# Approximation of Common Fixed Points of Nonexpansive Semigroups in Hilbert Spaces

Authors: Dan Zhang; Xiaolong Qin; Feng Gu
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

LetH be a real Hilbert space. Consider on H a nonexpansive semigroup S={T(s):0≤s<∞} with a common fixed point, a contraction f with the coefficient 0<α<1, and a strongly positive linear bounded self-adjoint operator A with the coefficient γ¯>  0. Let 0<γ<γ¯/α. It is proved that the sequence {xn} generated by the iterative method x0∈H,xn+1=αnγf(xn)+βnxn+((1-βn)I-αnA)(1/sn)∫0snT(s)xnds,n≥0 converges strongly to a common fixed point x*∈F(S), where F(S) denotes the common fixed point of the nonexpansive semigroup. The point x* solves the variational inequality 〈(γf-A)x*,x-x*〉≤0 for all x∈F(S).


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## Body

## 1. Introduction and Preliminaries

LetH be a real Hilbert space and T be a nonlinear mapping with the domain D(T). A point x∈D(T) is a fixed point of T provided Tx=x. Denote by F(T) the set of fixed points of T; that is, F(T)={x∈D(T):Tx=x}. Recall that T is said to be nonexpansive if(1.1)‖Tx-Ty‖≤‖x-y‖,∀x,y∈D(A).Recall that a familyS={T(s)∣s≥0} of mappings from H into itself is called a one-parameter nonexpansive semigroup if it satisfies the following conditions:(i)
T(0)x=x,
forallx∈H;(ii)
T(s+t)x=T(s)T(t)x, foralls,t≥0 and forallx∈H;(iii)
∥T(s)x-T(s)y∥≤∥x-y∥, foralls≥0 and forallx,y∈H;(iv)
for allx∈C, s↦T(s)x is continuous.We denote byF(S) the set of common fixed points of S, that is, F(S)=⋂0≤s<∞F(T(s)). It is known that F(S) is closed and convex; see [1]. Let C be a nonempty closed and convex subset of H. One classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping; see [2, 3]. More precisely, take t∈(0,1) and define a contraction Tt:C→C by
(1.2)Ttx=tu+(1-t)Tx,x∈C,
where u∈C is a fixed point. Banach’s contraction mapping principle guarantees that Tt has a unique fixed point xt in C. If T enjoys a nonempty fixed point set, Browder [2] proved the following well-known strong convergence theorem.Theorem B.
LetC be a bounded closed convex subset of a Hilbert space H and let  T be a nonexpansive mapping on C. Fix  u∈C and define zt∈Caszt=tu+(1-t)Tzt for t∈(0,1).  Then as t→0, {zt} converges strongly to a element of F(T) nearest to u.As motivated by Theorem B, Halpern [4] considered the following explicit iteration:
(1.3)x0∈C,xn+1=αnu+(1-αn)Txn,n≥0,
and proved the following theorem.Theorem H.
LetC be a bounded closed convex subset of a Hilbert space  H and let  T be a nonexpansive mapping on C. Define a real sequence  {αn} in  [0,1] by  αn=n-θ,0<θ<1. Define a sequence  {xn} by (1.3). Then  {xn} converges strongly to the element of  F(T) nearest to  u.In 1977, Lions [5] improved the result of Halpern [4], still in Hilbert spaces, by proving the strong convergence of {xn} to a fixed point of T where the real sequence {αn} satisfies the following conditions:(C1)
limn→∞αn=0;(C2)
∑n=1∞αn=∞;(C3)
limn→∞(αn+1-αn)/αn+12=0.It was observed that both Halpern’s and Lions’s conditions on the real sequence{αn} excluded the canonical choice αn=1/(n+1). This was overcome in 1992 by Wittmann [6], who proved, still in Hilbert spaces, the strong convergence of {xn} to a fixed point of T if {αn} satisfies the following conditions:(C1)
limn→∞αn=0;(C2)
∑n=1∞αn=∞;(C3)
∑n=1∞|αn+1-αn|<∞.Recall that a mappingf:H→H is an α-contraction if there exists a constant α∈(0,1) such that(1.4)‖f(x)-f(y)‖≤α‖x-y‖,∀x,y∈H.Recall that an operatorA is strongly positive on H if there exists a constant γ¯>0 such that(1.5)〈Ax,x〉≥γ¯‖x‖2,∀x∈H.Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [7–13] and the references therein. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping T on a real Hilbert space H:
(1.6)min⁡x∈F(T)12〈Ax,x〉-〈x,b〉,
where A is a linear bounded operator on H and b is a given point in H. In [11], it is proved that the sequence {xn} defined by the iterative method below, with the initial guess x0∈H chosen arbitrarily,
(1.7)xn+1=(I-αnA)Txn+αnb,n≥0,
strongly converges to the unique solution of the minimization problem (1.6) provided that the sequence {αn} satisfies certain conditions.Recently, Marino and Xu [9] studied the following continuous scheme:
(1.8)xt=tγf(xt)+(I-tA)Txt,
where f is an α-contraction on a real Hilbert space H, A is a bounded linear strongly positive operator and γ>0 is a constant. They showed that {xt} strongly converges to a fixed point x¯ of T. Also in [9], they introduced a general explicit iterative scheme by the viscosity approximation method:(1.9)xn∈H,xn+1=αnγf(xn)+(I-αnA)Txn,n≥0
and proved that the sequence {xn} generated by (1.9) converges strongly to a unique solution of the variational inequality(1.10)〈(A-γf)x*,x-x*〉≥0,∀x∈F(T),which is the optimality condition for the minimization problem(1.11)min⁡x∈F(T)12〈Ax,x〉-h(x),whereh is a potential function for γf (i.e., h′(x)=γf(x) for x∈H).In this paper, motivated by Li et al. [8], Marino and Xu [9], Plubtieng and Punpaeng [14], Shioji and Takahashi [15], and Shimizu and Takahashi [16], we consider the mapping Tt defined as follows:
(1.12)Ttx=tγf(x)+(I-tA)1λt∫0λtT(s)xds,
where γ>0 is a constant, f is an α-contraction, A is a bounded linear strongly positive self-adjoint operator and {λt} is a positive real divergent net. If γα<γ¯ for each 0<t<∥A∥-1, one can see that Tt is a (1-t(γ¯-γα))-contraction. So, by Banach’s contraction mapping principle, there exists an unique solution xt of the fixed point equation(1.13)xt=tγf(xt)+(I-tA)1λt∫0λtT(s)xtds.
We show that the sequence {xt} generated by above continuous scheme strongly converges to a common fixed point x*∈F(S), which is the unique point in F(S) solving the variational inequality 〈(γf-A)x*,x-x*〉≤0 for all x∈F(S). Furthermore, we also study the following explicit iterative scheme:
(1.14)x0∈H,xn+1=αnγf(xn)+βnxn+((1-βn)I-αnA)1sn∫0snT(s)xnds,n≥0.
We prove that the sequence {xn} generated by (1.14) converges strongly to the same x*.The results presented in this paper improve and extend the corresponding results announced by Marino and Xu [9], Plubtieng and Punpaeng [14], Shioji and Takahashi [15], and Shimizu and Takahashi [16].In order to prove our main result, we need the following lemmas.Lemma 1.1 (see [16]).
LetD be a nonempty bounded closed convex subset of a Hilbert space  H and let  S={T(t):0≤t<∞} be a nonexpansive semigroup on  D. Then, for any  0≤h<∞,
(1.15)lim⁡t→∞sup⁡x∈D‖1t∫0tT(s)xds-T(h)1t∫0tT(s)xds‖=0.Lemma 1.2 (see [17]).
LetH be a Hilbert space,  C a closed convex subset of  H, and  T:C→C a nonexpansive mapping with  F(T)≠∅. Then  I-T is demiclosed, that is, if  {xn} is a sequence in  C weakly converging to  x and if  {(I-T)xn} strongly converges to  y, then  (I-T)x=y.Lemma 1.3 (see [18]).
LetC be a nonempty closed convex subset of a real Hilbert space  H and let  PC be the metric projection from  H onto  C( i.e., for  x∈H,PCx is the only point in  C such that  ∥x-PCx∥=inf{∥x-z∥:z∈C}). Given  x∈H and  z∈C. Then  z=PCx if and only if there holds the relations
(1.16)〈x-z,y-z〉≤0,∀y∈C.Lemma 1.4.
LetH be a Hilbert space,  f a  α-contraction, and  A a strongly positive linear bounded self-adjoint operator with the coefficient  γ¯>0. Then, for  0<γ<γ¯/α,
(1.17)〈x-y,(A-γf)x-(A-γf)y〉≥(γ¯-γα)‖x-y‖2,x,y∈H.
That is, A-γf is strongly monotone with coefficient γ¯-αγ.Proof.
