# Hybrid Alternated Inertial Projection and Contraction Algorithm for Solving Bilevel Variational Inequality Problems

Authors: Jacob Ashiwere Abuchu; Austine Efut Ofem; Godwin Chidi Ugwunnadi; Ojen Kumar Narain; Azhar Hussain
Journal: Journal of Mathematics
 (2023)

---

## Abstract

This paper considers an alternated inertial-type extrapolation algorithm for solving bilevel pseudomonotone variational inequality problem in the framework of real Hilbert spaces with split variational inequality and fixed-point constraints of demimetric mapping. The algorithm which involves alternated inertial uses self-adjustment stepsize condition that depends solely on the information from previous iterative step. The bilevel problem considered in the work consists of upper-level problem with an underlying operator which is pseudomonotone, while the lower problem is associated with strongly monotone mapping and the fixed-point constraint of demimetric mapping. Our algorithm is anchored on modified projection and contraction techniques, fused with alternated inertial and relaxation. Under some suitable conditions on the algorithm control parameters, we obtain a strong convergence result of the proposed method without the prior knowledge of the operator norm or the coefficients of the underlying operators within the scope of real Hilbert spaces. Finally, some numerical examples are presented to illustrate the gain of our method in comparison to some related algorithms in the literature.


---

## Body

## 1. Introduction

LetD be a nonempty, closed and convex subset of a real Hilbert space ℋ that is embedded with the inner product ⋅,⋅, and associated norm ⋅. Let ℱ:D⟶ℋ and G:D⟶ℋ be real single-valued nonlinear mappings. Then, bilevel variational inequality problem (BVIP) introduced by Anh et al. [1] is formulated as follows:(1)Findu∗∈Ωsuch thatℱu∗,u−u∗≥0,∀u∈Ω,where Ω:=u∗∈VIG,D is the solution set of the variational inequality problem (VIP):(2)Search forv∗∈Dsuch thatGv∗,v−v∗≥0,∀v∈D.It is obvious from (2) that bilevel variational inequality problem is a classical problem containing another VIP as constraints. The variational inequality (2) is a very efficient tool in nonlinear analysis and mathematics in general. The concept of VIP can be first traced to Stampacchia [2] (it was also independently formulated by Fichera [3]) as a tool for modeling dynamical problems in mechanics. It provides a unified structure for investigating many problems arising in pure and applied sciences, structural analysis, and other areas (see [4–6] and references therein). Due to the importance associated with the problem (2), it has been extensively studied and extended in many directions. It is very interesting to note that finding the solution of VIP (2) is equivalent to solving the fixed-point problem (FPP):(3)Findu∗∈Dsuch thatu∗=PDu∗−ζGu∗,where ζ is strictly positive real number and PD is metric projection of ℋ onto D. It can be seen in [7], [Section 4, p. 314] that u∗∈D solves VIG,D if and only if u∗ solves (3).In 2012, Censor et al. [8] studied and extended the notion of VIP (2) to split variational inequality problem (SVIP) which entails:(4)Search foru∗∈Dsuch thatGu∗,u−u∗≥0,∀u∈D,(5)v∗=ℬu∗∈Qsuch thatVv∗,v−v∗≥0,∀v∈Q,where D and Q are nonempty closed and convex subsets of the Hilbert spaces ℋ1 and ℋ2, G:ℋ1⟶ℋ1, V:ℋ2⟶ℋ2 are nonlinear mappings and ℬ:ℋ1⟶ℋ2 is a bounded linear operator.From the knowledge of (3) emerged another interesting extension and generalization of VIP (2) known as split variational inequality and fixed-point problem (SVIFPP) introduced by Hai et al. in [9]. This notion involves(6)Findu∗∈Dwhich solvesGu∗,u−u∗≥0,∀u∈D,(7)v∗=ℬu∗∈Qsuch thatv∗=ℱv∗∈Q,where D and Q are nonempty closed and convex subsets of the Hilbert spaces ℋ1 and ℋ2, respectively, G:ℋ1⟶ℋ1, ℱ:ℋ2⟶ℋ2 are nonlinear mappings and ℬ:ℋ1⟶ℋ2 is bounded linear operator. If in (4) and (5) we let G=V=0, we obtain a special form of SVIP known as split feasibility problem (SFP):(8)Findu∗∈Dsuch thatℬu∗∈Q.Similarly, if in (6) and (7) we letℋ1=ℋ2 and ℬ:=ℐ (an identity operator), the split variational inequality and fixed-point problem reduces to finding the common elements of the variational inequality and fixed-point problems. That is(9)Findu∗∈Dwhich solvesGu∗,u−u∗≥0,∀u∈D,(10)u∗=ℱu∗∈D.From (9) and (10) we can deduce thatu∗∈Ω∩Fℱ, where Fℱ is the fixed point set of the operator ℱ defined by Fℱ:=u∗∈D:ℱu∗=u∗.In 2008, Mainge introduced and studied another special form of bilevel variational inequality problemBVIPℱ,Fℱ∩VIG,D:(11)Findv∗∈Fℱ∩VIG,Dwhich solvesℱv∗,v−v∗≥0,∀v∈Fℱ∩VIG,D.He [10] introduced an iterative algorithm for approximating common solution of the bilevel variational inequality problem and fixed-point problem (11). The algorithm entails(12)u0∈ℋ,wn=PDun−λnGun,qn=PDun−λnGwn,vn=qn−βnℱqn,un+1=ηvn+1−ηℱvn,n≥0,where G:ℋ⟶ℋ is monotone and l- Lipschitzian, ℱ:ℋ⟶ℋ is α− strongly monotone and k− Lipschitz continuous, ℱ:ℋ⟶ℋ is ζ-demicontractive, PD is a metric projection of ℋ onto D. Under certain conditions on the algorithm parameters, the author established a strong convergence result. But one noticeable defect of the algorithm is the double calculation of the projection onto the feasible set D per iteration which can be computationally very expensive if D is structurally complex.Inspired by the work of Mainge, the authors in [9] introduced and analyzed iterative algorithm for solving another special form of (1) as given in the following:(13)Findu∗∈Ωsuch thatℱu∗,u−u∗≥0,∀u∈Ω,where Ω:=u∗∈VIG,D:ℬu∗∈Fℱ, ℱ:ℋ1⟶ℋ1 is α− strongly monotone and 0<l− Lipschitz continuous on ℋ1, G:D⟶D is pseudomonotone and k− Lipschitzian on D, ℬ:ℋ1⟶ℋ2 is bounded linear operator and ℱ:ℋ2⟶ℋ2 is ϱ− demicontraction and demiclosed at the origin. Their iterative algorithm for solving (13) was based on subgradient extragradient technique with weaker condition on the cost operator compared to the one in [11]. They established a minimum-norm convergence result based on certain assumptions on the algorithm control parameters. Conspicuous among is the strict assumption that selection of stepsize depends on the operator norm ℬ which is not generally very easy to compute in most instances. This in turn will affect the efficiency of the iterative scheme. Stepsizes, as we know, play vital roles in the convergence properties of iterative algorithms, since the efficiency of iterative methods essentially depends on it and the knowledge of the operator norm or coefficient of the operator and otherwise.Recently, motivated by the work in [9], the authors in [12] introduced and studied an inertial iterative algorithm for solving BVIP (13) in real Hilbert spaces. The algorithm entails(14)u0,u1∈ℋ1,u¯n=un+αnun−un−1,z¯n=u¯n−ηnA∗I−VAu¯n,w¯n=PDz¯n−ζnGz¯n,vn=PΓnz¯n−ζnGw¯n,un+1=vn−β¯nℱvn,n≥1,where Γn is a half-space constructed as: Γn:=v¯∈ℋ1:v¯−ηnGz¯n−w¯n,v¯−w¯n≤0, G:ℋ1⟶ℋ1 is pseudomonotone and l− Lipschitzian on ℋ1, ℱ:ℋ1⟶ℋ1 is γ− strongly monotone and k− Lipschitz continuous on ℋ1, A:ℋ1⟶ℋ2 is a bounded linear operator, ℐ is an identity operator, V:ℋ2⟶ℋ2, is ϱ/1−ϱ− demicontractive mapping resulting from the composition of ϱ− demicontractive mapping V1:ℋ2⟶ℋ2 and directed mapping V2:ℋ2⟶ℋ2 such that ϱ/1−ϱ is restricted to set of points in 0,1/2, the stepsizes ηn and ζn are self-adaptive following some given rules. The inertial parameter α¯n satisfies the condition αn∈0,α¯n such that α¯n=minα¯,εn/maxn2un−un−12,nun−un−1 with α¯∈0,1. Under these control parameters, the authors obtained a strong convergence result.In investigating the evolving trend of research in solving the bilevel variational inequality (1) and the related problems (for instance (2), (4), (5), (11), (13)), many techniques and methods have been developed. These include gradient projection method [13], extragradient method [14, 15], subgradient extragradient method [9, 16, 17], and so on. It is worth mentioning here that these methods come with their associated defects ranging from stringent conditions on the cost operators, only weak convergence, double calculation of metric projections onto a feasible set per iteration (which is not computationally cost effective), strict conditions on algorithm control parameters like prior knowledge of Lipschitz constant, computation of coefficients of operators and so on. To effectively handle some of these shortcomings, the projection and contraction method (PCM) was introduced for solving variational inequalities and other related problems. He in [10] introduced and analyzed the following PCM algorithm:(15)vn=PDun−ζnGun,dn=un−vn−ζnGun−Gvn,un+1=un−ρβndn,n≥1,where ρ∈0,2, βn:=φun,vn/dn2 and φun,vn:=un−vn,dn, G:ℋ⟶ℋ is monotone and Lipschitz continuous and ζn is always updated by some certain self-adaptive rules. It was established that the sequence un generated by (15) converges weakly to the solution of VIP (2). The projection and contraction method has enjoyed several modifications and generalizations (see [18–20] and references therein).Recently, Viet Thong et al. in [21] introduced the following iterative algorithm for solving BVIP (1):(16)u1∈ℋ,vn=PDun−ζnGun,z¯n=un−ηξndn,dn=un−vn−ζnGun−Gvn,ξn=un−vn,dndn2,un+1=z¯n−α¯nρℱz¯n,n≥0,where G is l-Lipschitzian, pseudomonotone, and sequentially weakly continuous, ℱ is μ-strongly monotone and k-Lipschitz continuous, η∈0,2, β∈0,1,ζ0>0,ρ∈0,2μ/k2, α¯n⊂0,1 such that limn⟶∞αn¯=0 and ∑n=1∞αn=∞. The adaptive stepsize ζn+1 is updated as follows:(17)ζn+1=minβun−vnGun−Gvn,ζn,ifGun−Gvn≠0,ζn,otherwise.The authors established that under some appropriate assumptions, the sequenceun generated by (16) converged strongly to unique solution of BVIP (1).Also, Tan et al. [22] introduced and analyzed an accelerated projection and contraction extragradient algorithm for solving bilevel variational inequality problem (1). They proposed the iterative method:(18)u0,u1∈ℋ,wn=un+θnun−un−1,vn=PDwn−ζnGwn,z¯n=wn−ηξndn,dn=wn−vn−ζnGwn−Gvn,ξn=wn−vn,dndn2,un+1=z¯n−α¯nρℱz¯n,n≥0,where G is l-Lipschitzian, pseudomonotone, and sequentially weakly continuous, ℱ is μ-strongly monotone and k-Lipschitz continuous, η∈0,2, β∈0,1,ζ0>0,ρ∈0,2μ/k2, α¯n⊂0,1 such that limn⟶∞αn¯=0 and ∑n=1∞αn=∞. The stepsize ζn+1, is updated as follows:(19)ζn+1=minβwn−vnGwn−Gvn,ζn+σn,ifGwn−Gvn≠0,ζn+σn,otherwise.The authors under appropriate assumption on the algorithm parameters established that algorithm (18) converges strongly to the unique solution of BVIP (1). The PCM technique has enjoyed modifications in several ways. Inertial-type of projection and contraction algorithms for solving variational inequality problems is one of the most interesting and important extrapolation techniques which is geared toward improving the rate of convergence of the iterative schemes (see for example [23–25] and references therein). This is because the introduction of inertial into iterative algorithm has been shown to efficiently accelerate the rate of convergence of such iterative method.However, it has been observed that there are some attractive properties of projection and contraction method that do not hold in inertial projection method. For instance, the inertial term looses its Feje´r monotonicity for the real sequence un−u∗ for each u∗∈Ω (see for example [18, 25, 26]). This ensures that the sequence un generated swings back and forth within the solution set Ω. This has a negative effect on the rate of convergence of the inertial iterative algorithm comparatively to the noninertial ones. To mitigate this problem, Mu and Peng [27] introduced and studied alternated inertial methods which applies inertial every other iteration. Since then, variants of alternated inertial algorithms have been developed to solve variational inequalities and other related problems (check e. g [28–30]). The alternated inertial term is generated by the iterative process:(20)wn=un+αnun−un−1,ifn=odd,un,ifn=even.This form of inertial term (20) is quite unpopular compare to the usual vanilla inertial extrapolation step found in (14) (see also the inertial terms in [18, 31]). However, the alternated inertia has very good convergence properties and performance.Motivated by these results, we introduce and study a modified, relaxed, and alternated inertial extrapolation algorithm for finding the minimum-norm solution of bilevel variational inequality problem of the type:(21)Findu∗∈Ωsuch thatℱu∗,u−u∗≥0,∀u∈Ω,where(22)Ω≔u∗∈VIG,D:ℬu∗∈FT,where ℋ1, ℋ2 are real Hilbert spaces, ℱ:ℋ1⟶ℋ1 is γ− strongly monotone and k− Lipschitz continuous, G:ℋ1⟶ℋ1 is pseudomonotone and l− Lipschitzian, ℬ:ℋ1⟶ℋ2 is bounded linear operator, T:ℋ2⟶ℋ2 is ♭− demimetric and demiclosed at the origin. Our proposed iterative method is based on alternated inertial extrapolation and relaxation step under adaptive stepsize condition in real Hilbert spaces. Under some mild assumptions on the algorithm control parameters, we establish a strong convergence result without the prior knowledge of the operator norm or the coefficients of the underlying operators. We also present some numerical illustrations to show the applicability and efficiency of our method.


