λA

=H for all λ >0. If A is monotone, then we can define, for eachλ >0, a nonexpansive single-valued mapping J^A

λ :R I^H+λA

→D(A)by

J_λ^A= I^H+λA⁻¹

which is called the resolvent ofA. A monotone operatorAis said to satisfy the range condition ifD(A)⊂ R I^H+λA

for allλ >0, whereD(A)denotes the closure of the domain ofA. For a monotone operatorA, which satisfies the range condition, we haveA⁻¹(0) =Fix J^A

λ

for allλ >0.IfAis a maximal monotone operator, thenAsatisfies the range condition.

The following lemmas will be needed in what follows for the proof of the main results in this paper.

Lemma 2.1. [4] Let A:D(A)⊂H→2^H be a monotone operator. Then, we have the following statements:

(i) for r≥s>0, we have

x−J_s^Ax ≤2

x−J_r^Ax , for all x∈R I^H+rA

∩R I^H+sA

; (ii) for all r>0and for all x,y∈R I^H+rA

, we have x−y,J_r^Ax−J_r^Ay

≥

J_r^Ax−J_r^Ay

2; (iii) for all r>0and for all x,y∈R I^H+rA

, we have h I^H−J_r^A

x− I^H−J_r^A

y,x−yi ≥ k I^H−J_r^A

x− I^H−J_r^A yk²; (iv) ifΩ=A⁻¹(0)6=/0, then, for all x^∗∈Ωand for all x∈R I^H+rA

, J_r^Ax−x^∗

2≤ kx−x^∗k²−

x−J_r^Ax

2.

Lemma 2.2. [6] Let A:D(A)⊂H→2^Hbe a monotone operator. Then, forλ,µ>0, and x∈D(A), we have J_λ^Ax=J_µ^Aµ

λx+

1−µ λ

J_λ^Ax .

Lemma 2.3. [7] Let{s_n}be a sequence of nonnegative numbers,{α_n}be a sequence in(0,1)and{c_n}be a sequence of real numbers satisfying the conditions:

(i) s_n+1≤(1−α_n)s_n+α_nc_n, (ii) ∑^∞_n=0αn=∞,lim sup_n→∞cn≤0.

Thenlimn→∞s_n=0.

Lemma 2.4. [10] Let{x_n},{y_n}be bounded sequences in a Hilbert space H and{β_n}be a sequence in(0,1) with 0 < lim infn→∞βn ≤ lim sup_n→∞βn < 1.

Let xn+1= (1−β_n)y_n+βnxnfor all n≥0andlim sup_n→∞(ky_n+1−ynk − kx_n+1−xnk)≤0.

Thenlimn→∞kx_n−y_nk=0.

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3. Main Results

Theorem 3.1. Let H₁and H₂be two real Hilbert spaces. Let A:H₁→2^H1 and B:H₂→2^H2 be two maximal monotone operators on H1and H2, respectively. Let T:H1→H2be a bounded linear operator from H1onto H₂. Suppose thatΩ=A⁻¹0∩T⁻¹ B⁻¹0

6=/0. If the conditions(C1)-(C4) as follows are satisfied, (C1) min{inf_n{γ_n^A}, inf_n{γ_n^B}}=r>0,

limn→∞

γ_n+1^A −γ_n^A =0;

(C2) 0<lim inf_n→∞βn≤lim sup_n→∞βn<1;

(C3) ∑^∞n=1α_n=∞andlimn→∞α_n=0;

(C4) δ∈(0, ²

kTk²);

then for any u,x₀∈H₁, the sequence{x_n}generated by











y_n =J^A

γ_n^Ax_n, z_n =J^B

γ_n^B(Ty_n),

t_n =y_n+δT^∗(z_n−Ty_n),

x_n+1 =βnx_n+ (1−βn)[α_nu+ (1−αn)t_n],n≥0,

(5)

where γ_n⁴ ,

γ_n^B ,{β_n}, and{α_n}, converges strongly to x⁺=P_Ω^H1u as→∞.