From the definition of strongly positive linear bounded operator, we have(1.18)〈x-y,A(x-y)〉≥γ¯‖x-y‖2.
On the other hand, it is easy to see
(1.19)〈x-y,γfx-γfy〉≤γα‖x-y‖2.
Therefore, we have
(1.20)〈x-y,(A-γf)x-(A-γf)y〉=〈x-y,A(x-y)〉-〈x-y,γfx-γfy〉≥(γ¯-γα)‖x-y‖2
for all x,y∈H. This completes the proof.Remark 1.5.
Takingγ=1 and A=I, the identity mapping, we have the following inequality:
(1.21)〈x-y,(I-f)x-(I-f)y〉≥(1-α)‖x-y‖2,x,y∈H.
Furthermore, if f is a nonexpansive mapping in Remark 1.5, we have
(1.22)〈x-y,(I-f)x-(I-f)y〉≥0,x,y∈H.Lemma 1.6 (see [9]).
AssumeA is a strongly positive linear bounded self-adjoint operator on a Hilbert space  H with coefficient  γ¯>0 and  0<ρ≤∥A∥-1. Then  ∥I-ρA∥≤1-ργ¯.Lemma 1.7 (see [12]).
Let{αn} be a sequence of nonnegative real numbers satisfying the following condition:
(1.23)αn+1≤(1-γn)αn+γnσn,∀n≥0,
where {γn} is a sequence in (0,1) and {σn} is a sequence of real numbers such that(i)
limn→∞γn=0 and ∑n=0∞γn=∞,(ii)
eitherlimsupn→∞σn≤0 or ∑n=0∞|γnσn|<∞. Then {αn}n=0∞ converges to zero.


## 2. Main Results

Lemma 2.1.
LetH a real Hilbert space and  S={T(s):0≤s<∞} a nonexpansive semigroup on  H such that  F(S)≠∅. Let  {λt}0<t<1 be a continuous net of positive real numbers such that  limt→0λt=∞. Let  f:H→H be an  α-contraction,  A a strongly positive linear bounded self-adjoint operator of  H into itself with coefficient  γ¯>0. Assume that  0<γ<γ¯/α. Let  {xt} be a sequence defined by (1.13). Then (i)
{xt} is bounded for all t∈(0,∥A∥-1);(ii)
limt→0∥T(τ)xt-xt∥=0 for all 0≤τ<∞;(iii)
xt defines a continuous curve from (0,∥A∥-1) into H.Proof.
(i) Takingp∈F(S), we have
(2.1)‖xt-p‖≤‖tγf(xt)+(I-tA)1λt∫0λtT(s)xtds-p‖≤t‖γf(xt)-Ap‖+(1-tγ¯)1λt∫0λt‖T(s)xt-p‖ds≤t‖γf(xt)-Ap‖+(1-tγ¯)‖xt-p‖≤tγ‖f(xt)-f(p)‖+t‖γf(p)-Ap‖+(1-tγ¯)‖xt-p‖≤[1-t(γ¯-γα)]‖xt-p‖+t‖γf(p)-Ap‖.
It follows that
(2.2)‖xt-p‖≤1γ¯-αγ‖γf(p)-Ap‖.
This implies that {xt} is not only bounded, but also that {xt} is contained in B(p,1/(γ¯-γα)∥γf(p)-Ap∥) of center p and radius 1/(γ¯-γα)∥γf(p)-Ap∥, for all fixed p∈F(S). Moreover for p∈F(S) and t∈(0,∥A∥-1),
(2.3)‖1λt∫0λtT(s)xtds-p‖=‖1λt∫0λt(T(s)xt-T(s)p)ds‖≤‖xt-p‖≤1γ¯-γα‖γf(p)-Ap‖.
(ii) Observe that(2.4)‖T(τ)xt-xt‖≤‖T(τ)xt-T(τ)(1λt∫0λtT(s)xtds)‖+‖T(τ)(1λt∫0λtT(s)xtds)-1λt∫0λtT(s)xtds‖+‖1λt∫0λtT(s)xtds-xt‖≤2‖xt-1λt∫0λtT(s)xtds‖+‖T(τ)(1λt∫0λtT(s)xtds)-1λt∫0λtT(s)xtds‖=2t‖γf(xt)-A1λt∫0λtT(s)xtds‖+‖T(τ)(1λt∫0λtT(s)xtds)-1λt∫0λtT(s)xtds‖.