## 2. Preliminaries

In this section, we will present some definitions and results that will be useful in our main results.Letℋ be a real Hilbert space and D a nonempty, closed and convex subset of ℋ. For every ν∈ℋ there exists a unique nearest point in D denoted PDν such that(23)ν−PCν=infν−ω:ω∈D.The operatorPD is known as the metric projection of ℋ onto D and is known to be nonexpansive. Moreover, PD has the following characterization:Lemma 1 (see [32]).
LetPD:ℋ⟶D be metric projection of ℋ onto D. Then, the following hold:(i)
PDν−PDω2≤PDν−PDω,ν−ω∀ν,ω∈ℋ;(ii)
μ=PDνif and only ifν−μ,ω−μ≤0∀μ,ω∈D,ν∈ℋ;(iii)
ν−PDμ2+PDμ−μ2≤ν−μ2∀ν∈ℋ,μ∈D.Definition 1 (see [33, 34])..
Letℋ be real Hilbert space and D a nonempty, closed and convex subset of ℋ. Let ℳ:ℋ⟶ℋ be a real single-valued mapping and Fℳ:=υ∈ℋ:υ=ℳυ≠∅ denotes the set of all fixed points of ℳ. Then, ℳ is said to be as(a)
l-Lipschitz continuous on ℋ if there exists l>0 such that(24)ℳυ−ℳω≤lυ−ω∀υ,ω∈ℋ(b)
γ− strongly monotone on D if there exists γ>0 such that(25)ℳυ−ℳω,υ−ω≥γυ−ω2∀υ,ω∈D(c)
Monotone onD if(26)ℳυ−ℳω,υ−ω≥0∀υ,ω∈D(d)
Pseudomonotone onD if(27)ℳυ,ω−υ≥0⇒ℳω,ω−υ≥0∀υ,ω∈D(e)
η− demicontractive operator on ℋ if there exists η∈0,1 such that(28)ℳυ−p2≤υ−p2+ηυ−ℳυ2∀υ∈ℋ,p∈Fℳ.(f)
τ− demimetric on ℋ if there exists τ∈−∞,1 such that(29)υ−ℳυ,υ−p≥1−τ2υ−ℳυ2∀υ∈ℋ,p∈Fℳ.
It is obvious to see that (b)⇒ (c) ⇒ (d). But the converses are not generally true.Definition 2 (see [30])..
A sequenceun is said to be Feje'r monotone with respect to a set Γ if each point in the sequence is not strictly farther from any point in Γ than its predecessor. That is(30)un+1−u∗≤un−u∗∀u∗∈Γ.Lemma 2 (see [33, 35])..
Letℋ be a real Hilbert space with μ,ν∈ℋ and α∈ℝ, then the following holds(i)
2μ,ν=μ2+ν2−μ−ν2=μ+ν2−μ2−ν2;(ii)
μ−ν2≤μ2+2ν,ν−μ;(iii)
αμ+1−αν2=αμ2+1−αν2−α1−αμ−ν2.Lemma 3 (see [36])..
LetD be a nonempty, closed and convex subset of a real Hilbert space ℋ, and ℱ:ℋ⟶ℋ be a γ-strongly monotone and l− Lipschitz continuous mapping on ℋ. Let ρ∈0,1 and ξ∈0,2γ/l2. Then, for any nonexpansive mapping S:D⟶ℋ, we can associate a mapping Sξ:D⟶ℋ defined by Sξu=I−ρξℱSu,∀u∈D. Then, Sξ is a contraction mapping. That is(31)Sξu−Sξν≤1−ρτu−ν∀u,ν∈ℋ,where τ=1−1−ξ2γ−ξl2∈0,1.Lemma 4 (see ([37] Lemma 3.3))..
LetD be a nonempty, closed, and convex subsets of a real Hilbert space ℋ and A:ℋ⟶ℋ be a pseudomonotone and Lipschitz continuous operator. Suppose μn>0 is a real sequence and wnk is a subsequence of a sequence wn in ℋ such that wnk⟶x∗, and limn⟶∞wnk−νnk=0, where νnk is a subsequence of the sequence νn=PDwn−μnAwn∀n≥1, then x∗∈VIA,D.Lemma 5 (see [38])..
Letϖn be a sequence of positive real numbers. Let σn and ιn be sequences in 0,1 with ∑n=1∞σn=∞. Suppose that ϖn satisfies the inequality:(32)ϖn+1≤1−σnϖn+σnιn∀n∈ℕ.
If thelimsupk⟶∞ιnk≤0 for every subsequence of ϖnk of ϖn satisfying the condition liminfk⟶∞ϖnk+1−ϖnk≥0. Then, limn⟶∞ϖn=0.


## 3. Main Results

Throughout the remaining part of this work, we shall useun⟶u∗ (respectively, un⇀u∗) to denote that the sequence un converges strongly (respectively weakly) to a point u∗ as n⟶∞. We shall also make the following assumptions that will help us establish convergence result for our proposed iterative algorithm.Assumption 1.
Supposed that:(A1)
D is a nonempty, closed and convex subset of the real Hilbert space ℋ1;(A2)
G:ℋ1⟶ℋ1 is pseudomonotone operator and l− Lipschitz continuous on ℋ1, l>0;(A3)
ℱ:ℋ1⟶ℋ1 is γ− strongly monotone and k− Lipschitz continuous on ℋ1, k>0;(A4)
T:ℋ2⟶ℋ2 is ♭− demimetric mapping and demiclosed at the origin, ♭∈−∞,1;(A5)
ℬ:ℋ1⟶ℋ2 is a nonzero bounded linear operator with adjoint ℬ∗;(A6)
δ¯n is a real sequence defined in 0,1 and satisfying the property: limn⟶∞δ¯n=0,∑n=1∞δ¯n=∞, ιn a positive real sequence with ιn=°δ¯n, i.e., limn⟶∞ιn/δ¯n=0 and ∑n=1+∞ιn<∞, ∀n≥1, β¯n⊂b,1−δ¯n,b>0;(A7)
The solution set of BVIP (21),Ω≠∅.
Algorithm 1: An alternated inertial projection and contraction method.
Initialization: Choose ε>0,ζ∈0,2, ϱ∈ (0,2−ζ/ζ], σ∈0,1, ρ∈0,2γ/k2, and given the initial points u0,u1∈ℋ1 arbitrarily.Iterative Process Steps: Given the iterates un−1 and un, choose ϱn such that 0≤ϱn≤ϱ¯n, whereϱ¯n=minιn/maxn2un−un−12,nun−un−1,ϱ,ifun≠un−1;otherwise,setϱ¯n=ϱWe calculate the iterateun+1 as follows:Step 1: Setn=1, we compute the iterate un+1 as follows:νn=un+ϱnun−un−1ifn=oddunifn=evenand computewn=PDνn−κnGνnwhere the stepsizeκn is updated asφνn,wn=νn−wn−κnGνn−Gwnκn+1=minσνn−wn/Gwn−Gνn,κnifGwn≠GνnκnifGwn=GνnIfwn=νn, stop. wn is the required solution. Else, execute Step 2.Step 2: Computesn=νn−ζξnφνn,wnwhereξn=νn−wn,φνn,wn/φνn,wn2ifφνn,wn≠00otherwiseStep 3: Computetn=sn−λnℬ∗I−Tℬsnun+1=β¯nun+1−β¯ntn−δ¯nρℱtn,n≥1where for any fixedε>0, λn is chosen such that0<ε≤λn≤1−♭ℬsn−Tℬsn2/ℬ∗ℬsn−Tℬsn2−ε,ifℬsn≠Tℬsnotherwise setλn=λ≥0.Setn:=n+1 and return to Step 1.Remark 1.
It is pertinent we give some of the following highlights of Algorithm 1 in comparison to some methods in the literature.(1)
In [10–12], algorithms for solving two level variational inequality problems in the framework of real Hilbert spaces with a fixed-point constraint of demicontractive mapping were introduced by the authors. Their algorithms were based on the extragradient method and subgradient extragradient method with underlying class of monotone and inverse strongly monotone operators onℋ1 respectively. In this work we proposed a modified projection and contraction algorithm that is based on alternated inertial extrapolation and relaxation for solving bilevel variational inequality problem with a fixed-point constraint of demimetric mapping in the setting of real Hilbert spaces when the underlying operators are pseudomonotone and strongly monotone. It is worth mentioning here that the class of operator associated with the upper problem (pseudomonotone operators) and fixed-point constraint (demimetric mapping) considered in this work is more general than the ones in [10–12] and include several important classes of operators. Specifically, pseudomonotone operators enjoy the inclusion of strongly monotone operators, monotone operators, and several others. Also, a class of demimetric mappings includes nonexpansive mappings, demicontractive mappings, directed mappings, and others.(2)
The proposed method involves alternated inertia term as well as relaxation which are very key in improving the rate of convergence for solving BVIP (21). Notably, the alternated inertia unlike the usual vanilla inertia plays integral role in the recovery of monotonicity status of the sequenceun−u∗ for each u∗ in the solution set Ω. This helps in improving the rate of convergence of the iterative scheme.(3)
In our convergence analysis, the proof of the main result; Theorem 1 (strong convergence) does not follow the traditional way of “Two Cases Approach” found in many articles handling optimization problems (see [33, 39, 40]).(4)
In our proposed Algorithm 1, the predictive stepsize conditionsκn+1∈minσνn−wn/Gwn−Gνn,κn and λn+1∈1−♭ℬsn−Tℬsn2/ℬ∗ℬsn−Tℬsn2−ε,λn use self-adaptive methods which are very simple and do not require the priori knowledge of the operator norm ℬ unlike algorithm in [37] whose stepsize depends on operator norm which may not be easy to compute.


