A strong convergence theorem for a zero of the sum of a finite family of maximally monotone mappings

Getahun B. Wega; Habtu Zegeye; Oganeditse A. Boikanyo

doi:10.1515/dema-2020-0010

Article Open Access

A strong convergence theorem for a zero of the sum of a finite family of maximally monotone mappings

Getahun B. Wega , Habtu Zegeye and Oganeditse A. Boikanyo

Published/Copyright: July 8, 2020

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From the journal Demonstratio Mathematica Volume 53 Issue 1

Abstract

The purpose of this article is to study the method of approximation for zeros of the sum of a finite family of maximally monotone mappings and prove strong convergence of the proposed approximation method under suitable conditions. The method of proof is of independent interest. In addition, we give some applications to the minimization problems and provide a numerical example which supports our main result. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

Keywords: firmly nonexpansive; Hilbert spaces; maximally monotone mapping; strong convergence; zero points

MSC 2010: 47H04; 47H05; 47H10; 47J25

1 Introduction

Let H be a real Hilbert space with inner product 〈.,.〉 and induced norm ∥⋅∥. Recall that for a mapping A:H→2H, the domain of A, Dom(A), is given by Dom(A)={x∈H:Ax≠∅}, the graph of A, Gph(A), is given by Gph(A)={(x,y)∈H×H:y∈Ax} and the range of A, ran(A), is given by ran(A)={Ax:x∈Dom(A)}. The mapping A is called monotone if

(1)〈u−v,x−y〉≥0,∀(x,u),(y,v)∈Gph(A),

and it is called maximally monotone if it is monotone and the graph of A is not properly contained in the graph of any other monotone mapping. The resolvent of A with parameter λ>0 is JλA=(I+λA)−1, where I is the identity mapping on H, and it enjoys firmly nonexpansive property, that is, for any x,y∈ran(I+λA), we have

(2)∥JλAx−JλAy∥2≤〈x−y,JλAx−JλAy〉.

A monotone mapping A:H→H is called α-inverse strongly monotone if there exists a positive real number α such that for any x,y∈H we have

(3)〈Ax−Ay,x−y〉≥α∥Ax−Ay∥2.

Let Ak:H→2H,k=1,2,…,m, be maximally monotone mappings. Consider the inclusion problem of finding z∈H such that

(4)0∈A1z+A2z+⋯+Amz,

where m≥2. We denote the solution set of (4) by zero(A1+A2+⋯+Am)=(A1+A2+⋯+Am)−1(0). This problem, which includes variational inequality problems, equilibrium problems, complementary problems, minimization problems, nonlinear evolution equations, fixed point problems as special cases, is quite general. In fact, a number of problems arising in applied areas such as image recovery, machine learning and signal processing can be mathematically modeled as (4), see [1,2] and references therein. To be more precise, a stationary solution to the initial value problem of the evolution equation

(5)0∈F(t)+∂x∂t,x(0)=x0

can be formulated as (4) when the governing maximal monotone F is of the form F≔A1+A2+⋯+Am (see, e.g., [3]). Furthermore, optimization problems often need (see, e.g., [4]) to solve a minimization problem of the form

(6)minx∈H{g1(x)+g2(x)+⋯+gm(x)},

where gi,i=1,2,…,m are proper lower semicontinuous convex functions from H to the extended real line R¯≔(−∞,∞]. If in (6), we assume that Ai≔∂gi, for i=1,2,…,m, where ∂ of gi is the subdifferential operator of gi in the sense of convex analysis, then (6) is equivalent to (4). Consequently, considerable research efforts have been devoted to methods of finding approximate solutions (when they exist) of inclusions of the form (4) for a sum of a finite number of monotone mappings (see, e.g., [3,5]).

For the case where m=2, the inclusion problem (4) reduces to the problem of finding z∈H such that

(7)0∈Az+Bz,

where A and B are monotone mappings. For solving problem (7), several authors have studied different iterative schemes (see, e.g., [6,7,8,9,10,11,12,13,14,15,16] and references therein). The most attractive methods for solving the inclusion problem (7) are the Peaceman-Rachford and Douglas-Rachford iterative methods.

The nonlinear Peaceman-Rachford and Douglas-Rachford, splitting iterative methods, introduced by Lions and Mercier [3], are given by

(8)xn+1=(2JλA−I)(2JλB−I)xn,n ≥ 1,

and

(9)xn+1=JλA(2JλB−I)xn+(I−JλB)xn,n ≥ 1,

respectively, where λ>0 is a fixed scalar. The nonlinear Peaceman-Rachford algorithm (8) fails, in general, to converge (even in the weak topology in the infinite-dimensional setting). This is due to the fact that the generating mapping (2JλA−I)(2JλB−I) is merely nonexpansive. The nonlinear Douglas-Rachford algorithm (9) was initially proposed in [3] for finding a zero of the sum of two maximally monotone mappings and has been studied by many authors (see, e.g., [1,3,11,17,18] and references therein). This method always converges in the weak topology to a solution of (7), since the generating operator JλA(2JλB−I)+(I−JλB) for this algorithm is firmly nonexpansive (see, e.g., [11]).

