# Strong Convergence Theorems for Variational Inequalities and Split Equality Problem

Authors: Yu Jing Wu; Ru Dong Chen; Luo Yi Shi
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

LetH1,H2, and H3 be real Hilbert spaces, let C⊆H1,Q⊆H2 be two nonempty closed convex sets, and let A:H1→H3,B:H2→H3 be two bounded linear operators. The split equality problem (SEP) is to find x∈C,y∈Q such that Ax=By. Let H=H1×H2; consider f:H→H a contraction with coefficient 0<α<1, a strongly positive linear bounded operator T:H→H with coefficient γ̅>0, and M:H→H is a β-inverse strongly monotone mapping. Let 0<γ<γ̅/α, S=C×Q and G:H→H3 be defined by restricting to H1 is A and restricting to H2 is -B, that is, G has the matrix form G=[A,-B]. It is proved that the sequence {wn}={(xn,yn)}⊆H generated by the iterative method wn+1=PS[αnγf(wn)+(I-αnT)PS(I-γnG*G)PS(wn-λnMwn)] converges strongly to w̃ which solves the SEP and the following variational inequality: 〈(T-λf)w̃,w-w̃〉≥0 and 〈Mw̃,w-w̃〉≥0 for all w∈S. Moreover, if we take M=G*G:H→H,γn=0, then M  is a β-inverse strongly monotone mapping, and the sequence {wn} generated by the iterative method wn+1=αnγf(wn)+(I-αnT)PS(wn-λnG*Gwn) converges strongly to w̃ which solves the SEP and the following variational inequality: 〈(T-λf)w̃,w-w̃〉≥0 for all w∈S.


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## Body

## 1. Introduction and Preliminaries

Throughout this paper, letH be a real Hilbert space and S a nonempty closed convex subset of H, and let I denote the identity operator on Hilbert space H, Fix(T) the set of the fixed points of an operator T. An operator T on a Hilbert space H is nonexpansive if, for each x and y in H, ∥Tx-Ty∥≤∥x-y∥. T is said to be strongly positive, if there exists γ->0 such that 〈Tx,x〉≥γ-∥x∥2 for all x∈H. T is said to be inverse strongly monotone if there exists β>0 such that 〈x-y,Tx-Ty〉≥β∥Tx-Ty∥2 for all x,y∈H. A bounded linear operator T on a Hilbert space H is positive operator if T*=T and λ≥0 for all λ in the spectrum of T or equivalent to 〈Tx,x〉≥0 for all x∈H.LetPS denote the projection from H onto a nonempty closed convex subset S of H; that is,
(1)PS(w)=minx∈S∥x-w∥.
It is well known that PS(w) is characterized by the inequality
(2)〈w-PS(w),x-PS(w)〉≤0,∀x∈S,
and PS is nonexpansive.LetC and Q be nonempty closed convex subsets of real Hilbert spaces H1 and H2, respectively, and let A:H1→H2 be a bounded linear operator. The split feasibility problem (SFP) is to find a point x satisfying the property
(3)x∈C,Ax∈Q.
The SFP was first introduced by Censor and Elfving [1], which attracts many authors’ attention due to its application in signal processing [1]. Various algorithms have been invented to solve it (see [2–7]).Recently, Moudafi [8] proposes a new split equality problem (SEP): let H1, H2, and H3 be real Hilbert spaces, let C⊆H1, Q⊆H2 be two nonempty closed convex sets, and let A:H1→H3, B:H2→H3 be two bounded linear operators. Find x∈C, y∈Q satisfying
(4)Ax=By.
When B=I, SEP reduces to the well-known SFP. In the paper [8], Moudafi gives the alternating CQ-algorithm and relaxed alternating CQ-algorithm iterative algorithm to solve the split equality problem. We must point out that the above algorithms converge weakly to a solution of the SEP.We useΓ={(x,y)∈H1×H2,Ax=By,x∈C,y∈Q} to denote the solution set of SEP, and letting S=C×Q in H=H1×H2 define G:H→H3 by G=[A,-B], then G*G:H→H has the matrix form
(5)G*G=[A*A-A*B-B*AB*B].
The original problem can now be reformulated as finding w=(x,y)∈S with Gw=0 or, more generally, minimizing the function ∥Gw∥ over w∈S. Therefore solving SEP (4) is equivalent to solving the following minimization problem:
(6)minw∈Sf(w)=12∥Gw∥2.
