# Strong Convergence Theorems for a Pair of Strictly Pseudononspreading Mappings

Authors: Bin-Chao Deng; Tong Chen
Journal: Journal of Mathematics
 (2013)

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## Abstract

LetH be a real Hilbert space. Let T1,T2:H→H be k1-, k2-strictly pseudononspreading mappings; let αn and βn be two real sequences in (0,1). For given x0∈H, the sequence xn is generated iteratively by xn+1=βnxn+1-βnTw1αnγfxn+I-μαnBTw2xn, ∀n∈N, where Twi=1−wiI+wiTi with i=1,2 and B:H→H is strongly monotone and Lipschitzian. Under some mild conditions on parameters αn and βn, we prove that the sequence xn converges strongly to the set FixT1∩FixT2 of fixed points of a pair of strictly pseudononspreading mappings T1 and T2.


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## Body

## 1. Introduction

LetH be a real Hilbert space whose inner product and norm are denoted by 〈·,·〉 and ∥·∥, and let C be a nonempty, closed, and convex subset of H, respectively. Recall the following definitions.Definition 1.
LetA:C→H be a nonlinear mapping.  (1)
Ais said to be monotone if
(1)〈Ax-Ay,x-y〉≥0,∀x,y∈C.(2)
A is said to be strongly monotone if there exists a constant α>0 such that
(2)〈Ax-Ay,x-y〉≥α∥x-y∥2,∀x,y∈C.
For such a case, A is said to be α-strongly-monotone.(3)
A is said to be inverse strongly if there exists a constant α>0 such that
(3)〈Ax-Ay,x-y〉≥α∥Ax-Ay∥2,∀x,y∈C.
For such a case, A is said to be α-inverse-strongly monotone.
The classical variational inequality which is denoted byVI(A,C) is to find x∈C such that
(4)〈Ax,y-x〉≥0,∀y∈C.
The variational inequality has been extensively studied in the literature; see, for example, [1, 2] and the reference therein. Recall that T is a nonexpansive mapping of C into itself; that is,
(5)∥Tx-Ty∥≤∥x-y∥,∀x,y∈C.
The set of fixed points of   T is the set Fix(T)={x∈H,Tx=x}.
In 2011, Osilike and Isiogugu [3] introduced a new class of mappings, the so-called k-strictly pseudononspreading; that is, a mapping T:D(T)⊆H→H is k-strictly pseudononspreading if there exists k∈[0,1) such that
(6)∥Tx-Ty∥2≤∥x-y∥2+k∥x-Tx-(y-Ty)∥2+2〈x-Tx,y-Ty〉,
for all x,y∈D(T). They showed that the class of nonspreading mappings is properly contained in the class of strictly pseudononspreading mappings.
The iteration procedure of Mann’s type for approximating fixed points of a nonexpansive mappingT is the following: x1∈C and
(7)xn+1=αnxn+(1-αn)Txn,∀n∈N,
where {αn} is a sequence in [0,1]; see [4]. For two nonexpansive mappings S and T, Takahashi and Tamura [5] considered the following iteration procedure: x1∈C and
(8)xn+1=αnxn+(1-αn)S(βnxn+(1-βn)Txn),∀n∈N,
where {αn} and {βn} are sequences in [0,1].
In 2010, Tian [6] introduced the following general viscosity iterative scheme for finding an element of set of solutions to the fixed point of nonexpansive mapping in Hilbert space. Define sequence {xn} by
(9)xn+1=αnγf(xn)+(I-μαnB)Txn,n≥0,
where B is k-Lipschitzian and η-strongly monotone operator and T is a nonexpansive mapping on H; then he proved that if the sequence {αn} satisfies appropriate conditions, the sequence {xn} generated by (9) converges strongly to the unique solution x*∈C of the variational inequality
(10)〈(γf-μB)x*,x-x*〉≤0,∀x∈C,
where C=Fix(T).
In this paper, motivated by Takahashi and Tamura [5] and Tian [6], we introduce the following iteration scheme for finding a common point of the set Fix(T1)⋂‍Fix(T2) of fixed points of a pair of strictly pseudononspreading mappings T1 and T2:
(11)xn+1=βnxn+(1-βn)Tω1×({Tω2xn}αnγf(xn)+(I-μαnB)Tω2xn),∀n∈N,
where Tωi=(1-ωi)I+ωiTi with i=1,2 and B:H→H  is  η-strongly monotone and k-Lipschitzian on H with k>0, η>0. Under suitable conditions, we prove a strong convergence theorem, which is different from the results of general viscosity iterative scheme in [6].