## 4. Convergence Analysis

Remark 2.
Note from Algorithm 1 Step 1, it can be observe thatκn+1≤κn∀n≥1. Thus, for Gwn≠Gνn, then we have(33)σνn−wnGwn−Gνn≥σwn−νnlwn−νn=σl.
This implies thatminκ1,σ/l≤κn∀n≥1. Since the sequence κn is monotone decreasing, the limit exists. Thus, there exists κ>0 such that limn⟶∞κn=κ.Lemma 6.
Supposeun is a sequence generated by Algorithm 1 under Assumption 1. Then, un is boundedProof.
Letu∗∈Ω. Then, if n is odd, since w2n+1=PDν2n+1−κ2n+1Gν2n+1∈D, we have(34)Gu∗,w2n+1−u∗≥0.
Since the operatorG is pseudomonotone, we have(35)Gw2n+1,w2n+1−u∗≥0.
Since the stepsizeκ2n+1>0, it follows that(36)κ2n+1Gw2n+1,w2n+1−u∗≥0.
By Lemma 1, we have(37)ν2n+1−w2n+1−κ2n+1Gν2n+1,w2n+1−u∗≥0.
By adding (36) and (37) we obtain(38)ν2n+1−w2n+1−κ2n+1Gν2n+1−Gw2n+1,w2n+1−u∗≥0.
From Algorithm 1, Step 2 and (38), we get(39)φν2n+1,w2n+1,w2n+1−u∗≥0.
But we observe that(40)ν2n+1−u∗,φν2n+1,w2n+1=ν2n+1−w2n+1+w2n+1−u∗,φν2n+1,w2n+1=ν2n+1−w2n+1,φν2n+1,w2n+1+w2n+1−u∗,φν2n+1,w2n+1=ν2n+1−w2n+1,φν2n+1,w2n+1+w2n+1−u∗,ν2n+1−w2n+1−κ2n+1Gν2n+1−Gw2n+1.
Sincew2n+1 is a projection onto D, it follows that(41)w2n+1−u∗,ν2n+1−w2n+1−κ2n+1Gν2n+1−Gw2n+1≥0.
Thus, from (39), (40), and (41) we get(42)ν2n+1−u∗,φν2n+1,w2n+1≥ν2n+1−w2n+1,φν2n+1,w2n+1.
But we observe from the definition of the sequencesn in Algorithm 1 Step 2 that for each u∗∈Ω,(43)s2n+1−u∗2=ν2n+1−ηξ2n+1φν2n+1,w2n+1−u∗2=ν2n+1−u∗2−2ηξ2n+1ν2n+1−u∗,φν2n+1,w2n+1+ηξ2n+12φν2n+1,w2n+12=ν2n+1−u∗2−2ηξ2n+1ν2n+1−w2n+1,φν2n+1,w2n+1−2ηξ2n+1w2n+1−u∗,φν2n+1,w2n+1+ηξ2n+12φν2n+1,w2n+12≤ν2n+1−u∗2−2ηξ2n+1ν2n+1−w2n+1,φν2n+1,w2n+1+ηξ2n+12φν2n+1,w2n+12.
From the definition ofξn, we know that(44)ξ2n+1=ν2n+1−w2n+1,φν2n+1,w2n+1φν2n+1,w2n+12ifφν2n+1,w2n+1≠0.
It follows that(45)ξ2n+1φν2n+1,w2n+12=ν2n+1−w2n+1,φν2n+1,w2n+1.
Combining (43) and (45) we obtain(46)s2n+1−u∗2≤ν2n+1−u∗2−2ζξ2n+1ν2n+1−w2n+1,φν2n+1,w2n+1+η2ξ2n+1ν2n+1−w2n+1,φν2n+1,w2n+1=ν2n+1−u∗2−ζξ2n+12−ζν2n+1−w2n+1,φν2n+1,w2n+1=ν2n+1−u∗2−ζξ2n+12−ζξ2n+1φν2n+1,w2n+12=ν2n+1−u∗2−ζ2−ζξ2n+1φν2n+1,w2n+12.
From Algorithm 1 Step 2 we have(47)s2n+1=ν2n+1−ζξ2n+1φν2n+1,w2n+1.
Implying that(48)s2n+1−ν2n+1ζ=−ξ2n+1φν2n+1,w2n+1.
Thus,(49)s2n+1−ν2n+12ζ2=ξ2n+1φν2n+1,w2n+12.
From (46) and (49) we get(50)s2n+1−u∗2≤ν2n+1−u∗2−2−ζζs2n+1−ν2n+12(51)≤ν2n+1−u∗2.
Using the definition of the sequencetn in Algorithm 1 Step 3, Lemma 2 we get(52)t2n+1−u∗2=s2n+1−λ2n+1ℬ∗I−Tℬs2n+1−u∗2=s2n+1−u∗2−2λ2n+1s2n+1−u∗,ℬ∗I−Tℬs2n+1+λ2n+12ℬ∗I−Tℬs2n+12=s2n+1−u∗2−2λ2n+1ℬs2n+1−ℬu∗,I−Tℬs2n+1+λ2n+12ℬ∗I−Tℬs2n+12.
Since the operatorT is ♭− demimetric, from (52) we get(53)t2n+1−u∗2≤s2n+1−u∗2−λ2n+11−♭I−Tℬs2n+12+λ2n+12ℬ∗I−Tℬs2n+12=s2n+1−u∗2+λ2n+1λ2n+1ℬ∗I−Tℬs2n+12−1−♭I−Tℬs2n+12.
But we know from the definition of the adaptive stepsizeλ2n+1 in (34) that for ℬsn≠Tℬsn we have(54)λ2n+1+ε≤1−♭ℬs2n+1−Tℬs2n+12ℬ∗ℬs2n+1−Tℬs2n+12.
It follows that(55)λ2n+1+εℬ∗ℬs2n+1−Tℬs2n+12≤1−♭ℬs2n+1−Tℬs2n+12.
This implies(56)λ2n+1ℬ∗ℬs2n+1−Tℬs2n+12−1−♭ℬs2n+1−Tℬs2n+12≤−εℬ∗ℬs2n+1−Tℬs2n+12.
Implying,(57)λ2n+1λ2n+1ℬ∗ℬs2n+1−Tℬs2n+12−1−♭ℬs2n+1−Tℬs2n+12≤−ελ2n+1ℬ∗ℬs2n+1−Tℬs2n+12.
Hence,(58)λ2n+1λ2n+1ℬ∗ℬs2n+1−Tℬs2n+12−1−♭ℬs2n+1−Tℬs2n+12≤−ε2ℬ∗ℬs2n+1−Tℬs2n+12.
Combining (53) and (58) we obtain(59)t2n+1−u∗2≤s2n+1−u∗2−ελ2n+1ℬ∗ℬs2n+1−Tℬs2n+12≤s2n+1−u∗2−ε2ℬ∗ℬs2n+1−Tℬs2n+12,(60)≤s2n+1−u∗2.
Also using the definition of the sequenceνn in Algorithm 1 Step 1 and for each u∗∈Ω we have(61)ν2n+1−u∗=u2n+1+ϱ2n+1u2n+1−u2n−u∗≤u2n+1−u∗+δ¯2n+1ϱ2n+1δ¯2n+1u2n+1−u2n.