Taking B(p,1/(γ¯-γα)∥γf(p)-Ap∥) as D in Lemma 1.1 and passing to limt→0 in (2.4), we can obtain (ii) immediately.
(iii) Takingt1,t2∈(0,∥A∥-1) and fixing p∈F(S), we see that
(2.5)‖xt1-xt2‖≤‖(t1-t2)γf(xt1)+t2γ(f(xt1)-f(xt2))-(t1-t2)A1λt1∫0λt1T(s)xt1ds+(I-t2A)(1λt1∫0λt1T(s)xt1ds-1λt2∫0λt2T(s)xt2ds)‖≤|t1-t2|γ‖f(xt1)‖+t2γα‖xt1-xt2‖+|t1-t2|‖A‖‖1λt1∫0λt1T(s)xt1ds‖+(1-t2γ¯)‖1λt1∫0λt1T(s)xt1ds-1λt2∫0λt1T(s)xt2ds-1λt2∫λt1λt2T(s)xt2ds‖≤|t1-t2|γ‖f(xt1)‖+t2γα‖xt1-xt2‖+|t1-t2|‖A‖‖1λt1∫0λt1T(s)xt1ds‖+(1-t2γ¯)(‖xt1-xt2‖+|1λt1-1λt2|‖∫0λt1T(s)xt2ds‖+1λt2‖∫λt1λt2T(s)xt2ds‖).
Thus applying (2.3), we arrive at
(2.6)‖xt1-xt2‖≤|t1-t2|γ‖f(xt1)‖+t2γα‖xt1-xt2‖+|t1-t2|‖A‖(1γ¯-γα‖γf(p)-Ap‖+‖p‖)+(1-t2γ¯)(‖xt1-xt2‖+2λt2|λt2-λt1|(1γ¯-γα‖γf(p)-Ap‖+‖p‖))≤|t1-t2|(γ‖f(xt1)‖+‖A‖(1γ¯-γα‖γf(p)-Ap‖+‖p‖))+(1-t2(γ¯-γα))‖xt1-xt2‖+2λt2|λt2-λt1|(1γ¯-γα‖γf(p)-Ap‖+‖p‖).
It follows that
(2.7)‖xt1-xt2‖≤M1|t1-t2|+M2|λt2-λt1|,
where
(2.8)M1=γ(γ¯-γα)‖f(xt1)‖+‖A‖‖γf(p)-Ap‖+(γ¯-γα)‖A‖‖p‖t2(γ¯-γα)2
and
(2.9)M2=2(‖γf(p)-Ap‖+(γ¯-γα)‖p‖)λt2t2(γ¯-γα)2.
This inequality, together with the continuity of the net {λt}, gives the continuity of the curve {xt}.Theorem 2.2.
LetH be a real Hilbert space  H and  S={T(s):0≤s<∞} a nonexpansive semigroup such that  F(S)≠∅. Let  {λt}0<t<1 be a net of positive real numbers such that  limt→0λt=∞. Let  f be an  α-contraction and let  A be a strongly positive linear bounded self-adjoint operator on  H with the coefficient  γ¯>0. Assume that  0<γ<γ¯/α. Then sequence  {xt} defined by (1.13) strongly converges as  t→0 to  x*∈F(S), which solves the following variational inequality:
(2.10)〈(γf-A)x*,p-x*〉≤0,∀p∈F(S).
Equivalently, one has
(2.11)PF(S)(I-A+γf)x*=x*.Proof.
The uniqueness of the solution of the variational inequality (2.10) is a consequence of the strong monotonicity of A-γf (Lemma 1.4) and it was proved in [9]. Next, we will use x*∈F(S) to denote the unique solution of (2.10). To prove that xt→x*(t→0), we write, for a given p∈F(S),
(2.12)xt-p=t(γf(xt)-Ap)+(I-tA)(1λt∫0λtT(s)xtds-p).
Using xt-p to make inner product, we obtain that
(2.13)‖xt-p‖2=〈(I-tA)(1λt∫0λtT(s)xtds-p),xt-p〉+t〈γf(xt)-Ap,xt-p〉≤(1-tγ¯)‖xt-p‖2+t〈γf(xt)-Ap,xt-p〉.
It follows that
(2.14)‖xt-p‖2≤1γ¯(γ〈f(xt)-f(p),xt-p〉+〈γf(p)-Ap,xt-p〉)≤γαγ¯‖xt-p‖2+1γ¯〈γf(p)-Ap,xt-p〉,
which yields that
(2.15)‖xt-p‖2≤1γ¯-αγ〈γf(p)-Ap,xt-p〉.
