Iterative methods for solving variational inclusions and fixed-point problems have been considered and investigated by many scholars. In this paper, we use the Halpern-type method for finding a common solution of variational inclusions and fixed-point problems of pseudocontractive operators. We show that the proposed algorithm has strong convergence under some mild conditions.
Ministry of Science and Technology of the People's Republic of China109-2115-M-037-0011. Introduction
Let H be a real Hilbert space with inner product ⋅,⋅ and induced norm ⋅. Let C be a nonempty closed and convex subset of H. Let f:C⟶H and g:H⟶2H be two nonlinear operators. Recall that the variational inclusion ([1]) is to solve the following problem of finding x‡∈2H verifying(1)0∈f+gx‡.Here, use f+g−10 to denote the set of solutions of (1).
Special Case 1.
Let δC:H⟶0,+∞ be defined by(2)δC=0,x∈C,+∞,x∉C.
Setting g=∂δC, variational inclusion (1) reduces to find x‡∈C such that(3)fx‡,x−x‡≥0,∀x∈C.
Problem (3) is the well-known variational inequality which has been studied, extended, and developed in a broad category of jobs (see, e.g., [2–14]).
Special Case 2.
Let φ:H⟶ℝ∪+∞ be a proper lower semicontinuous convex function and ∂φ be the subdifferential of φ. Setting g=∂φ, variational inclusion (1) reduces to find x‡∈H such that(4)fx‡,x−x‡+φx−φx‡≥0,∀x∈H.
Problem (4) is called the mixed quasi-variational inequality [15] which is a very significant extension of variational inequality (3) involving the nonlinear function φ. It is well known that a large number of practical problems arising in various branches of pure and applied sciences can be formulated as the model of mixed quasi-variational inequality (4).
Problem (1) plays a key role in minimization, convex feasibility problems, machine learning, and others. A popular algorithm for solving problem (1) is the forward-backward algorithm [16] generated by(5)xn+1=I+λg−1I−λfxn,n≥1,where I−λf is a forward step and I+λg−1 is a backward step with λ>0. This algorithm is a splitting algorithm which solves the difficulty of calculating of the resolvent of f+g.
Recently, there has been increasing interest for studying common solution problems relevant to (1) (see for example, [17–27]). Especially, Zhao, Sahu, and Wen [28] presented an iterative algorithm for solving a system of variational inclusions involving accretive operators. Ceng and Wen [29] introduced an implicit hybrid steepest-descent algorithm for solving generalized mixed equilibria with variational inclusions and variational inequalities. Li and Zhao [30] considered an iterate for finding a solution of quasi-variational inclusions and fixed points of nonexpansive mappings.
Motivated by the results in this direction, the main purpose of this paper is to research a common solution problem of variational inclusions and fixed point of pseudocontractions. We suggest a Halpern-type algorithm for solving such problem. We show that the proposed algorithm has strong convergence under some mild conditions.
2. Preliminaries
Let H be a real Hilbert space. Let g:H⟶2H be an operator. Write domg=x∈H:gx≠∅. Recall that g is called monotone if ∀x,y∈domg, u∈gx and v∈gy, x−y,u−v≥0.
A monotone operator g is maximal monotone if and only if its graph is not strictly contained in the graph of any other monotone operator on H.
For a maximal monotone operator g on H,
Set g−10=x∈H:0∈gx
Denote its resolvent by Jλg=I+λg−1 which is single-valued from H into domg
It is known that g−10=FixJλg,∀λ>0 and Jλg is firmly nonexpansive, i.e.,(6)Jλgx−Jλgy2≤Jλgx−Jλgy,x−y,for all x,y∈C.
Let C be a nonempty closed convex subset of a real Hilbert space H. Recall that an operator T:C⟶C is said to be
L-Lipschitz if there exists a positive constant L such that(7)Tx−Ty≤Lx−y,∀x,y∈C.
If L=1, T is nonexpansive.
Pseudocontractive if(8)Tx−Ty,x−y≤x−y2,∀x,y∈C.
Inverse-strongly monotone if
(9)Tx−Ty,x−y≥αTx−Ty2,∀x,y∈C,where α>0 is a constant and T is also called α-ism.
Recall that the projection PC is an orthographic projection from H onto C, which is defined by x−PCx=miny∈Cx−y. It is known that PC is nonexpansive.
Lemma 1 (see [23, 31]).
Let C be a nonempty closed convex subset of a real Hilbert space H. Let T:C⟶C be an L-Lipschitz pseudocontractive operator. Then,
T is demiclosed, i.e., xn⇀p and Txn⟶q⇒Tp=q
For 0<ζ<1/1+L2+1, ∀x∈C and y∈FixT, we have(10)T1−ζx+ζTx−y2≤x−y2+1−ζx−T1−ζx+ζTx2.
Lemma 2 (see [16, 32]).
Let H be a real Hilbert space and let g be a maximal monotone operator on H. Then, we have(11)Jsgx−Jtgx2≤s−ttJsgx−Jtgx,Jsgx−x,for all s,t>0 and x∈H.