In the paper [9], we use the well-known Tychonov regularization to get some algorithms converge strongly to the minimum-norm solution of the SEP. Moreover, in the paper [9], we obtain that, for 0<γ<2/∥G∥2, Fix(PS(I-γG*G))=Fix(PS)∩Fix(I-γG*G)=Γ. That is to say, the set of the solution of SEP is equal to the fixed points set of the nonexpansive mapping PS(I-γG*G).Recall that the variational inequality problem is to findp∈H such that 〈Mp,x-p〉≥0, for all x∈S, where S is closed, convex, and nonempty, and M is a mapping on H. The set of solutions of the variational inequality is denoted by VI(S,M). In the paper [9], we also get that w∈Fix(PS(I-γG*G)) if and only if w is a solution of the variational inequality 〈G*Gw,v-w〉≥0, for all v∈S.In this paper, we consider a contractionf on H=H1×H2 with coefficient 0<α<1, a strongly positive linear bounded operator T with coefficient γ->0, and M is a β-inverse strongly monotone mapping. Let 0<γ<γ-/α, S=C×Q. We prove that the sequence {wn}={(xn,yn)}⊆H generated by the iterative method wn+1=PS[αnγf(wn)+(I-αnT)PS(I-γnG*G)PS(wn-λnMwn)] converges strongly to w~ which solves the SEP and the following variational inequality: 〈(A-λf)w~,w-w~〉≥0 and 〈Mw~,w-w~〉≥0 for all w∈S under some mild conditions for αn, γn, and λn.We now collect some elementary facts which will be used in the proofs of our main results.Lemma 1.
AssumingT is a positive operator, then T is 1/∥T∥-inverse strongly monotone.Proof.
SinceT is a positive operator on H, by the functional calculus of positive operator (for more details see [10, Chapter 4]), there exists positive operator T1 such that T=T1T1 and ∥T1∥=∥T∥1/2.
Hence, for anyx,y∈H, ∥Tx-Ty∥2=∥T1T1(x-y)∥2≤∥T1∥2∥T1(x-y)∥2=∥T∥〈T1(x-y),T1(x-y)〉=∥T∥〈x-y,T1T1(x-y)〉=∥T∥〈x-y,Tx-Ty〉. It follows that 〈x-y,Tx-Ty〉≥(1/∥T∥)∥Tx-Ty∥2. That is to say, T is 1/∥T∥-inverse strongly monotone.Lemma 2 (see [11, 12]).
LetX be a Banach space, C a closed convex subset of X, and T:C→C a nonexpansive mapping with Fix(T)≠∅. If {xn} is a sequence in C weakly converging to x and if {(I-T)xn} converges strongly to y, then (I-T)x=y.Lemma 3 (see [13]).
Let{sn} be a sequence of nonnegative real numbers, {αn} a sequence of real numbers in [0,1] with ∑n=1∞‍αn=∞, {un} a sequence of nonnegative real numbers with ∑n=1∞‍un<∞, and {tn} a sequence of real numbers with limsupntn≤0. Suppose that
(7)sn+1=(1-αn)sn+αntn+un,∀n∈ℕ.
Then limn→∞sn=0.Lemma 4 (see [14]).
AssumingT is a strongly positive linear bounded operator on a Hilbert space H with coefficient γ->0, then ∥I-ρT∥≤1-ργ- for 0<ρ≤∥T∥-1.Lemma 5 (see [13]).
LetS be a nonempty and closed subset of a Banach space X, and let {Rn} be a family of mappings of S into itself which satisfies the condition: for each bounded subset E of S,
(8)∑n=1∞‍sup{∥Rn+1w-Rnw∥,w∈E}<∞.
Then, for each w∈S, {Rnw} converges strongly to a point in S. Moreover, R is defined by
(9)Rw=limn→∞Rnw,w∈E.
Then, for each bounded subset E of S,
(10)limsupn→∞{∥Rw-Rnw∥,w∈E}=0,
and Fix(R)=∩Fix(Rn), if ∩Fix(Rn)≠∅.


## 2. Main Results

Theorem 6.
