# Forcing Strong Convergence of a Mann-Based Iteration for Nonexpansive and Monotone Operators in a Hilbert Space

Authors: Songtao Lv
Journal: Journal of Mathematics
 (2022)

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## Abstract

Mann iteration is weakly convergent in infinite dimensional spaces. We, in this paper, use the nearest point projection to force the strong convergence of a Mann-based iteration for nonexpansive and monotone operators. A strong convergence theorem of common elements is obtained in an infinite dimensional Hilbert space. No compact conditions are needed.


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## Body

## 1. Introduction: Preliminaries

In the real world, there are a lot of nonlinear phenomena, which can be modelled into variational inequalities and variational inclusions, such as signal processing, image recovery, and machine learning; see, e.g., [1–7] and the references therein. Fixed point methods are powerful and popular for dealing various nonlinear operator equations and inequalities in abstract spaces, in particular, for variational inequalities and variational inclusions. Recently, various efficient fixed point methods have been introduced and investigated; see, e.g., [8–13] and the references therein. Let T be a nonlinear operator on a Hilbert space H, which is endowed with inner product ·,· and induced norm ·. The fixed point set of T is presented by FixT. Recall that T is said to be contractive iff there is a real number a∈0,1 such that(1)Tx−Ty≤ax−y,∀x,y∈H.Recall thatT is said to be nonexpansive iff(2)Tx−Ty≤x−y,∀x,y∈H.Recall thatT is said to be firmly nonexpansive iff(3)Tx−Ty2≤x−y,Tx−Ty,∀x,y∈H.It is clear that the class of firmly nonexpansive mappings is a special class of nonexpansive mappings. One knows the projection operator (see below) is firmly nonexpansive. The class of nonexpansive operators is significant in various nonlinear equations and mathematical programming computation. It also has wide real applications in applied and industrial fields. For various iterative methods, Mann iteration is popular for dealing with fixed points of nonexpansive operators. It reads(4)xn+1=1−αnTxn+αnxn,where αn is a real number sequence in the interval 0,1. However, the Mann iteration is weakly convergent only in infinite dimensional spaces; see, e.g., [14] and the references therein. To force the strong convergence without possible compact assumptions, various regularized methods have been investigated in Hilbert spaces and Banach spaces recently; see, e.g., [15–19] and the references therein. One of the efficient regularized methods is the Halpern iteration, which reads(5)xn+1=1−αnTxn+αnx,where αn is a real number sequence in the interval 0,1 and x is a fixed anchor. With some conditions on αn, it was proved that xn converges to x, which is a special fixed point of T, that is, the nearest point in FixT to x. Halpern [20] pointed out that conditions (c1) αn⟶0 as n⟶∞ and (c2) ∑n=1∞αn=∞ are necessary if the Halpern iteration scheme converges in norm. In view of (c2), the Halpern iteration may not be a fast iteration. Recently, a number of researchers investigated the problem of removing (c2) with the aid of projections; see, e.g., [21–24] and the references therein. In 2000, Moudafi [25] further proposed the viscosity approximation iteration, which reads as follows:(6)xn+1=1−αnTxn+αnSxn,where S is a contraction. This approximation method, which improves the property of the class of nonexpansive mappings, is popular from the viewpoint of variational inequalities. Indeed, the fixed point also solves a monotone variational inequality with S. Another popular regularized method is the