# Weak and Strong Convergence Theorems for Strictly Pseudononspreading Mappings and Equilibrium Problem in Hilbert Spaces

Authors: Yun He Zhao; Shih-sen Chang
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

The purpose of this paper is to propose an iterative algorithm for equilibrium problem and a class of strictly pseudononspreading mappings which is more general than the class of nonspreading mappings studied recently in Kurokawa and Takahashi (2010). We explored an auxiliary mapping in our theorems and proofs and under suitable conditions, some weak and strong convergence theorems are proved. The results presented in the paper extend and improve some recent results announced by some authors.


---

## Body

## 1. Introduction and Preliminaries

Throughout this paper, we assume thatH is a real Hilbert space and C is a nonempty and closed convex subset of H. In the sequel, we denote by “xn→x” and “xn⇀x” the strong and weak convergence of {xn}, respectively. Denote by F(T) the set of fixed points of a mapping T.Definition 1.
LetT:C→C be a mapping.(1)
T is said to be nonexpansive, if ∥Tx-Ty∥≤∥x-y∥,∀x, y∈C.(2)
T is said to be quasinonexpansive, if F(T) is nonempty and
(1)∥Tx-p∥≤∥x-p∥,∀x∈C,p∈F(T).(3)
T is said to be nonspreading [1, 2], if
(2)2∥Tx-Ty∥2≤∥Tx-y∥2+∥Ty-x∥2,∀x,y∈C.It is easy to prove that T:C→C is nonspreading if and only if
(3)∥Tx-Ty∥2≤∥x-y∥2+2〈x-Tx,y-Ty〉∀x,y∈C.(4)
T:C→H is said to be k-strictly pseudononspreading in the terminology of Browder-Petryshyn [3], if there exists k∈[0,1) such that
(4)∥Tx-Ty∥2≤∥x-y∥2+k∥x-Tx-(y-Ty)∥2+2〈x-Tx,y-Ty〉∀x,y∈C.Remark 2.
(1) IfT:C→C is a nonspreading mapping with F(T)≠∅, then T is quasinonexpansive and F(T) is closed and convex.
(2) Clearly every nonspreading mapping isk-strictly pseudononspreading with k=0, but the inverse is not true. This can be seen from the following example.Example 3.
Letℛ denote the set of all real numbers. Let T:ℛ→ℛ be a mapping defined by
(5)Tx={x,x∈(-∞,0),-2x,x∈[0,∞).
It is easy to see that T is a k-strictly pseudononspreading mapping with k∈[0,1), but it is not nonspreading (see, [4]).Definition 4.
(1) LetT:H→H be a mapping. I-T is said to be demiclosed at 0, if for any sequence {xn}⊂H with xn⇀x* and ∥(I-T)xn∥→0, we have x*=Tx*.
(2) A Banach spaceE is said to have Opial's property, if for any sequence {xn}⊂E with xn⇀x*, we have
(6)liminfn→∞∥xn-x*∥<liminfn→∞∥xn-y∥,∀y∈Ewithy≠x*.
It is well known that each Hilbert space processes opial property.
(3) A mappingS:C→C is said to be semicompact, if for any bounded sequence {xn}⊂C with limn→∞∥xn-Sxn∥=0, then there exists a subsequence {xni}⊂{xn} such that {xni} converges strongly to some point x*∈C.Lemma 5 (see [5]).
LetE be a uniformly convex Banach space and let Br(0):={x∈E:∥x∥≤r} be a closed ball with center 0 and radius r>0. For any given sequence {x1,x2,…,xn,…}⊂Br(0) and any given number sequence {λ1,λ2,…,λn,…} with λi≥0, ∑i=1∞‍λi=1, there exists a continuous strictly increasing and convex function g:[0,2r)→[0,∞) with g(0)=0 such that for any i,j∈𝒩, i<j the following holds:
(7)∥∑n=1∞‍λnxn∥2≤∑n=1∞‍λn∥xn∥2-λiλjg(∥xi-xj∥).Lemma 6.
LetH be a real Hilbert space, C be a nonempty and closed convex subset of H, and let T:C→C be a k-strictly pseudononspreading mapping.(i)
IfF(T)≠∅, then it is closed and convex.(ii)
(I-T) is demiclosed at origin.Lemma 7.
LetT:C→C be a k-strictly pseudononspreading mapping with k∈[0,1). Denote by Tβ:=βI+(1-β)T, where β∈[k,1), then(i)
F(T)=F(Tβ);(ii)
the following inequality holds:(8)∥Tβx-Tβy∥2≤∥x-y∥2+21-β〈x-Tβx,y-Tβy〉,∀x,y∈C;(iii)
Tβ is a quasinonexpansive mapping, that is,
(9)∥Tβx-p∥2≤∥x-p∥2,∀x∈C,p∈F(T).Proof.
The conclusion (i) is obvious. Now we prove the conclusion (ii). SinceT is k-strictly pseudononspreading, for any x,y∈C we have
(10)∥Tβx-Tβy∥2=∥β(x-y)+(1-β)(Tx-Ty)∥2=β∥x-y∥2+(1-β)∥Tx-Ty∥2-β(1-β)∥x-Tx-(y-Ty)∥2≤β∥x-y∥2+(1-β)×{∥x-y∥2+k∥x-Tx-(y-Ty)∥2+2〈x-Tx,y-Ty〉∥x-y∥2}-β(1-β)∥x-Tx-(y-Ty)∥2=∥x-y∥2+2(1-β)〈x-Tx,y-Ty〉-(1-β)(β-k)∥x-Tx-(y-Ty)∥2≤∥x-y∥2+2(1-β)〈x-Tx,y-Ty〉=∥x-y∥2+2(1-β)〈x-Tβx,y-Tβy〉.
Takey∈F(T) in (8), then y∈F(Tβ). Hence, conclusion (iii) is proved.
This completes the proof.In the sequel, we assume thatϕ:C×C→ℛ is a bifunction satisfying the following conditions:(A1)
ϕ(x,x)=0,∀x∈C;(A2)
ϕ is monotone, that is, ϕ(x,y)+ϕ(y,x)≤0,∀x,y∈C;(A3)
limt↓0ϕ(tz+(1-t)x,y)≤ϕ(x,y),∀x,y,z∈C;(A4)
for eachx∈C, y↦ϕ(x,y) is convex and lower semicontinuous.Recalled that the “so-called”equilibrium problem for a bifunction function  ϕ is to find a point x*∈C, such that
(11)ϕ(x*,y)≥0,∀y∈C.Lemma 8 (see [6, 7]).
LetC be a nonempty and closed convex subset of a Hilbert space H and let ϕ:C×C→ℛ be a bi-function satisfying conditions: (A1), (A2), (A3), and (A4). Then, for any r>0 and x∈C, there exists z∈C such that
(12)ϕ(z,y)+1r〈y-z,z-x〉≥0,∀y∈C.
Furthermore, if for given r>0, we define a mapping Tr:C→C by
(13)Trx:={z∈C:ϕ(z,y)+1r〈y-z,z-x〉≥0,∀y∈C},
then the following hold:(1)
Tr is single-valued;(2)
Tr is firmly nonexpansive, that is, ∥Trx-Try∥2≤〈Trx-Try,x-y〉;(3)
F(Tr)=Ω, where Ω is the set of solutions of the equilibrium problem (11);(4)
Ω is a closed and convex subset of C.Concerning the weak and strong convergence problem for some kinds of iterative algorithms for nonspreading mappings,k-strictly pseudononspreading mappings and other kind of nonlinear mappings have been considered in Osilike and Isiogugu [4], Igarashi et al. [8], Iemoto and Takahashi [9], Kurokawa and Takahashi [10], and Kim [11–28]. The purpose of this paper is to propose an iterative algorithm for an infinite family of strictly pseudononspreading mappings and equilibrium problem. Under suitable conditions, some weak and strong convergence theorems are proved. The results presented in the paper extend and improve the corresponding results in [4, 8–11].