## 2. Preliminaries

We need some facts and tools in Hilbert spaceH which are listed as in the following lemmas.Definition 2.
A mappingT is said to be demiclosed, if for any sequence {xn} which weakly converges to y, and if the sequence {Txn} strongly converges to z, then T(y)=z.Definition 3.
T
:
H
→
H is called demicontractive on H, if there exists a constant α<1 such that
(12)∥Tx-q∥2≤∥x-q∥2+α∥x-Tx∥2,∀(x,q)∈H×Fix(T).Remark 4.
Everyk-strictly pseudononspreading mapping with a nonempty fixed point set Fix(T) is demicontractive (see [7, 8]).Remark 5 (see [9]).
LetT be a α-demicontractive mapping on H with Fix(T)≠∅ and Tω=(1-ω)I+ωT for ω∈(0,∞). (A1)
Tα-demicontractive is equivalent to
(13)〈x-Tx,x-q〉≥12(1-α)∥x-Tx∥2,∀(x,q)∈H×Fix(T).(A2)
Fix(T)=Fix(Tω) if ω≠0.Remark 6.
According toI-Tω=ω(I-T) with T being a k-strictly pseudononspreading mapping, we obtain
(14)〈x-Tωx,x-q〉≥ω(1-k)2∥x-Tx∥2,∀(x,q)∈H×Fix(T).Proposition 7 (see [3]).
LetC be a nonempty closed convex subset of a real Hilbert space H, and let T:C→C be a k-strictly pseudononspreading mapping. If Fix(T)≠∅, then it is closed and convex.Proposition 8 (see [3]).
LetC be a nonempty closed convex subset of a real Hilbert space H, and let T:C→C be a k-strictly pseudononspreading mapping. Then (I-T) is demiclosed at 0.Lemma 9.
LetH be a real Hilbert space. The following expressions hold:  (i)
∥tx+(1-t)y∥2=t∥x∥2+(1-t)∥y∥2-t(1-t)∥x-y∥2, ∀x,y∈H, ∀t∈[0,1].(ii)
∥x+y∥2≤∥x∥2+2〈y,x+y〉,  ∀x,y∈H.Lemma 10 (see [10]).
LetC be a closed convex subset of a Hilbert space H, and let T:H→H be a k-strictly pseudononspreading mapping with a nonempty fixed point set. Let k≤ω<1 be fixed, and define Tω:C→C by
(15)Tωx=(1-ω)x+ωTx,∀x∈C.
Then Fix(Tω)=Fix(T).Lemma 11 (see [11]).
Assume{αn} is a sequence of nonnegative real numbers such that
(16)αn+1≤(1-γn)αn+δn,n≥0,
where {γn} is a sequence in (0,1) and {δn} is a sequence in ℝ such that  (i)
∑n=1∞‍γn=∞,(ii)
limsupn→∞(δn/γn)=0 or ∑n=1∞‍|δn|<∞. Then limn→∞αn=0.Lemma 12 (see [12]).
Let{𝒯n} be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence {𝒯nj}j≥0 of {𝒯n} which satisfies 𝒯nj<𝒯nj+1 for all j≥0. Also consider that the sequence of integers {δ(n)}n≥n0 is defined by
(17)δ(n)=max{k≤n∣𝒯k<𝒯k+1}.
Then {δ(n)}n≥n0 is a nondecreasing sequence verifying limn→∞δ(n)=∞,  ∀n≥n0; it holds that 𝒯δ(n)<𝒯δ(n)+1 and one has
(18)𝒯n<𝒯δ(n)+1.Lemma 13.
LetK be a closed convex subset of a real Hilbert space H, given x∈H and y∈K. Then y=PKx if and only if there holds the inequality
(19)〈x-y,y-z〉≥0,∀z∈K.