In 1979, Passty [11] studied the forward-backward splitting method which is given by

(10)xn+1=(I+λnB)−1(I−λnA)xn,n ≥ 1,

where {λn} is a sequence of positive scalars, A and B are maximal monotone mappings. He proved that the sequence in (10) converges weakly to the solution of problem (7). Different authors have used algorithm (10), for the inclusion problem (7), when A is a single-valued α-inversely strong monotone (or α-strongly monotone) mapping and B is a maximal monotone mapping defined in real Hilbert spaces (see, e.g., [18,19]).

We remark that the aforementioned results provide weak convergence. But we also indicate that several authors have studied different iterative methods (see, e.g., [21,22,23,24] and references therein) and proved strong convergent results to approximate zeros of the sum of monotone mappings A and B, where A:H→H is an α-inverse strongly monotone mapping and B:H→2H is a maximally monotone mapping under certain conditions (see, e.g., [19,25,26,27]).

In 2012, Takahashi et al. [19] studied the following Halpern-type iteration in a Hilbert space setting: for any x0∈H,

(11)xn+1=βnxn+(1−βn)(αnu+(1−αn)JrnB(xn−rnAxn)),∀n≥0,

where u∈H is a fixed vector and A is an α-inverse strongly monotone and single-valued mapping on H, and B is a maximally monotone mapping on H. They proved that the sequence {xn} generated by (11) converges strongly to a point x∈(A+B)−1(0) provided that the control sequences {βn}, {αn} and {rn} satisfy appropriate conditions.

Recently, Bot et al. [25] studied the following classical Douglas-Rachford method:

(12)yn=JγB(βnxn);zn=JγA(2yn−βnxn);xn+1=βnxn+λn(zn−yn),n≥1,

where A:H→2H and B:H→2H are maximally monotone mappings and {λn} and {βn} are real sequences satisfying certain conditions and showed that {xn} converges strongly to x¯=PF(RγARγB)(0), as n→∞, where RγA=2JγA−I while {yn} and {zn} converge strongly to JγB(x¯)∈zer(A+B), as n→∞.

More recently, Wega and Zegeye [15] constructed an algorithm that converges strongly to a solution of the sum of two maximally monotone mappings using a different technique.

Question 1

A natural question arises whether we can obtain a strong convergence result for approximating zeros of the sum of a finite family of maximally monotone mappings via the extended solution set of the sum of maximally monotone mappings?

In 2009, Svaiter [28] constructed a new approach for splitting algorithms, which starts by reformulating (4) as the problem of locating a point in a certain extended solution setSe(A1,A2,…,Am) subset of H×Hm, which is defined by:

Se(A1,…,Am)={(z,w1,…,wm)∈V:z∈H,wk∈Ak(z),k=1,2,…,m},

where

V={(z,w1,…,wm)∈H×Hm:w1+⋯+wm=0}.

He proved weak convergence results provided that H has a finite dimension or A1+A2+⋯+Am is maximal monotone.

We remark that the extended solution set is associated with the common fixed points of a countable family of nonexpansive mappings and so the methods of approximating fixed points are used to approximate the solution of problem (4).

Motivated and inspired by the above results, our purpose in this article is to construct a viscosity-type algorithm for finding zeros of the sum of a finite family of maximally monotone mappings via the extended solution set Se(A1,A2,…,Am) and discuss its strong convergence. The viscosity method introduced by Moudafi [30] involves a contraction mapping f in the procedure and it can be regarded as a regularization process for the solution of problem (4), which is supposed to induce the convergence in norm of the iterates. Another advantage of this method is that it allows one to select a particular solution point of (4), which satisfies some variational inequality. The assumption that one of the mappings is α-inverse strongly monotone is dispensed with. Our results provide an affirmative answer to our question. Our method of proof is of independent interest. Our results improve and generalize several results in the literature.

2 Preliminaries

In this section, we recall some definitions and known results that will be used in the sequel.

Let C be a nonempty, closed and convex subset of a real Hilbert space H. A mapping T:C→C is said to be a contraction if there exists α∈[0,1) such that ∥Tx−Ty∥≤α∥x−y∥, ∀x,y∈C and it is said to be nonexpansive if α=1. The set of fixed points of T is defined by F(T)≔{x∈C:Tx=x}.

Lemma 2.1

[29] Let C be a closed and convex subset of a real Hilbert space H, andT:C→Cbe a nonexpansive mapping. Let{xn}⊆Candx∈Csuch thatxn⇀xandxn−Txn→0asn→∞. Then,x∈F(T).

The following lemmas shall be used in the later section.

Lemma 2.2

[28] Finding a point inSe(A1,…,Am)is equivalent to solving (4) in the sense that

0∈A1(z)+⋯+Am(z)⇔∃w1,…,wm∈H:(z,w1,…,wm)∈Se(A1,…,Am).