Proof. Let p∈Ω, fromp∈A⁻¹0 and Lemma2.1(iv), we have ky_n−pk²≤ kx_n−pk²− kx_n−J^A

γ_n^Ax_nk². (6)

SinceT p∈B⁻¹0,Bis a maximal monotone operator, thenT p=J^B

γ_n^B(T p), using Lemma2.1(iv), we get kz_n−T pk²≤ kTy_n−T pk²− kTy_n−J^B

γ_n^B(Ty_n)k². (7)

Based on Lemma2.1(iii), we have the following evaluations

kt_n−pk²=ky_n−pk²+δ²kT^∗(z_n−Ty_n)k²+2δhy_n−p,T^∗(z_n−Ty_n)i

=ky_n−pk²+δ²kT^∗(z_n−Ty_n)k²+2δhT(y_n−p),z_n−Ty_ni

=ky_n−pk²+δ²kT^∗(z_n−Ty_n)k²−2δhTy_n−T p,−J^B

γ_n^B(Ty_n) +Ty_n+T p−J^B

γ_n^B(T p)i

≤ ky_n−pk²−δ(2−δkTk²)kz_n−Ty_nk²

≤ kx_n−pk²− kx_n−J^A

γ_n^Ax_nk²−δ(2−δkTk²)kz_n−Ty_nk². (8) Letd_n=αnu+ (1−α_n)t_n.From (8), we have

kd_n−pk ≤α_nku−pk+ (1−α_n)kt_n−pk ≤α_nku−pk+ (1−α_n)kx_n−pk. (9) Hence, from (9) we can deduce that

kx_n+1−pk ≤βnkx_n−pk+ (1−β_n)kd_n−pk

≤βnkx_n−pk+ (1−β_n)kd_n−pk

≤[1−α_n(1−βn)]kx_n−pk+α_n(1−βn)ku−pk

≤max{kx_n−pk,ku−pk} ≤. . .≤

≤max{kx₀−pk,ku−pk}.

Therefor, sequence{x_n} is bounded. We also deduce from (6)-(8) that sequences{y_n},{z_n} v`a{t_n}are bounded.

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Now, we are showing that limn→∞kxn+1−x_nk=0. In deed, from Lemma2.2, we have ky_n+1−y_nk=

J^A

γ_n+1^A x_n+1−J^A

γ_n+1^A

γ_n+1^A

γ_n^A x_n+ (1−γ_n+1^A γ_n^A )J^A

γ_n^Ax_n

≤ kx_n+1−x_nk+|γ_n+1^A −γ_n^A|

γ_n^A kx_n−J^A

γ_n^Ax_nk

≤ kx_n+1−x_nk+K₁|γ_n+1^A −γ_n^A|, (10) whereK1=

sup_n{kxn−J^A

γA n

xnk}

r <∞.Similarly, we also have

kz_n+1−z_nk ≤Ty_n+1−Ty_nk+K₂|γ_n+1^B −γ_n^B|, (11) withK₂=

sup_n{kTy_n−J^B

γB n

(Tyn)k}

r <∞.From Lemma2.1(iii), we get

ktn+1−t_nk²=kyn+1−y_nk²+δ²kT^∗[(zn+1−Tyn+1)−(z_n−Ty_n)]k² +2δhyn+1−y_n,T^∗[(z_n+1−Tyn+1)−(z_n−Ty_n)]i

≤ ky_n+1−y_nk²+δ²kTk²k(z_n+1−Ty_n+1)−(z_n−Ty_n)k² +2δhTy_n+1−Ty_n,(z_n+1−Ty_n+1)−(z_n−Ty_n)i

≤ ky_n+1−y_nk²−δ(2−δkTk²)k(z_n+1−Ty_n+1)−(z_n−Ty_n)k². (12) It follows fromdn=αnu+ (1−α_n)t_nthat

kd_n+1−d_nk ≤ |α_n+1−αn|kuk+k(1−α_n+1)t_n+1−(1−αn)t_nk

≤ |α_n+1−αn|kuk+|α_n+1−αn|kt_n+1k+ (1−α_n)kt_n+1−t_nk

≤ kt_n+1−t_nk+K₃|α_n+1−α_n|, (13) whereK₃=kuk+sup_n{kt_nk}<∞.From (10)-(12) v`a (13), we obtain

kd_n+1−d_nk ≤ kx_n+1−x_nk+K₁|γ_n+1^A −γ_n^A|+K₃|α_n+1−αn|.

