# Yosida Approximation Iterative Methods for Split Monotone Variational Inclusion Problems

Authors: Mohammad Dilshad; Abdulrahman F. Aljohani; Mohammad Akram; Ahmed A. Khidir
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we present two iterative algorithms involving Yosida approximation operators for split monotone variational inclusion problems (SpMVIP). We prove the weak and strong convergence of the proposed iterative algorithms to the solution of SpMVIP in real Hilbert spaces. Our algorithms are based on Yosida approximation operators of monotone mappings such that the step size does not require the precalculation of the operator norm. To show the reliability and accuracy of the proposed algorithms, a numerical example is also constructed.


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## Body

## 1. Introduction

Variational inequality which was brought into existence by Hartman and Stampacchia [1] plays an important role as mathematical model in physics, economics, optimization, networking structural analysis, and medical images. In 1994, Censor and Elfving [2] first presented the split feasibility problems (in short, SFP) for modeling in medical image reconstruction. From the last two decades, SFP has been implemented widely in intensity-modulation therapy treatment planning and other branches of applied sciences (see, e.g., [3–5]). Censor et al. [6] combined the VIP and SFP and presented a new type of variational inequality problem called split variational inequality problem (in short, SVIP) as follows:
(1)Findx∗∈Csuchthatx∗∈VIPV1;CandAx∗∈VIPV2;Q,where C and Q are closed, convex subsets of Hilbert spaces H1 and H2, respectively, A:H1⟶H2 is a bounded linear operator, V1:H1⟶H1 and V2:H2⟶H2 are two operators, VIPV1;C=y∈C:V1y,x−y≥0,∀x∈C and VIPV2;Q=z∈Q:gz,x−z≥0,∀x∈Q.Moudafi [7] generalized SVIP into split monotone variational inclusion problem (in short, SpMVIP) as follows:
(2)Findx∗∈H1suchthatx∗∈VIV1,G1;H1andAx∗∈VIV2,G2;H2,where G1:H1⟶2H1 and G2:H2⟶2H2 are set-valued mappings on Hilbert spaces H1 and H2, respectively, VIV1,G1;H1=y∈H1:0∈V1y+G1y and VIV2,G2;H2=z∈H2:0∈V2z+G2z.Moudafi [7] formulated the following iterative algorithm to find the solution of SpMVIP. Let λ>0, select an arbitrary starting point x0∈H1, and compute
(3)xn+1=Uxn+γA∗W−IAxn,where A∗ is an adjoint operator of A, γ∈0,1/L with L being a spectral radius of operator A∗A, U=RλG1I−λV1=I+λG1−1I−λV1 and W=RλG2I−λV2=I+λG2−1I−λV2.LetNCx=z∈H1:z,y−x≤0,∀y∈C and NQx=w∈H2:w,y−x≤0,∀y∈Q be normal cones to the closed and convex sets C and Q, respectively. If G1=NC and G2=NQ, then SpMVIP reduces to SpVIP. If V1=V2=0, then SpMVIP reduces to the split variational inclusion problem (in short, SpVIP) for set-valued maximal monotone mappings, introduced and studied by Byrne et al. [8]:
(4)Findx∗∈H1suchthatx∗∈VIG1;H1andAx∗∈VIG2;H2,where VIG1;H1=y∈H1:0∈G1y and VIG2;H2=z∈H2:0∈G2z, G1,G2 are the same as in (2). We denote the solution set of SpVIP by Δ. Moreover, Byrne et al. [8] presented the following iterative algorithm to find the solution of SpVIP. Let λ>0, and select a starting point x0∈H1. Then, compute
(5)xn+1=RλG1xn+γA∗RλG2−IAxn,where A∗ is the adjoint operator of A, L=A∗A, γ∈0,2/L and RλG1, RλG2 are the resolvents of monotone mappings G1,G2, respectively. It can be easily seen that x∗ solves SpVIP if and only if x∗=RλG1x∗+γA∗I−RλG2Ax∗. Kazmi and Rizwi [9] proposed the following iterative method for approximating the common solutions of SpVIP and fixed point problem of a nonexpansive mapping:
(6)yn=RλG1xn+γA∗RλG2−IAxn,xn+1=αnfxn+1−αnSyn,where f is a contraction and S is nonexpansive mapping. Later, Sitthithakerngkiet et al. [10] studied the common solutions of SpVIP and a fixed point of an infinite family of nonexpansive mappings and introduced the following iterative method:
(7)yn=RλG1xn+γA∗RλG2−IAxn,xn+1=αnξu+βnxn+1−βnI−αnDWnyn,∀n≥1,whereu∈H1 is a given point and Wn is W-mapping which is generated by an infinite family of nonexpansive mappings. Similar results related to SpVIP can be found in [11–17].The common figure among the above-explained iterative methods is that they used the resolvent of associated monotone mappings; secondly, the step size depends on the operator normA∗A. To avoid this obstacle, self-adaptive step size iterative algorithms have been introduced (see, for example, [18–24]). Lopez et al. [20] introduced a relaxed method for solving split feasibility problem with self-adaptive step size. Recently, Dilshad et al. [25] proposed two iterative algorithms to solve SpVIP in which the precalculation of the operator norm A∗A is not required. They studied the weak and strong convergence of the proposed methods to approximate the solution of SpVIP with the step size γn=xn−RλG1xn2+A∗I−RλG2Axn2/xn−RλG1xn+A∗I−RλG2Axn2, which do not depend upon the precalculated operator norm.The resolvent of a maximal monotone operatorG is defined as JλG=I+λG−1, where λ is a positive real number. A resolvent operator of maximal monotone operator is single valued and firmly nonexpansive. Due to the fact that the zeros of maximal monotone operator are the fixed point sets of resolvent operator, the resolvent associated with a set-valued maximal monotone operator plays an important role to find the zeros of monotone operators. Following Byrne’s iterative method (5), which is mainly based on the resolvents of monotone mappings, many researchers introduced and studied various iterative methods for SpVIP (see, for example, [7–9, 18, 25, 26] and references therein).Yosida approximation operator for a monotone mappingG and parameter λ>0 is defined as JλG=1/λI−RλG. It is well known that set-valued monotone operator can be regularized into a single-valued monotone operator by the process known as the Yosida approximation. Yosida approximation is a tool to solve a variational inclusion problem using nonexpansive resolvent operator and has been used to solve various variational inclusions and system of variational inclusions in linear and nonlinear spaces (see, for example, [18, 25–30]).Due to the fact that the zero of Yosida approximation operator associated with monotone operatorG is the zero of inclusion problem 0∈Gx and inspired by the work of Moudafi, Byrne, Kazmi, and Dilshad et al., our motive is to propose two iterative methods to solve SpMVIP. The rest of the paper is organized as follows.The next section contains some fundamental results and preliminaries. In Section3, we describe two iterative algorithms using Yosida approximation of monotone mappings G1 and G2. Section 4 is devoted to the study of weak and strong convergence of the proposed iterative methods to the solution of SpMVIP. In the last section, we give a numerical example in support of our main results and show the convergence of sequence obtained from the proposed algorithm to the solution of SpMVIP.