hybrid projection method, which was considered by Nakajo and Takahashi [18] for fixed points of nonexpansive mappings first. Indeed, they studied the following algorithm:(7)x0∈C,yn=1−αnTxn+αnxn,Qn=x∈C:xn−x,xn−x0≤0,Cn=x∈C:x−yn≤x−xn,xn+1=ProjQn∩Cnx0,where C is a closed, convex, and nonempty subset of H and ProjQn∩Cn is the nearest point projection onto the intersection set. They obtained a strong convergence theorem for nonexpansive mappings in a real Hilbert spaces without compact assumption on T. For more general nonlinear mappings though the projection-based method, we refer to [26–30] and the references therein.LetC be a convex and closed subset of a real Hilbert space H. From now on, ProjC is borrowed to denote the nearest projection onto subset C, i.e., ProjCx≔argminx−y,y∈C. Let A be a nonlinear mapping on H. Recall that A is said to be(1)
Strongly monotone iff there exists a positive constant ξ such that Ax−Ay,x−y≥ξx−y2, ∀x,y∈H(2)
Monotone iff Ax−Ay,x−y≥0, ∀x,y∈H(3)
Cocoercive iff there exists a positive constant ∀x,y∈H such that Ax−Ay,x−y≥ξAx−Ay2, ∀x,y∈HLetB:H⇉H be a multivalued nonlinear mapping. Next, we turn our attention to the class of multivalued mappings. B is said to be a monotone mapping if and only if for all x,y∈H, f∈By, and e∈Bx⇒e−f,x−x>0. The symbol B−10 is used to stand for the set of zero points of B. Mapping B is said to be a maximally monotone mapping iff the graph of B, GraphB, is not contained in the graph of any other monotone mapping properly. Let JβB=Id+βB−1, where Id is the identity mapping and β is a constant. This operator is called the resolvent of B. Its domain is denoted by DomB in this paper. It is clear B−10=FixJβB.Consider the following variational inclusion problem, which finds a pointx∈C such that x∈B+A−10, where B is a multivalued maximally monotone mapping and A is a ξ-cocoercive mapping. For the inclusion problem, splitting methods (FB, PR, and DR) are popular for zero points of the sum of the monotone mappings. Splitting methods were considered by many authors for image recovery, signal processing, and machine learning. The FB-type splitting method means an iterative method for which each iteration involves only with the individual operators not the sum. In this paper, with the condition that the solution set is nonempty, we consider finding a θ∈C such that θ∈FT∩B+A−10, where T is a nonexpansive mapping with a nonempty fixed point set, B is a multivalued maximally monotone mapping, and A is a ξ-cocoercive mapping. We establish a strong convergence with the aid of hybrid projection and FB splitting in a Hilbert space. Our strong convergence theorem requires less restriction on parameter sequences and the operators.To show our main findings, we also need the following necessary tools.The nearest point projection operatorProjC has the following property:(8)ProjCy−ProjCx2≤y−x,ProjCy−ProjCx,∀x,y∈H.Lemma 1 (see [31]).
LetH be a Hilbert space, and let C be a convex, closed, and nonempty subset of H. Let T be a nonexpansive mapping on C. Then, FixT is convex and closed.Remark 1.
LetH be a Hilbert space, and let C be a convex, closed, and nonempty subset of H. Let A:C⟶H be a ξ-cocoercive mapping, and let B:H⇉H be a multivalued maximally monotone operator. Then, FixJβBId−βA=B+A−10, where β is some constant and Id is the identity mapping. Besides, the resolvent is firmly nonexpansive. From Lemma 1, we have that B+A−10 is convex and closed.Lemma 2 (see [31]).
LetH be a Hilbert space, and let C be a convex, closed, and nonempty subset of H. Let T be a nonexpansive mapping on C. Then, Id−T is demiclosed (let xn be a sequence weakly converging to x, and let Txn−xn⟶∞ be n⟶∞. Then, x is a fixed point of T).