From Algorithm 1, we know thatϱ2n+1/δ¯2n+1u2n+1−u2n≤ι2n+1/2n+1δ¯2n+1≤ι2n+1/δ¯2n+1. By (A6) it follows that(62)limn⟶∞ϱ2n+1δ¯2n+1u2n+1−u2n=0.
This implies that the sequenceϱ2n+1/δ¯2n+1u2n+1−u2n is bounded. And so, there exists a positive constant K0>0 such that ϱ2n+1/δ¯2n+1u2n+1−u2n≤K0 for all n≥1. Thus, from (61) we get(63)ν2n+1−u∗≤u2n+1−u∗+δ¯2n+1K0.
Using the definition ofun+1 in Algorithm 1 (52), (60), (63), and Lemma 3 we obtain(64)u2n+2−u∗=β¯2n+1u2n+1+1−β¯2n+1t2n+1−δ¯2n+1ρℱt2n+1−u∗=β¯2n+1u2n+1−u∗+1−β¯2n+1t2n+1−1−β¯2n+1u∗−δ¯2n+1ρℱt2n+1+δ¯2n+1ρℱu∗−δ¯2n+1ρℱu∗=1−β¯2n+1−δ¯2n+1ρℱt2n+1−1−β¯2n+1−δ¯2n+1ρℱu∗+β¯2n+1u2n+1−u∗−δ¯2n+1ρℱu∗≤1−β¯2n+1−δ¯2n+1ρℱt2n+1−1−β¯2n+1−δ¯2n+1ρℱu∗+β¯2n+1u2n+1−u∗+δ¯2n+1ρℱu∗≤1−β¯2n+1−τδ¯2n+1t2n+1−u∗+β¯2n+1u2n+1−u∗+δ¯2n+1ρℱu∗≤1−β¯2n+1−τδ¯2n+1s2n+1−u∗+β¯2n+1u2n+1−u∗+δ¯2n+1ρℱu∗≤1−β¯2n+1−τδ¯2n+1ν2n+1−u∗+β¯2n+1u2n+1−u∗+δ¯2n+1ρℱu∗≤1−β¯2n+1−τδ¯2n+1u2n+1−u∗+δ¯2n+1K0+β¯2n+1u2n+1−u∗+δ¯2n+1ρℱu∗≤1−τδ¯2n+1u2n+1−u∗+δ¯2n+1K0+δ¯2n+1ρℱu∗≤1−τδ¯2n+1u2n+1−u∗+δ¯2n+1K0+ρℱu∗≤1−τδ¯2n+1u2n+1−u∗+τδ¯2n+1K0+ρℱu∗τ≤maxu2n+1−u∗,τ−1K0+ρℱu∗,where τ:=1−1−ρ2γ−ρk2.
Following similar line of argument we get(65)u2n+1−u∗=β¯2nu2n+1−β¯2nt2n−δ¯2nρℱt2n−u∗=β¯2nu2n−u∗+1−β¯2nt2n−1−β¯2nu∗−δ¯2nρℱt2n+δ¯2nρℱu∗−δ¯2nρℱu∗=1−β¯2n−δ¯2nρℱt2n−1−β¯2n−δ¯2nρℱu∗+β¯2nu2n−u∗−δ¯2nρℱu∗≤1−β¯2n−δ¯2nρℱt2n−1−β¯2n−δ¯2nρℱu∗+β¯2nu2n−u∗+δ¯2nρℱu∗≤1−β¯2n−τδ¯2nt2n−u∗+β¯2nu2n−u∗+δ¯2nρℱu∗≤1−β¯2n−τδ¯2ns2n−u∗+β¯2nu2n−u∗+δ¯2nρℱu∗≤1−β¯2n−τδ¯2nν2n−u∗+β¯2nu2n−u∗+δ¯2nρℱu∗=1−β¯2n−τδ¯2nu2n−u∗+β¯2nu2n−u∗+δ¯2nρℱu∗≤1−β¯2n−τδ¯2nu2n−u∗+δ¯2nϱ2nδ¯2nu2n−u2n−1+β¯2nu2n−u∗+δ¯2nρℱu∗≤1−τδ¯2nu2n−u∗+δ¯2nK0+δ¯2nρℱu∗≤1−τδ¯2nu2n−u∗+δ¯2nK0+ρℱu∗≤1−τδ¯2nu2n−u∗+τδ¯2nK0+ρℱu∗τ≤maxu2n−u∗,τ−1K0+ρℱu∗.
Hence, by induction(66)un−u∗≤maxu1−u∗,τ−1K0+ρℱu∗,∀n≥1.
Sinceun−u∗ is bounded, it implies that un is bounded. Consequently, νn, wn, sn, tn, and ℱtn are all bounded.Lemma 7.
Letun be sequence generated by Algorithm 1 under Assumption 1 such that limn⟶∞ϱ2n+1/δ¯2n+1u2n+1−u2n=0. Then, un satisfies the inequality(67)ψ2n+2≤1−τδ¯2n+1ψ2n+1+τδ¯2n+1jn,where(68)ψ2n+2≔u2n+2−u∗2,ψ2n+1≔u2n+1−u∗2,jn≔τ−1K0K1+2τ−1ρℱu∗,u∗−u2n+2,with δ¯n as defined in (A6), τ:=1−1−ρ2γ−ρk2,K0>0 is a real constant and K1:=sup2u2n+1−u∗+δ¯2n+1u2n+1−u2n.Proof.
Letu∗∈Ω. Then, from the definition of un+1 in Step 3 of Algorithm 1 under Assumption 1, using Lemma 2, Lemma 3, (51), and (59) we obtain(69)u2n+2−u∗2=β¯2n+1u2n+1+1−β¯2n+1t2n+1−δ¯2n+1ρℱt2n+1−u∗2=β¯2n+1u2n+1−u∗+1−β¯2n+1t2n+1−1−β¯2n+1u∗−δ¯2n+1ρℱt2n+1+δ¯2n+1ρℱu∗−δ¯2n+1ρℱu∗2=1−β¯2n+1−δ¯2n+1ρℱt2n+1−1−β¯2n+1−δ¯2n+1ρℱu∗+β¯2n+1u2n+1−u∗−δ¯2n+1ρℱu∗2≤1−β¯2n+1−δ¯2n+1ρℱt2n+1−1−β¯2n+1−δ¯2n+1ρℱu∗+β¯2n+1u2n+1−u∗2+2δ¯2n+1ρℱu∗,u∗−u2n+2≤1−β¯2n+1−δ¯2n+1ρℱt2n+1−1−β¯2n+1−δ¯2n+1ρℱu∗+β¯2n+1u2n+1−u∗2+2δ¯2n+1ρℱu∗,u∗−u2n+2≤1−β¯2n+1−τδ¯2n+1t2n+1−u∗2+β¯2n+1u2n+1−u∗2+2δ¯2n+1ρℱu∗,u∗−u2n+2≤1−β¯2n+1−τδ¯2n+1s2n+1−u∗2−ε2ℬ∗ℬs2n+1−Tℬs2n+12+β¯2n+1u2n+1−u∗2+2δ¯2n+1ρℱu∗,u∗−u2n+2=1−β¯2n+1−τδ¯2n+1s2n+1−u∗2−1−β¯2n+1−τδ¯2n+1ε2ℬ∗ℬs2n+1−Tℬs2n+12+β¯2n+1u2n+1−u∗2+2δ¯2n+1ρℱu∗,u∗−u2n+2≤1−β¯2n+1−τδ¯2n+1ν2n+1−u∗2−2−ζζs2n+1−ν2n+12−1−β¯2n+1−τδ¯2n+1ε2ℬ∗ℬs2n+1−Tℬs2n+12+β¯2n+1u2n+1−u∗2+2δ¯2n+1ρℱu∗,u∗−u2n+2=1−β¯2n+1−τδ¯2n+1ν2n+1−u∗2−2−ζζ1−β¯2n+1−τδ¯2n+1s2n+1−ν2n+12−1−β¯2n+1−τδ¯2n+1ε2ℬ∗ℬs2n+1−Tℬs2n+12+β¯2n+1u2n+1−u∗2+2δ¯2n+1ρℱu∗,u∗−u2n+2.