Then, from the conditions (C1) and (C3), we have

n→∞limsup(kd_n+1−d_nk − kx_n+1−x_nk)≤0.

So, it follows from Lemma2.4that

n→∞limkx_n−d_nk=0. (14)

Hence, we havekx_n+1−d_nk=βnkx_n−d_nk →0,which together with (14) yields that

n→∞limkxn+1−x_nk=0 (15)

Next, we prove that the set of weak cluster points of the sequence{x_n}is contained inΩ. Indeed, we denote ω(x_n)the set of weak cluster points of the sequence{x_n}and suppose thatx^∗is an arbitrarily inω(x_n). Then there is a subsequence

x_nk of{x_n}such thatx_nk→x^∗. From the convexity of the functionk.k²onH₁and (8), we deduce that

kx_n+1−pk²≤βnkx_n−pk²+ (1−βn)kα_nu+ (1−αn)t_n−pk²

≤β_nkx_n−pk²+ (1−β_n)[α_nku−pk²+ (1−α_n)kt_n−pk²]

≤ kx_n−pk²+α_nku−pk²− kx_n−J^A

γ_n^Ax_nk²−δ(2−δkTk²)kz_n−Ty_nk². Thus, we gain

kx_n−J^A

γ_n^Ax_nk²+δ(2−δkTk²)kz_n−Ty_nk²≤(kx_n−pk²− kx_n+1−pk²) +αnku−pk²

≤(kx_n−pk − kx_n+1−pk)kx_n+1−x_nk+αnku−pk².

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It follows fromkxn+1−x_nk →0 and the conditions (C3)-(C4) that

n→∞limkx_n−J^A

γ_n^Ax_nk²=lim

n→∞kz_n−Ty_nk²=0.

So, we have

n→∞limkx_n−J^A

γ_n^Ax_nk=lim

n→∞kJ^B

γ_n^B(Ty_n)−Ty_nk= lim

n→∞kt_n−y_nk=0. (16) From Lemma2.1(i), we get limn→∞kx_n−J_r^Ax_nk=limn→∞kJ_r^B(Ty_n)−Ty_nk=0.

Sincex_nk*x^∗, v`a limn→∞kx_n−y_nk=0, one hasy_n*x^∗. BecauseTis a bounded linear operator,Ty_n*T x^∗. By the use of Lemma2.4, we obtainx^∗∈A⁻¹0, andT x^∗∈B⁻¹0, sox^∗∈A⁻¹0∩T⁻¹(B⁻¹0). Consequently, ω(x_n)⊆Ω.

Finally, we showx_n→x⁺=P_Ω^H1u. By puttingx⁺=P_Ω^H1u, from (7) we get that

kx_n+1−x⁺k=βnhx_n−x⁺,xn+1−x⁺i+ (1−βn)hα_nu+ (1−αn)t_n−x⁺,xn+1−x⁺i

≤βn

kx_n−x⁺k²+kx_n+1−x⁺k²

2 + (1−βn)(1−α_n)kt_n−x⁺k²+kx_n+1−x⁺k² 2

+αn(1−βn)hu−x⁺,x_n+1−x⁺i

≤β_nkx_n−x⁺k²+kxn+1−x⁺k²

2 + (1−β_n)(1−α_n)kx_n−x⁺k²+kxn+1−x⁺k² 2

+α_n(1−β_n)hu−x⁺,xn+1−x⁺i.