Lemma 3 (see [33]).
Assume that a real number sequence an⊂0,∞ satisfies(12)an+1≤1−γnan+δnγn,where γn⊂0,1 and δn⊂−∞,+∞ satisfy the following conditions:
∑n=1∞γn=∞
lim supn⟶∞δn≤0 or ∑n=1∞δnγn<∞
Then, limn⟶∞an=0.
Lemma 4 (see [8]).
Let sn⊂0,∞ be a sequence. Assume that there exists at least a subsequence sni of sn verifying sni≤sni+1 for all i≥0. Let τn be an integer sequence defined as τn=maxi≤n:sni<sni+1. Then τn⟶∞ as n⟶∞ and(13)maxsτn,sn≤sτn+1.
3. Main Results
Let C be a nonempty closed convex subset of a real Hilbert space H. Let the operator f:C⟶H be an α-ism. Let g:H⟶2H be a maximal monotone operator with domg⊂C. Let T:C⟶C be an L−Lipschitz pseudocontractive operator with L>1. Let αn⊂0,1 and λn⊂0,∞ be two sequences. Let ν and ζ be two constants.
Next, we introduce a Halpern-type algorithm for finding a common solution of variational inclusion (1) and fixed point of pseudocontractive operator T.
Algorithm 1.
Let u∈C be a fixed point. Choose x0∈C. Set n=0.
Step 1. For given xn, compute yn by(14)yn=1−νxn+νT1−ζxn+ζTxn.
Suppose that Γ:=FixT∩f+g−10≠∅. Assume that the following conditions are satisfied:
limn⟶∞αn=0 and ∑n=1∞αn=∞
0<d1<λn<d2<2α and 0<ν<ζ<1/1+L2+1
Then, the sequence xn generated by Algorithm 1 converges strongly to PΓu.
Proof.
Let x∗∈FixT∩f+g−10. Set un=JλngI−λnfyn,∀n≥0. Since f is α-ism, we have(16)fyn−fx∗,yn−x∗≥αfyn−fx∗2.
By the nonexpansivity of Jλng, we have(17)un−x∗2=JλngI−λnfyn−JλngI−λnfx∗2≤yn−x∗−λnfyn−fx∗2=yn−x∗2−2λnfyn−fx∗,yn−x∗+λn2fyn−fx∗2≤yn−x∗2−2λnαfyn−fx∗2+λn2fyn−fx∗2=yn−x∗2−λn2α−λnfyn−fx∗2≤yn−x∗2−d12α−d2fyn−fx∗2by condition r2≤yn−x∗2.
Using Lemma 1, we get(18)T1−ζI+ζTxn−x∗2≤xn−x∗2+1−ζxn−T1−ζI+ζTxn2.
This together with (14) implies that(19)yn−x∗2=1−νxn+νT1−ζI+ζTxn−x∗2=1−νxn−x∗+νT1−ζI+ζTxn−x∗2=1−νxn−x∗2+νT1−ζI+ζTxn−x∗2−ν1−νT1−ζI+ζTxn−xn2≤xn−x∗2−νζ−νT1−ζI+ζTxn−xn2≤xn−x∗2.
According to (15)-(19), we obtain(20)xn+1−x∗=αnu−x∗+1−αnun−x∗≤αnu−x∗+1−αnxn−x∗≤⋯≤maxu−x∗,x0−x∗.
Then, the sequence xn is bounded. The sequences un and yn are also bounded.
Again, by (15)-(19), we deduce(21)xn+1−x∗2=αnu−x∗+1−αnun−x∗2≤αnu−x∗2+1−αnun−x∗2by the convexity of ⋅2≤αnu−x∗2+un−x∗2≤αnu−x∗2+xn−x∗2−d12α−d2fyn−fx∗2−νζ−νT1−ζI+ζTxn−xn2.
It follows that(22)d12α−d2fyn−fx∗2+νζ−νT1−ζI+ζTxn−xn2≤αnu−x∗2+xn−x∗2−xn+1−x∗2.
Since Jλng is firmly nonexpansive, using (6), we have(23)un−x∗2=JλngI−λnfyn−JλngI−λnfx∗2≤I−λnfyn−I−λnfx∗,un−x∗=yn−x∗,un−x∗−λnun−x∗,fyn−fx∗=12yn−x∗2+un−x∗2−yn−un2−λnyn−x∗,fyn−fx∗−λnun−yn,fyn−fx∗≤12yn−x∗2+un−x∗2−yn−un2+λnun−ynfyn−fx∗,which leads to(24)un−x∗2≤yn−x∗2−yn−un2+2λnun−ynfyn−fx∗≤xn−x∗2−yn−un2+2λnun−ynfyn−fx∗.
Combining (21) with (24), we obtain(25)xn+1−x∗2≤αnu−x∗2+un−x∗2≤αnu−x∗2+xn−x∗2−yn−un2+2λnun−ynfyn−fx∗.which results in that(26)yn−un2≤αnu−x∗2+xn−x∗2−xn+1−x∗2+2λnun−ynfyn−fx∗.