But we observe that(70)ν2n+1−u∗2=u2n+1−u∗+ϱ2n+1u2n+1−u2n2=u2n+1−u∗2+2ϱ2n+1u2n+1−u∗,u2n+1−u2n+ϱ2n+12u2n+1−u2n2≤u2n+1−u∗2+2ϱ2n+1u2n+1−u∗u2n+1−u2n+ϱ2n+12u2n+1−u2n2=u2n+1−u∗2+ϱ2n+1u2n+1−u2n2u2n+1−u∗+ϱ2n+1u2n+1−u2n=u2n+1−u∗2+ϱ2n+1u2n+1−u2nK1,where K1:=sup2u2n+1−u∗+ϱ2n+1u2n+1−u2n.
By combining (69) and (70) we have(71)u2n+2−u∗2≤1−β¯2n+1−τδ¯2n+1u2n+1−u∗2+ϱ2n+1u2n+1−u2nK1−2−ζζ1−β¯2n+1−τδ¯2n+1s2n+1−ν2n+12−1−β¯2n+1−τδ¯2n+1ε2ℬ∗ℬs2n+1−Tℬs2n+12+β¯2n+1u2n+1−u∗2+2δ¯2n+1ρℱu∗,u∗−u2n+2≤1−β¯2n+1−τδ¯2n+1u2n+1−u∗2+δ¯2n+1K1ϱ2n+1δ¯2n+1u2n+1−u2n−2−ζζ1−β¯2n+1−τδ¯2n+1s2n+1−ν2n+12−1−β¯2n+1−τδ¯2n+1ε2ℬ∗ℬs2n+1−Tℬs2n+12+β¯2n+1u2n+1−u∗2+2δ¯2n+1ρℱu∗,u∗−u2n+2,(72)≤1−τδ¯2n+1u2n+1−u∗2+δ¯2n+1K0K1−2−ζζ1−β¯2n+1−τδ¯2n+1s2n+1−ν2n+12−1−β¯2n+1−τδ¯2n+1ε2ℬ∗ℬs2n+1−Tℬs2n+12+2δ¯2n+1ρℱu∗,u∗−u2n+2≤1−τδ¯2n+1u2n+1−u∗2−2−ζζ1−β¯2n+1−τδ¯2n+1s2n+1−ν2n+12−1−β¯2n+1−τδ¯2n+1ε2ℬ∗ℬs2n+1−Tℬs2n+12+τδ¯2n+1τ−1K0K1+2τ−1ρℱu∗,u∗−u2n+2,(73)≤1−τδ¯2n+1u2n+1−u∗2+τδ¯2n+1τ−1K0K1+2τ−1ρℱu∗,u∗−u2n+2,which yields the desired result.Theorem 1.
Supposeun is the sequence generated by Algorithm 1 under Assumption 1. Then, un converges strongly to u∗ which solves BVIP (21), where(74)u∗=minν∗:ν∗∈Ω.Proof.
Letu∗∈Ω. Using the definition of un+1 in Algorithm 1 under Assumption 1 and following the same technique in Lemma 7 we obtain(75)u2n+1−u∗2=β¯2nu2n+1−β¯2nt2n−δ¯2nρℱt2n−u∗2=β¯2nu2n−u∗+1−β¯2nt2n−1−β¯2nu∗−δ¯2nρℱt2n+δ¯2nρℱu∗−δ¯2nρℱu∗2=1−β¯2n−δ¯2nρℱt2n−1−β¯2n−δ¯2nρℱu∗+β¯2nu2n−u∗−δ¯2nρℱu∗2≤1−β¯2n−δ¯2nρℱt2n−1−β¯2n−δ¯2nρℱu∗+β¯2nu2n−u∗2+2δ¯2nρℱu∗,u∗−u2n+1≤1−β¯2n−δ¯2nρℱt2n−1−β¯2n−δ¯2nρℱu∗+β¯2nu2n−u∗2+2δ¯2nρℱu∗,u∗−u2n+1≤1−β¯2n−τδ¯2nt2n−u∗2+β¯2nu2n−u∗2+2δ¯2nρℱu∗,u∗−u2n+1≤1−β¯2n−τδ¯2ns2n−u∗2−ε2ℬ∗ℬs2n−Tℬs2n2+β¯2nu2n−u∗2+2δ¯2nρℱu∗,u∗−u2n+1=1−β¯2n−τδ¯2ns2n−u∗2−1−β¯2n−τδ¯2nε2ℬ∗ℬs2n−Tℬs2n2+β¯2nu2n−u∗2+2δ¯2nρℱu∗,u∗−u2n+1≤1−β¯2n−τδ¯2nν2n−u∗2−2−ζζs2n−ν2n2−1−β¯2n−τδ¯2nε2ℬ∗ℬs2n−Tℬs2n2+β¯2nu2n−u∗2+2δ¯2nρℱu∗,u∗−u2n+1=1−β¯2n−τδ¯2nu2n−u∗2−2−ζζ1−β¯2n−τδ¯2ns2n−u2n2−1−β¯2n−τδ¯2nε2ℬ∗ℬs2n−Tℬs2n2+β¯2nu2n−u∗2+2δ¯2nρℱu∗,u∗−u2n+1≤1−β¯2n−τδ¯2nu2n−u∗2+ϱ2nu2n−u2n−1K1−2−ζζ1−β¯2n−τδ¯2ns2n−u2n2−1−β¯2n−τδ¯2nε2ℬ∗ℬs2n−Tℬs2n2+β¯2nu2n−u∗2+2δ¯2nρℱu∗,u∗−u2n+1=1−β¯2n−τδ¯2nu2n−u∗2+δ¯2nK1ϱ2nδ¯2nu2n−u2n−1−2−ζζ1−β¯2n−τδ¯2ns2n−u2n2,(76)−1−β¯2n−τδ¯2nε2ℬ∗ℬs2n−Tℬs2n2+β¯2nu2n−u∗2+2δ¯2nρℱu∗,u∗−u2n+1≤1−τδ¯2nu2n−u∗2+δ¯2nK0K1−2−ζζ1−β¯2n−τδ¯2ns2n−u2n2−1−β¯2n−τδ¯2nε2ℬ∗ℬs2n−Tℬs2n2+2δ¯2nρℱu∗,u∗−u2n+1≤1−τδ¯2nu2n−u∗2−2−ζζ1−β¯2n−τδ¯2ns2n−u2n2−1−β¯2n−τδ¯2nε2ℬ∗ℬs2n−Tℬs2n2+τδ¯2nτ−1K0K1+2τ−1ρℱu∗,u∗−u2n+1≤1−τδ¯2nu2n−u∗2+τδ¯2nτ−1K0K1+2τ−1ρℱu∗,u∗−u2n+1.
Using Lemma 5 and (76) we need establish that(77)limsupℓ⟶∞τ−1K0K1+2τ−1ρℱu∗,u∗−u2nℓ+1≤0,for each subsequence u2nℓ−u∗ of the sequence u2n−u∗ satisfying the condition:(78)liminfℓ⟶∞u2nℓ+1−u∗−u2nℓ−u∗≥0.
Letu2nℓ−u∗ be the subsequence of u2n−u∗ satisfying condition (78), then(79)liminfℓ⟶∞u2nℓ+1−u∗2−u2nℓ−u∗2=liminfℓ⟶∞u2nℓ+1−u∗−u2nℓ−u∗u2nℓ+1−u∗+u2nℓ−u∗≥0.
From (75) we have(80)limsupℓ⟶∞2−ζζ1−β¯2nℓ−τδ¯2nℓs2nℓ−u2nℓ2+1−β¯2nℓ−τδ¯2nℓε2ℬ∗ℬs2nℓ−Tℬs2nℓ2≤limsupℓ⟶∞1−τδ¯2nℓu2nℓ−u∗2+τδ¯2nℓτ−1K0K1+2τ−1ρℱu∗u∗−u2nℓ+1−u2nℓ+1−u∗2≤limsupℓ⟶∞u2nℓ−u∗2−u2nℓ+1−u∗2=−liminfℓ⟶∞u2nℓ+1−u∗2−u2nℓ−u∗2≤0.
This follows that(81)limℓ⟶∞2−ζζ1−β¯2nℓ−τδ¯2nℓs2nℓ−u2nℓ2+1−β¯2nℓ−τδ¯2nℓε2ℬ∗ℬs2nℓ−Tℬs2nℓ2=0.
Thus,(82)limℓ⟶∞s2nℓ−u2nℓ=0,(83)limℓ⟶∞ℬ∗ℬs2nℓ−Tℬs2nℓ=0.
Observe from the definition of the sequenceφνnℓ,wnℓ that(84)φν2nℓ,w2nℓ=ν2nℓ−w2nℓ−κ2nℓGν2nℓ−Gw2nℓ≤ν2nℓ−w2nℓ+κ2nℓGν2nℓ−Gw2nℓ≤ν2n−w2nℓ+κ2nℓκ2nℓ+1σν2nℓ−w2nℓ=1+κ2nℓκ2nℓ+1σν2nℓ−w2nℓ.
This follows that(85)1φν2nℓ,w2nℓ≥11+κ2nℓ/κ2nℓ+1σν2nℓ−w2nℓ.
