# An Iterative Algorithm for Solving Fixed Point Problems and Quasimonotone Variational Inequalities

Authors: Tzu-Chien Yin; Yan-Kuen Wu; Ching-Feng Wen
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we survey a common problem of the fixed point problem and the quasimonotone variational inequality problem in Hilbert spaces. We suggest an iterative algorithm for finding a common element of the solution of a quasimonotone variational inequality and the fixed point of a pseudocontractive operator. Convergence theorems are shown under some mild conditions. Several corollaries are also obtained.


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## Body

## 1. Introduction

LetH be a real Hilbert space with an inner product ⋅,⋅ and an induced norm ⋅. Let C be a nonempty closed and convex subset of H. Let f:C⟶H be a nonlinear operator. In this paper, our work is closely related to a classical variational inequality of finding a point x†∈C such that(1)fx†,x−x†≥0,∀x∈C.We useSolC,f to denote the solution set of (1).It is well known that variational inequality problems provide a general mathematical framework for a large number of problems arising in optimization [1–8]. For example, constrained optimization problems such as LP and NLP are special cases of variational inequalities, and systems of equations and complementarity problems can be cast as variational inequalities. Thus, variational inequality problems have many applications, including those in transportation networks [9], signal processing [10, 11], regression analysis [12], equilibrium problems [13, 14], fixed point problems [15–19], and complementarity problems [1, 20]. There are numerous iterative algorithms for solving variational inequalities and related problems, (see for examples [21–31]).Letφ:C⟶ℝ be a convex function. Letting fx=∇φx, the variational inequality (1) is equivalent to the following minimization problem:(2)minx∈Cφx,which implies that we can use the following projection-gradient algorithm [32–35] to solve variational inequality (1), i.e., an iterative sequence un generated by the recursive form:(3)un+1=projCun−ςnfun,where ςn>0 is the step size, and projC:H⟶C is the metric projection.The sequenceun generated by the projection-gradient algorithm is the convergent provided. f is strongly (pseudo) monotone (see [25, 36]), or f is inverse strongly monotone (see [10, 35]). However, if f is plain monotone, then the sequence un generated by (3) does not necessarily converge. To overcome this flaw, many iterative methods have been proposed, such as the proximal point method [37, 38], Korpelevich’s extragradient method [39–41] and its variant forms [42–44], the subgradient extragradient method [45, 46], and Tseng’s method [47]. Especially, Bot et al. [48] suggested the following Tseng-type forward-backward-forward algorithm:(4)vn=PCun−λfun,un+1=μkvn+λfun−fvn+1−μkun,∀n≥0.Bot et al. [48] proved that the sequence un generated by (4) converges weakly to an element in Sol C,f provided f is pseudomonotone and sequentially weakly continuous.LetSoldC,f be the solution set of the dual variational inequality of (1), that is,(5)SoldC,f≔u∈C|fx,x−u≥0,∀x∈C.where SoldC,f is the closed convex. If C is convex and f is continuous, then SoldC,f⊂SolC,f.To show the convergence of the sequenceun, a common condition SolC,f⊂SoldC,f has been added, that is,(6)fx,x−u≥0,∀u∈SolC,fandx∈C,which is a direct consequence of the pseudomonotonicity of f. But this conclusion (that is, SolC,f⊂SoldC,f) is false, if f is quasimonotone.The main purpose of this paper is to introduce a self-adaptive forward-backward-forward algorithm to solve quasimonotone variational inequalities (1) and the fixed point problem of pseudocontractive operators. The algorithm is designed such that the step-sizes are dynamically chosen and its convergence is guaranteed without prior knowledge of the Lipschitz constant of f. We prove that the proposed algorithm converges weakly to a common element of the solution of a quasimonotone variational inequality and the fixed point of a pseudocontractive operator under some additional conditions.