## 2. Preliminaries

LetH be a real Hilbert space endowed with norm · and inner product ·,·. The strong and weak convergence of a sequence xn to x is denoted by xn⟶x and xn⇀x, respectively. The operator T:H⟶H is said to be a contraction if ∀x,y∈H,Tx−Ty≤κx−y,κ∈0,1; if κ=1, then T is called nonexpansive and firmly nonexpansive if ∀x,y∈H,Tx−Ty2≤x−y,Tx−Ty; T is called τ-inverse strongly monotone if there exists τ>0 such that Tx−Ty,x−y≥τTx−Ty2.For somex∈H1, there exists a unique nearest point in C denoted by PCx such that
(8)x−PCx≤x−y,∀y∈C.PCx is called the projection of x onto C⊂H, which satisfies
(9)x−y,PCx−PCy≥PCx−PCy2,∀x,y∈H.Moreover,PCx is also characterized by the fact that
(10)PCx=z⇔x−z,y−z≥0,y∈C.In Hilbert spaces, the following equality and inequality hold for allx,y,z∈H,α,β,γ∈0,1 such that α+β+γ=1(11)αx+βy+γz2=αx2+βy2+γz2−αβx−y2−βγy−z2−γαx−z2,(12)x+y2≤x2+2y,x+y.LetG:H⟶2H be a set-valued operator. The graph of G is defined by x,y:y∈Gx, and inverse of G is denoted by G−1=y,x:y∈Gx. A set-valued mapping G is said to be monotone if u−v,x−y≥0, for all u∈Gx,v∈Gy. A monotone operator G is called a maximal monotone if there exists no other monotone operator such that its graph properly contains the graph of G.Lemma 1 (see [31]).
Ifan is a sequence of nonnegative real numbers such that
(13)an+1≤1−βnan+δn,n≥0,where βn is a sequence in 0,1 and δn is a sequence in ℝ such that
(i)
∑n=1∞βn=∞(ii)
limsupn⟶∞δn/βn≤0 or limsupn⟶∞δn<∞
thenlimn⟶∞an=0.Lemma 2 (see [32]).
LetH be a Hilbert space. A mapping F:H⟶H is τ-inverse strongly monotone if and only if I−τF is firmly nonexpansive, for τ>0.Lemma 3 (see [33]).
LetH be a Hilbert space and xn be a bounded sequence in H. Assume there exists a nonempty subset C⊂H satisfying the properties
(i)
limn⟶∞xn−z exists for every z∈C(ii)
ωwxn⊂C
Then, there existsx∗∈C such that xn converges weakly to x∗.Lemma 4 (see [34]).
LetΓn be a sequence of real numbers that does not decrease at infinity in the sense that there exists a subsequence ΓnkofΓn such that Γnk<Γnk+1 for all k≥0. Also, consider the sequence of integers σnn≥n0 defined by
(14)σn=maxk≤n:Γk≤Γk+1.
Then,σnn≥n0 is a nondecreasing sequence verifying limn⟶∞σn=∞ and for all n≥n0,
(15)maxΓσn,Γn≤Γσn+1.


## 3. Yosida Approximation Iterative Methods

LetV1:H1⟶H1,V2:H2⟶H2 be single-valued monotone mappings and G1:H1⟶2H1,G2:H2⟶2H2 be set-valued mappings such that V1+G1:H1⟶2H1 and V2+G2:H2⟶2H2 are set-valued maximal monotone mappings; Rλ1V1+G1, Rλ2V2+G2 and Jλ1V1+G1, Jλ2V2+G2 are the resolvents and Yosida approximation operators of V1+G1 and V2+G2, respectively. We propose the following iterative methods to approximate the solution of SpMVIP.Algorithm 1.
For an arbitraryx0, compute the n+1th iteration as follows:
(16)un=xn−γnJλ1V1+G1xn,xn+1=un−μnA∗Jλ2V2+G2Aun,where γn and μn are defined as
(17)γn=τnJλ1V1+G1xnJλ1V1+G1xn+A∗Jλ2V2+G2Axn,ifJλ1V1+G1xn+A∗Jλ2V2+G2Axn≠0,0,otherwise,(18)μn=τnJλ2V2+G2AunJλ1V1+G1un+A∗Jλ2V2+G2Aun,ifJλ1V1+G1un+A∗Jλ2V2+G2Aun≠0,0,otherwise,where λ1>0, λ2>0 and θ=min2λ1,2λ2 such that τn∈0,θ.Algorithm 2.
For an arbitraryx0, compute the n+1th iteration as follows:
(19)un=xn−γnJλ1V1+G1xn,vn=un−μnA∗Jλ2V2+G2Aun,xn+1=1−βnun+αnvn−un.where γn and μn are defined as
(20)γn=τnJλ1V1+G1xnJλ1V1+G1xn+A∗Jλ2V2+G2Axn,ifJλ1V1+G1xn+A∗Jλ2V2+G2Axn≠0,0,otherwise,μn=τnJλ2V2+G2AunJλ1V1+G1un+A∗Jλ2V2+G2Aun,ifJλ1V1+G1un+A∗Jλ2V2+G2Aun≠0,0,otherwise,where αn, βn∈0,1, λ1>0, λ2>0, and θ=min2λ1,2λ2 such that τn∈0,θ.