LetΓ denote the solutions set of the SEP, S=C×Q, f a contraction on H=H1×H2 with coefficient 0<α<1, a strongly positive linear bounded operator T with coefficient γ->0, and M a β-inverse strongly monotone mapping. Assume that 0<γ<γ-/α, Γ∩VI(S,M)≠∅, and {wn}={(xn,yn)}⊆H is the sequence generated by the following algorithm: w0∈H,
(11)wn+1=PS[αnγf(wn)+(I-αnT)PS(I-γnG*G)PS(wn-λnMwn)]
for all n=0,1,2,…, where αn∈(0,1),λn∈(0,2β), and γn∈(0,2/∥G∥2). If αn, λn, and γn are chosen such that λn∈[a,b] for some a,b with 0<a<b<2β and(i)
limnαn=0,(ii)
∑n=0∞αn=∞,(iii)
∑n=0∞|αn+1-αn|<∞,(v)
∑n=0∞|λn+1-λn|<∞,(iv)
∑n=0∞|γn+1-γn|<∞,then the sequence {wn} converges strongly to w~ which solves the SEP and the following variational inequality: 〈(T-λf)w~,w-w~〉≥0 and 〈Mw~,w-w~〉≥0 for all w∈S.Proof.
First, we show that the sequence{wn} is bounded. Consider the mapping I-λnM. Since M is a β-inverse strongly monotone mapping, we have that, for all w,v∈S,
(12)∥(I-λnM)w-(I-λnM)v∥2=∥(w-v)-λn(Mw-Mv)∥2=∥w-v∥2-2λn〈w-v,Mw-Mv〉+λn2∥Mw-Mv∥2≤∥w-v∥2+λn(λn-2β)∥Mw-Mv∥2.
Since 0<λn<2β, we deduce that ∥(I-λnM)w-(I-λnM)v∥2≤∥w-v∥2. That is to say, I-λnM is nonexpansive.
Lettingvn=PS(wn-λnMwn), Rn=PS(I-γnG*G), choosing u∈Γ∩VI(S,M), then u=PS(I-γG*G)u and u=PS(I-λnM)u. PS is nonexpansive, which implies that
(13)∥vn-u∥=∥PS(wn-λnMwn)-PS(u-λnMu)∥≤∥(wn-λnMwn)-(u-λnMu)∥=∥(I-λnM)wn-(I-λnM)u∥≤∥wn-u∥,∥wn+1-u∥=∥PS[αnγf(wn)+(I-αnT)Rnvn]-u∥≤∥αnγf(wn)+(I-αnT)Rnvn-u∥=∥αn(γf(wn)-Tu)+(I-αnT)(Rnvn-u)∥.
Since T is strongly positive linear bounded operator, Rn is nonexpansive, and f is contraction, we have
(14)∥wn+1-u∥≤αn∥γf(wn)-Tu∥+(1-αnγ-)∥Rnvn-u∥≤αn∥γf(wn)-γf(u)∥+αn∥γf(u)-Tu∥+(1-αnγ-)∥Rnvn-u∥≤αγαn∥wn-u∥+αn∥γf(u)-Tu∥+(1-αnγ-)∥Rnvn-u∥=αγαn∥wn-u∥+αn∥γf(u)-Tu∥+(1-αnγ-)∥Rnvn-Rnu∥≤αγαn∥wn-u∥+αn∥γf(u)-Tu∥+(1-αnγ-)∥vn-u∥≤αγαn∥wn-u∥+αn∥γf(u)-Tu∥+(1-αnγ-)∥wn-u∥=(1-α(γ--γαn))∥wn-u∥+αn∥γf(u)-Tu∥≤max{∥wn-u∥,1γ--γαn∥γf(u)-Tu∥}.
It follows from induction that
(15)∥wn-u∥≤max{∥w0-u∥,1γ--γαn∥γf(u)-Tu∥},hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhn>0.
Hence, {wn} is bounded, so are {vn}, {Rnvn}, {Mwn}, and {f(wn)}.
Next, we show that∥wn+1-wn∥→0 and ∥vn-Rnvn∥→0. In fact, in view that PS and I-λnM are nonexpansive, we can get that
(16)∥vn+1-vn∥=∥PS(wn+1-λn+1Mwn+1)-PS(wn-λnMwn)∥≤∥(wn+1-λn+1Mwn+1)-(wn-λnMwn)∥≤∥(wn+1-λn+1Mwn+1)-(wn-λn+1Mwn)∥+|λn+1-λn|∥Mwn∥≤∥wn+1-wn∥+|λn+1-λn|∥Mwn∥.