## 2. Preliminaries

LetC be a nonempty closed convex subset of a real Hilbert space H. Let T:C⟶C be a nonlinear operator. FixT is used to denote the set of fixed points of T, i.e., FixT:=x∈C|x=Tx. “ ⇀ ” and “ ⟶ ” is used to denote weak convergence and strong convergence, respectively. Let un be a sequence in H. ωwun is used to denote the set of all weak cluster points of un, i.e., ωwun=u†:∃uni⊂unsuch thatuni⇀u†i⟶∞.Letf:C⟶H be a nonlinear operator. We recall that f is said to be(i)
pseudomonotone if(7)fx†,x−x†≥0impliesfx,x−x†≥0,∀x,x†∈C(ii)
quasimonotone if(8)fx†,x−x†>0impliesfx,x−x†≥0,∀x,x†∈C(iii)
L-Lipschitz continuous if there exists some constant L>0 such that(9)fx−fx†≤Lx−x†, for allx,x†∈C(iv)
sequently weakly continuous ifun⇀x˜ implies that fun⇀fx˜.We recall that an operatorT:C⟶C is said to be pseudocontractive if(10)Tx−Tx†2≤x−x†2+I−Tx−I−Tx†2,for all x,x†∈C.For fixedx∈H, there exists a unique x†∈C satisfying x−x†=infx−x˜:x˜∈C. x† is denoted by projCx. The projection projC has the following basic property: for given x∈H,(11)x−projCx,y−projCx≤0,∀y∈C.Applying this characteristic inequality, we have the following equivalence relation:(12)x†∈Solf,C⇔x†=projCx†−ςfx†,∀ς>0.In a Hilbert spaceH, we have(13)ζu+1−ζu†2=ζu2+1−ζu†2−ζ1−ζu−u†2,∀u,u†∈H and ∀ζ∈0,1.Lemma 1 (see [44]).
LetC be a nonempty, convex, and closed subset of a Hilbert space H. We assume that T:C⟶C is an L-Lipschitz pseudocontractive operator. Then, for all u˜∈C and u†∈FixT, we have(14)u†−T1−ϖu˜+ϖTu˜2≤u˜−u†2+1−ϖu˜−T1−ϖu˜+ϖTu˜2,where 0<ϖ<1/1+L2+1.Lemma 2 (see [14]).
LetC be a nonempty, convex, and closed subset of a Hilbert space H. Let T:C⟶C be a continuous pseudocontractive operator. Then,(i)
FixT⊂C is closed and convex(ii)
T is a demiclosedness, i.e., un⇀z˜ and Tun⟶z† imply that Tz˜=z†