Lemma 2.3

[28] If the monotone operatorsA1,…,Amare maximally monotone, the corresponding extended solution setSe(A1,…,Am)is closed and convex inHm+1.

Lemma 2.4

[28] Given(xk,yk)∈Gph(Ak),k=1,…,m, defineφ:V→ℝvia

(13)φ(z,w1,…,wm)=∑k=1m〈z−xk,yk−wk〉.

Then, for any(z,w1,…,wm)∈Se(A1,…,Am), one hasφ(z,w1,…,wm)≤0, that is,

Se(A1,…,Am)⊆{(z,w1,…,wm)∈V:φ(z,w1,…,wm)≤0}.

In addition, φis affine on V, with

(14)∇φ=∑k=1myk,x1−x¯,x2−x¯,…,xm−x¯,wherex¯=1m∑k=1mxk

and

(15)∇φ=0⇔(x1,y1,…,ym)∈Se(A1,…,Am),x1=x2=⋯=xm⇔φ(z,w1,…,wm)=0, ∀(z,w1,…,wm)∈V.

The function φ in Lemma 2.4 is called decomposable separators.

Lemma 2.5

[31] Let{an}be a sequence of nonnegative real numbers satisfying the following relation:

an+1≤(1−αn)an+αnδn,n≥n0,

where{αn}⊂(0,1) and {δn}⊂ℝsatisfying the following conditions: ∑n=1∞αn=∞, andlimsupn→∞δn≤0or∑n=1∞|αnδn|<∞. Then, limn→∞an=0.

Lemma 2.6

[32] Letx,y∈H. If H is a real Hilbert space, then the following inequality holds:

∥x+y∥2≤∥x∥2+2〈y,x+y〉.

Lemma 2.7

[33] Let C be a closed and convex subset of a real Hilbert space H andx∈Hbe given. The metric projection of H onto C,PC, is characterized by the following:

PCx∈C,〈x−PCx,PCx−z〉≥0, for allz∈C;
PCis firmly nonexpansive and hence nonexpansive.

3 Main results

In this section, we introduce an algorithm for finding a point in Se(A1,…,Am), which will lead us to a solution of the sum of a finite family of maximally monotone mappings in a Hilbert H space and discuss its strong convergence.

In what follows, let H be a real Hilbert space and Ak:H→2H,k=1,…,m, where m≥2, be a family of maximal monotone mappings satisfying Se(A1,…,Am)≠∅. Let {αn}⊂(0,1) such that limn→∞αn=0, ∑n=1∞αn=∞ and ∑n=1∞|αn−αn−1|<∞, and let {βn} be a decreasing sequence in (0,1] with β0=1.

We now propose the following algorithm which basically uses Algorithm 3 of [28].

Algorithm 3.1

Step 0: Select initial guess u0=(z0,w1,0,…,wm,0)∈V.

Step 1: Given nth iterates (n≥0), compute xk,n and yk,n satisfying:

xk,n=(I+λk,nAk)−1(zn+λk,nwk,n),k=1,2,…,m,yk,n=wk,n+zn−xk,nλk,n,k=1,2,…,m,

where the real sequence λk,n>0, for each k=1,2,…,m and n≥0.

Step 2: Define φi:V→ℝ by

(16)φi(z,w1,…,wm)=∑k=1m〈z−xk,i,yk,i−wk〉,

and compute

(17)Tiun=(zn,w1,n,…,wm,n)−max0,φi(zn,w1,n,…,wm,n)∥∇φi∥2∇φi,

where ∇φi=(∑k=1myk,i,x1,i−xi¯,x2,i−xi¯,…,xm,i−xi¯) and ||∇φi||2=∑k=1mxk,i−xi¯2+∑k=1myk,i2 with xi¯=1m∑k=1mxk,i, which is the projection of un=(zn,w1,n,…,wm,n) onto the half space

Hi={(z,w1,…,wm)∈V:φi(z,w1,…,wm)≤0}.

Step 3: Compute

(18)vn=βnun+∑i=1n(βi−1−βi)Tiun,un+1=αnf(un)+(1−αn)vn,∀n≥0,

where f:V→V is a contraction mapping with constant α. Set n≔n+1 and go to step 1.

Remark 3.1

The following points indicate that Algorithm 3.1 is well defined.

By maximality of Ak the resolvent mapping (I+λk,nAk)−1 is single-valued and everywhere defined for each k=1,2,…,m (see, e.g., [34]). By rearranging the equation in step 1, one has the following equation:
(19)xk,n+λk,nyk,n=zn+λk,nwk,n,∀n≥0,
where for each k=1,2,…,m and n≥0, λk,n>0 and yk,n∈Ak(xk,n). Hence, for each k=1,2,…,m, (xk,n,yk,n)∈Gph(Ak) exists and is unique. Thus, the decomposable separator function φ in Lemma 2.4 is well defined.
Since φi:V→ℝ is affine on V (see Lemma 2.4) and the half space Hi is closed and convex subspace of V for all i≥1, the projection of un onto Hi given by (17) exists and is firmly nonexpansive (see, e.g., [20,28]).