## 3. Main Results

LetC be a nonempty closed convex subset of a real Hilbert space H, and let T1,T2:C→C be k1-, k2-strictly pseudononspreading mappings with nonempty fixed point set Ω=Fix(T1)⋂‍Fix(T2). Let f be a L-Lipschitz mapping on H with coefficient L>0. Assume that the set Ω is nonempty. Since Ω is closed and convex, the nearest point projection from C onto Ω is well defined. Recall that B:H→H is η-strongly monotone and k-Lipschitzian on H with k>0, η>0.Lemma 14 (see [13]).
LetH be a real Hilbert space and let B be a k-Lipschitzian and η-strongly monotone operator with k>0, η>0. Let 0<μ<(2η/k2) and τ=μ(η-(ηk2/2)). Then for t∈min{1,(1/τ)}, I-tμB is in contradiction with constant 1-tτ.Lemma 15.
LetT be a k-strictly pseudononspreading mapping on H, and Tωx=(1-ω)x+ωTx with 0<k<ω<1/2. Then
(20)∥Tωx-Tωy∥2≤∥x-y∥2+2ω〈x-Tωx,y-Tωy〉,∀x,y∈C.Proof.
Forx,y∈H, we have
(21)∥Tωx-Tωy∥2=(1-ω)∥x-y∥2+ω∥Tx-Ty∥2-ω(1-ω)∥x-Tx-(y-Ty)∥2≤(1-ω)∥x-y∥2+ω[∥x-y∥2+k∥x-Tx-(y-Ty)∥2+2〈x-Tx,y-Ty〉{∥x-y∥2}]-ω(1-ω)∥x-Tx-(y-Ty)∥2=∥x-y∥2+2ω〈x-Tx,y-Ty〉-ω(1-ω)(1-k)∥x-Tx-(y-Ty)∥2≤∥x-y∥2+2ω〈x-Tx,y-Ty〉=∥x-y∥2+2ω〈x-Tωx,y-Tωy〉.Remark 16.
Ify=p∈Fix(T) and from Lemma 15, we can easily claim that ∥Tωx-p∥≤∥x-p∥, ∀x∈C.Theorem 17.
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let T1,T2:C→C be k1-, k2-strictly pseudononspreading mappings and demiclosed. Let f be L-Lipschitz mapping on H with coefficient L>0 and let B:H→H be η-strongly monotone and k-Lipschitzian on H with k>0, η>0, 0<k1, and k2<ω<1/2. Let the sequence of {αn} and {βn} satisfy the following conditions:  (c1)
limn→∞αn=0,(c2)
∑n=0∞‍αn=∞ and limn→∞αn+1/αn=1,(c3)
0<c≤βn≤d<1.Let 0<μ<2η/k2, 0<γ<μ(η-(μk2/2))/L=τ/L, and let the following sequence {xn}n=0∞ be a sequence in C generated from an arbitrary x1∈C by
(22)xn+1=βnxn+(1-βn)×Tω1(αnγf(xn)+(I-μαnB)Tω2xn),
where Tωi=(1-ωi)I+ωiTi with 0<ki<ωi<1, i=1,2. Then {xn} converges strongly to the unique element x* in Ω=Fix(T1)⋂‍Fix(T2) verifying
(23)x*=(PΩ·f)x*,
which equivalently solves the following variational inequality problem:
(24)x*∈Ω,〈(γf-μB)x*,v-x*〉≤0,∀v∈Ω.Proof.
Letx*∈Ω. According to Remark 16, we have
(25)∥Tωxn-x*∥≤∥xn-x*∥.
Let yn=αnγf(xn)+(I-μαnB)Tω2xn. From (22), we obtain
(26)∥xn+1-x*∥=∥βnxn+(1-βn)Tω1yn-x*∥≤βn∥xn-x*∥+(1-βn)∥Tω1yn-x*∥≤βn∥xn-x*∥+(1-βn)∥yn-x*∥,
And, from Lemma 14, we also obtain that
(27)∥yn-x*∥=∥αnγ(f(xn)-f(x*))+(I-μαnB)(Tω2xn-x*)+αn(γf(x*)-μBx*)∥≤αnγ∥f(xn)-f(x*)∥+(1-αnτ)∥Tω2xn-x*∥+αn∥γf(x*)-μBx*∥≤αnγL∥xn-x*∥+(1-αnτ)∥xn-x*∥+αn∥γf(x*)-μBx*∥.
Together with (26) and (27), we obtain