## 3. Main Results

In this section, we introduce our main results. LetC be a nonempty closed convex subset of a real Hilbert space H. We assume that the following conditions are satisfied:(C1): the operatorf:H⟶H is quasimonotone; κ-Lipschitz continuous and satisfies the following property (P):(15)H∈xn⇀x†∈Hasn⟶∞liminfn⟶+∞fxn=0imply thatfx†=0(C2): the operatorT:H⟶H is pseudocontractive and L-Lipschitz continuous(C3):Γ:=SoldC,f∩FixT≠∅ and x∈C:fx=0∖SoldC,f is a finite setRemark 1.
If the operatorf is sequentially weakly continuous, then f satisfies the property (P).Next, we present an iterative algorithm for finding a common point inΓ. Let ζn, αn, and ϖn be three sequences in 0,1. Let β∈0,1 and ς0>0 be two constants.Algorithm 1.
Initialization: letu0∈H be an initial guess. We set n=0.Step 1. Let the n-th iterate un be given. We compute(16)v^n=1−ϖnun+ϖnTunvn=1−αnun+αnTv^n.Step 2. Let the n-th step size ςn be known. We compute(17)wn=projCvn−ςnfvn,and(18)un+1=1−ζnvn+ζnwn+ζnςnfvn−fwn.  Step 3. We update the n+1-th step size by the following form:(19)ςn+1=minςn,βwn−vnfwn−fvn,iffwn≠fvn,ςn,else.We setn:=n+1 and return to step 1.Based on Algorithm1, we have the following remark.Remark 2.
(i) By (17), if at some step wn=vn=projCvn−ςnfvn, then vn∈SolC,f. (ii) By (19), ςn+1≤ςn and ςn≥minς0,β/κ for all n, so limn⟶∞ςn=ς† exists, and ς†≥minς0,β/κ>0.Next, we prove the convergence of Algorithm1.Theorem 1.
Suppose that0<α¯<αn<α¯<ϖn<ϖ¯<1/1+L2+1∀n≥0and0<liminfn⟶∞ζn≤limsupn⟶∞ζn<1. Then, the sequence un generated by Algorithm 1 converges weakly to some point in Γ.Proof.
Letx˜∈Γ. Since x˜∈SoldC,f⊂C, from (11) and (17), we have(20)wn−vn+ςnfvn,wn−x˜≤0,which yields that(21)wn−vn,wn−x˜≤ςnfvn,x˜−wn.Noting that wn∈C and x˜∈SoldC,f, we have(22)fwn,x˜−wn≤0.
Combining (21) and (23), we obtain(23)wn−vn,wn−x˜+ςnfvn−fwn,wn−x˜≤0.
In Hilbert spaceH, we have x−y,x−z=1/2x−y2+x−z2−y−z2 for all x,y,z∈H. Setting x=wn, y=vn, and z=x˜, we deduce wn−vn,wn−x˜=1/2wn−vn2+wn−x˜2−vn−x˜2. This together with (1) implies that(24)12wn−vn2+wn−x˜2−vn−x˜2+ςnfvn−fwn,wn−x˜≤0,
and it follows that(25)wn−x˜2≤vn−x˜2−2ςnfvn−fwn,wn−x˜−wn−vn2.
Based on (18), we have(26)un+1−x˜2=1−ζnvn−x˜+ζnwn−x˜+ζnςnfvn−fwn2=1−ζnvn−x˜+ζnwn−x˜2+ζn2ςn2fvn−fwn2+2ζn1−ζnςnvn−x˜,fvn−fwn+2ζn2ςnwn−x˜,fvn−fwn.
Using (13) and from (26), we deduce(27)un+1−x˜2=1−ζnvn−x˜+ζnwn−x˜2−ζn1−ζnvn−wn2+ζn2ςn2fvn−fwn2+2ζn2ςnwn−x˜,fvn−fwn+2ζn1−ζnςnvn−x˜,fvn−fwn.
According to (25) and (27), we obtain(28)un+1−x˜2≤vn−x˜−ζn2−ζnvn−wn2+ζn2ςn2fvn−fwn2+2ζn1−ζnςnvn−wn,fvn−fwn≤vn−x˜−ζn2−ζnvn−wn2+ζn2ςn2fvn−fwn2+2ζn1−ζnςnvn−wnfvn−fwn.
Thanks to (19), fwn−fvn≤βwn−vn/ςn+1. This together with (28) implies that(29)un+1−x˜2≤vn−x˜−ζn2−ζnvn−wn2+ζn2β2ςn2ςn+12wn−vn2+2ζn1−ζnβςnςn+1vn−wn2=vn−x˜2−ζn2−ζn−ζnβ2ςn2ςn+12−21−ζnβςnςn+1vn−wn2.
It is noted that0<liminfn⟶∞ζn≤limsupn⟶∞ζn<1 and limn⟶∞ςn/ςn+1=1. Then, we have liminfn⟶∞ζn2−ζn−ζnβ2ςn2/ςn+12−21−ζnβςn/ςn+1>0. So, there exists a positive constant θ and a positive integer N such that when n≥N,\(30)ζn2−ζn−ζnβ2ςn2ςn+12−21−ζnβςnςn+1≥θ.
In combination with (29), we get(31)un+1−x˜2≤vn−x˜−θvn−wn2,n≥N.
By (13) and (16), we obtain(32)vn−x˜2=1−αnun−x˜+αnTv^n−x˜2=1−αnun−x˜2+αnTv^n−x˜2−αn1−αnun−Tv^n2.
Using Lemma1, we have(33)Tv^n−x˜2=T1−ϖnun+ϖnTun−x˜2≤un−x˜2+1−ϖnun−Tv^n2.