## 2. Main Results

Theorem 1.
Assume thatH is a Hilbert space and C is a convex and closed subset in space H. Assume that A is a single-valued ξ-cocoercive mapping from set C to space H and B is a set-valued maximally monotone mapping from H to H. Assume that T is a nonexpansive mapping from C to C, and CSSB,A,T=B+A−10∩FixT is nonempty. Assume that αn and βn are positive real number sequences. Let xn be a sequence in set C generated in the following iterative process:(9)x0∈C,yn=1−αnTxn+αnJβnBxn−βnAxn,Qn=x∈C:xn−x,xn−x0≤0,Cn=x∈C:x−yn≤x−xn,xn+1=ProjQn∩Cnx0,where JβnB is the resolvent mapping Id+βnB−1. Assume that αn and βn satisfy the conditions (i) 1>αn≥α>0 with α being a fixed real number and (ii) 0<β≤βn≤β′<2ξ with β and β′ being two fixed real numbers. Then, the sequence xn converges strongly to ProjCSSA,B,Tx0.Proof.
From Lemma1, we have that FixT is convex and closed. From Remark 1, we have that B+A−10 is convex and closed. Hence, CSSB,A,T is convex and closed. This shows that the metric (nearest point) projection onto the set is well-defined.
Note thatx−yn2≤x−xn2 is equivalent to 2x,xn−yn≤xn2−yn2. Let x and x′ be the points in Cn. Then,(10)2rx,xn−yn≤rxn2−yn2,21−rx′,xn−yn≤1−rxn2−yn2,where r is a real number in 0,1. Adding the two inequalities above, we have(11)2rx+1−rx′,xn−yn≤xn2−yn2,that is,(12)rx+1−rx′−yn≤rx+1−rx′−xn.
It shows thatrx+1−rx′∈Cn. Cn is convex. The closedness of Cn is obvious. The definition of ξ-cocoercive mappings send us to the situation Id−βnA is a nonexpansive mapping for each n. Indeed, for any w,v∈C,(13)Id−βnAw−Id−βnAv2=βn2Aw−Av2−2βnAw−Av,w−v+w−v2≤βnβn−2ξAw−Av2+w−v2.
This indicatesId−βnA is a mapping of nonexpansive. Observe that CSSB,A,T⊂Cn. Indeed, from the nonexpansivity of the resolvent, we have(14)yn−p≤1−αnTxn−p+αnJβnBxn−βnAxn−p=1−αnTxn−Tp+αnJβnBxn−βnAxn−JβnBp−βnAp≤1−αnxn−p+αnId−βnAxn−Id−βnAp≤xn−p,∀p∈CSSA,B,T.
So, we complete the proofCSSA,B,T⊂Cn.
On the contrary, it is obvious thatQn is convex and closed. Next, one shows that CSSB,A,T⊂Qn∩Cn. Borrowing C0=C, we have CSSB,A,T⊂Q0∩C0. Let xm be a given vector, and CSSB,A,T⊂Qm∩Cm for some positive integer m. There is a vector xm+1∈Qm∩Cm with xm+1=ProjQm∩Cmx0. There holds x0−xm+1,xm+1−j≥0 for all j∈Qm∩Cm. Borrowing CSSB,A,T⊂Qm∩Cm, we get CSSB,A,T⊂Qm+1. Thus, CSSB,A,T⊂Qm+1∩Cm+1. Hence, CSSB,A,T⊂Qn∩Cn for all n.
One next observes thatxn is a bounded sequence. As we have showed that CSSB,A,T is convex and closed set in C, a unique vector μ∈CSSA,B,T with μ=ProjCSSA,B,Tx0 is guaranteed. We have the construction of xn+1, that is, ProjQn∩Cnx0=xn+1. So,(15)x0−xn+1≤x0−ν,for each ν∈Qn∩Cn. By μ∈CSSA,B,T⊂Qn∩Cn, we obtain(16)x0−xn+1≤x0−μ,that infers xn is a bounded sequence. Our next step shows xn+1−xn⟶0 as n⟶∞. Because xn=ProjQnx0 and xn+1∈Qn∩Cn⊂Qn, one infers that(17)x0−xn≤x0−xn+1.
Borrowing the conclusion (xn is a bounded sequence), one infers that the limit of x0−xn exists. We may suppose that limn⟶∞x0−xn=d>0. Observe(18)xn+1−x02−x0−xn2≥xn+1−x02−x0−xn2−2x0−xn,xn−xn+1=xn−x02+x0−xn+12+2xn−x0,x0−xn+1=xn−xn+12≥0,thanks to x0−xn,xn−xn+1≥0 (xn+1∈Qn and the property of the metric projection). By the limit of the limit of ‖x0−xn‖, one infers limn⟶∞xn+1−xn2=0.
Note thatxn+1 is in Cn. So,(19)xn+1−yn≤xn+1−xn.
