# Strong Convergence Theorems for Maximal Monotone Operators with Nonspreading Mappings in a Hilbert Space

Authors: Hongjie Liu; Junqing Wang; Qiansheng Feng
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We prove the strong convergence theorems for finding a common element of the set of fixed points of a nonspreading mappingT and the solution sets of zero of a maximal monotone mapping and an α-inverse strongly monotone mapping in a Hilbert space. Manaka and Takahashi (2011) proved weak convergence theorems for maximal monotone operators with nonspreading mappings in a Hilbert space; there we introduced new iterative algorithms and got some strong convergence theorems for maximal monotone operators with nonspreading mappings in a Hilbert space.


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## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈·,·〉, and let C be a nonempty closed convex subset of H. We denote by F(T) the set of fixed point of T. Then, a mapping T:C→C is said to be nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all x,y∈C. The mapping T:C→C is said to be firmly nonexpansive if ∥Tx-Ty∥2≤〈x-y,Tx-Ty〉 for all x,y∈C; see, for instance, Browder [1] and Goebel and Kirk [2]. The mapping T:C→C is said to be firmly nonspreading [3] if
(1.1)2∥Tx-Ty∥2≤∥Tx-y∥2+∥x-Ty∥2,
for all x,y∈C. Iemoto and Takahashi [4] proved that T:C→C is nonspreading if and only if
(1.2)∥Tx-Ty∥2≤∥x-y∥2+2〈x-Tx,y-Ty〉,
for all x,y∈C. It is not hard to know that a nonspreading mapping is deduced from a firmly nonexpansive mapping; see [5, 6], and a firmly nonexpansive mapping is a nonexpansive mapping.Many studies have been done for structuring the fixed point of nonexpansive mappingT. In 1953, Mann [7] introduced the iteration as follows: a sequence {xn} defined by
(1.3)xn+1=αnxn+(1-αn)Txn,
where the initial guess x1∈C is arbitrary and {an} is a real sequence in [0,1]. It is known that under appropriate settings, the sequence {xn} converges weakly to a fixed point of T. However, even in a Hilbert space, Mann iteration may fail to converge strongly, for example see [8].Some attempts to construct iteration method guaranteeing the strong convergence have been made. For example, Halpern [9] proposed the following so-called Halpern iteration:
(1.4)xn+1=αnu+(1-αn)Txn,
where u,x1∈C are arbitrary and {an} is a real sequence in [0,1] which satisfies αn→0, ∑n=1∞‍αn=∞ and ∑n=1∞‍|αn-αn+1|<∞. Then, {xn} converges strongly to a fixed point of T; see [9, 10].In 1975, Baillon [11] first introduced the nonlinear ergodic theorem in Hilbert space as follows:
(1.5)Snx=∑k=0n-1‍Tkx
converges weakly to a fixed point of T for some x∈C.Recently, in the case whenT:C→C is a nonexpansive mapping, A:C→H is an α-inverse strongly monotone mapping, and B∈H×H is a maximal monotone operator, Takahashi et al. [12] proved a strong convergence theorem for finding a point of F(T)∩(A+B)-1(0), where F(T) is the set of fixed points of T and (A+B)-1(0) is the set of zero points of A+B.In 2011, Manaka and Takahashi [13] for finding a point of the set of fixed points of T and the set of zero points of A+B in a Hilbert space, they introduced an iterative scheme as follows:
(1.6)xn+1=βnxn+(1-βn)T(Jλn(I-λnA)xn),
where T is a nonspreading mapping, A is an α-inverse strongly monotone mapping, and B is a maximal monotone operator such that Jλ=(I-λB)-1; {βn} and {λn} are sequences which satisfy 0<c≤βn≤d<1 and 0<a≤λn≤b<2α. Then they proved that {xn} converges weakly to a point p=limn→∞PF(T)∩(A+B)-1(0)xn.Motivated by above authors, we generalize and modify the iterative algorithms (1.5) and (1.6) for finding a common element of the set of fixed points of a nonspreading mapping T and the set of zero points of monotone operator A+B (A is an α-inverse strongly monotone mapping, and B is a maximal monotone operator). First, we prove that the sequence generated by our iterative method is weak convergence under the property conditions. Then, we prove that the strong convergence in a Hilbert space. As expected, we get some weak and strong convergence theorems about the common element of the set of fixed points of a nonspreading mapping and the set of zero points of an α-inverse strongly monotone mapping and a maximal monotone operator in a Hilbert space.