(28)∥xn+1-x*∥≤[1-αn(1-βn)(τ-γL)]∥xn-x*∥+αn(1-βn)∥γf(x*)-μBx*∥.
Putting M1=max{∥x0-x*∥,∥γf(x*)-x*∥}, we clearly obtain ∥xn-x*∥≤M1. By induction, we can deduce that {xn} is bounded and the sequences {f(xn)} and {Tωixn},  i=1,2, are also bounded.
Next, we show the following estimation:(29)𝒯n+1≤(1-αnδ)𝒯n-δ(∥xn-T2xn∥2+∥Tω1yn-xn∥2)+2αn(1-βn)〈γf(xn)-μBx*,yn-x*〉,
where 𝒯n=∥xn-x*∥2 and δ>0 is chosen so that
(30)δ≤min{ω2(1-αnτ)(1-βn)(1-ω2)(1-k2),βn(1-βn)}.
From Lemma 15, we obtain
(31)∥Tω2xn-x*∥2≤∥xn-x*∥2-ω2(1-ω2)(1-k2)∥xn-T2xn∥2.
By (25) and (31), we get
(32)∥yn-x*∥2=∥αn(γf(xn)-μBx*)+(I-μαnB)(Tω2xn-x*)∥2≤(1-αnτ)∥Tω2xn-x*∥2+2αn〈γf(xn)-μBx*,yn-x*〉≤(1-αnτ)∥xn-x*∥2-ω2(1-αnτ)(1-ω2)(1-k2)∥xn-T2xn∥2+2αn〈γf(xn)-μBx*,yn-x*〉.
Consequently,
(33)∥xn+1-x*∥2=βn∥xn-x*∥2+(1-βn)∥Tω1yn-x*∥2-βn(1-βn)∥Tω1yn-xn∥2≤βn∥xn-x*∥2-βn(1-βn)∥Tω1yn-xn∥2+(1-βn)∥yn-x*∥2≤βn∥xn-x*∥2-βn(1-βn)∥Tω1yn-xn∥2+(1-βn)(1-αnτ)∥xn-x*∥2-ω2(1-αnτ)(1-βn)(1-ω2)(1-k2)×∥xn-T2xn∥2+2αn(1-βn)〈γf(xn)-μBx*,yn-x*〉≤(1-αnτ(1-βn))∥xn-x*∥2-βn(1-βn)∥Tω1yn-xn∥2-ω2(1-αnτ)(1-βn)(1-ω2)(1-k2)×∥xn-T2xn∥2+2αn(1-βn)〈γf(xn)-μBx*,yn-x*〉,
and the desired inequality (29) follows.
Finally, we showlimn→∞𝒯n=0 by considering two possible cases.Case 1.
{
𝒯
n
} is nonincreasing sequence with some n≥n0. In this case, {𝒯n} is then convergent because it is also nonnegative (hence it is bounded form blow). From (29), we have
(34)δ(∥xn-T2xn∥2+∥Tω1yn-xn∥2)≤αnM+(𝒯n-𝒯n+1),
where 0<M=2〈γf(xn)-μBx*,yn-x*〉. Consequently, both ∥xn-T2xn∥ and ∥Tω1yn-xn∥ converge to zero. From (22) and (25), limn→∞αn=0 and the sequence {xn} is bounded, we obtain
(35)∥yn-xn∥=∥αn(γf(xn)-μBxn)+(I-μαnB)(Tω2xn-xn)∥≤αn∥γf(xn)-μBxn∥+(1-αnτ)∥Tω2xn-xn∥≤αn∥γf(xn)-μBxn∥+(1-αnτ)ω2∥T2xn-xn∥⟶0,
which implies
(36)∥xn-Tω1xn∥2≤∥xn-Tω1yn∥2+∥Tω1yn-Tω1xn∥2≤∥xn-Tω1yn∥2+∥xn-yn∥2+2ω1〈xn-Tω1xn,yn-Tω1yn〉≤∥xn-Tω1yn∥2+∥xn-yn∥2+2ω1〈xn-Tω1xn,yn-xn〉+2ω1〈xn-Tω1xn,xn-Tω1yn〉.
From limn→∞∥Tω1yn-xn∥=0, limn→∞∥yn-xn∥=0, and xn-Tω1xn=ω1(xn-T1xn), we obtain
(37)limn→∞∥xn-T1xn∥=limn→∞∥xn-Tω1xn∥=0.
Using the demiclosedness principle (Proposition 8) and (35), we know that ωw(xn)∈Ω; hence
(38)limsupn→∞〈γf(xn)-μBx*,yn-x*〉=limsupn→∞〈γf(xn)-μBx*,xn-x*〉=maxw∈ωw(xn)〈γf(w)-μBx*,w-x*〉≤0.
It then follows from (29) that
(39)𝒯n+1≤(1-αnδ)𝒯n+2αn(1-βn)〈γf(xn)-μBx*,yn-x*〉.
According to Lemma 11, we obtain limn→∞𝒯n=0.Case 2.
Suppose there exists a subsequence{𝒯nk}k≥0 of {𝒯n}n≥0 such that 𝒯nk≤𝒯nk+1 for all k≥0. In this situation, we consider the sequence of indices {δ(n)} as defined in Lemma 12. It follows that 𝒯δ(n+1)-𝒯δ(n)>0, which by (29) amounts to