Hence,
(17)∥wn+1-wn∥=∥PS[αnγf(wn)+(I-αnT)Rnvn]iii-PS[αn-1γf(wn-1)+(I-αn-1T)Rn-1vn-1]∥≤∥[αnγf(wn)+(I-αnT)Rnvn]hhh-[αn-1γf(wn-1)+(I-αn-1T)Rn-1vn-1]∥=∥αnγ(f(wn)-f(wn-1))+γ(αn-αn-1)hhh×f(wn-1)+(I-αnT)(Rnvn-Rn-1vn-1)hhh+(αn-αn-1)TRn-1vn-1∥≤αnαγ∥wn-wn-1∥+γ|αn-αn-1|∥f(wn-1)∥+(1-αnγ-)∥Rnvn-Rn-1vn-1∥+|αn-αn-1|∥TRn-1vn-1∥≤αnαγ∥wn-wn-1∥+γ|αn-αn-1|∥f(wn-1)∥+(1-αnγ-)∥Rnvn-Rnvn-1∥+(1-αnγ-)∥Rnvn-1-Rn-1vn-1∥+|αn-αn-1|∥TRn-1vn-1∥≤αnαγ∥wn-wn-1∥+γ|αn-αn-1|∥f(wn-1)∥+(1-αnγ-)∥vn-vn-1∥+(1-αnγ-)×∥Rnvn-1-Rn-1vn-1∥+|αn-αn-1|∥TRn-1vn-1∥≤αnαγ∥wn-wn-1∥+γ|αn-αn-1|∥f(wn-1)∥+(1-αnγ-)∥wn-wn-1∥+(1-αnγ-)|λn-λn-1|×∥Mwn-1∥+(1-αnγ-)∥Rnvn-1-Rn-1vn-1∥+|αn-αn-1|∥TRn-1vn-1∥=(1-αn(γ--αγ))∥wn-wn-1∥+γ|αn-αn-1|∥f(wn-1)∥+(1-αnγ-)×|λn-λn-1|∥Mwn-1∥+(1-αnγ-)×∥Rnvn-1-Rn-1vn-1∥+|αn-αn-1|∥TRn-1vn-1∥.
Based on
(18)∥Rnvn-1-Rn-1vn-1∥=∥PS(I-γnG*G)vn-1-PS(I-γn-1G*G)vn-1∥≤∥(I-γnG*G)vn-1-(I-γn-1G*G)vn-1∥≤|γn-γn-1|∥G∥2∥vn-1∥,
we deduce that
(19)∥wn+1-wn∥≤(1-αn(γ--αγ))∥wn-wn-1∥+γ|αn-αn-1|∥f(wn-1)∥+(1-αnγ-)|λn-λn-1|∥Mwn-1∥+(1-αnγ-)|γn-γn-1|∥G∥2∥vn-1∥+|αn-αn-1|∥TRn-1vn-1∥.
By Lemma 3, we have
(20)∥wn+1-wn∥⟶0.
On the other hand, since u=PS(u-λnMu) and u=Rnu we have
(21)∥wn+1-u∥2=∥PS[αnγf(wn)+(I-αnT)Rnvn]-PS(u)∥2≤∥αnγf(wn)+(I-αnT)Rnvn-u∥2=∥αn(γf(wn)-Tu)+(I-αnT)(Rnvn-u)∥2≤(∥αn(γf(wn)-Tu)∥hh+∥(I-αnT)(Rnvn-u)∥)2≤αn∥γf(wn)-Tu∥2+∥(I-αnγ-)Rnvn-u∥2+2αn(I-αnγ-)∥γf(wn)-Tu∥∥Rnvn-u∥≤αn∥γf(wn)-Tu∥2+(I-αnγ-)∥vn-u∥2+2αn(I-αnγ-)∥γf(wn)-Tu∥∥vn-u∥≤αn∥γf(wn)-Tu∥2+(I-αnγ-)×∥(I-λnM)wn-(I-λnM)u∥2+2αn(I-αnγ-)∥γf(wn)-Tu∥∥vn-u∥≤αn∥γf(wn)-Tu∥2+(I-αnγ-)×(∥wn-u∥2-2λn〈wn-u,Mwn-Mu〉i+λn2∥Mwn-Mu∥2)+2αn(I-αnγ-)×∥γf(wn)-Tu∥∥vn-u∥≤αn∥γf(wn)-Tu∥2+(I-αnγ-)(∥wn-u∥2-2λnβ∥Mwn-Mu∥2+λn2∥Mwn-Mu∥)+2αn(I-αnγ-)∥γf(wn)-Tu∥∥vn-u∥≤αn∥γf(wn)-Tu∥2+(I-αnγ-)×(∥wn-u∥2+λn(λn-2β)∥Mwn-Mu∥2)+2αn(I-αnγ-)∥γf(wn)-Tu∥∥vn-u∥≤αn∥γf(wn)-Tu∥2+∥wn-u∥2h+(I-αnγ-)b(b-2β)∥Mwn-Mu∥2h+2αn(I-αnγ-)∥γf(wn)-Tu∥∥vn-u∥.
