# Strong Convergence Theorems for a Countable Family of Total Quasi-ϕ-Asymptotically Nonexpansive Nonself Mappings

Authors: Liang-cai Zhao; Shih-sen Chang
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

The purpose of this paper is to introduce a class of total quasi-ϕ-asymptotically nonexpansive-nonself mappings and to study the strong convergence under a limit condition only in the framework of Banach spaces. As an application, we utilize our results to study the approximation problem of solution to a system of equilibrium problems. The results presented in the paper extend and improve the corresponding results announced by some authors recently.


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## Body

## 1. Introduction

Throughout this paper, we assume thatE is a real Banach space, C is a nonempty closed and convex subset of E, E* is the dual space of E, and J:E→2E* is the normalized duality mapping defined by
(1.1)J(x)={f*∈E*,〈x,f*〉=‖x‖2=‖f‖2},x∈E.Recall that a Banach spaceE is said to be strictly convex if ∥x+y∥/2<1 for all x,y∈U={z∈E:||z||=1} with x≠y. E is said to be uniformly convex, if for each ϵ∈(0,2], there exists δ>0 such that ∥x+y∥/2<1-δ for all x,y∈U with ∥x-y∥≥ϵ. E is said to be smooth, if the limit
(**)lim⁡t→0‖x+ty‖-‖x‖t
exists for all x,y∈U. And E is said to be uniformly smooth, if the above limit is exists uniformly for x,y∈U.In the sequel, we shall denote the fixed point set of a mappingT by F(T). When {xn} is a sequence in E, then xn→x (xn⇀x) will denote strong (weak) convergence of the sequence {xn} to x.A mappingT:C→C is said to be nonexpansive, if
(1.2)‖Tx-Ty‖≤‖x-y‖,∀x,y∈C.A mappingT:C→C is said to be asymptotically nonexpansive if there exists a sequence {kn}⊂[1,∞) such that
(1.3)‖Tnx-Tny‖≤kn‖x-y‖,∀x,y∈C.Recall that a subsetC of E is said to be retract of E, if there exists a continuous mapping P:E→C such that Px=x, for all x∈C.It is well known that every nonempty closed and convex subset of a uniformly convex Banach space is a retract ofE. A mapping P:E→C is said to be a retraction, if P2=P. It follows that if a mapping P is a retraction, then Py=y for all y in the range of P. A mapping P:E→C is said to be a nonexpansive retraction, if it is nonexpansive and it is a retraction from E to C.In the sequel, we assume thatE is a smooth, strictly convex, and reflexive Banach space and C is a nonempty closed convex subset of E. Throughout this paper  we assume that ϕ:E×E→ℛ+ is the Lyapunov function which is defined by
(1.4)ϕ(x,y)=‖x‖-2〈x,Jy〉+‖y‖2,∀x,y∈E.It is obvious from the definition ofϕ that
(1.5)(‖x‖-‖y‖)2≤ϕ(x,y)≤(‖x‖+‖y‖)2,∀x,y∈E,(1.6)ϕ(x,J-1(λJy+(1-λ)Jz))≤λϕ(x,y)+(1-λ)ϕ(x,z),∀x,y∈E.Following Alber [1], the generalized projection ΠC:E→C is defined by
(1.7)ΠC(x)=arginf⁡y∈Cϕ(y,x),∀x∈E.Lemma 1.1 (see [1]).
LetE be a smooth, strictly convex, and reflexive Banach space and C be a nonempty closed convex subset of E. Then the following conclusions hold: (1)
ϕ(x,ΠCy)+ϕ(Πcy,y)≤ϕ(x,y) for all x∈C and y∈E;(2)
Ifx∈E and z∈C, then z=ΠCx⇔〈z-y,Jx-Jz〉≥0, for all y∈C;(3)
Forx,y∈E,ϕ(x,y)=0 if and only if x=y.Remark 1.2.
IfE is a real Hilbert space H, then ϕ(x,y)=||x-y||2 and ΠC=PC (the metric projection of H onto C).A mappingT:C→C is said to be closed, if for any sequence {xn}⊂C with xn→x and Txn→y, then Tx=y.Definition 1.3.