## 2. Main Results

Throughout this section, we assume that the following conditions are satisfied.(1)
H is a real Hilbert spaces, C is a nonempty and close convex subset of H.(2)
For eachSi:C→C, i=1,2,… is a ki-strictly pseudononspreading mapping with k:=supi≥1ki∈(0,1). For given β∈[k,1), denoted by Si,β:=βI+(1-β)Si, for each i=1,2,…, it follows from (8) that
(14)∥Si,βx-Si,βy∥2≤∥x-y∥2+21-β〈x-Si,βx,y-Si,βy〉,∀x,y∈C.(3)
ϕ:C×C→ℛ is a bifunction satisfying the conditions (A1)–(A4). Then it follows from Lemma 8 that the mapping Tr defined by (13) is single valued, z=Trx, F(Tr)=Ω (where Ω is the solution set of the equilibrium problem (11)), and Ω is a closed and convex subset of C.We are now in a position to give the following result.Theorem 9.
LetH,C, {Si}, k,β, {Si,β}, ϕ, Tr, and Ω be the same as above. Let {xn} and {un} be the sequences defined by
(15){x1∈Cchosenarbitrarily,ϕ(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=α0,nun+∑i=1∞‍αi,nSi,βun,
where {αi,n}⊂(0,1) and {rn} satisfy the following conditions:(a)
∑i=0∞‍αi,n=1, for each n≥1;(b)
for eachi≥1,  liminfn→∞α0,nαi,n>0;(c)
{rn}⊂(0,∞) and liminfn→∞rn>0.(I)
Ifℱ:=(⋂i=1∞‍F(Si))⋂‍Ω≠∅, then both {xn} and {un} converge weakly to some point x*∈ℱ;(II)
in addition, if there exists some positive integerm such that Sm is semicompact, then both {xn} and {un} converge strongly to x*∈ℱ.Proof.
First, we prove the conclusion (I). The proof is divided into three steps.
Step  1. We prove that the sequences {xn}, {un}, {Si,βun}, and {Si,βxn},i≥1 all are bounded, and for each p∈ℱ the limits limn→∞∥xn-p∥, limn→∞∥un-p∥ exist and
(16)limn→∞∥xn-p∥=limn→∞∥un-p∥.
In fact, it follows from Lemma8 that un=Trnxn, p=Trnp, and
(17)∥un-p∥=∥Trnxn-Trnp∥≤∥xn-p∥,∀n≥1.
Since p∈ℱ, by Lemma 7(i), p∈⋂i=1∞‍F(Si,β). Hence, it follows from (17) and (9) that
(18)∥xn+1-p∥=∥α0,nun+∑i=1∞‍αi,nSi,βun-p∥≤α0,n∥un-p∥+∑i=1∞‍αi,n∥Si,βun-p∥≤α0,n∥un-p∥+∑i=1∞‍αi,n∥un-p∥=∥un-p∥≤∥xn-p∥,∀n≥1.
This implies that for each p∈ℱ, the limits limn→∞∥xn-p∥ and limn→∞∥un-p∥ exist. And so {xn} and {un} are bounded and (16) holds.
Furthermore, by (9), it is easy to see that for each i≥1, {Si,βun} and {Si,βxn} are also bounded.
Step  2. Next we prove that for each i≥1 the following holds:
(19)limn→∞∥xn-Sixn∥=limn→∞∥un-Siun∥=0.
In fact, by Lemma 5 for any positive integer i≥1 and p∈ℱ, we have
(20)∥xn+1-p∥2=∥α0,n(un-p)+∑i=1∞‍αi,n(Si,βun-p)∥2≤α0,n∥un-p∥2+∑i=1∞‍αi,n∥Si,βun-p∥2-α0,nαi,ng(∥un-Si,βun∥)≤α0,n∥un-p∥2+∑i=1∞‍αi,n∥un-p∥2-α0,nαi,ng(∥un-Si,βun∥)≤∥un-p∥2-α0,nαi,ng(∥un-Si,βun∥)≤∥xn-p∥2-α0,nαi,ng(∥un-Si,βun∥).
This shows that
(21)α0,nαi,ng(∥un-Si,βun∥)≤∥xn-p∥2-∥xn+1-p∥2⟶0(asn⟶∞).
Since g is a continuous and strictly increasing function with g(0)=0. By condition (b), it yields that
(22)limn→∞∥un-Si,βun∥=0.
Therefore, we have
(23)limn→∞∥un-Siun∥=limn→∞11-β∥un-Si,βun∥=0.
On the other hand, it follows from Lemma 8 that un=Trnxn and for each p∈ℱ(24)∥un-p∥2=∥Trnxn-Trnp∥2≤〈Trnxn-Trnp,xn-p〉=〈un-p,xn-p〉=12{∥un-p∥2+∥xn-p∥2-∥xn-un∥2}.
This shows that
(25)∥un-p∥2≤∥xn-p∥2-∥xn-un∥2.
In view of (20) and (25)