Since H is a Hilbert space and {xt} is bounded as t→0, we have that if {tn} is a sequence in (0,1) such that tn→0 and xtn⇀x¯. By (2.15), we see xtn→x¯. Moreover, by (ii) of Lemma 2.1 we have x¯∈F(S). We next prove that x¯ solves the variational inequality (2.10). From (1.13), we arrive at
(2.16)(A-γf)xt=-1t(I-tA)[xt-1λt∫0λtT(s)xtds].
For p∈F(S), it follows from (1.22) that
(2.17)〈(A-γf)xt,xt-p〉=-1t〈(I-tA)[xt-1λt∫0λtT(s)xtds],xt-p〉=-1t〈1λt∫0λt[(I-T(s))xt-(I-T(s))p]ds,xt-p〉+〈A1λt∫0λt(I-T(s))xtds,xt-p〉=-1tλt∫0λt〈(I-T(s))xt-(I-T(s))p,xt-p〉ds+〈A1λt∫0λt(I-T(s))xtds,xt-p〉≤〈A1λt∫0λt(I-T(s))xtds,xt-p〉=〈A(tγf(xt)-tA1λt∫0λtT(s)xtds),xt-p〉=t〈A(γf(xt)-A1λt∫0λtT(s)xtds),xt-p〉.
Passing to limt→0, since {xt} is a bounded sequence, we obtain
(2.18)〈(A-γf)x¯,x¯-p〉≤0,
that is, x¯ satisfies the variational inequality (2.10). By the uniqueness it follows x¯=x*. In a summary, we have shown that each cluster point of {xt} (as t→0) equals x*. Therefore, xt→x* as t→0. The variational inequality (2.10) can be rewritten as
(2.19)〈[(I-A+γf)x*]-x*,x*-p〉,p∈F(S).
This, by Lemma 1.3, is equivalent to
(2.20)PF(S)(I-A+γf)x*=x*.
This completes the proof.Remark 2.3.
Theorem2.2 which include the corresponding results of Shioji and Takahashi [15] as a special case is reduced to Theorem 3.1 of Plubtieng and Punpaeng [14] when A=I, the identity mapping and γ=1.Theorem 2.4.
LetH be a real Hilbert space  H and  S={T(s):0≤s<∞} a nonexpansive semigroup such that  F(S)≠∅. Let  {sn} be a positive real divergent sequence and let  {αn} and  {βn} be sequences in  (0,1) satisfying the following conditions  limn→∞αn=limn→∞βn=0 and  ∑n=0∞αn=∞. Let  f be an  α-contraction and let  A be a strongly positive linear bounded self-adjoint operator with the coefficient  γ¯>0. Assume that  0<γ<γ¯/α. Then sequence  {xn} defined by (1.14) strongly converges to  x*∈F(S), which solves the variational inequality (2.10).Proof.
We divide the proof into three parts.Step 1.
Show the sequence{xn} is bounded.
Noticing thatlimn→∞αn=limn→∞βn=0, we may assume, with no loss of generality, that αn/(1-βn)<∥A∥-1 for all n≥0. From Lemma 1.6, we know that ∥(1-βn)I-αnA∥≤(1-βn-αnγ¯). Picking p∈F(S), we have
(2.21)‖xn+1-p‖=‖αn(γf(xn)-Ap)+βn(xn-p)+((1-βn)I-αnA)(1sn∫0snT(s)xnds-p)‖≤αn‖γf(xn)-Ap‖+βn‖xn-p‖+(1-βn-αnγ¯)‖1sn∫0snT(s)xnds-p‖≤αnγ‖f(xn)-f(p)‖+αn‖γf(p)-Ap‖+βn‖xn-p‖+(1-βn-αnγ¯)‖xn-p‖≤[1-αn(γ¯-γα)]‖xn-p‖+αn‖γf(p)-Ap‖.
By simple inductions, we see that(2.22)‖xn-p‖≤max⁡{‖x0-p‖,‖Ap-γf(p)‖γ¯-γα},
which yields that the sequence {xn} is bounded.Step 2.
Show that(2.23)limsup⁡n→∞〈(γf-A)x*,yn-x*〉≤0,
where x* is obtained in Theorem 2.2 and yn=(1/sn)∫0snT(s)xnds.