Therefore, we have
(22)(I-αnγ-)b(2β-b)∥Mwn-Mu∥2≤αn∥γf(wn)-Tu∥2+∥wn-u∥2-∥wn+1-u∥2+2αn(I-αnγ-)∥γf(wn)-Tu∥∥vn-u∥=αn∥γf(wn)-Tu∥2+2αn(I-αnγ-)×∥γf(wn)-Tu∥∥vn-u∥+(∥wn-u∥-∥wn+1-u∥)×(∥wn-u∥-∥wn+1-u∥)≤αn∥γf(wn)-Tu∥2+2αn(I-αnγ-)∥γf(wn)-Tu∥∥vn-u∥+∥wn-wn+1∥(∥wn-u∥-∥wn+1-u∥).
Since αn→0, ∥wn-wn+1∥→0, we can get that
(23)∥Mwn-Mu∥⟶0.
Furthermore, by the property of the projection PS, we have
(24)∥vn-u∥=∥PS(I-λnM)wn-PS(I-λnM)u∥≤〈(wn-λnMwn)-(u-λnMu),vn-u〉=12(∥(wn-λnMwn)-(u-λnMu)∥2+∥vn-u∥2-∥(wn-λnMwn)-(u-λnMu)-(vn-u)∥2)≤12(∥wn-u∥2+∥vn-u∥2-∥(wn-vn)-λn(Mwn-Mu)∥2)≤∥wn-u∥2-∥wn-vn∥2+2λn〈wn-vn,Mwn-Mu〉-λn2∥Mwn-Mu∥2.
Then, we have
(25)∥wn+1-u∥2=∥PS[αnγf(wn)+(I-αnT)Rnvn]-PS(u)∥2≤αn∥γf(wn)-Tu∥2+(1-αnγ-)∥vn-u∥2+2αn(I-αnγ-)∥γf(wn)-Tu∥∥vn-u∥≤αn∥γf(wn)-Tu∥2+(1-αnγ-)∥wn-u∥2-(1-αnγ-)∥wn-vn∥2+2(1-αnγ-)×λn〈wn-vn,Mwn-Mu〉-(1-αnγ-)×λn2∥Mwn-Mu∥2+2αn(I-αnγ-)×∥γf(wn)-Tu∥∥vn-u∥,
which yields that
(26)(1-αnγ-)∥wn-vn∥2≤αn∥γf(wn)-Tu∥2+(∥wn-u∥+∥wn+1-u∥)∥wn-wn+1∥+2(1-αnγ-)λn〈wn-vn,Mwn-Mu〉-(1-αnγ-)λn2∥Mwn-Mu∥2+2αn(I-αnγ-)∥γf(wn)-Tu∥∥vn-u∥.
Since αn→0, ∥wn-wn+1∥→0, and ∥Mwn-Mu∥→0, we can get that
(27)∥wn-vn∥⟶0.
Next, since αn→0 we can get that
(28)∥wn+1-Rnvn∥=∥PS[αnγf(wn)+(I-αnT)Rnvn]-PS(Rnvn)∥≤∥αnγf(wn)+(I-αnT)Rnvn-Rnvn∥=αn∥γf(wn)-TRnvn∥⟶0.
Hence,
(29)∥wn-Rnvn∥≤∥wn-wn+1∥+∥wn+1-Rnvn∥⟶0,∥wn-Rnwn∥≤∥wn-Rnvn∥+∥Rnvn-Rnwn∥≤∥wn-Rnvn∥+∥vn-wn∥⟶0,∥vn-Rnvn∥≤∥vn-wn∥+∥wn-Rnvn∥⟶0.
Let the mappingR be defined by
(30)Rw=limn→∞Rnw,∀w∈H.