Thus,

[1+α_n(1−β_n)]kxn+1−x⁺k²≤[1−α_n(1−β_n)]kx_n−x⁺k²+2(1−β_n)α_nhu−x⁺,xn+1−x⁺i.

The last inequalities imply that

kx_n+1−x⁺k²≤[1−α_n(1−βn)]kx_n−x⁺k²+ (1−β_n)α_n 2

1+ (1−β_n)α_nhu−x⁺,x_n+1−x⁺i. (17) Letsn=kx_n−x⁺k²andcn=_1+α ²

n(1−β_n)hu−x⁺,xn+1−x⁺i. Then the inequality (17) can be rewritten in the following form

s_n+1≤[1−αn(1−βn)]s_n+αn(1−βn)c_n. (18) Now, we will show that lim sup_n→∞c_n≤0.Indeed, suppose that{x_nk}is a subsequence of{x_n}such that

lim sup

n→∞

hu−x⁺,x_n−x⁺i=lim

k→∞hu−x⁺,x_nk−x⁺i.

Since{x_nk} is bounded, there exists a subsequence{x_n

kl}of{x_nk}such that x_nkl *x^∗. Without loss of generality, we writexn_k*x^∗. Fromω(x_n)⊆Ωsox^∗∈Ω. Fromx⁺=P_Ω^H1uand (5), it is deduced that

lim sup

n→∞

hu−x⁺,x_n−x₊i=hu−x⁺,x^∗−x⁺i ≤0,

which together with (15) and conditions (C2)-(C3), we get lim sup_n→∞c_n≤0. Since∑^∞n=1αn=∞ and condition (C2), we have∑^∞_n=1αn(1−βn) =∞. Hence, all conditions of Lemma2.3are sastified. Therefore, we immediately deduce thatsn→0, that isxn→x⁺=P_Ω^H1. This completes the proof.

In the next method, we use the viscocity approximation one to solve a common null point problem and a variational inequality.

Theorem 3.2. If the conditions (C1)-(C4) are satisfied, then the sequence{e_n}generated by











u_n =J^A

γ_n^Ae_n, v_n =J^B

γ_n^B(Tu_n),

w_n =u_n+δT^∗(v_n−Tu_n),

en+1 =β_ne_n+ (1−β_n)[α_nf(e_n) + (1−α_n)w_n],n≥0,

(19)

where f :H₁→H₁is a contractive mapping from H₁into itself with the contraction coefficient c∈(0,1), converges strongly to x^∗∈Ω=A⁻¹0∩T⁻¹(B⁻¹0)which is the unique solution of the variational inequality

h(I−f)x^∗,y−x^∗i ≥0, ∀x∈Ω. (20)

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226(15): 20 - 27

25

Proof. BecauseP_Ω^H1f is contractive mapping, Banach contraction mapping principle guarantees thatP_Ω^H1f has a unique fixed pointx^∗which is also the unique solution of the variational inequality (20).

From Theorem3.1, replacinguby f(x^∗)in (5), we have the sequence{x_n}converging strongly to x^∗=P_Ω^H1f(x^∗).

Now we only need to prove thatke_n−x_nk →0, asn→∞. Note that

ke_n+1−x_n+1k ≤βnke_n−x_nk+ (1−βn)[α_ncke_n−x^∗k+ (1−αn)kw_n−t_nk].

From Lemma2.1(iii), we have

kw_n−t_nk²=ku_n−y_nk²+δ²kT^∗[(v_n−Tu_n)−(z_n−Ty_n)]k²+2δhu_n−y_n,T^∗[(v_n−Tu_n)−(z_n−Ty_n)]i

≤ ku_n−y_nk²−δ(2−δkTk²)k(v_n−Tu_n)−(z_n−Ty_n)k²

≤ ku_n−y_nk². (21)

From the nonexpansiveness ofJ^A

γ_n^A, we have

ku_n−y_nk ≤ ke_n−x_nk. (22) Thanks to (21)-(22), we obtain

ken+1−xn+1k ≤β_nke_n−x_nk+ (1−β_n)[α_ncke_n−x^∗k+ (1−α_n)ke_n−x_nk]

≤[1−α_n(1−β_n)]ke_n−x_nk+α_n(1−β_n)c(ke_n−x_nk+kx_n−x^∗k])

= [1−α_n(1−β_n)(1−c)]ke_n−x_nk+α_n(1−β_n)c(kx_n−x^∗k]).