Substituting (33) into (32), we get(34)vn−x˜2≤un−x˜2+αn−ϖnαnun−Tv^n2,which results, together with (31), that(35)un+1−x˜2≤un−x˜2−ϖn−αnαnun−Tv^n2−θvn−wn2,n≥N,which implies that(36)ϖn−αnαnun−Tv^n2+θvn−wn2≤un−x˜2−un+1−x˜2,n≥N.
By assumption,liminfn⟶∞ϖn−αnαn>0. From (35), we conclude that un+1−x˜≤un−x˜,n≥N. Therefore, limn⟶∞un−x˜ exists, and the sequence un is bounded.
In combination with (36), we derive(37)limn⟶∞un−Tv^n=0,(38)limn⟶∞vn−wn=0.
By (16), vn−un=αnTv^n−un, it follows from (37) that(39)limn⟶∞vn−un=0.
From (38) and the Lipschitz continuity of f, we have(40)limn⟶∞fvn−fwn=0.
According to the boundedness of the sequenceun, we conclude that the sequence vn is bounded by (34) and the sequence wn is bounded because of wn≤vn+ςnfvn by (17).
SinceT is L-Lipschitz continuous, we have(41)un−Tun≤un−Tv^n+Tv^n−Tun≤un−Tv^n+Lϖnun−Tun.
It follows that(42)un−Tun≤11−Lϖnun−Tv^n.
This together with (37) implies that(43)limn⟶∞un−Tun=0.
By virtue of (18), (38), and (40), we have(44)limn⟶∞un+1−vn=0.
Next, we show thatωwun⊂Γ. Selecting any x∗∈ωwun and letting uni to be a subsequence of un such that uni⇀x∗ as i⟶∞, from (38) and (39), we have vni⇀x∗ and wni⇀x∗. Taking into account (43) and Lemma 2, we obtain that x∗∈FixT. Next, we show that x∗∈SolC,f. Based on (11) and wni=projCvni−ςnifvni, we receive(45)wni−vni+ςnifvni,wni−x†≤0,∀x†∈C,which yields(46)1ςnivni−wni,u−wni+fvni,wni−vni≤fvni,x†−vni,∀x†∈C.Owing to (39), limi⟶∞vni−wni=0. It follows from (46) that(47)liminfi⟶∞fvni,x†−vni≥0,∀x†∈C.
There are two possible cases:liminfi⟶+∞fvni=0 and liminfi⟶+∞fvni>0.
Ifliminfi⟶+∞fvni=0, by vni⇀x∗ and f satisfying (16), we obtain that fx∗=0. If liminfi⟶+∞fvni>0, then there exists an integer ℐ>0 satisfying fvni≠0 for all i≥ℐ. By (47), we achieve(48)liminfi⟶+∞fvnifvni,x†−vni≥0,∀x†∈C.
Letξj be a positive strictly decreasing sequence such that ξj⟶0 as j⟶+∞. By virtue of (48), there exists a strictly increasing subsequence nij satisfying nij≥ℐ and ∀j≥0,(49)fvnijfvnij,x†−vnij+ξj>0,∀x†∈C,which results that(50)fvnij,x†−vnij+ξjfvnij>0,∀x†∈C,∀j≥0.
We setv˜j=fvnij/fvnij2 for all j≥0. Then, fvnij,v˜j=1 for each j≥0. Owing to (50), we have(51)fvnij,x†+ξjfvnijv˜j−vnij>0,∀x†∈C,∀j≥0.
Sincef is quasimonotone on H, by (51), we get(52)fx†+ξjfvnijv˜j,x†+ξjfvnijv˜j−vnij≥0,∀x†∈C,∀j≥0.
Sincelimj⟶+∞ξjfvnijv˜j=limj⟶+∞ξj=0 and f is Lipschitz continuous, limj⟶∞fx+ξjfvnijv˜j=fx. Letting j⟶+∞ in (52), we deduce(53)fx†,x†−x∗≥0,∀x†∈C,which means x∗∈SoldC,f.
Next, we show thatx∗ is the unique weak cluster point of un in SoldC,f. Let x¯∈SoldC,f be another weak cluster point of un. Then, there exists a sequence un of un satisfying unj⇀x¯ as j⟶+∞. We note that for all k≥0,(54)2un,x∗−x¯=un−x¯2−un−x∗2+x∗2−x¯2.We note that limn⟶+∞un−x∗ and limn⟶+∞un−x¯ exist. From (54), limn⟶+∞un,x∗−x¯ exists. Hence,(55)limi⟶+∞uni,x∗−x¯=limj⟶+∞unj,x∗−x¯.Since uni⇀x∗ and unj⇀x¯, from (55), we have(56)x∗,x∗−x¯=x¯,x∗−x¯,which implies that x∗−x¯2=0, and hence x∗=x¯. Therefore, un has the unique weak cluster point in SoldC,f. By the condition (C3), x∈C,fx=0∖SoldC,f is a finite set. Therefore, un has finite weak cluster points in SolC,f denoted by q1,q2,…,qm. We set N0=1,2,…,m and ν=minqj−qk/3,j,k∈N0,j≠k. Let qj,j∈N0 be any weak cluster point in SolC,f and unij be a subsequence of un satisfying unij⇀qj as i⟶+∞. Then, we have(57)limi⟶+∞unij,qj−qkqj−qk=qj,qj−qkqj−qk,∀k∈N0andk≠j.