(40)δ(∥xδ(n)-T2xδ(n)∥2+∥Tω1yδ(n)-xδ(n)∥2)≤αδ(n)M.
In a similar way to Case 1, we obtain
(41)limsupn→∞〈γf(xδ(n))-μBx*,yδ(n)-x*〉≤0.
From Lemma 12 and (29)
(42)𝒯δ(n)≤2(1-βδ(n))〈γf(xδ(n))-μBx*,yδ(n)-x*〉,
for all n>n0. Taking limsup in this inequality, we obtain limn→∞𝒯δ(n)=0. Moreover, it follows from (22) that
(43)𝒯δ(n)+12=∥(xδ(n)-x*)-(xδ(n)-xδ(n)+1)∥2≤𝒯δ(n)2+∥xδ(n)-xδ(n)+1∥2≤𝒯δ(n)2+∥xδ(n)-Tω1yδ(n)∥2,
which together with (40) implies limn→∞𝒯δ(n)+1=0. Consequently, from Lemma 12, we obtain limn→∞𝒯n=0.
In addition, from (38), we obtain
(44)limsupn→∞〈γf(xn)-μBx*,xn-x*〉=limsupn→∞〈γf(xn)-γf(x*),xn-x*〉+limsupn→∞〈γf(x*)-μBx*,xn-x*〉=γLlimn→∞𝒯n+limsupn→∞〈γf(x*)-μBx*,xn-x*〉=〈γf(x*)-μBx*,y--x*〉,y-∈Ω;
then variational inequality (44) can be written as
(45)〈(I-μB+γf)x*-x*,y--x*〉≥0,y-∈Ω.
So, by Lemma 13, it is equivalent to the fixed point equation
(46)PFix(T)(I-μB+γf)x*=x*.Remark 18.
For a nonspreading mappingT, we have ϱ=0 in Theorem 17 to obtain the following corollary.Corollary 19.
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let T1,T2:H→H be nonspreading mappings and demiclosed. Let f be L-Lipschitz mapping on H with coefficient L>0 and let B:H→H be η-strongly monotone and k-Lipschitzian on H with k>0, η>0. Let the sequence of {αn} and {βn} satisfy the following conditions:  (c1)
limn→∞αn=0,(c2)
∑n=0∞‍αn=∞ and limn→∞αn+1/αn=1,(c3)
0<c≤βn≤d<1.Let the following sequence {xn}n=0∞ be a sequence in C generated from an arbitrary x1∈C by
(47)xn+1=βnxn+(1-βn)Tω1(αnγf(xn)+(I-μαnB)Tω2xn),
where Tωi=(1-ωi)I+ωiTi, ωi∈(0,1), i=1,2. Then {xn} converges strongly to the unique element x* in Ω=Fix(T1)⋂‍Fix(T2) verifying
(48)x*=(PΩ·f)x*,
which equivalently solves the following variational inequality problem:
(49)x*∈Ω〈(γf-μB)x*,v-x*〉≤0,∀v∈Ω.Remark 20.
IfB=I and μ=1 in Corollary 19, we obtain the following corollary.Corollary 21.
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let T1,T2:H→H be nonspreading mappings and demiclosed. Let f be L-Lipschitz mapping on H with coefficient L>0. Let the sequence of {αn} and {βn} satisfy the following conditions:  (c1)
limn→∞αn=0,(c2)
∑n=0∞‍αn=∞ and limn→∞αn+1/αn=1,(c3)
0<c≤βn≤d<1.Let the following sequence {xn}n=0∞ be a sequence in C generated from an arbitrary x1∈C by
(50)xn+1=βnxn+(1-βn)Tω1(αnγf(xn)+(1-αn)Tω2xn),
where Tωi=(1-ωi)I+ωiTi, ωi∈(0,1), i=1,2. Then {xn} converges strongly to the unique element x* in Ω=Fix(T1)⋂‍Fix(T2) verifying
(51)x*=(PΩ·f)x*,
which equivalently solves the following variational inequality problem:
(52)x*∈Ω,〈(γf-I)x*,v-x*〉≤0,∀v∈Ω.