## 4. Main Results

We assume that the problemSpMVIP is consistent and the solution set is denoted by Δ.First, we prove following lemmas, which are used in the proof of our main results.Lemma 5.
LetV1:H1⟶H1 be single-valued monotone mappings and G1:H1⟶2H1 be set-valued mappings such that V1+G1:H1⟶2H1 be set-valued maximal monotone mapping. If Rλ1V1+G1 and Jλ1V1+G1 are the resolvent and Yosida approximation operators of V1+G1, respectively, then for λ1>0, following are equivalent:
(i)
x∗∈H1 is the solution of V1+G1−10(ii)
Rλ1V1+G1x∗=x∗(iii)
Jλ1V1+G1x∗=0Proof.
The proof is trivial which is an immediate consequence of definitions of resolvent and Yosida approximation operator of maximal monotone mappingV1+G1.☐Theorem 6.
LetH1, H2 be real Hilbert spaces; V1:H1⟶H1, V2:H2⟶H2 be single-valued monotone mappings, G1:H1⟶2H1, G2:H2⟶2H2 be set-valued maximal monotone mappings such that V1+G1 and V2+G2 are maximal monotone, and A:H1⟶H2 be a bounded linear operator. Assume that θ=min2λ1,2λ2 such that infτnθ−τn>0. Then, the sequence xn generated by Algorithm 1 converges weakly to a point z∈Δ.Proof.
Letz∈Δ. Since the Yosida approximation operator Jλ1V1+G1 is λ1-inverse strongly monotone, for λ1>0, then by Algorithm 1 and (12), we have
(21)un−z2=xn−γnJλ1V1+G1xn−z2=xn−z2+γn2Jλ1V1+G1xn2−2γnJλ1V1+G1xn,xn−z≤xn−z2+γn2Jλ1V1+G1xn2−2γnλ1Jλ1V1+G1xn2=xn−z2+γn2−2γnλ1Jλ1V1+G1xn2.
Now, using (17), we estimate that
(22)γn2−2γnλ1Jλ1V1+G1xn2=Jλ1V1+G1xn2τn2Jλ1V1+G1xn2Jλ1V1+G1xn+A∗Jλ2V2+G2Axn2−2τnλ1Jλ1V1+G1xnJλ1V1+G1xn+A∗Jλ2V2+G2Axn=Jλ1V1+G1xn3τn2Jλ1V1+G1xn−2λ1τnJλ1V1+G1xn+A∗Jλ2V2+G2AxnJλ1V1+G1xn+A∗Jλ2V2+G2Axn2≤Jλ1V1+G1xn3τn2−2λ1τnJλ1V1+G1xn+A∗Jλ2V2+G2AxnJλ1V1+G1xn+A∗Jλ2V2+G2Axn2=τn2−2λ1τnJλ1V1+G1xn3Jλ1V1+G1xn+A∗Jλ2V2+G2Axn.
From (21) and (22), we get
(23)un−z2≤xn−z2+τn2−2λ1τnJλ1V1+G1xn3Jλ1V1+G1xn+A∗Jλ2V2+G2Axn.
SinceJλ2V2+G2 is λ2-inverse strongly monotone and using (12), we estimate
(24)xn+1−z2=un−μnJλ2V2+G2Aun−z2≤un−z2+μn2Jλ2V2+G2Aun2−2μnJλ2V2+G2Aun,un−z=un−z2+μn2Jλ2V2+G2Aun2−2μnJλ1V1+G1Aun2=un−z2+μn2−2μnλ2Jλ1V2+G2Aun2.
By (18), it turns out that
(25)μn2−2μnλ2Jλ2V2+G2Aun2=Jλ2V2+G2Aun2τn2Jλ2V2+G2Aun2Jλ1V1+G1un+A∗Jλ2V2+G2Aun2−2τnλ2Jλ2V2+G2AunJλ1V1+G1un+A∗Jλ2V2+G2Aun=Jλ2V2+G2Aun3τn2Jλ2V2+G2Aun−2λ2τnJλ1V1+G1un+A∗Jλ2V2+G2AunJλ1V1+G1un+A∗Jλ2V2+G2Aun2=Jλ2V2+G2Aun3τn2−2λ2τnJλ1V1+G1un+A∗Jλ2V2+G2AunJλ1V1+G1un+∥A∗Jλ2V2+G2Aun2=τn2−2λ2τnJλ2V2+G2Aun3Jλ1V1+G1un+A∗Jλ2V2+G2Aun.
It follows from (24) and (25) that
(26)xn+1−z2≤un−z2+τn2−2λ2τnJλ2V2+G2Aun3Jλ1V1+G1un+A∗Jλ2V2+G2Aun.
Combining (23) and (26), we get
(27)xn+1−z2≤xn−z2−τn2λ1−τnJλ1V1+G1xn3Jλ1V1+G1xn+A∗Jλ2V2+G2Axn−τn2λ2−τnJλ2V2+G2Aun3Jλ1V1+G1un+A∗Jλ2V2+G2Aun,(28)≤xn−z,which implies that xn is Fejér monotone with respect to Δ and hence bounded, which assures that limn⟶∞xn−z exists for all z∈Δ. Keeping in mind that θ=min2λ1,2λ2, from (27), we have