From Lemma2.3, we get limn→∞ke_n−x_nk=0. Thus, limn→∞ke_n−x^∗k=0, we obtain that{e_n}generated by (19) converges strongly tox^∗=P^H1

Ω f(x^∗).

4. Numerical Results

In the following example, for the convergence illustration of iterative methods (5) and (19) studied in Theorems3.1and3.2, respectively, we present a numerical example which is finding a common solution of a common null point problem and a variational inequality in Euclidean spaces. We perform the iterative schemes in MATLAB R2016a running on a laptop with Intel(R) Core(TM) i3-5200U CPU @ 2.20 GHz, RAM 10 GB.

LetH₁=R²,H₂=R⁵.Mapping f :R²→R², where f(x) = 1

2xis a contractive mapping with the contration coefficientc=¹₂.InR², the maximal monotone operatorAis defined as

A(x₁,x₂) = (2x₁+2x₂,2x₁+2x₂)^T, and inH₂, the maximal monotone operatorBis defined as

Bx=











0 0 0 0 0

0 1 1 0 0

0 −1 1 0 0

0 0 0 1 1

0 0 0 0 1









 x,

wherex= (x₁,x₂,x₃,x₄,x₅)∈R⁵.LetT:R²→R⁵be a bounded linear operator defined by T(x₁,x2) = (x₁+x2,2x₁,3x₁+4x₂,0,x1+x2)^T,

such thatΩ=A⁻¹0∩T⁻¹(B⁻¹0)6=/0.

We will use method (5) to solve the null point problem which is finding ax⁺∈Ωsuch thatx⁺=P_Ω^H1ufor any u,x₀∈H₁, knowing that the exact solution of the considered problem isx^∗= (0,0)∈R². The following table shows the approximate solutionsx_n= (x¹_n,x²_n)∈R²of the above problem with the corresponding parameters.

Next, we findx⁺∈Ωthat is also the solution of the variational inequalityh(I−f)x^∗,y−x^∗i ≥0, ∀y∈Ω.

Using (19), we have the approximate solution shown in following table.

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Iter(n) x¹_n x²_n Iter(n) x¹_n x²_n

1 -2.0000 1.0000 50 -0.052869 0.039805

2 -1.4022 0.79506 100 -0.0066537 0.0049955 3 -1.0927 0.68717 300 -6.6336e-06 -1.2403e-06 10 -0.45866 0.34394 500 -1.7653e-06 -2.4079e-06

Table 1: x0= (−2,1)^T,u= (−1/100,−1/100)^T,δ=0.05,γ_n^A=γ_n^B=1,βn=¹₂,αn=_n+1¹ Iter(n) x¹_n+1 x²_n+1 Iter(n) x¹_n+1 x²_n+1

1 -2.0000 1.0000 50 -0.13956 0.10517

2 -1.6519 0.92031 100 -0.020987 0.015816

3 -1.4221 0.87588 300 -1.5558e-05 1.1725e-05 10 -0.80129 0.6003 500 -1.3146e-08 9.9068e-09 Table 2: x0= (−2,1)^T,δ =0.05, γ_n^A=γ_n^B=1,β_n=¹₂,αn=_n+1¹

5. Conclusion

We have presented in this paper the two iterative methods based on Halpern method and the viscosity one to solve a common null point problem and a variational inequality in Hilbert spaces. The strong convergence of the methods is proven under some certain assumptions and a numerical example for the convergence illustration of the proposed method is given.

Acknowledgments

The two authors would like to thank the refrees for their useful suggestions and comments that help to improve the presentation of this paper.

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