LetP:E→C be the nonexpansive retraction. (1)
T:C→E is said to be quasi-ϕ-nonexpansive nonself mapping, if F(T)≠∅ and
(1.8)ϕ(u,Tx)≤ϕ(u,x),∀x∈C,u∈F(T).(2)
T:C→E is said to be quasi-ϕ-asymptotically nonexpansive nonself mapping, if F(T)≠∅ and there exists a real sequence {kn}⊂[1,∞) with kn→1 such that
(1.9)ϕ(u,T(PT)n-1x)≤knϕ(u,x),∀x∈C,u∈F(T),n≥1.(3)
T:C→E is said to be total quasi-ϕ-asymptotically nonexpansive nonself mapping, if F(T)≠∅ and there exists nonnegative real sequence {νn},{μn} with νn→0,μn→0(as n→∞) and a strictly increasing continuous function ρ:ℛ+→ℛ+ with ρ(0)=0 such that for all x∈C,u∈F(T)(1.10)ϕ(u,T(PT)n-1x)≤ϕ(u,x)+νnρ(ϕ(u,x))+μn,∀n≥1.(4)
A countable family of nonself mappings{Ti}:C→E is said to be uniformly total quasi-ϕ-asymptotically nonexpansive, if ⋂i=1∞‍F(Ti)≠∅ and there exists nonnegative real sequence {νn},{μn} with νn→0,μn→0 (as n→∞) and a strictly increasing continuous function ρ:ℛ+→ℛ+ with ρ(0)=0 such that for each i≥1 and all x∈C,u∈⋂i=1∞‍F(Ti)(1.11)ϕ(u,Ti(PTi)n-1x)≤ϕ(u,x)+νnρ(ϕ(u,x))+μn,∀n≥1.Remark 1.4.
From the definitions, it is easy to know that(1)
IfT is a quasi-ϕ-nonexpansive nonself mapping, then it must be a quasi-ϕ-asymptotically nonexpansive nonself mapping with {kn=1}.(2)
Takingρ(t)=t,t>0, νn=(kn-1) and μn=0, then (1.9) can be rewritten as
(1.12)ϕ(u,T(PT)n-1x)≤ϕ(u,x)+νnρ(ϕ(u,x))+μn,∀n≥1,x∈C,u∈F(T).This implies that each quasi-ϕ-asymptotically nonexpansive nonself mapping must be a total quasi-ϕ-asymptotically nonexpansive nonself mapping, but the converse is not true.A nonself mappingT:C→E is said to be uniformly L-Lipschitz continuous, if there exists a constant L>0 such that
(1.13)‖T(PT)n-1x-T(PT)n-1y‖≤L‖x-y‖,∀x,y∈C,n≥1.Lemma 1.5 (see [2]).
LetE be a smooth and uniformly convex Banach space and let {xn}, {yn} be two sequences of E. If ϕ(xn,yn)→0(as n→∞) and either {xn} or {yn} is bounded, then ||xn-yn||→0(as n→∞).Lemma 1.6.
LetE be a smooth, strictly convex, and reflexive Banach space and C be a nonempty closed and convex subset E. Let T:C→E be a closed and total quasi-ϕ-asymptotically nonexpansive nonself mapping with nonnegative real sequence {νn},{μn} and a strictly increasing continuous function ρ:ℛ+→ℛ+ such that νn→0,μn→0 and ρ(0)=0. Then the fixed point set F(T) is a closed and convex subset of C.Proof.
Let{xn} be a sequence in F(T) such that xn→u(as n→∞). Since Txn=xn→u, by the closeness of T, we have u=Tu, that is, u∈F(T). This shows that F(T) is a closed set in C.
Next, we prove thatF(T) is convex. For any x,y∈F(T),t∈(0,1), putting q=tx+(1-t)y, we prove that q∈F(T). Indeed, let {un} be a sequence generated by
(1.14)u1=Tq,u2=TPTq=TPu1,u3=T(PT)2q=TPu2,…,un=T(PT)n-1q=TPun-1,…,
we have(1.15)ϕ(q,un)=‖q‖2-2〈q,Jun〉+‖un‖2=‖q‖2-2t〈x,Jun〉-2(1-t)〈y,Jun〉+‖un‖2=‖q‖2+tϕ(x,un)+(1-t)ϕ(y,un)-t‖x‖2-(1-t)‖y‖2.