Puttingz0=PF(S)x0, from (2.22) we see that the closed ball M of center z0 and radius max{∥z0-p∥,∥Az0-γf(z0)∥/(γ¯-γα)} is T(s)-invariant for each s∈[0,∞) and contain {xn}. Therefore, we assume, without loss of generality, S={T(s):0≤s<∞} is a nonexpansive semigroup on M. It follows from Lemma 1.1 that
(2.24)lim⁡n→∞‖yn-T(h)yn‖=0
for all 0≤h<∞. Taking a suitable subsequence {yni} of {yn}, we see that
(2.25)limsup⁡n→∞〈(γf-A)x*,yn-x*〉=lim⁡i→∞〈(γf-A)x*,yni-x*〉.
Since the sequence {yn} is also bounded, we may assume that yni⇀x¯. From the demiclosedness principle, we have x¯∈F(S). Therefore, we have
(2.26)limsup⁡n→∞〈(γf-A)x*,yn-x*〉=〈(γf-A)x*,x¯-x*〉≤0.
On the other hand, we have
(2.27)‖xn+1-yn‖≤αn‖γf(xn)-Axn‖+βn‖xn-yn‖.
From the assumption limn→∞αn=limn→∞βn=0, we see that
(2.28)lim⁡n→∞‖xn+1-yn‖=0,
which combines with (2.26) gives that
(2.29)limsup⁡n→∞〈(γf-A)x*,xn+1-x*〉≤0.Step 3.
Showxn→x* as n→∞.
Note that(2.30)‖xn+1-x*‖2=〈αn(γf(xn)-Ax*)+βn(xn-x*)+((1-βn)I-αnA)(yn-x*),xn+1-x*〉=αn〈γf(xn)-Ax*,xn+1-x*〉+βn〈xn-x*,xn+1-x*〉+〈((1-βn)I-αnA)(yn-x*),xn+1-x*〉≤αn(γ〈f(xn)-f(x*),xn+1-x*〉+〈γf(x*)-Ax*,xn+1-x*〉)+βn‖xn-x*‖‖xn+1-x*‖+‖(1-βn)I-αnA‖‖yn-x*‖‖xn+1-x*‖≤αnαγ‖xn-x*‖‖xn+1-x*‖+αn〈γf(x*)-Ax*,xn+1-x*〉+βn‖xn-x*‖‖xn+1-x*‖+(1-βn-αnγ¯)‖xn-x*‖‖xn+1-x*‖=[1-αn(γ¯-γα)]‖xn-x*‖‖xn+1-x*‖+αn〈γf(x*)-Ax*,xn+1-x*〉≤1-αn(γ¯-γα)2(‖xn-x*‖2+‖xn+1-x*‖2)+αn〈γf(x*)-Ax*,xn+1-x*〉.≤1-αn(γ¯-γα)2‖xn-x*‖2+12‖xn+1-x*‖2+αn〈γf(x*)-Ax*,xn+1-x*〉.
It follows that
(2.31)‖xn+1-x*‖2≤[1-αn(γ¯-γα)]‖xn-x*‖2+2αn〈γf(x*)-Ax*,xn+1-x*〉.By using Lemma 1.7, we can obtain the desired conclusion easily.Remark 2.5.
Ifγ=1 and A=I, the identity mapping, then Theorem 2.4 is reduced to Theorem 3.3 of Plubtieng and Punpaeng [14].
If the sequence{βn}≡0, then Theorem 2.4 is reduced to the following.Corollary 2.6.
LetH be a real Hilbert space  H and  S={T(s):0≤s<∞} a nonexpansive semigroup such that  F(S)≠∅. Let  {sn} be a positive real divergent sequence and let  {αn} be a sequence in  (0,1) satisfying the following conditions  limn→∞αn=0 and  ∑n=0∞αn=∞. Let  f be a  α-contraction and let  A be a strongly positive linear bounded self-adjoint operator with the coefficient  γ¯>0. Assume that  0<γ<γ¯/α. Let  {xn} be a sequence generated by the following manner:
(2.32)x0∈H,xn+1=αnγf(xn)+(I-αnA)1sn∫0snT(s)xnds,n≥0.
Then the sequence {xn} defined by above iterative algorithm converges strongly to x*∈F(S), which solves the variational inequality (2.10).Remark 2.7.
Corollary2.6 includes Theorem 2 of Shioji and Takahashi [15] as a special case.Remark 2.8.
Theorem2.2 and Corollary 2.6 improve Theorem 3.2 and Theorem 3.4 of Marino and Xu [9] from a single nonexpansive mapping to a nonexpansive semigroup, respectively.



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*Source: 417234-2012-02-20.xml*