(29)∑n=1∞τnθ−τnJλ1V1+G1xn3Jλ1V1+G1xn+A∗Jλ2V2+G2Axn+Jλ2V2+G2Aun3Jλ1V1+G1un+A∗Jλ2V2+G2Aun<∞.
Due to the assumption thatinfτnθ−τn>0 and the properties of convergent series, we conclude that
(30)limn⟶∞Jλ1V1+G1xn=limn⟶∞Jλ2V2+G2Aun=0.
Hence, there exist constantsK1 and K2 such that
(31)Jλ1V1+G1xn≤K1,Jλ2V2+G2Aun≤K2.
By Algorithm1 and (30), we get
(32)xn+1−xn≤xn+1−un+un−xn≤K1γn+K2μn⟶0,asn⟶∞.
Letx⋆∈ωwxnandxnkbe a subsequence ofxnthat converges weakly tox⋆, which implies thatxnkandunkalso converge tox⋆. Recall that Jλ1V1+G1 is λ1-inverse strongly monotone and xnk converges to x⋆, and using (30), we get
(33)Jλ1V1+G1xnk−Jλ1V1+G1x⋆,xnk−x⋆≥λ1Jλ1V1+G1xnk−Jλ1V1+G1x⋆2.
Taking limitk⟶∞, we obtain Jλ1V1+G1x⋆=0.
ReplacingJλ1V1+G1 by Jλ2V2+G2A, xnk by Aunk with the same arguments, we get Jλ2V2+G2Ax⋆=0. This completes the proof.☐Theorem 7.
LetH1, H2 be real Hilbert spaces; V1:H1⟶H1, V2:H2⟶H2 be single-valued monotone mappings, G1:H1⟶2H1, G2:H2⟶2H2 be set-valued maximal monotone mappings such that V1+G1 and V2+G2 are maximal monotone, and A:H1⟶H2 be a bounded linear operator. If αn, βn are real sequences in 0,1 and θ=min2λ1,2λ2 such that τn∈0,θ and
(34)limn⟶∞βn=0,∑n=0∞βn=∞,limn⟶∞1−αnαn>0,infnτnθ−τn>0,then the sequence xn generated by Algorithm 2 converges strongly to z=PΔ0.Proof.
Letz=PΔ0; then, from (23) and (26) of the proof of Theorem 6, we have
(35)un−z2≤xn−z2+τn2−2λ1τnJλ1V1+G1xn3Jλ1V1+G1xn+A∗Jλ2V2+G2Axn,(36)vn−z2≤un−z2+τn2−2λ2τnJλ2V2+G2Aun3Jλ1V1+G1un+A∗Jλ2V2+G2Aun.
Sinceτn≤min2λ1,2λ2, we get ∥vn−z∥≤∥un−z∥≤∥xn−z∥. From Algorithm 2, we have
(37)xn+1−z=1−βnun+αnvn−un−z≤1−αn−βnun−z++αnvn−z+βn−z≤1−βnxn−z+βnz≤maxxn−z,z≤⋯≤maxx0−z,z,which implies that the sequence xn is bounded and hence, the sequences un,vn,Jλ1V1+G1un and A∗Jλ2V2+G2Aun are also bounded. Now,
(38)xn+1−z2=1−βnun+αnvn−un−z2≤1−αn−βnun−z2+αnvn−z2+βnz2−αn1−αn−βnvn−un2.
Combining (35), (36), and (38), we obtain
(39)xn+1−z2≤1−αn−βnxn−z2+τn2−2λ1τnJλ1V1+G1xn3Jλ1V1+G1xn+A∗Jλ2V2+G2Axn+αnun−z2+τn2−2λ2τnJλ2V2+G2Aun3Jλ1V1+G1un+A∗Jλ2V2+G2Axn+βnz2−αn1−αn−βnvn−un2≤xn−z2+βn−xn−z2+z2−αn1−αn−βnvn−un2−1−αn−βnτn2−2λ1τnJλ1V1+G1xn3Jλ1V1+G1xn+A∗Jλ2V2+G2Axn−αnτn2−2λ2τnJλ2V2+G2Aun3Jλ1V1+G1un+A∗Jλ2V2+G2Aun.
We discuss the two possible cases.
Case 1. If the sequencexn−z is nonincreasing, then there exists a number k≥0 such that xn+1−z≤xn−z, for each n≥k. Then, limn⟶∞xn−z exists and hence, limn⟶∞xn+1−z−xn−z=0. Thus, it follows from (39) that
(40)limn⟶∞vn−un=0,limn⟶∞Jλ1V1+G1xn=0,limn⟶∞Jλ2V2+G2Aun=0.
From (40), we conclude that limn⟶∞γn=limn⟶∞μn=0 and limn⟶∞un−xn=0. We observe from Algorithm 2 that xn+1−un=αnvn−un+γnun⟶0; thus,
(41)xn+1−xn≤xn+1−un+un−xn≤vn−un+γnun+un−xn⟶0asn⟶∞.
This shows that the sequencexn is asymptotically regular. By Theorem 6, we have that ωwxn⊂Δ. Settingzn=1−αnun+αnvnand rewritingxn+1=1−βnzn+αnβnvn−un, we have