Lemma 3.2

The sequence{un}generated by Algorithm 3.1 is bounded.

Proof

Let u∗=(z∗,w1∗,w2∗,…,wm∗)∈Se(A1,…,Am), then by Lemma 2.4 we have u∗∈Hi for all i≥0, which implies that Se(A1,…,Am)⊂⋂i=1∞F(Ti)=ℱ≠∅. Now, the fact that Ti is nonexpansive and (18) yields

(20)||vn−u∗||=βnun+∑i=1n(βi−1−βi)Tiun−u∗=βnun+∑i=1n(βi−1−βi)Tiun−∑i=1n(βi−1−βi)u∗+∑i=1n(βi−1−βi)u∗−u∗=βn(un−u∗)+∑i=1n(βi−1−βi)(Tiun−u∗)≤βn∥un−u∗∥+∑i=1n(βi−1−βi)∥un−u∗∥=∥un−u∗∥.

Thus, it follows that

∥un+1−u∗∥=∥αnf(un)+(1−αn)vn−u∗∥≤αn∥f(un)−u∗∥+(1−αn)∥vn−u∗∥≤αn∥f(un)−f(u∗)∥+αn∥f(u∗)−u∗∥+(1−αn)∥vn−u∗∥≤(1−αn(1−α))∥un−u∗∥+αn∥f(u∗)−u∗∥≤max∥un−u∗∥,∥f(u∗)−u∗∥1−α.

Then, we have

∥un−u∗∥≤max∥u0−u∗∥,∥f(u∗)−u∗∥1−α,

which implies that {un} is bounded. Hence, we can obtain that {Tiun}, {vn} and {f(un)} are bounded.□

Lemma 3.3

The sequence{un}generated by Algorithm 3.1 converges strongly tou∗=(z∗,w1∗,…,wm∗)∈ℱ=⋂i=1∞F(Ti).

Proof

We proceed with the following steps.

Step 1. First we show that limn→∞∥un+1−un∥=0.

From (18), we have

for some M>0 as the sequences {un} and {Tiun−1} are bounded. Thus, the inequalities in (18) and (21) imply that

Since {βn} is strictly decreasing, it implies that ∑n=1∞(βn−1−βn)<∞. Hence, from (22), conditions of {αn} and Lemma 2.5, we immediately obtain that

(23)||un+1−un||→0asn→ ∞.

Step 2. We show that ||Tiun−un||→0 as n→∞. Take u∗=Pℱ(f(u∗)). Note that

(24)∥un−u∗∥2≥∥Tiun−Tiu∗∥2=∥Tiun−un+un−Tiu∗∥2=∥Tiun−un∥2+∥un−u∗∥2+2〈Tiun−un,un−u∗〉,

which yields

(25)12∥Tiun−un∥2≤〈un−Tiun,un−u∗〉.

Furthermore, from (18) and (25), we immediately obtain

(26)12∑i=1n(βi−1−βi)∥Tiun−un∥2≤∑i=1n(βi−1−βi)〈un−Tiun,un−u∗〉=〈(1−βn)un−∑i=1n(βi−1−βi)Tiun,un−u∗〉=〈(1−βn)un−vn+βnun,un−u∗〉=〈un−vn,un−u∗〉=〈un−un+1,un−u∗〉+〈un+1−vn,un−u∗〉,

and from (18) and (26), we have

(27)12∑i=1n(βi−1−βi)||Tiun−un||2≤〈un−un+1,un−u∗〉+〈αnf(un)+(1−αn)vn−vn,un−u∗〉≤∥un−un+1∥∥un−u∗∥+〈αnf(un)−αnvn,un−u∗〉≤∥un−un+1∥∥un−u∗∥+αn∥f(un)−vn∥∥un−u∗∥.

Thus, from Lemma 3.2, (23), (27) and the condition on {αn}, we conclude that

(28)limn→∞∑i=1n(βi−1−βi)∥Tiun−un∥2=0.

Since {βn} is strictly decreasing, for every i∈ℕ, equality (28) yields

(29)limn→∞∥Tiun−un∥=0andlimn→∞∥Tnun−un∥=0.

Step 3. We show that limsupn→∞〈f(u∗)−u∗,un−u∗〉≤0.

Since {un} is bounded, there exist u∈H×Hm and a subsequence {unk} of {un} such that

limsupn→∞f(u∗)−u∗,un−u∗=limk→∞f(u∗)−u∗,unk−u∗

and unk⇀u. From (29) and Lemma 2.1, we obtain that u∈F(Ti) for each i≥1 and hence u∈ℱ. Since u∗=Pℱ(f(u∗)), by Lemma 2.7, we have

(30)limsupn→∞〈f(u∗)−u∗,un−u∗〉=limk→∞〈f(u∗)−u∗,unk−u∗〉=〈f(u∗)−u∗,u−u∗〉≤0,asrequired.