Also, we observe that(86)ν2nℓ−w2nℓ,φν2nℓ,w2nℓ=ν2nℓ−w2nℓ,ν2nℓ−w2nℓ−κ2nℓGν2nℓ−Gw2nℓ=ν2nℓ−w2nℓ2−ν2nℓ−w2nℓ,κ2nℓGν2nℓ−Gw2nℓ≥ν2nℓ−w2nℓ2−κ2nℓκ2nℓ+1σν2nℓ−w2nℓ2=1−κ2nℓκ2nℓ+1σν2nℓ−w2nℓ2.
But from Algorithm 1 Step 2 (settingν2nℓ=u2nℓ since n is even), we have(87)s2nℓ−u2nℓ=−ζξ2nφu2nℓ,w2nℓ.
It implies(88)s2nℓ−u2nℓ=ζξ2nφu2nℓ,w2nℓ=ζu2nℓ−w2nℓ,φu2nℓ,w2nℓφu2nℓ,w2nℓ.
Combining (85), (86), and (88) we obtain(89)s2nℓ−u2nℓ≥ζ1−κ2nℓ/κ2nℓ+1σu2nℓ−w2nℓ21+κ2nℓ/κ2nℓ+1σu2nℓ−w2nℓ.
It follows(90)u2nℓ−w2nℓ≤κ2nℓ+1+κ2nℓσζκ2nℓ+1−κ2nℓσs2nℓ−u2nℓ.
From Remark 2, we know that(91)limℓ⟶∞κ2nℓ+1+κ2nℓσκ2nℓ+1−κ2nℓσ=1+σ1−σ.
This shows that the sequenceκ2nℓ+1+κ2nℓσ/κ2nℓ+1−κ2nℓσ is bounded. Thus, by combining (82) and (90) we get(92)limℓ⟶∞u2nℓ−w2nℓ=0.
Since the sequences2nℓ is bounded and the operator T is ♭− demimetric, there exists a real positive constant M>0 such that(93)s2nℓ−u∗ℬ∗ℬs2nℓ−Tℬs2nℓ≥s2nℓ−u∗,ℬ∗ℬs2nℓ−Tℬs2nℓ=ℬs2nℓ−ℬu∗,ℬs2nℓ−Tℬs2nℓ≥1−♭2ℬs2nℓ−Tℬs2nℓ2.
Buts2nℓ−u∗≤M, for any M>0 since s2nℓ is bounded. Therefore, we have,(94)ℬs2nℓ−Tℬs2nℓ2≤2M1−♭ℬ∗ℬs2nℓ−Tℬs2nℓ.
Consequently,(95)ℬs2nℓ−Tℬs2nℓ2≤2M1−♭ℬ∗ℬs2nℓ−Tℬs2nℓ2.
Combining (83) and (95) we get(96)limℓ⟶∞ℬs2nℓ−Tℬs2nℓ=0.
Similarly, using (82) and (92)(97)s2nℓ−w2nℓ≤s2nℓ−u2nℓ+u2nℓ−w2nℓ⟶0asℓ⟶∞.
Thus,(98)limℓ⟶∞s2nℓ−w2nℓ=0.
Using (83) we get(99)t2nℓ−s2nℓ=λ2nℓℬ∗ℬs2nℓ−Tℬs2nℓ⟶0asℓ⟶∞.
From (82) and (99) we obtain(100)t2nℓ−u2nℓ≤t2nℓ−s2nℓ+s2nℓ−u2nℓ⟶0asℓ⟶∞.
Sinceδ¯n⟶0 as n⟶∞, using (100) and condition (A6) we have(101)u2nℓ+1−u2nℓ≤1−β¯2nℓt2nℓ−u2nℓ+δ¯2nℓρℱt2nℓ⟶0asℓ⟶∞.
Sinceu2nℓ is bounded and limℓ⟶∞u2nℓ−w2nℓ=0, there exists subsequence u2nℓk of u2nℓ such that u2nℓk⇀z∗∈ℋ1 as k⟶∞. From Lemma 4 we obtain that z∗∈VIG,D.
Also, sinceu2nℓk⇀z∗, from (82) we have s2nℓk⇀z∗ as k⟶∞. We know that the operator ℬ is bounded and so, we have that ℬs2nℓk converges weakly to ℬz∗∈ℋ2ask⟶∞. Thus, combining (110) and demiclosedness of the operator T, we get that ℬz∗∈FT.
Lastly, we need establish thatu2nℓ⟶u∗ as ℓ⟶∞ which solves BVIP (21). But we know that u2nℓk converge weakly to z∗, hence(102)limsupℓ⟶∞ℱu∗,u∗−u2nℓ=limk⟶∞ℱu∗,u∗−u2nℓk=ℱu∗,u∗−z∗.
Sincez∗ is a unique solution of (21), then z∗∈Ω and it follows that(103)limsupℓ⟶∞ℱu∗,u∗−u2nℓ=ℱu∗,u∗−z∗≤0.
Thus, using (101) and (103) we obtain,(104)limsupℓ⟶∞ℱu∗,u∗−u2nℓ+1=limsupℓ⟶∞ℱu∗,u∗−u2nℓ=ℱu∗,u∗−z∗≤0.
Applying Lemma 7, we obtain(105)limℓ⟶∞u2nℓ−u∗=0.
Therefore,u2n⟶u∗∈Ω as n⟶∞ establishing strong convergence result for sequence of even terms.
Next we establish convergence for sequence of odd termsu2n+1. From (71) and (101) and the condition on the control sequences in (A6), we get(106)limℓ⟶∞ν2nℓ+1−u2nℓ≤limℓ⟶∞u2nℓ+1−u2nℓ+limℓ⟶∞δ¯2nℓ+1ϱ2nℓ+1δ¯2nℓ+1u2nℓ+1−u2nℓ=0.
Hence,(107)limℓ⟶∞ν2nℓ+1−u2nℓ=0.
Following the same argument in obtaining (82), (83), and (96) we get,(108)limℓ⟶∞s2nℓ+1−u2nℓ+1=0,limℓ⟶∞ℬ∗ℬs2nℓ+1−Tℬs2nℓ+1=0,(109)limℓ⟶∞ℬs2nℓ+1−Tℬs2nℓ+1=0.
Also, following similar approach of getting (98), (99), and (100) we obtain(110)limℓ⟶∞s2nℓ+1−w2nℓ+1=0,limℓ⟶∞t2nℓ+1−s2nℓ+1=0,(111)limℓ⟶∞t2nℓ+1−u2nℓ+1=0.
Using (111) and Assumption (A6) we get(112)u2nℓ+2−u2nℓ+1≤1−β¯2nℓ+1t2nℓ+1−u2nℓ+1+δ¯2nℓ+1ρℱt2nℓ+1⟶0asℓ⟶∞.
Thus, following the same argument used even case, we show that(113)limsupℓ⟶∞ℱu∗,u∗−u2nℓ+1=ℱu∗,u∗−z∗≤0.
And so by Lemma 5 we obtain(114)limℓ⟶∞u2nℓ+1−u∗=0.
Therefore, the sequenceun converges strongly to u∗∈Ω completing the proof.