(42)zn−z=1−αnun+αnvn−z≤1−αnun−z+αnvn−z≤xn−z.
From (42) and Algorithm 2, we get
(43)xn+1−z2=1−βnzn−z+βnαnvn−un−z2≤1−βn2zn−z2+2βnαnvn−un−z,xn+1−z≤1−βnxn−z2+2βnαnvn−un,xn+1−z+−z,xn+1−z,or
(44)an+1=1−βnan+bn,where an=xn−z,bn=2βnαnvn−un,xn+1−z+−z,xn+1−z.
Sinceωwxn⊂Δ and z=PΔ0, then using (40), we get
(45)limsupn⟶∞bnβn=limsupn⟶∞2αnvn−un,xn+1−z+−z,xn+1−z=limsupn⟶∞−z,xn+1−z≤0.
Thus, by Lemma1, we obtain xn⟶z.
Case 2. If the sequencexn−z is not nonincreasing, we can select a subsequence xnk−z of xn−z such that xnk−z≤xn−z for all k∈ℕ. In this case, we define a subsequence of positive integers σn⟶∞ with the properties
(46)xσn−z<xσn+1−z,maxxσn−z,xn−z≤xσn+1−z.
Ifxn+1−z>xn−z for some n≥0, then it follows from (39) that
(47)αn1−αn−βnvn−un2+1−αn−βnτn2−2λ1τnJλ1V1+G1xn3Jλ1V1+G1xn+A∗Jλ2V2+G2Axn+αnτn2−2λ2τnJλ2V2+G2Aun3Jλ1V1+G1un+A∗Jλ2V2+G2Aun≤βnz2−xn−z2.
Replacingn by σn and taking limit n⟶∞, we get the following relation for the subsequences xσn, uσn, and vσn:
(48)limn⟶∞vσn−uσn=0,limn⟶∞Jλ1V1+G1xσn=0,limn⟶∞Jλ2V2+G2Auσn=0
Thus, we havexσn+1−xσn⟶0, as n⟶∞ and ωwxσn⊂Δ. It is remaining to show that xn⟶z.
Replacingn by σn in (47), using xσn−z<xσn+1−z and boundedness of xn−z, we have
(49)xσn−z2≤Mvσn−uσn+2−z,xσn+1−z.
Sincez=PΔ0, ωxσn⊂Δ with using vσn−uσn⟶0 and xσn+1−xσn⟶0, we have
(50)limsupn⟶∞−z,xσn+1−z=limsupn⟶∞−z,xσn−z=maxr∈ωwxσn−z,r−z≤0.
From (49) and (52), we conclude that xσn⟶z and
(51)xn−z≤xσn+1−z≤xσn+1−xσn+xσn−z⟶0,that is, xn⟶z. This complete the proof.☐Forτn=1, we have the following result for the convergence of Algorithm 2.Corollary 8.
LetH1, H2, V1,V2, G1,G2, and A, A∗ be the same as defined in Theorem 7. If αn, βn are sequences in 0,1 and assuming that λ1>1/2 and λ2>1/2 satisfying
(52)limn⟶∞βn=0,∑n=0∞βn=∞,limn⟶∞1−αnαn>0,then the sequence xn generated by Algorithm 2 (with τn=1) converges strongly to z=PΔ0.Forβn=0, we have the following corollary for the convergence of Algorithm 2.Corollary 9.
LetH1, H2, V1,V2, G1,G2, and A, A∗ be the same as defined in Theorem 7. If αn is a sequence in 0,1 and assuming that θ=min2λ1,2λ2 such that τn∈0,θ and
(53)limn⟶∞1−αnαn>0,infnτnθ−τn>0,then the sequence xn generated by the iterative method
(54)un=xn−γnJλ1V1+G1xn,vn=un−μnA∗Jλ2V2+G2Aun,xn+1=1−αnun+αnvn,where γn and μn are defined as in Algorithm 2 (with τn=1), converges strongly to z∈Δ.Forτn=1 and βn=0, we have the following corollary for the convergence of Algorithm 2.Corollary 10.
LetH1, H2, V1,V2, G1,G2, and A, A∗ be the same as defined in Theorem 7. If αn be a sequence in 0,1 and assuming that
(55)limn⟶∞1−αnαn>0,λ1>12,λ2>12,then the sequence xn generated by the iterative method
(56)un=xn−γnJλ1V1+G1xn,vn=un−μnA∗Jλ2V2+G2Aun,xn+1=1−αnun+αnvn,where γn and μn are defined in Algorithm 2 (with τn=1), converges strongly to z∈Δ.