Step 4. Finally, we show that un→u∗ as n→∞. From Lemma 2.6, (18) and (20), we have

(31)∥un+1−u∗∥2=∥αnf(un)+(1−αn)vn−u∗∥2≤(ααn∥un−u∗∥+(1−αn)∥un−u∗∥)2+2αn〈f(u∗)−u∗,un+1−u∗〉≤(1−(1−α)αn)||un−u∗||2+2αn〈f(u∗)−u∗,un+1−u∗〉≤(1−(1−α)αn)∥un−u∗∥2+2αn〈f(u∗)−u∗,un−u∗〉+2αn∥f(u∗)−u∗∥∥un+1−un∥.

Therefore, from (30), (31) and Lemma 2.5, we conclude that un→u∗ as n→∞.□

Lemma 3.4

If in Algorithm 3.1, there existλ̲>λ̲>0such that the sequence{λk,n}⊂[λ̲,λ̲], for eachk=1,2,…,mandn≥0, then there existsη>0such that

(32)φn(un)≥η∥∇φn(un)∥2 ∀ n≥0,

where∇φn(un)=(∑k=1myk,n,x1,n−xn¯,x2,n−xn¯,…,xm,n−xn¯)and∥∇φn(un)∥2=∑k=1mxk,n−xn¯2+∑k=1myk,n2withxn¯=1m∑k=1mxk,n.

Proof

From Eq. (19), we have

(33)zn−xk,nλk,n=yk,n−wk,n.

From (16), (33) and condition of λk,n, we get

(34)φn(un)=∑k=1mzn−xk,n,yk,n−wk,n=∑k=1mzn−xk,n,zn−xk,nλk,n=∑k=1m1λk,n∥zn−xk,n∥2≥1λ¯∑k=1m∥zn−xk,n∥2.

By rearranging equation (33), we can also get

(35)yk,n=zn−xk,nλk,n+wk,n,

which implies that

(36)∑k=1myk,n=∑k=1mzn−xk,nλk,n+∑k=1mwk,n=∑k=1mzn−xk,nλk,n,

and hence

(37)∑k=1myk,n2=∑k=1mzn−xk,nλk,n2≤1λ̲2∑k=1mzn−xk,n2=1λ̲2mzn−xn¯2=m2λ̲2zn−xn¯2,

which is equivalent to

(38)λ̲2m∑k=1myk,n2≤mzn−xn¯2.

In addition, from the properties of inner product, we have

(39)∑k=1mxk,n−xn¯2=∑k=1mxk,n−zn+zn−xn¯2=∑k=1m(xk,n−zn)−(xn¯−zn)2=∑k=1mxk,n−zn2−2∑k=1m(xk,n−zn),xn¯−zn+∑k=1mxn¯−zn2=∑k=1mxk,n−zn2−2mxn¯−zn,xn¯−zn+mxn¯−zn2=∑k=1mxk,n−zn2−2mxn¯−zn2+mxn¯−zn2=∑k=1m∥xk,n−zn∥2−m∥xn¯−zn∥2,

which implies that

(40)∑k=1mxk,n−xn¯2+mxn¯−zn2=∑k=1mxk,n−zn2.

From (38) and (40), we get

(41)∑k=1mxk,n−xn¯2+λ̲2m∑k=1myk,n2≤∑k=1mxk,n−zn2

and hence

(42)1λ̲∑k=1mxk,n−xn¯2+λ̲2m∑k=1myk,n2≤1λ̲∑k=1mxk,n−zn2.

Thus, from (42), (34) and setting η=min{1λ̲,λ̲2mλ̲}, we get

(43)η∑k=1mxk,n−xn¯2+∑k=1myk,n2=η∥∇φn(un)∥2≤φn(un).

□

Next, we state and prove our main theorem.

Theorem 3.5

Let H be a real Hilbert space and letAk:H→2H, fork∈{1,2,…,m}andm≥2, be maximally monotone mappings satisfyingΩ≔{z∈H:0∈A1z+A2z+⋯+Amz}≠∅. Letf:V→Vbe a contraction mapping with constantα. Let the real sequence{λk,n}⊂[λ̲,λ̲]for someλ̲>λ̲>0and for eachk=1,2,…,mandn≥0. Then, the sequence{un}generated by Algorithm 3.1 converges strongly to an elementu∗=(z∗,w1∗,…,wm∗)∈Se(A1,A2,…,Am), satisfying the variational inequality

(44)〈(f−I)u∗,u∗−x〉≥0,∀x∈Se(A1,A2,…,Am),

wherez∗∈Ω.

Proof

By Lemma 3.4, there exists η>0 such that

(45)φn(un)≥η∇φn2 ∀n≥0.