Since
(1.16)tϕ(x,un)+(1-t)ϕ(y,un)≤t(ϕ(x,q)+νnρ(ϕ(x,q))+μn)+(1-t)(ϕ(y,q)+νnρ(ϕ(y,q))+μn)=t(‖x‖2-2〈x,Jq〉+‖q‖2+νnρ(ϕ(x,q))+μn)+(1-t)(‖y‖2-2〈y,Jq〉+‖q‖2+νnρ(ϕ(y,q))+μn)=t‖x‖2+(1-t)‖y‖2-‖q‖2+tνnρ(ϕ(x,q))+(1-t)νnρ(ϕ(y,q))+μn.
Substituting (1.16) into (1.15), and simplifying we have
(1.17)ϕ(q,un)≤tνnρ(ϕ(x,q))+(1-t)νnρ(ϕ(y,q))+μn⟶0(n⟶∞).
By Lemma 1.5, we have un→q(n→∞). This implies that un+1→q(n→∞).
Sinceun+1=T(PT)nq=TPT(PT)n-1q=TPun and T is closed, we have q=TPq. Since q∈C, Pq=q, thus q=Tq. this implies that F(T) is a convex set in C.Concerning the strong and weak convergence of asymptotically nonexpansive self or nonself mappings, relatively nonexpansive, quasi-ϕ-nonexpansive and quasi-ϕ-asymptotically nonexpansive self or nonself mappings have been considered extensively by several authors in the setting of Hilbert or Banach spaces (see e.g., [2–19]).The purpose of this paper is to modify the Halpern and Mann-type iteration algorithm for a family of of total quasi-ϕ-asymptotically nonexpansive nonself mappings and to have the strong convergence under removing F(T) is a convex set of condition and a limit condition only in the framework of Banach spaces. As an application, we utilize our results to study the approximation problem of solution to a system of equilibrium problems. The results presented in the paper extend and improve the corresponding results of Chang et al. [4–7], W. P. Guo and W. Guo [8], Hao et al. [9], Kamimura and Takahashi [10], Kiziltunc and Temir [11], Nilsrakoo and Saejung [2], Pathak et al. [12], Qin et al. [13], Su et al. [14], Thianwan [15], Wang et al. [16], Yıldırım and Özdemir [17], Yang and Xie [18], Zegeye et al. [19], Kanjanasamranwong et al. [20], Saewan and Kumam [21–24] and Wattanawitoon and Kumam [25].


## 2. Main Results

Theorem 2.1.
LetE be a real uniformly convex and uniformly smooth Banach space, and C be a nonempty closed convex subset E. Let Ti:C→E,i=1,2,… be a family of closed and uniformly total quasi-ϕ-asymptotically nonexpansive nonself mappings with nonnegative real sequence {νn},{μn} and a strictly increasing continuous function ρ:ℛ+→ℛ+ such that νn→0,μn→0 and ρ(0)=0, and for each i≥1, Ti be uniformly Li-Lipschitz continuous. Let {αn} be a sequence in [0,1] and {βn} be a sequence in (0,1) satisfying the following conditions: (a)
limn→∞αn=0;(b)
0<liminfn→∞βn≤limsupn→∞βn<1.
Let{xn} be a sequence generated by
(2.1)x1∈Echosenarbitrarily;C1=C,yn,i=J-1[αnJx1+(1-αn)(βnJxn+(1-βn)JTi(PTi)n-1xn)],i≥1,Cn+1={z∈Cn:sup⁡i≥1ϕ(z,yn,i)≤αnϕ(z,x1)+(1-αn)ϕ(z,xn)+θn},xn+1=ΠCn+1x1,∀n≥1.
where θn=νnsupu∈ℱρ(ϕ(u,xn))+μn, for all n≥1, ℱ:=⋂i=1∞‍F(Ti). If ℱ is a nonempty-bounded subset in C, then {xn} converges strongly to Πℱx1.Proof.
We divide the proof of Theorem2.1 into five steps.
(I)
ℱ  and  Cn,n≥1  are closed and convex subset inC.
In fact, it follows from Lemma1.6 that F(Ti), i≥1 is closed and convex subset of C. Therefore ℱ is a closed and convex subset in C.