## 5. Numerical Example

LetH1=H2=ℝ and V1=V2=0; G1:ℝ⟶ℝ, G2:ℝ⟶ℝ are defined as G1x=2x+3 and G2x=2x+1, respectively. One can easily check that G1 and G2 are monotone and the Yosida approximation operator of G1 and G2 for λ1=λ2=1 is computed as
(57)Jλ1V1+G1x=2x+33,Jλ2V2+G2x=2x+23.LetA:H1⟶H2 be defined as Ax=2x/3, then, for τn=2−e1/n/2∈0,2, we compute the step size as
(58)γn=τnJλ1V1+G1xnJλ1V1+G1xn+A∗Jλ2V2+G2Axn=2−e1/n2913,μn=τnJλ2V2+G2AunJλ1V1+G1un+A∗Jλ2V2+G2Aun=2−e1/n2613.Then, forαn=2−e1/n/3 and two different values of βn (for example, βn=1/n+5 and βn=1/n+10) and for arbitrary x0 (for example, x0=−2 and x0=0), we compute the iterative sequences from Algorithm 2 as follows:
(59)un=xn−2−e1/n23132xn+3,vn=un−2−e1/n281172xn+3,xn+1=1−1n+5un+2−e1/n3vn−un.In Figures1 and 2, we show that the obtained sequences un,vn, and xn converge to z=−3/2 for randomly selected arbitrary values of x0=−2 and 0.Figure 1
Convergence of iterative sequencesun,vn, and xn to z=−1.5 for βn=1/n+5 and x0=−2.Figure 2
Convergence of iterative sequencesun,vn, and xn to z=−1.5 for βn=1/n+10 and x0=0.


## 6. Conclusions

We have proposed two iterative algorithms forSpMVIP which are mainly based on the Yosida approximation operators. Since the zero of Yosida approximation of monotone mapping V1+G1 is the solution of V1+G1−10, we used the Yosida approximations of monotone mappings V1+G1 and V2+G2 to solve SpMVIP. We proved the weak and strong convergence of the composed iterative algorithms to investigate the solution of SpMVIP under some suitable assumptions such that the estimation of step size does not require any prior calculation of the operator norm A∗A. To show the accuracy and efficiency of our algorithms, we have present a numerical example and showed the convergence using different parameters.



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*Source: 3665713-2022-01-25.xml*