By the definition ofν, we have ∀k≠j,(58)qj,qj−qkqj−qk=qj−qk2+qj2−qk22qj−qk>ν+qj2−qk22qj−qk.
In the light of (57) and (58), there exists an integer intij such that when i≥intij,(59)unij∈x:x,qj−qkqj−qk>ν+qj2−qk22qj−qk,k∈N0,k≠j.We write(60)Sbj=∩k=1,k≠jmx:x,qj−qkqj−qk>ν+qj2−qk22qj−qk.
Taking into account (59) and (60), we have unij∈Sbj when i≥maxintij,j∈N0.
Now, we show thatun∈∪j=1mSbj for a large enough n. If not, there exists a subsequence unl of un such that unl∉∪j=1mSbj. By the boundedness of unl, there exists a subsequence of unl convergent weakly to x∗. Without the loss of generality, we still denote the subsequence as unl. According to assumptions, unl∉∪j=1mSbj, so unl∉Sbj for any j∈N0. Therefore, there exists a subsequence unls of unl such that when ∀s≥0,(61)unls∉x:x,qj−qkqj−qk>ν+qj2−qk22qj−qk,k∈N0,k≠j.Thus,(62)x∗∉x:x,qj−qkqj−qk>ν+qj2−qk22qj−qk,k∈N0,k≠j,which implies that x∗≠qjandj∈N0. This is impossible. So, for a large enough positive integer N1, un∈∪j=1mSbj when n≥N1.
Next, we show thatun has the unique weak cluster point in SolC,f. First, there exists a positive integer N2≥N1 such that un+1−un<ν for all n≥N2. We assume that un has at least two weak cluster points in SolC,f. Then, there exists n^≥N2 such that un^∈Sbj and un^+1∈Sbk, where j,k∈N0 and m≥2, that is,(63)un^∈Sbj=∩k=1,k≠jmx:x,qj−qkqj−qk>ν+qj2−qk22qj−qk,and(64)un^+1∈Sbk=∩j=1,j≠kmx:x,qk−qjqk−qj>ν+qk2−qj22qk−qj.Therefore,(65)un^,qj−qkqj−qk>ν+qj2−qk22qj−qk,and(66)un^+1,qk−qjqk−qj>ν+qk2−qj22qk−qj.Combining (65) and (66), we achieve(67)un^−un^+1,qj−qkqj−qk>2ν.At the same time, we have(68)‖un^+1−un^‖<ν.Based on (67) and (68), we deduce(69)2ν<un^−un^+1,qj−qkqj−qkg≤un^−un^+1<ν.This leads to a contradiction. Then, un has the unique weak cluster point in SolC,f. So, the sequence un has the unique weak cluster point x∗∈Γ. Therefore, the sequence un converges weakly to x∗∈Γ. This completes the proof.Based on Algorithm1 and Theorem 1, we can obtain the following algorithms and the corresponding corollaries.Algorithm 2.
Initialization: letu0∈H be an initial guess. We set n=0.Step 1. Let the n-th iterate un and the n-th step size ςn be given. We compute(70)wn=projCun−ςnfun,and(71)un+1=1−ζnun+ζnwn+ζnςnfun−fwn.Step 2. We update the n+1-th step size by the following form:(72)ςn+1=minςn,βwn−unfwn−fun,iffwn≠fun,ςn,else.We setn:=n+1 and return to step 1.Corollary 1.
We assume that the operatorf:H⟶H is quasimonotone, κ-Lipschitz continuous and satisfies the property (P). Suppose that SoldC,f≠∅, x∈C:fx=0∖SoldC,f is a finite set and 0<liminfn⟶∞ζn≤limsupn⟶∞ζn<1. Then, the sequence un generated by Algorithm 2 converges weakly to some point in SolC,f.Algorithm 3.
Initialization: letu0∈C and ς0>0. We set n=0.Step 1. For known un, we compute(73)un+1=1−αnun+αnT1−ϖnun+ϖnTun.Step 2. We set n:=n+1 and return to step 1.Corollary 2.
We assume thatT:C⟶C is a pseudocontractive and L-Lipschitz continuous operator. We suppose that FixT≠∅ and 0<α¯<αn<α¯<ϖn<ϖ¯<1/1+L2+1∀n≥0. Then, the sequence un generated by Algorithm 3 converges weakly to some point in FixT.



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