Again by the assumption thatC1=C is closed and convex. Suppose that Cn is closed and convex for some n≥2. In view of the definition of ϕ we have that
(2.2)Cn+1={z∈Cn:sup⁡i≥1ϕ(z,yn,i)≤αnϕ(z,x1)+(1-αn)ϕ(z,xn)+θn}=⋂i≥1‍{z∈Cn:ϕ(z,yn,i)≤αnϕ(z,x1)+(1-αn)ϕ(z,xn)+θn}∩Cn=⋂i≥1‍{‖yn,i‖2z∈Cn:2αn〈z,Jx1〉+2(1-αn)〈z,Jxn〉-2〈z,Jyn,i〉≤αn‖x1‖2+(1-αn)‖xn‖2-‖yn,i‖2+θn}∩Cn.
This implies that Cn+1 is closed and convex. The conclusion is proved.
(II)   Now we prove that  ℱ⊂Cn,n≥1.
In fact, it is obvious thatℱ⊂C1=C. Suppose that ℱ⊂Cn for some n≥2. Letting
(2.3)wn,i=J-1(βnJxn+(1-βn)JTi(PTi)n-1xn),
it follows from (1.6) that for any u∈ℱ⊂Cn we have
(2.4)ϕ(u,yn,i)=ϕ(u,J-1(αnJx1+(1-αn)Jwn,i))≤αnϕ(u,x1)+(1-αn)ϕ(u,wn,i),(2.5)ϕ(u,wn,i)=ϕ(u,J-1(βnJxn+(1-βn)JTi(PTi)n-1xn))≤βnϕ(u,xn)+(1-βn)ϕ(u,Ti(PTi)n-1xn)≤βnϕ(u,xn)+(1-βn){ϕ(u,xn)+νnρ(ϕ(u,xn))+μn}=ϕ(u,xn)+(1-βn)(νnρ(ϕ(u,xn))+μn)≤ϕ(u,xn)+νnρ(ϕ(u,xn))+μn,
therefore we have
(2.6)sup⁡i≥1ϕ(u,yn,i)≤αnϕ(u,x1)+(1-αn){ϕ(u,xn)+νnρ(ϕ(u,xn))+μn}≤αnϕ(u,x1)+(1-αn)ϕ(u,xn)+νnsup⁡u∈Fρ(ϕ(u,xn))+μn≤αnϕ(u,x1)+(1-αn)ϕ(u,xn)+θn,
where θn=νnsupu∈ℱρ(ϕ(u,xn))+μn. This shows that u∈Cn+1, and so ℱ⊂Cn+1. The conclusion is proved.
(III)   Next we prove that  {xn}  is a Cauchy sequence inC.
In fact, sincexn=ΠCnx1, from Lemma 1.1(2) we have
(2.7)〈xn-y,Jx1-Jxn〉≥0,∀y∈Cn.
Again since ℱ⊂Cn, for all n≥1, we have
(2.8)〈xn-u,Jx1-Jxn〉≥0,∀u∈F.
It follows from Lemma 1.1(1) that for each u∈ℱ and for each n≥1(2.9)ϕ(xn,x1)=ϕ(ΠCnx1,x1)≤ϕ(u,x1)-ϕ(u,xn)≤ϕ(u,x1).
Therefore {ϕ(xn,x1)} is bounded. By virtue of (1.5), {xn} is also bounded.
Sincexn=ΠCnx1 and xn+1=ΠCn+1x1∈Cn+1⊂Cn, we have ϕ(xn,x1)≤ϕ(xn+1,x1), for all n≥1. This implies that {ϕ(xn,x1)} is nondecreasing. Hence the limit limn→∞ϕ(xn,x1) exists. By the construction of Cn, for any positive integer m≥n, we have Cm⊂Cn and xm=ΠCmx1∈Cn. This shows that
(2.10)ϕ(xm,xn)=ϕ(xm,ΠCnx1)≤ϕ(xm,x1)-ϕ(xn,x1)⟶0,asn,m⟶∞.
It follows from Lemma 1.5 that limm,n→∞||xm-xn||=0. Hence {xn} is a Cauchy sequence in C. Since C is a nonempty closed subset of Banach space E, it is complete, without loss of generality, we can assume that xn→x*(n→∞).
By the assumption, it is easy to see that(2.11)lim⁡n→∞θn=lim⁡n→∞(νnsup⁡u∈Fρ(ϕ(u,xn))+μn)=0.
(IV)   Now we prove that  x*∈ℱ.
In fact, sincexn+1∈Cn+1 and αn→0, it follows from (2.1) and (2.11) that
(2.12)sup⁡i≥1ϕ(xn+1,yn,i)≤αnϕ(xn+1,x1)+(1-αn)ϕ(xn+1,xn)+θn⟶0(asn⟶∞).