That indicates thatxn+1−yn⟶0 as n⟶∞. Furthermore, xn−yn⟶0 as n⟶∞. Let zn=JβnBxn−βnAxn. For any p∈CSSB,A,T, ξ-cocoercive and resolvent operators send us to(20)p−zn2≤JβnBp−βnAp−JβnBxn−βnAxn2≤p−βnAp−xn−βnAxn2≤p−xn2−2ξ−βnβnAp−Axn2.
So,(21)p−yn2≤1−αnTxn−p2+αnzn−p2≤1−αnTxn−Tp2+αnp−xn2−2ξ−βnβnAp−Axn2≤p−xn2−αn2ξ−βnβnAp−Axn2.
That is,(22)αn2ξ−βnβnAp−Axn2≤yn−xnp−xn+p−yn.
By the fact thatyn−xn⟶0 as n⟶∞, we have Axn−Ap⟶0 as n⟶∞. By the firm nonexpansivitity of the resolvent operator, we also have(23)p−zn2≤p−zn,p−βnAp−xn−βnAxn=12p−xn2+p−zn2−xn−zn−βnAp−Axn2≤12p−xn2+p−zn2−xn−zn2−βn2Ap−Axn2+2βnxn−znAp−Axn≤12p−xn2+p−zn2−xn−zn2+2βnxn−znAp−Axn,which holds that(24)p−zn2≤p−xn2−xn−zn2+2βnxn−znAp−Axn,p−yn2≤1−αnTxn−Tp2+αnJβnBxn−βnAxn−p2≤1−αnxn−p2+αnzn−p2≤xn−p2−αnxn−zn2+2βnαnxn−znAp−Axn.
So,αnxn−zn2≤xn−ynC+2βnαnxn−znAp−Axn, where C is some constant. By the requirement on the control parameter and the result that Ap−Axn⟶∞ as n⟶∞ and xn−zn⟶∞ as n⟶∞. With a simple calculation, we have xn−Txn⟶∞ as n⟶∞. We have the fact that Id−βnAxn∈Id+βnBzn. It holds(25)xn−znβn−Axn∈Bzn.
By the assumption thatB is maximally monotone,(26)xn−znβn−Axn−u,zn−v≥0,for any u∈Bv. By the result that xn is a bounded sequence, there is a subsequence xnm converges to θ weakly. The ξ-cocoercive mappings yield Axnm⟶Aθ. It holds −Aθ−u,θ−v≥0. It shows 0∈B+Aθ. Note that Id−T is demiclosed (Lemma 2). One asserts θ∈FixT. One next shows that θ=ProjCSSB,A,Tx0 and xn converges to it strongly. Set x¯=ProjCSSB,A,Tx0. Since the functional · is weakly lower semicontinuous, one has(27)x¯−x0≤θ−x0≤liminfm⟶∞x0−xnm≤limsupm⟶∞x0−xnm≤x¯−x0.
One getsx¯−x0=limm⟶∞x0−xnm=θ−x0. Since the framework is a Hilbert space, one gets xn⟶θ as n⟶∞. This finishes this theorem.Let(28)∂fx=z∈H:fx+y−x,z≤fy,∀y∈H,∀,x∈H,where f:H⟶−∞,∞ is a proper, convex, and lower semicontinuous function. Rockfellar [32] proved that ∂f is a multivalued maximally monotone operator. Let C be a closed, convex, and nonempty subset of H and iC be the indicator function of C, that is,(29)iCx=0,x∈C,∞,x∉C.Furthermore, we define the normal coneNCv of C at v as follows:(30)NCv=z∈H:z,y−v≤0,∀y∈H,for any v∈C. Then, iC:H⟶−∞,∞ is proper, convex, and lower semicontinuous on H. ∂iC is a maximally monotone operator. Let Resλx=Id+λ∂iC−1x. So, ∂iCx=NCx and x∈C; we obtain(31)v=Jλ∂iCx⟺v=ProjCx,where ProjC∂iC is the metric projection onto C. This yields x∈A+∂iC−10⟺x∈VIA,C, where VIA,C denotes the classical variational inequality, that is, find a point x∈C such that Ax,y−x≥0 for all y∈C.Corollary 1.
Assume thatH is a Hilbert space and C is a convex and closed subset in space H. Assume that A is a single-valued ξ-cocoercive mapping from set C to space H. Assume that T is a nonexpansive mapping from C to C and VIA,C∩FixT is nonempty. Assume that αn and βn are positive real number sequences. Let xn be a sequence in set C generated in the following iterative process:(32)x0∈C,yn=1−αnTxn+αnProjCxn−βnAxn,Qn=x∈C:xn−x,xn−x0≤0,Cn=x∈C:x−yn≤x−xn,xn+1=ProjQn∩Cnx0.Assume thatαn and βn satisfy the conditions (i) 1>αn≥α>0 with α being a fixed real number and (ii) 0<β≤βn≤β′<2ξ with β and β′ being two fixed real numbers. Then, the sequence xn converges strongly to ProjVIA,C∩FixTx0.



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