## 2. Preliminaries

LetH be a real Hilbert space with inner product 〈·,·〉, and let C be a nonempty closed convex subset of H. A set-valued mapping B:D(B)⊆H→H is said to be monotone if for any x,y∈D(B) and x*∈Bx and y*∈By, it holds that
(2.1)〈x-y,x*-y*〉≥0.A monotone operatorB on H is said to be maximal if B has no monotone extension, that is, its graph is not properly contained in the graph of any other monotone operator on H. For a maximal monotone operator B on H and r>0, we may define a single-valued operator Jr=(I+rB)-1:2H→D(B), which is called the resolvent of B for r>0. Let B be a maximal monotone operator on H, and let B-1(0)={x∈H:0∈Bx}. For a constant α>0, the mapping A:C→H is said to be an α-inverse strongly monotone if for any for all x,y∈C,
(2.2)〈x-y,Ax-Ay〉≥α∥Ax-Ay∥2.Remark 2.1.
It is not hard to know that ifA is an α-inverse strongly monotone mapping, then it is 1/α-Lipschitzian and hence uniformly continuous. Clearly, the class of monotone mappings include the class of an α-inverse strongly monotone mappings.Remark 2.2.
It is well known that ifT:C→C is a nonexpansive mapping, then I-T is 1/2-inverse strongly monotone, where I is the identity mapping on H; see, for instance, [14]. It is known that the resolvent Jr is firmly nonexpansive and B-1(0)=F(Jr) for all r>0.
For a single-valued mappingT, a point p is called a fixed point of T if p=Tp. For a multivalued mapping T, a point p is called a fixed point of T if p∈Tp. The set of fixed points of T is denoted by F(T).
LetE be a uniformly convex real Banach space, K be a nonempty closed convex subset of E. A multivalued mapping T:K→CB(K) is said to be as follows.(i)
Contraction if there exists a constantk∈[0,1) such that
(2.3)H(Tx,Ty)≤k∥x-y∥,∀x,y∈K.(ii)
Nonexpansive if(2.4)H(Tx,Ty)≤∥x-y∥,∀x,y∈K.(iii)
Quasinonexpansive ifF(T)≠∅ and
(2.5)H(Tx,Tp)≤∥x-p∥,∀x∈K,∀p∈F(T).
It is well known that every nonexpansive multivalued mappingT with F(T)≠∅ is multivalued quasinonexpansive. But there exist multivalued quasi-nonexpansive mappings that are not multivalued nonexpansive. It is clear that if T is a quasi-nonexpansive multivalued mapping, then F(T) is closed.
A Banach spaceE is said to satisfy Opials condition if whenever {xn} is a sequence in E which converges weakly to x, then
(2.6)liminfn→∞∥xn-x∥<liminfn→∞∥xn-y∥,∀y∈E,x≠y.Lemma 2.3 (Manaka and Takahashi [13]).
LetH be a real Hilbert space, and let C be a nonempty closed convex subset of H. Let α>0. Let A be an α-inverse strongly monotone mapping of C into H, and let B be a maximal monotone operator on H such that the domain of B is included in C. Let Jλ=(I+λB)-1 be the resolvent of B for any λ>0. Then, the following hold(i)
ifu,v∈(A+B)-1(0), then Au=Av;(ii)
for anyλ>0, u∈(A+B)-1(0) if and only if u=Jλ(I-λA)u.Lemma 2.4 (Schu [15]).
Suppose thatE is a uniformly convex Banach space and 0<p≤tn≤q<1 for all positive integers n. Also suppose that {xn} and {yn} are two sequences of E such that limsupn→∞∥xn∥≤r, limsupn→∞∥yn∥≤r, and limn→∞∥tnxn+(1-tn)yn∥=r hold for some r≥0. Then, limn→∞∥xn-yn∥=0.Lemma 2.5 (Liu [16] and Xu [17]).
Let{an} be a sequence of nonnegative real numbers satisfying the property as follows
(2.7)an+1≤(1-tn)an+bn+tncn,
where {tn}, {bn}, and {cn} satisfy the restrictions as follows(i)
∑n=0∞‍tn=∞,(ii)
∑n=0∞‍bn<∞,(iii)
limsupn→∞cn≤0.
Then,{an} converges to zero as n→∞.