Since xn→x*, by virtue of Lemma 1.5 for each i≥1,we have
(2.13)lim⁡n→∞yn,i=x*.
Since {xn} is bounded, {Ti}i=1∞ is uniformly total quasi-ϕ-asymptotically nonexpansive nonself mappings with nonnegative real sequence {νn},{μn} and a strictly increasing continuous function ρ:ℛ+→ℛ+ such that νn→0,μn→0, and ρ(0)=0, for any given u∈ℱ, we have
(2.14)ϕ(u,Ti(PTi)n-1xn)≤ϕ(u,xn)+νnρ(ϕ(u,xn))+μn.
This implies that {Ti(PTi)n-1xn} is uniformly bounded. Since
(2.15)‖wn,i‖=‖J-1(βnJxn+(1-βn)JTi(PTi)n-1xn)‖≤βn‖xn‖+(1-βn)‖Ti‖PTi‖n-1xn‖≤‖xn‖+‖Ti(PTi)n-1xn‖.
This implies that {wn,i} is also uniformly bounded.
Sinceαn→0, from (2.1), for each i≥1 we have
(2.16)lim⁡n→∞‖Jyn,i-Jwn,i‖=lim⁡n→∞αn‖Jx1-Jwn,i‖=0.
Since J-1 is uniformly continuous on each bounded subset of E*, it follows from (2.13) and (2.16) that
(2.17)lim⁡n→∞wn,i=x*foreachi≥1.
Since J is uniformly continuous on each bounded subset of E, we have
(2.18)0=lim⁡n→∞‖Jwn,i-Jx*‖=lim⁡n→∞‖βnJxn+(1-βn)JTi(PTi)n-1xn-Jx*‖=lim⁡n→∞‖βn(Jxn-Jx*)+(1-βn)(JTi(PTi)n-1xn-Jx*)‖=lim⁡n→∞(1-βn)‖JTi(PTi)n-1xn-Jx*‖.
By condition (b), we have that
(2.19)lim⁡n→∞‖JTi(PTi)n-1xn-Jx*‖=0.
Since J is uniformly continuous, this shows that limn→∞Ti(PTi)n-1xn=x* uniformly in i≥1.
Again by the assumptions that for eachi≥1, Ti is uniformly Li-Lipschitz continuous, thus we have
(2.20)‖Ti(PTi)nxn-Ti(PTi)n-1xn‖≤‖Ti(PTi)nxn-Ti(PTi)nxn+1‖+‖Ti(PTi)nxn+1-xn+1‖+‖xn+1-xn‖+‖xn-Ti(PTi)n-1xn‖≤(Li+1)‖xn-xn+1‖+‖Ti(PTi)nxn+1-xn+1‖+‖xn-Ti(PTi)n-1xn‖.
Since limn→∞Ti(PTi)n-1xn=x* and xn→x*, these together with (2.20) imply that limn→∞||Ti(PTi)nxn-Ti(PTi)n-1xn||=0 and limn→∞Ti(PTi)nxn=x*, that is,
(2.21)lim⁡n→∞TiP(PTi)n-1xn=x*.
In view continuity of TiP, it yields that TiPx*=x*. Since x*∈C,Px*=x*. This shows that Tx*=x*. By the arbitrariness of i≥1, we have x*∈ℱ.
(V)  Finally we prove that  xn→x*=Πℱx1.
Letw=Πℱx1. Since w∈ℱ⊂Cn and xn=ΠCnx1, we have ϕ(xn,x1)≤ϕ(w,x1), for all n≥1. This implies that
(2.22)ϕ(x*,x1)=lim⁡n→∞ϕ(xn,x1)≤ϕ(w,x1).
In view of the definition of Πℱx1, from (2.22) we have x*=w. Therefore xn→x*=Πℱx1. This completes the proof of Theorem 2.1.Theorem 2.2.