This implies that φn(un) is always nonnegative, and from (17), we obtain

(46)Tnun−un=−φn(un)∥∇φn∥2∇φn,

which implies

(47)∥Tnun−un∥=−φn(un)∥∇φn∥2∇φn=φn(un)∥∇φn∥,

for all n such that ∇φn≠0. Thus, dividing both sides of inequality (45) by ∥∇φn∥, we obtain

(48)∥Tnun−un∥≥η∥∇φn∥,

which is also true for n having ∇φn=0. From (29), we have ∥Tnun−un∥→0 as n→∞, so (48) implies ∇φn→0 as n→∞. Thus, it follows from (47) that

(49)limn→∞φn(un)=0.

From the expression for ∇φn, we have ∑k=1myk,n→0 and xk,n−xn¯→0 for k=1,2,…,m as n→∞. Moreover, subtracting xk,n+λk,nwk,n from both sides of Eq. (19), we obtain

(50)zn−xk,n=λk,n(yk,n−wk,n).

This and the definition of φn imply that

(51)φn(un)=∑k=1mzn−xk,n,yk,n−wk,n=∑k=1mλk,n(yk,n−wk,n),yk,n−wk,n=∑k=1mλk,n∥yk,n−wk,n∥2.

Hence, from (49), (51) and the fact that λk,n>λ̲, we have

(52)limn→∞∥yk,n−wk,n∥=0,forallk=1,2,…,m,

and from (50) and (52), we obtain

(53)limn→∞∥zn−xk,n∥=0,forallk=1,2,…,m.

Moreover, from Lemma 3.3 the sequence {un}={(zn,w1,n,…,wm,n)} converges strongly to a point u∗=(z∗,w1∗,…,wm∗)∈ℱ=⋂i=1∞F(Ti). In addition, Eqs. (52) and (53) imply that yk,n→wk∗ and xk,n→z∗, for each k=1,2,…,m. Since Gph(Ak) is closed and (xk,n,yk,n)∈Gph(Ak) for all k=1,2,…,m and n≥0, we get wk∗∈Ak(z∗). Furthermore, since {un}⊂V and V is a closed subspace, we also have u∗∈V and hence u∗=(z∗,w1∗,…,wm∗)∈Se(A1,A2,…,Am) and by Lemma 2.2 we obtain z∗∈(A1+A2+⋯+Am)−1(0). Moreover, since u∗=Pℱ(f(u∗)) by Lemma 2.7, we obtain the variational inequality

(54)〈(f−I)u∗,u∗−x〉≥0,∀x∈Se(A1,A2,…,Am),

as Se(A1,A2,…,Am)⊂ℱ. The proof is complete.□

Remark 3.6

We observe that Algorithm 3.1 is equivalent to the following scheme:

(55)u0=(z0,w1,0,…,wm,0)∈Vchosenarbitrarly,xk,n=(I+λk,nAk)−1(zn+λk,nwk,n),k=1,2,…,m,yk,n=wk,n+zn−xk,nλk,n,k=1,2,…,m,un+1=αnf(zn,w1,n,…,wm,n)+(1−αn)(cn,d1,n,…,dm,n),n≥0,

where f:V→V is a contraction mapping with constant α, λk,n>0 for k=1,2,…,m, and n≥0, and

cn=βnzn+∑i=1m(βi−1−βi)cˆi,

dk,n=βnwk,n+∑i=1m(βi−1−βi)dˆk,i,

cˆi=zn,ifφi(un)≤ 0,zn−δi∑k=1myk,i,ifφi(un) > 0,

dˆk,i=wk,n,ifφi(un) ≤ 0,wk,n−δi(xk,i−xi¯),ifφi(un) > 0,

with

δi=∑k=1m〈zn−xk,i,yk,i−wk,n〉∑k=1myk,i2+∑k=1mxk,i−xi¯2,

for xi¯=1m∑k=1mxk,i.

Remark 3.7

At this point, we know that Se(A1,A2,…,Am)⊂ℱ and one can show that ℱ⊂Se(A1,A2,…,Am) and hence Se(A1,A2,…,Am)=ℱ.

If in (55) we assume λk,n=λ>0, for k=1,2,…,m and n≥0, then we get the following corollary for the sum of a finite family of maximally monotone mappings in Hilbert spaces.

Corollary 3.8

(56)xk,n=(I+λAk)−1(zn+λwk,n),k=1,2,…,m,yk,n=wn,k+zn−xk,nλ,k=1,2,…,m,un+1=αnf(zn,w1,n,…,wm,n)+(1−αn)(cn,d1,n,…,dm,n),n≥0,

where{cn}and{dk,n}are as in (55) andλ>0. Then,{un}converges strongly to an elementu∗=(z∗,w1∗,…,wm∗)ofSe(A1,A2,…,Am), wherez∗∈Ω.

If in Theorem 3.5 we replace the contraction mapping f by constant u∈V, then we get the following corollary for the sum of a finite family of maximally monotone mappings in Hilbert spaces.