By Lemma 5 we can get that Fix(R)=∩Fix(Rn) and
(31)∥Rwn-wn∥≤∥Rwn-Rvn∥+∥Rvn-Rnvn∥+∥Rnvn-wn∥≤∥wn-vn∥+sup{∥Rnz-Rz∥,z∈{vn}}+∥Rnvn-wn∥⟶0.
Note that it is easy to check that PΓ∩VI(S,M)(γf+I-T) is a contraction; by Banach contraction mapping principle, PΓ∩VI(S,M)(γf+I-T) has a unique fixed point p∈H. By the property of the projection PΓ∩VI(S,M), we can get that
(32)〈(γf-T)p,w-p〉≤0,∀w∈Γ∩VI(S,M).
Choose a subsequence vnk of vn such that
(33)limsupn→∞〈(γf-T)p,Rnvn-p〉=limn→∞〈(γf-T)p,Rnkvnk-p〉.
Since {vn} is bounded, there exists a subsequence {vnki} of {vnk} which converges weakly q. Without loss of generality, we may assume that vnk converges weakly q. Note that, by ∥vn-Rnvn∥→0, we can get Rnkvnk converges weakly q. Furthermore, ∥wn-Rwn∥→0,∥vn-wn∥→0; by Lemmas 2 and 5, we obtain that q∈Fix(R)=∩Fix(Rn).
Next we will prove thatq∈VI(S,M).
Choosev∈S, and let wv∈{w∈H,〈v-u,w〉≥0,∀u∈S}. Since vn∈S, we can get 〈v-vn,wv〉≥0. On the other hand, by the property of the projection PS and vn=PS(wn-λnMwn), we have
(34)〈v-vn,vn-(wn-λnMwn)〉≥0,〈v-vn,vn-wnλn+Mwn〉≥0.
It follows that
(35)〈v-vnk,Mv+wv〉≥〈v-vnk,Mv〉≥〈v-vnk,Mv〉-〈v-vnk,vnk-wnkλnk+Mwnk〉=〈v-vnk,Mv-Mwnk-vnk-wnkλnk〉=〈v-vnk,Mv-Mvnk〉+〈v-vnk,Mvnk-Mwnk〉-〈v-vnk,vnk-wnkλnk〉≥〈v-vnk,Mvnk-Mwnk〉-〈v-vnk,vnk-wnkλnk〉.
Since ∥Mvn-Mwn∥≤∥vn-wn∥→0 and vnk converges weakly q, we can get that 〈v-q,Mv+wv〉≥0.
Note thatM is a β-inverse strongly monotone mapping, and if wv∈{w∈H,〈v-u,w〉≥0,∀u∈S}, then αwv∈{w∈H,〈v-u,w〉≥0,∀u∈S} for all α>0. Hence, we can choose wv∈{w∈H,〈v-u,w〉≥0,∀u∈S} such that 〈v-q,wv〉≥〈v-q,Mv-Mq〉. By
(36)〈v-q,Mv+wv〉≥0,
we can get that
(37)〈v-q,Mv-Mq〉+〈v-q,Mq-wv〉≥0.
Hence
(38)〈v-q,Mq〉≥〈v-q,wv〉-〈v-q,Mv-Mq〉≥0.
It follows that q∈VI(S,M). We obtain that q∈PΓ∩VI(S,M) and 〈(γf-T)p,q-p〉≤0, for all w∈Γ∩VI(S,M).