## 3. Strong Convergence Theorem

In this section, we prove the strong convergence theorems for finding a common element in common set of the fixed sets of a nonspreading mapping and the solution sets of zero of a maximal monotone operator and anα-inverse strongly monotone operator and in a Hilbert space.Theorem 3.1.
LetC be a nonempty convex closed subset of a real Hilbert space H, let A:C→H be an α-inverse strongly monotone, let B:D(B)⊆C→2H be maximal monotone, let Jλ=(I+λB)-1 be the resolvent of B for any λ>0, and let T:C→C be a nonspreading mapping. Assume that F:=F(T)∩(A+B)-1(0)≠∅. We define
(3.1)x1=x∈C,arbitrarily,zn=Jλn(I-λnA)xn,yn=1n∑k=1n‍Tkzn,xn+1=αnu+(1-αn)yn,
where {αn} is sequences in [0,1] such that limn→∞αn=0,∑n=1∞‍αn=∞. There exists a,b such that 0<a≤λn≤b<2α for each n∈N. Then, {xn} converges strongly to Pu, and P is the metric projection of H onto F.Proof.
First, we prove that{xn} is bounded and limn→∞∥xn-p∥ exists for each p∈F(T). In fact, from Lemma 2.3, we have p=Jλn(I-λnA)p, together with (3.1) and A is an α-inverse strongly monotone, we get that
(3.2)∥zn-p∥2=∥Jλn(I-λnA)xn-Jλn(I-λnA)p∥2≤∥(I-λnA)xn-(I-λnA)p∥2=∥xn-p∥2-2λn〈xn-p,Axn-Ap〉+λn2∥Axn-Ap∥2≤∥xn-p∥2-2λnα∥Axn-Ap∥2+λn2∥Axn-Ap∥2=∥xn-p∥2-λn(2α-λn)∥Axn-Ap∥2≤∥xn-p∥2.
From the definition ofyn and T is nonspreading mapping, we obtain that
(3.3)∥yn-p∥=∥1n∑k=0n-1‍Tkzn-p∥≤1n∑k=0n-1∥Tkzn-p∥‍≤1n∑k=0n-1‍∥zn-p∥=∥zn-p∥≤∥xn-p∥.
Together with (3.1), we have that
(3.4)∥xn+1-p∥=∥αnu+(1-αn)yn-p∥≤αn∥u-p∥+(1-αn)∥yn-p∥≤αn∥u-p∥+(1-αn)∥xn-p∥.
Hence, we get that(3.5)∥xn+1-p∥≤max{∥u-p∥,∥xn-p∥},
for all n∈N. This means that {xn-p} is bounded, so {xn} is bounded. From T is nonspreading, (3.3), and (3.2), we get that {yn}, {zn}, and {Tnzn} are all bounded.
Since{xn} is bounded, there exists a subsequence {xnk} of {xn} such that limk→∞∥xnk-p∥ exists. Since {xnk} is bounded, there exists a subsequence {xnki} of {xnk} such that xnki⇀w∈C as i→∞. Now, we prove that w∈F. First, we prove that w∈F(T). Since ∥xn+1-yn∥=αn∥u-yn∥, replacing n by nki, we have ∥xnki+1-ynki∥=αnki∥u-ynki∥. Together with αn→0 and {yn} is bounded, we obtain that limi→∞∥xnki+1-ynki∥=0, so we have ynki⇀w.
Letn∈N. Since T is nonspreading, we have that for all y∈C and k=0,1,2,…,n-1,
(3.6)∥Tk+1zn-Ty∥2≤∥Tkzn-y∥2+2〈Tkzn-Tk+1zn,y-Ty〉=∥Tkzn-Ty∥2+∥Ty-y∥2+2〈Tkzn-Ty,Ty-y〉+2〈Tkzn-Tk+1zn,y-Ty〉.
Summing these inequalities fromk=0 to n-1 and dividing by n, we have
(3.7)1n(∥Tnzn-Ty∥2-∥zn-Ty∥2)≤∥Ty-y∥2+2〈yn-Ty,Ty-y〉+2n〈zn-Tnzn,y-Ty〉.
Replacingn by nki, we have
(3.8)1nki(∥Tnkiznki-Ty∥2-∥znki-Ty∥2)≤∥Ty-y∥2+2〈ynki-Ty,Ty-y〉+2nki〈znki-Tnkiznki,y-Ty〉.
Since{zn} and {Tnzn} are bounded, we have that
(3.9)0≤∥Ty-y∥2+2〈w-Ty,Ty-y〉
as i→∞. Putting y=w, we have
(3.10)0≤∥Tw-w∥2+2〈w-Tw,Tw-w〉=-∥Tw-w∥2.
Hence,w∈F(T).