LetE, C, {αn}, {βn} be the same as in Theorem 2.1. Let Ti:C→E,i=1,2,…  be a family of closed and uniformly quasi-ϕ-asymptotically nonexpansive nonself mappings with sequence {kn}⊂[1,∞),kn→1, and for each i≥1, Ti be uniformly Li-Lipschitz continuous. Let {xn} be a sequence generated by
(2.23)x1∈Echosenarbitrarily;C1=C,yn,i=J-1[αnJx1+(1-αn)(βnJxn+(1-βn)JTi(PTi)n-1xn)],i≥1,Cn+1={z∈Cn:sup⁡i≥1ϕ(z,yn,i)≤αnϕ(z,x1)+(1-αn)ϕ(z,xn)+θn},xn+1=ΠCn+1x1,∀n≥1,
where θn=(kn-1)supu∈ℱϕ(u,xn),  ℱ:=⋂i=1∞‍F(Ti). If ℱ is a nonempty bounded subset in C, then {xn} converges strongly to Πℱx1.Proof.
By Remark1.4Ti:C→E,i=1,2,… be a family of closed and uniformly quasi-ϕ-asymptotically nonexpansive nonself mappings that it is a family of closed and uniformly total quasi-ϕ-asymptotically nonexpansive nonself mappings with taking ρ(t)=t,t>0, νn=(kn-1) and μn=0. Therefore all conditions in Theorem 2.1 are satisfied. By the similar methods as given in the proof of Theorem 2.1, we can prove that the sequence {xn} defined by (2.23) converges strongly to Πℱx1.Theorem 2.3.
LetE, C, {αn}, {βn} be the same as in Theorem 2.2. Let Ti:C→E,i=1,2,… be a family of quasi-ϕ-nonexpansive nonself mappings such that ℱ:=⋂i=1∞‍F(Ti)≠∅ and for each i≥1, Ti be uniformly Li-Lipschitz continuous. Let {xn} be a sequence generated by
(2.24)x1∈Echosenarbitrarily;C1=C,yn,i=J-1[αnJx1+(1-αn)(βnJxn+(1-βn)JTixn)],i≥1,Cn+1={z∈Cn:sup⁡i≥1ϕ(z,yn,i)≤αnϕ(z,x1)+(1-αn)ϕ(z,xn)},xn+1=ΠCn+1x1,∀n≥1.
Then {xn} converges strongly to Πℱx1.Proof.
By Remark1.4Ti:C→E,i=1,2,… be a family of quasi-ϕ-nonexpansive nonself mappings that it is a family of uniformly quasi-ϕ-asymptotically nonexpansive nonself mappings with sequence {kn}={1}. Hence θn=(kn-1)supu∈ℱϕ(u,xn)=0 Therefore all conditions in Theorem 2.2 are satisfied. By the similar methods, we can prove that the sequence {xn} defined by (2.24) converges strongly to Πℱx1.


## 3. Application and Example

In this section we utilize the results presented in Section2 to prove a strong convergence theorem concerning maximal monotone operators in Hilbert spaces.LetE be a real Hilbert space and let A be a maximal monotone operator from E to E. For each r>0, we can define a single valued mapping JrA:E→E by JrA=(I+rA)-1 and such a mapping JrA is called the resolvent ofA. It is easy to prove that JrA is a nonexpansive mapping and A-1(0)=F(JrA) for all r>0. Therefore it is a uniformly 1-Lipschitz continuous and quasi-ϕ-nonexpansive mapping. Hence for each p∈F(JrA) and w∈E, we have
(3.1)ϕ(p,JrAw)≤ϕ(p,w),
and F(JrA)=A-1(0). These show that all conditions in Theorem 2.3 are satisfied. Hence from Theorem 2.3 we have the following.Theorem 3.1.
LetE be a real Hilbert space. Let A1,A2 be two maximal monotone operators from E to E such that ℱ=A1-1(0)∩A2-1(0)≠∅. Let JrA1 and JrA2 be the resolvent of A1 and A2, respectively, where r>0. Let {αn}, {βn} be the same as in Theorem 2.3 and {xn} be the sequence defined by
(3.2)x1∈Echosenarbitrarily;C1=E,yn,i=J-1[αnJx1+(1-αn)(βnJxn+(1-βn)JJrAixn)],i=1,2,Cn+1={z∈Cn:max⁡i=1,2ϕ(z,yn,i)≤αnϕ(z,x1)+(1-αn)ϕ(z,xn)},xn+1=PCn+1x1,∀n≥1,
where PC is the metric projection from H onto the subset C⊂H. Then the sequence {xn} defined by (3.2) converges strongly to Pℱx1.



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*Source: 136134-2012-08-28.xml*