## 4. Application

In this section, we constructed a numerical example to illustrate that our main results are well defined. The following example is introduced by Deng et al. [14]. They prove that T is (1/3)-strictly pseudononspreading.Example 22.
LetX=l2 with the norm ∥·∥ defined by
(53)∥x∥=∑i=1∞‍xi2,∀x=(x1,x2,…,xn,…)∈X,C={x=(x1,x2,…,xn,…)∣xi∈R1,i=1,2,…}, and let C be an orthogonal subspace of X (i.e.,  ∀x,y∈C, we have 〈x,y〉=0). Then it is obvious that C is a nonempty closed convex subset of X. Now, for any x=(x1,x2,…,xn,…)∈C, define a mapping T:C→C as follows:
(54)Tx={(x1,x2,…,xn,…),∏i=1∞‍xi<0,(-2x1,-2x2,…,-2xn,…),∏i=1∞‍xi≥0.From the definition of this example, letx=(x1,x2)∈R2 and f(x)=8(sin(x1),cos(x2)) be a Lipschitz mapping on R2. Assume that T1(x)=(0,4x1) and T2(x)=(4x2,0) be  (1/3)-strictly pseudononspreading mappings. Let αn=βn=1/5, γ=1/8, μ=ω1=ω2=1, and B=I. According to (22), we can obtain the following algorithm:
(55)xn+1=15xn+4200T1(f(xn))+1625T1(T2(xn)).
From Theorem 17, we can easily know that algorithm (55) converges to the unique point in R2. Let x0=(5,4) and |xn+1-xn|≤10-5, and let x* be the fixed point of the algorithm (55). Using the software of MATLAB, we obtain x*=(0,0).



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