Finally, we prove thatlimn→∞∥wn-q∥=0. Since ∥vn-u∥≤∥wn-u∥, we have
(39)∥wn+1-p∥2=∥PS[αnγf(wn)+(I-αnT)Rnvn]-PS(p)∥2≤∥αn(γf(wn)-Tp)+(I-αnT)(Rnvn-p)∥2≤∥(I-αnT)(Rnvn-p)∥2+αn2∥γf(wn)-Tq∥2+2αn〈(I-αnT)(Rnvn-p),γf(wn)-Tq〉≤(1-αnγ-)2∥vn-p∥2+αn2∥γf(wn)-Tq∥2+2αn〈Rnvn-p,γf(wn)-Tp〉-2αn2〈T(Rnvn-p),γf(wn)-Tp〉≤(1-αnγ-)2∥vn-p∥2+αn2∥γf(wn)-Tq∥2+2αn〈Rnvn-p,γf(wn)-γf(p)〉+2αn〈Rnvn-p,γf(p)-Tp〉-2αn2〈T(Rnvn-p),γf(wn)-Tp〉≤(1-αnγ-)2∥vn-p∥2+αn2∥γf(wn)-Tq∥2+2αn∥Rnvn-p∥∥γf(wn)-γf(p)∥+2αn〈Rnvn-p,γf(p)-Tp〉-2αn2〈T(Rnvn-p),γf(wn)-Tp〉≤(1-αnγ-)2∥vn-p∥2+αn2∥γf(wn)-Tq∥2+2γααn∥vn-p∥∥wn-p∥+2αn〈Rnvn-p,γf(p)-Tp〉-2αn2〈T(Rnvn-p),γf(wn)-Tp〉≤(1-αnγ-)2∥vn-p∥2+αn2∥γf(wn)-Tq∥2+2γααn∥wn-p∥2+2αn〈Rnvn-p,γf(p)-Tp〉-2αn2〈T(Rnvn-p),γf(wn)-Tp〉≤[(1-αnγ-)2+2γααn]∥wn-p∥2+αn[2〈Rnvn-p,γf(p)-Tp〉hhhhhi+αn∥γf(wn)-Tq∥2hhhhhi+2αn∥T(Rnvn-p)∥∥γf(wn)-Tp∥]=[1-2(γ--γα)αn]∥wn-p∥2+αn[αnγ-22〈Rnvn-p,γf(p)-Tp〉+αn∥γf(wn)-Tq∥2+2αn∥T(Rnvn-p)∥×∥γf(wn)-Tp∥+αnγ-2∥wn-p∥2].
Since {wn}, {f(wn)}, {Rnvn}, and limsupn→∞〈(γf-T)p,Rnvn-p〉≤0, using Lemma 3, we can conclude that
(40)limn→∞∥wn-q∥=0,
which completes the proof.By Lemma1, we know that G*G is 1/∥G∥2-inverse strongly monotone. Hence, letting M=G*G, we can get the following Corollary.Corollary 7.
LetΓ denote the solutions set of the SEP, S=C×Q, f a contraction on H=H1×H2 with coefficient 0<α<1, a strongly positive linear bounded operator T with coefficient γ->0. Assume that 0<γ<γ-/α, Γ≠∅, and {wn}={(xn,yn)}⊆H is the sequence generated by the following algorithm: w0∈H,
(41)wn+1=PS[αnγf(wn)+(I-αnT)PSiiihhh×(I-γnG*G)PS(wn-λnG*Gwn)]
for all n=0,1,2,…, where αn∈(0,1), λn,γn∈(0,2/∥G∥2). If αn, λn, and γn are chosen so that λn∈[a,b] for some a,b with 0<a<b<2/∥G∥2 and(i)
limnαn=0,(ii)
∑n=0∞αn=∞,(iii)
∑n=0∞|αn+1-αn|<∞,(v)
∑n=0∞|λn+1-λn|<∞,(iv)
∑n=0∞|γn+1-γn|<∞,then the sequence {wn} converges strongly to w~ which solves the SEP and the following variational inequality: 〈(T-λf)w~,w-w~〉≥0 for all w∈S.Furthermore, ifM=G*G and γn=0, changed slightly for Theorem 6, we get the following result.Corollary 8.
LetΓ denote the solutions set of the SEP, S=C×Q, f a contraction on H=H1×H2 with coefficient 0<α<1, a strongly positive linear bounded operator T with coefficient γ->0. Assume that 0<γ<γ-/α, Γ≠∅, and {wn}={(xn,yn)}⊆H is the sequence generated by the following algorithm: w0∈H,
(42)wn+1=αnγf(wn)+(I-αnT)PS(wn-λnG*Gwn)
for all n=0,1,2,…, where αn∈(0,1), λn∈(0,2/∥G∥2). If αn, λn are chosen so that λn∈[a,b] for some a,b with 0<a<b<2/∥G∥2 and(i)
limnαn=0,(ii)
∑n=0∞αn=∞,(iii)
∑n=0∞|αn+1-αn|<∞,(v)
∑n=0∞|λn+1-λn|<∞,then the sequence {wn} converges strongly to w~ which solves the SEP and the following variational inequality: 〈(T-λf)w~,w-w~〉≥0 for all w∈S.



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*Source: 151906-2013-10-23.xml*