Corollary 3.9

Let H be a real Hilbert space and letAk:H→2H, fork∈{1,2,…,m}andm≥2, be maximally monotone mappings satisfyingΩ≔{z∈H:0∈A1z+A2z+⋯+Amz}≠∅. For arbitraryu0=(z0,w1,0,…,wm,0)∈Vdefine an iterative algorithm by

(57)xk,n=I+λk,nAk−1zn+λk,nwk,n,k=1,2,…,m,yk,n=wk,n+zn−xk,nλk,n,k=1,2,…,m,un+1=αnu+(1−αn)cn,d1,n,…,dm,n,n≥0,

where{cn}and{dk,n}are as in (55), {λk,n}⊂[λ̲,λ̲]for someλ̲>λ̲>0and for eachk=1,2,…,mandn≥0, andu=(z,w1,…,wm)∈V⊂Hm+1. Then,{un}converges strongly to an elementu∗=(z∗,w1∗,…,wm∗)ofSe(A1,A2,…,Am), wherez∗∈Ω.

We note that if in Corollary 3.9 we assume that u=(0,0,…,0)∈V and we get the following theorem for approximating the minimum-norm point of the extended solution set of the sum of a finite family of maximally monotone mappings in Hilbert spaces.

Theorem 3.10

Let H be a real Hilbert space and letAk:H→2H, fork∈{1,2,…,m}andm≥2, be maximally monotone mappings satisfyingΩ≔{z∈H:0∈A1z+A2z+⋯+Amz}≠∅. For arbitraryu0=(z0,w0,1,…,w0,m)∈Vdefine an iterative algorithm by

(58)xk,n=(I+λk,nAk)−1(zn+λk,nwk,n),k=1,2,…,m,yk,n=wk,n+zn−xk,nλk,n,k=1,2,…,m,un+1=(1−αn)(cn,d1,n,…,dm,n),n≥0,

where{cn}and{dk,n}are as in (55) and{λk,n}⊂[λ̲,λ̲]for someλ̲>λ̲>0and for eachk=1,2,…,mandn≥0. Then,{un}converges strongly to the minimum-norm pointu∗=(z∗,w1∗,…,wm∗)ofSe(A1,A2,…,Am), wherez∗∈Ω.

Proof

We note that since u∗=(z∗,w1∗,…,wm∗)=Pℱ(0), where u∗∈Se(A1,A2,…,Am)⊂ℱ, we obtain that u∗ is the minimum-norm point of Se(A1,A2,…,Am).□

4 Application to convex minimization problem

In this section, we apply Theorem 3.5 to study the convex minimization problem.

Let gk:H→ℝ, k=1,2,…,m, where m≥2, be a finite family of convex and lower semicontinuous functions. We consider the problem of finding z∗∈H such that

(59)g1(z∗)+g2(z∗)+⋯+gm(z∗)=minz∈H{g1(z)+g2(z)+⋯+gm(z)}.

We note that problem (59) is equivalent, by Fermat’s rule, to the problem of finding z∗∈H such that

(60)0∈∂g1(z∗)+∂g2(z∗)+⋯+∂gm(z∗),

where ∂g is a subdifferential of g, which is maximally monotone (see, e.g., [35]). So, we obtain the following theorem from Theorem 3.5.

Theorem 4.1

Let H be a real Hilbert space. Letgk:H→ℝ, for eachk=1,2,3,…,mandm≥2, be a convex and lower semicontinuous function such thatΩ=minz∈H{g1(z)+g2(z)+⋯+gm(z)}≠∅. Letf:V→Vbe a contraction mapping with constantα. For arbitraryu0=(z0,w0,1,…,w0,m)∈Vdefine an iterative algorithm by

(61)xk,n=(I+λk,n∂gk)−1(zn+λk,nwk,n),k=1,2,…,m,yk,n=wk,n+zn−xk,nλk,n,k=1,2,…,m,un+1=αnf(zn,w1,n,…,wm,n)+(1−αn)(cn,d1,n,…,dm,n),n≥0,

where for eachk=1,2,…,mandn≥0, {λk,n}⊂[λ̲,λ̲]for someλ̲>λ̲>0,

cn=βnzn+∑i=1n(βi−1−βi)cˆi,

dk,n=βnwk,n+∑i=1n(βi−1−βi)dˆk,i,

cˆi=zn,ifφi(un)≤ 0,zn−δi∑k=1myk,i,ifφi(un) > 0,

dˆk,i=wk,n,ifφi(un) ≤ 0,wk,n−δixk,i−xi¯,ifφi(un) > 0,

with

δi=∑k=1m〈zn−xk,i,yk,i−wk,n〉∑k=1myk,i2+∑k=1m∥xk,i−xi¯∥2,

forxi¯=1m∑k=1mxk,i. Then,{un}converges strongly to an elementu∗=(z∗,w1∗,…,wm∗)ofSe(∂g1,∂g2,…,∂gm), wherez∗∈Ω.