Next, we prove thatw∈(A+B)-1(0). From (3.2) and (3.3) we have that
(3.11)∥xn+1-p∥2≤αn∥u-p∥2+(1-αn)∥yn-p∥2≤αn∥u-p∥2+(1-αn)∥zn-p∥2≤αn∥u-p∥2+(1-αn)(∥xn-p∥2-λn(2α-λn)∥Axn-Ap∥2)=αn(∥u-p∥2-∥xn-p∥2)+∥xn-p∥2-(1-αn)λn(2α-λn)∥Axn-Ap∥2.
We rewrite above inequality as follows:(3.12)(1-αn)λn(2α-λn)∥Axn-Ap∥2≤αn(∥u-p∥2-∥xn-p∥2)+∥xn-p∥2-∥xn+1-p∥2.
Replacingn by nk, we have
(3.13)(1-αnk)λnk(2α-λnk)∥Axnk-Ap∥2≤αnk(∥u-p∥2-∥xnk-p∥2)+∥xnk-p∥2-∥xnk+1-p∥2.
Together withlimn→∞αn=0, 0<a≤λn≤b<2α and since limk→∞∥xnk-p∥ exists, we obtain that
(3.14)limk→∞∥Axnk-Ap∥=0.
SinceJλn is firmly nonexpansive, and from (3.2), we have that
(3.15)∥zn-p∥2=∥Jλn(I-λnA)xn-Jλn(I-λnA)p∥2≤〈zn-p,(I-λnA)xn-(I-λnA)p〉=12{∥zn-p∥2+∥(I-λnA)xn-(I-λnA)p∥2-∥zn-p-(I-λnA)xn+(I-λnA)p∥2}≤12{∥zn-p∥2+∥xn-p∥2-∥zn-p-(I-λnA)xn+(I-λnA)p∥2}=12{∥zn-p∥2+∥xn-p∥2-∥zn-xn∥2-2λn〈zn-xn,Axn-Ap〉-λn2∥Axn-Ap∥2}.
This means that(3.16)∥zn-p∥2≤∥xn-p∥2-∥zn-xn∥2-2λn〈zn-xn,Axn-Ap〉-λn2∥Axn-Ap∥2.
Together with (3.1) and (3.3), we have
(3.17)∥xn+1-p∥2≤αn∥u-p∥2+(1-αn)∥yn-p∥2≤αn∥u-p∥2+(1-αn)∥zn-p∥2≤αn∥u-p∥2+(1-αn)×{∥xn-p∥2-∥zn-xn∥2-2λn〈zn-xn,Axn-Ap〉-λn2∥Axn-Ap∥2}≤αn∥u-p∥2+∥xn-p∥2-∥zn-xn∥2-2λn〈zn-xn,Axn-Ap〉-λn2∥Axn-Ap∥2.
Therefore, we have(3.18)∥zn-xn∥2≤αn∥u-p∥2+∥xn-p∥2-∥xn+1-p∥2-2λn〈zn-xn,Axn-Ap〉-λn2∥Axn-Ap∥2.
Replacingn by nk, we have
(3.19)∥znk-xnk∥2≤αnk∥u-p∥2+∥xnk-p∥2-∥xnk+1-p∥2-2λnk〈znk-xnk,Axnk-Ap〉-λnk2∥Axnk-Ap∥2.
Sincelimk→∞∥xnk-p∥ exists, from (3.14) and limn→∞αn=0, we obtain
(3.20)limn→∞∥znk-xnk∥=0.
SinceA is Lipschitz continuous, we also obtain
(3.21)limn→∞∥Aznk-Axnk∥=0.
By the definition ofJλn and (3.1), we have that
(3.22)zn=(I-λnB)-1(I-λnA)xn⟺(I-λnA)xn∈(I-λnB)zn=zn+λnBzn⟺xn-zn-λnAxn∈λnBzn⟺1λn(xn-zn-λnAxn)∈Bzn.
SinceB is monotone, so for (e,f)∈B, we have that
(3.23)〈zn-e,1λn(xn-zn-λnAxn)-f〉≥0,
and hence
(3.24)〈zn-e,xn-zn-λn(Axn+f)〉≥0.
Replacingn by nki, we have that
(3.25)〈znki-e,xnki-znki-λnki(Axnki+f)〉≥0.
SinceA is an α-inverse strongly monotone, we have
(3.26)〈xnki-w,Axnki-Aw〉≥α∥Axnki-Aw∥2.
This means that Axnki→Aw as i→∞. From (3.20) and xnki⇀w, we get that znki⇀w, together with (3.25), we have that
(3.27)〈w-e,-Aw-f〉≥0.
SinceB is maximal monotone, so (-Aw)∈Bw. That is, w∈(A+B)-1(0).
Now, we prove thatxn→Pu as n→∞. Without loss of generality, we may assume that there exists a subsequence {xnki+1} of {xn+1} such that
(3.28)limsupn→∞〈u-Pu,xn+1-Pu〉=limi→∞〈u-Pu,xnki+1-Pu〉.
SinceP is the metric projection of H onto F and xnki+1⇀w∈F, we have
(3.29)limi→∞〈u-Pu,xnki+1-Pu〉=〈u-Pu,w-Pu〉≤0.
This implies that(3.30)limn→∞〈u-Pu,xn+1-Pu〉≤0.
From (2.1), (3.1), and (3.3), we have
(3.31)∥xn+1-Pu∥2=∥(1-αn)(yn-Pu)+αn(u-Pu)∥2≤(1-αn)2∥yn-Pu∥2+2αn〈u-Pu,xn+1-Pu〉≤(1-αn)∥xn-Pu∥2+2αn〈u-Pu,xn+1-Pu〉.
From Lemma2.5 and (3.30), we have
(3.32)limn→∞∥xn-Pu∥=0.
This means thatxn→Pu as n→∞.



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