Next, we analyze two cases. (i) ∃n0∈ℕ such that xn+1−x∗≤xn−x∗,∀n≥n0. (ii) For any n0∈ℕ, ∃m≥n0 such that xm−x∗≤xm+1−x∗.
In case of (i), limn⟶∞xn−x∗ exists. From (22), we deduce(27)limn⟶∞fyn−fx∗=0and(28)limn⟶∞T1−ζI+ζTxn−xn=0.
It follows from (14) that(29)limn⟶∞yn−xn=limn⟶∞νT1−ζI+ζTxn−xn=0.
On the basic of (26) and (27), we have(30)limn⟶∞yn−un=limn⟶∞yn−JλngI−λnfyn=0.
Note that xn+1−xn≤αnu−xn+1−αnun−xn. Thanks to (29) and (30), we derive that(31)limn⟶∞xn+1−xn=0.
This together with (28) implies that(34)limn⟶∞xn−Txn=0.
Set p=PΓu. Next, we prove that(35)limsupn⟶∞u−p,xn+1−p≤0.
Since xn+1 is bounded, there exists a subsequence xni+1 of xn+1 satisfying
xni+1⇀x˜ (hence, xni⇀x˜ by (31))
limsupn⟶∞u−p,xn+1−p=limi⟶∞u−p,xni+1−p
From (34) and Lemma 1, we obtain x˜∈FixT.
Owing to (29) and (30), we have that yni⇀x˜ and(36)limi⟶∞JλnigI−λnifyni−yni=0.
Since λn∈d1,d2, without loss of generality, we assume that λni⟶λ†>0i⟶∞. Observe that(37)JλnigI−λnifyni−Jλ†gI−λ†fyni≤JλnigI−λnifyni−Jλ†gI−λnifyni+Jλ†gI−λnifyni−Jλ†gI−λ†fyni≤JλnigI−λnifyni−Jλ†gI−λnifyni+λni−λ†fyni.
Applying Lemma 2, we obtain(38)JλnigI−λnifyni−Jλ†gI−λnifyni2≤λni−λ†λ†JλnigI−λnifyni−Jλ†gI−λnifyni,JλnigI−λnifyni−I−λnifyni≤λni−λ†λ†JλnigI−λnifyni−Jλ†gI−λnifyniJλnigI−λnifyni−I−λnifyni.
It follows that(39)JλnigI−λnifyni−Jλ†gI−λnifyni≤λni−λ†λ†JλnigI−λnifyni−I−λnifyni.
Thanks to (37) and (39), we get(40)JλnigI−λnifyni−Jλ†gI−λ†fyni≤λni−λ†fyni+λni−λ†λ†JλnigI−λnifyni−I−λnifyni.
Noting that λni⟶λ†i⟶∞, from (36) and (40), we get(41)limi⟶∞yni−Jλ†gI−λ†fyni=0.
By Lemma 1, we deduce that x˜∈FixJλ†gI−λ†f=f+g−10. Therefore, x˜∈Γ and(42)limsupn⟶∞u−p,xn+1−p=limi⟶∞u−p,xni+1−p=u−p,x˜−p≤0.
From (15), we have(43)xn+1−p2=αnu−p+1−αnun−p2≤1−αnun−p2+2αnu−p,xn+1−p≤1−αnxn−p2+2αnu−p,xn+1−p.
Applying Lemma 3 to (43) to deduce xn⟶p.
In case of (ii), let sn=xn−x∗. So, we have sn0≤sn0+1. Define an integer sequence τn,∀n≥n0, by τn=maxi∈ℕ|n0≤i≤n,si≤si+1. It is obvious that limn⟶∞τn=∞ and sτn≤sτn+1 for all n≥n0. Similarly, we can prove that limn⟶∞xτn−Txτn=0 and limn⟶∞JλτngI−λτnfxτn=0. Therefore, all weak cluster points ωwxτn⊂Γ. Consequently,(44)limsupn⟶∞u−p,xτn−p≤0.
Note that sτn≤sτn+1. From (43), we deduce(45)sτn2≤sτn+12≤1−ατnsτn2+2ατnu−p,xτn+1−p.
It follows that(46)sτn2≤2u−p,xτn+1−p.
Combining (44) and (46), we have limsupn⟶∞sτn≤0 and hence(47)limk⟶∞sτk=0.
From (45), we deduce that limsupn⟶∞sτn+12≤limsupn⟶∞sτn2. This together with (47) implies that limn⟶∞sτn+1=0. According to Lemma 4, we get 0≤sn≤maxsτn,sτn+1. Therefore, sn⟶0 and xn⟶p. This completes the proof.
Remark 1.
Since the pseudocontractive operator is nonexpansive, Theorem 1 still holds if T is nonexpansive.
Remark 2.
Assumption r1 imposed on parameter αn is essential and we do not add any other assumptions.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
Zhangsong Yao was partially supported by the Grant 19KJD100003. Ching-Feng Wen was partially supported by the Grant of MOST 109-2115-M-037-001.
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