Proof

Set Ak=∂gk, for k=1,2,…,m. Then, we have that Ak, k=1,2,…,m, is maximally monotone with Ω≔{z∈H:0∈A1z+A2z+...+Amz}≠∅. Hence, the conclusion follows from Theorem 3.5.□

5 Numerical example

In this section, we present some numerical experiment results to explain the conclusion of our result. The following numerical example verifies the conclusion of Corollary 3.8.

Example 5.1

Let H=l2, where l2 is the space of sequences. Let A1,A2,A3:l2→l2 be defined by A1x=x+(1,2,3,0,0,…), A2x=2x+(3,4,5,0,0,…) and A3x=3x−(2,2,2,0,0,…), where x=(x1,x2,…)∈l2. We see that A1, A2 and A3 are maximally monotone with R(I+λA1)=R(I+λA2)=R(I+λA3)=l2 for each λ>0. Now, by direct calculation we get that

x1,n=zn+λw1,n−λ(1,2,3,0,0,…)1+λ,y1,n=zn+λw1,n+(1,2,3,0,0,…)1+λ,x2,n=zn+λw2,n−λ(3,4,5,0,0,…)1+2λ,y2,n=2zn+2λw2,n+(3,4,5,0,0,…)1+2λ,x3,n=zn+λw3,n+λ(2,2,2,0,0,…)1+3λandy3,n=3zn+3λw3,n−(2,2,2,0,0,…)1+3λ,

for some λ>0. Thus, if we assume λ=1, αn=1100(n+100), βn=1n, for all n≥1 and f(x)=x1000 then Algorithm (56) reduces to the following:

(62)un+1=1105(n+102)zn,w1,n,w2,n,w3,n+1−1102(n+102)cn,d1,n,d2,n,d3,n,

where {cn} and {dk,n}, for k=1,2,3, are as in (55).

Now, if we take initial point u1=(z1,w1,1,w2,1,w3,1), where z1=(2,−1,−2,0,0,…), w1,1=(1,1,1,0,0,…), w2,1=(0,0,0,0,0,…) and w3,1=(−1,−1,−1,0,0,…), then the numerical experiment results using MATLAB provide that the first component {zn} of {un}={(zn,w1,n,w2,n,w3,n)} generated by (62) converges strongly to the solution z∗=−13,−23,−1,0,0,…∈(A1+A2+A3)−1(0) (Table 1).

Table 1

Convergence of the first component {zn} of {un} generated by (62)

N	zn	∥zn+1−zn∥l2
1	(2.0000,−1.0000,−2.0000,0,0,…)
2	(2.1662,−1.3413,−2.5015,0,0,…)	0.690
3	(1.3286,−1.4059,−2.3823,0,0,…)	0.8485
4	(0.7003,−1.3819,−2.2615,0,0,…)	0.6403
5	(0.4264,−1.3101,−2.1714,0,0,…)	0.2971
10	(0.0554,−1.0218,−1.7395,0,0,…)	0.0999
100	(−0.3081,−0.5944,−0.9294,0,0,…)	9.4868 × 10⁻⁴
200	(−0.2977,−0.6367,−0.9519,0,0,…)	3.6056 × 10⁻⁴
300	(−0.3064,−0.6500,−0.9776,0,0,…)	2.4495 × 10⁻⁴
400	(−0.3141,−0.6539,−0.9969,0,0,…)	1.4142 × 10⁻⁴
500	(−0.3196,−0.6549,−0.9969,0,0,…)	1.4142 × 10⁻⁴
1,000	(−0.3300−0.6554,−1.0006,0,0,…)	0
2,000	(−0.3324,−0.6597,−0.9987,0,0,…)	0
3,000	(−0.3330,−0.6635,−1.0001,0,0,…)	0
	⋮	⋮
	↓	↓
	−13,−23,−1,0,0,…	0

6 Conclusion

In this article, we constructed and studied algorithms which start by reformulating (4) as the problem of locating a point in a certain extended solution setSe(A1,A2,…,Am)⊂H×Hm, which converges strongly to a zero of the sum of a finite family of maximally monotone mappings in Hilbert spaces. The assumption that one of the mappings is single-valued, α-inverse strongly monotone or α-strongly monotone is dispensed with. In addition, we applied our main results to study the convex minimization problem. Finally, we provided a numerical example to support our results. Our results extend the results of [28] in the sense that our theorems provide strong convergence in arbitrary Hilbert spaces. In particular, Theorem 3.5 extends Proposition 7 of Svaiter [28] from weak to strong convergence. Moreover, our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

Acknowledgements

The authors express their deep gratitude to the referees and the editor for their valuable comments and suggestions.

Funding: Getahun B. Wega and Habtu Zegeye gratefully acknowledge the funding received from Simons Foundation based at Botswana International University of Science and Technology (BIUST).

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Received: 2019-07-22

Revised: 2020-05-02

Accepted: 2020-05-10

Published Online: 2020-07-08

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2020-0010

Keywords for this article

firmly nonexpansive; Hilbert spaces; maximally monotone mapping; strong convergence; zero points

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