## 5. Numerical Examples

Under this section, we shall present numerical illustrations to show the behavior of our proposed iterative method, Algorithm 1 in comparison to algorithms of Mainge´ (12), Akutsah et al. (14), and Thong et al. (16). All codes are written in Matlab R2018a and executed on a PC Desktop Intel (R) Core (TM) i5-6300U CPU @ 2.40 GHz 2.50 GHz, RAM 8.00 GB. In the numerical results presented in the table below, “No of Iter.” denotes the number of iterations, and “Sec.” denotes the running time of CPU in seconds. Also, for the purpose of these numerical implementations, we shall choose κ1=1, σ=0.5, ρ=0.2, ζ=1.5, ϱ=0.3, λn=2, δ¯n=1/n+10.3, β¯n=0.8−δ¯n, ιn=δ¯n/n0.03, and ϱn=1/n+1 for all n≥1. It is obvious that, these parameters satisfy the conditions in Assumption 1. Similarly, we choose λn=1/2n+5,η=0.5,αn=1/n+1 for algorithm of Mainge´, ζ0=0.5,β¯n=n0.5,αn=n−1/n+2,ϱ=0.2 for algorithm of Akutsah et al., and η=0.5,ζn=n+1/2n+5,αn=n+1/n2+3 for algorithm of Thong et al.Example 1.
Letℋ1=ℋ2=l2ℝ be a linear spaces of square summable sequences. That is(115)l2ℝ≔=u≔u1,u2,u3,…,uk∈ℝ:∑k=1∞uk2<+∞,with the associated norm ul2=∑k=1∞uk21/2 and inner product u,ν=∑k=1∞ukνk where u=ukk=1∞ and ν=νkk=1∞. Let G:l2ℝ⟶l2ℝ and ℱ:l2ℝ⟶l2ℝ be mappings defined by Gu=max0,u/2 with u=ukk=1∞ and ℱν=3ν with ν=νkk=1∞, respectively. Consider a mapping ℬ:l2ℝ⟶l2ℝ such that ℬu=0,u1,u2/2,u3/3,u4/4,…,∀u∈l2ℝ. ℬ as defined is a bounded linear operator with adjoint ℬ∗ν=ν2,ν3/2,ν4/3,…∀ν∈l2ℝ. Let the feasible set D be defined by D:=u∈l2ℝ:ul2≤1. Finally, let T:l2ℝ⟶l2ℝ be a mapping defined by Tu=2usinu/3. We shall take En=un−un−1≤10−4 as the stopping criterion and choose the following as starting points:Case A:u0=0.7,0.1,2,…,0,…′ and u1=1,1,−3,…0…′.Case B:u0=0,0,0,…,0,…′ and u1=3,3,−6,…,0,…′.Case C:u0=0.1,0.1,3,…,0,…′ and u1=1,1,−1,…,0,…′.Case D:u0=0.1,0.3,2,…,0,…′ and u1=1,0,0,…,0,…′.Example 2.
Letℋ1=ℋ1=L20,1 be linear spaces of square integrable functions embedded with inner product u,ν=∫01uxνxdx∀u,ν∈L20,1,x∈0,1 and associated norm uL2=∫01ux2|dx1/2∀u∈L20,1. Consider the feasible D:=u∈L20,1:u,ν≤a,a>0. Define a projection from L20,1 onto D by(116)PDu=a−u,νν2,ifu,ν>a,u,otherwise.
Consider operatorℳ:D⟶ℝ given by ℳw=1/1+w2 and Volterra integral operator W:L20,1⟶L20,1 defined by Wzx=∫0xztdt∀z∈L20,1 and x∈0,1. W is linear, monotone and π/2− Lipschitzian (see [41]). Let G:L20,1⟶L20,1 be a mapping defined by Gzx:=ℳzWzx. Given that Gz,ν−z≥0∀ν,z∈D, it implies that(117)Gν,ν−z=ℳνWν,ν−z≥ℳνWν,ν−z−Wz,ν−z=ℳνWν−Wz,ν−z≥0.
This shows thatG is pseudomonotone. Define T:L20,1⟶L20,1 by Tux=ux/3. T is 0− demimetric. Also let ℬ:L20,1⟶L20,1 be a bounded linear mapping defined by ℬux=ux/2 with adjoint ℬ∗νx=νx/2∀u,ν∈L20,1 and x∈0,1. Taking En=un−un−1≤10−5 as the stopping criterion we obtain the following results using the following as starting points:Case A:u0x=2x2+3 and u1x=sinx+1.Case B:u0x=2ex+x and u1x=sin2x.Case C:u0x=2ex+3x and u1x=sinx2.Case D:u0x=ex+x and u1x=cos3x+x.Example 3.
Letℋ1=ℝn and ℋ2=ℝm. Consider a mapping G:ℝn⟶ℝn defined by Gu=DTDu+b, where D is an n×n matrix and b is a column vector in ℝn. It can be easily seen that G is monotone and Lipschitzian with Lipschitz constant k=D. Let ℱ:ℝm⟶ℝm be a mapping defined by ℱu=Eu+q, where E is m×m symmetric matrix which is positive definite and q is vector in ℝm. ℱ defined above is strongly monotone and Lipschitz continuous. Let D:=u∈ℋ1:u<1. Consider the bounded linear operator ℬ generated randomly in the interval (0, 1) and mapping T:ℝm⟶ℝm define by Tu=2u. We shall consider different cases of dimensions m,n=100,10,150,15,200,20, and (250, 25) with u0=1,1,1,…,1T and u1 randomly generated in ℝm. Also we take En=un−un−1≤10−5 as the stopping criterion. Using the appropriate parameters we obtain the following results:
With the examples (Example 1–Example 3, and cases (Case A–Case D) for each, we compare Algorithm 1 with Algorithm (12) of Mainge in [11], Algorithm (16) of Thong et al. in [21] and Algorithm (14) of Akutsah et al. in [12], Tables 1–3 and Figures 1–3 show that our algorithm performs better than the Algorithms in Mainge, Thong et al. and Akutsah et al. in terms of number of iterations steps and CPU running time. Also, in Example 3 we compare Algorithm 1 with Algorithm (16) of Thong et al. in [21] and Algorithm (14) of Akutsah et al. in [12] which the results validate the fact that our proposed algorithm performs better.Table 1
Numerical result for Example 1.
CasesAbuchu et al.MaingeThong et al.Akutsah et al.Case ANo of Iter.9161612Sec.0.00150.00220.008320.0015Case BNo of Iter.12171813Sec.0.00043130.0005360.00061790.0008362Case CNo of Iter.9181512Sec.0.00034440.0004720.00039830.0003874Case DNo of Iter.8231412Sec.0.00025660.0029060.00026970.005884Table 2
Numerical result for Example 2.
CasesAbuchu et al.MaingeThong et al.Akutsah et al.Case ANo of Iter.8211414Sec.40.446042.541558.9468119.5821Case BNo of iter.9191815Sec.46.725452.353040.040897.1884Case CNo of Iter.9181815Sec.55.312757.0295371.6369114.0089Case DNo of Iter.10181915Sec.60.940163.0712152.7729130.0836Table 3
Numerical result for Example 3.
(m, n)Abuchu et al.Thong et al.Akutsah et al.(100, 10)No of Iter132514Sec0.01260.03230.0138(150, 15)No of iter94913Sec0.02300.08430.0262(200, 20)No of Iter104613Sec0.06870.15170.0623(250, 25)No of Iter95010Sec0.11600.21590.1708Figure 1
Plot of errors against iterations number n for Example 1; (a): case A; (b): case B; (c): case C; (d): case D.
(a)(b)(c)(d)Figure 2
Plot of errors against iterations numbern for Example 2; (a): case A; (b): case B; (c): case C; (d): case D.
(a)(b)(c)(d)Figure 3
Plot of errors against iterations number n for Example 3; (a): case A; (b): case B; (c): case C; (d): case D.
(a)(b)(c)(d)


## 6. Conclusion

We introduce and study a modified relaxed alternated inertial projection and contraction algorithm for solving bilevel split variational inequality problem when the underlying mapping is pseudomonotone in real Hilbert spaces with a fixed-point constraint of demimetric mapping. Our proposed iterative algorithm is based on alternated inertial extrapolation fused with relaxation under adaptive stepsize conditions in the framework of real Hilbert spaces. Under some certain assumptions on the algorithm parameters, we establish that the sequence generated by the algorithm converges strongly to a unique solution of bilevel pseudomonotone variational inequality problem without the prior knowledge of the operator norm or the coefficients of the underlying of operators. Finally, some numerical examples were presented which validate the applicability and efficiency of our iterative method.



---

*Source: 3185746-2023-03-22.xml*