(26)∥xn+1-p∥2≤∥un-p∥2≤∥xn-p∥2-∥xn-un∥2,
that is,
(27)∥xn-un∥2≤∥xn-p∥2-∥xn+1-p∥2⟶0(asn⟶∞).
In view of (27), (22), (14), and noting that {xn-Si,βxn} is bounded, we have
(28)∥xn-Si,βxn∥≤∥xn-un∥+∥un-Si,βun∥+∥Si,βun-Si,βxn∥≤∥xn-un∥+∥un-Si,βun∥+{∥xn-un∥2+21-β|〈un-Si,βun,xn-Si,βxn〉|}1/2⟶0(asn⟶∞).
Therefore, we have
(29)limn→∞∥xn-Sixn∥=limn→∞11-β∥xn-Si,βxn∥=0.
The conclusion is proved.
Step  3. Next we prove that the weak-accumulation point set Wω(xn) of the sequence {xn} is a singleton and Wω(xn)⊂ℱ.
In fact, for anyw∈Wω(xn), their exists a subsequence {xni}⊂{xn} such that xni⇀w. It follows from (27) that uni⇀w. Since un=Trnxn, from (15) and condition (A2) we have
(30)〈y-uni,1rni(uni-xni)〉≥ϕ(y,uni),∀y∈C.
Since (1/rni)(uni-xni)→0(asni→∞) and uni⇀w, it follows from condition (A4) that
(31)ϕ(y,w)≤0,∀y∈C.
For any t∈(0,1),y∈C, letting yt=ty+(1-t)w, then yt∈C. By condition (A1) and (A4), we have
(32)0=ϕ(yt,yt)≤tϕ(yt,y)+(1-t)ϕ(yt,w)≤tϕ(yt,y).
This implies that ϕ(yt,y)≥0. Letting t→0, by condition (A3) we have
(33)ϕ(w,y)≥0,∀y∈C.
This shows that w∈C is a solution to the equilibrium (11), that is, w∈Ω.
On the other hand, by Lemma6, for each i≥1, I-Si is demiclosed at 0. In view of (19), we know that w∈ℱ. Due to the arbitrariness of w∈Wω(xn), we have Wω(xn)⊂ℱ.
Now we prove thatWω(xn) is a singleton. Suppose to the contrary that there exist x*,y*∈Wω(xn) with x*≠y*. Therefore, there exist subsequences {xni} and {xnj} in {xn} such that xni⇀x* and xnj⇀y*. Since x*,y*∈ℱ, by (16), the limits limn→∞∥xn-x*∥ and limn→∞∥xn-y*∥ exist. By using the opial property of H, we have
(34)liminfni→∞∥xni-x*∥<liminfni→∞∥xni-y*∥=limn→∞∥xn-y*∥=limnj→∞∥xnj-y*∥<liminfnj→∞∥xnj-x*∥=limn→∞∥xn-x*∥=liminfni→∞∥xni-x*∥.
This is a contradiction. Therefore, Wω(xn) is a singleton. Without loss of generality, we can assume that Wω(xn)={x*} and xn⇀x*. By using (15) and (19), we have un⇀x*.
This completes the proof of the conclusion (I).
Next we prove the conclusion (II).
Without loss of generality, we can assume thatS1 is semicompact. From (19) we have that
(35)∥xn-S1xn∥⟶0(asn⟶∞).
Therefore, there exists a subsequence of {xni}⊂{xn} such that xni→u*∈C. Since xni⇀x*, we have x*=u* and so xni→x*∈ℱ. By virtue of (16), we have
(36)limn→∞∥un-x*∥=0,limn→∞∥xn-x*∥=0,
This completes the proof of Theorem9.Takingϕ≡0 and rn=1,foralln≥1 in Theorem 9, we have xn=un,for alln≥1, Therefore, the following theorem can be obtained from Theorem 9 immediately.Theorem 10.
LetH,C, {Si}, k,β and {Si,β} be the same as in Theorem 9. Let {xn} be the sequences defined by
(37){x1∈C,chosenarbitrarily,xn+1=α0,nxn+∑i=1∞‍αi,nSi,βxn,
where {αi,n}⊂(0,1) satisfies the following conditions:(a)
∑i=0∞‍αi,n=1, for each n≥1;(b)
for eachi≥1,  liminfn→∞α0,nαi,n>0.(I)
Ifℱ:=(⋂i=1∞‍F(Si))≠∅, then both {xn} converge weakly to some point x*∈ℱ;(II)
in addition, if there exists some positive integerm such that Sm is semicompact, then {xn} converge strongly to x*∈ℱ.Remark 11.
Theorems9 and 10 improve and extend the corresponding recent results of [4, 8–11].



---

*Source: 169206-2013-09-24.xml*