# Hybrid Algorithm for Common Fixed Points of Uniformly Closed Countable Families of Hemirelatively Nonexpansive Mappings and Applications

Authors: Sumei Ai; Yongfu Su
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

The authors have obtained the following results: (1) the definition of uniformly closed countable family of nonlinear mappings, (2) strong convergence theorem by the monotone hybrid algorithm for two countable families of hemirelatively nonexpansive mappings in a Banach space with new method of proof, (3) two examples of uniformly closed countable families of nonlinear mappings and applications, (4) an example which is hemirelatively nonexpansive mapping but not weak relatively nonexpansive mapping, and (5) an example which is weak relatively nonexpansive mapping but not relatively nonexpansive mapping. Therefore, the results of this paper improve and extend the results of Plubtieng and Ungchittrakool (2010) and many others.


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## Body

## 1. Introduction and Preliminaries

LetE be a Banach space with the dual E*. We denote by J the normalized duality mapping from E to 2E* defined by (1.1)Jx={f∈E*:〈x,f〉=‖x‖2=‖f‖2},where 〈·,·〉 denotes the generalized duality pairing. It is well known that the normalized duality  J has the following properties: (1) if E is smooth, then J is single valued; (2) if E is strictly convex, then J is one-to-one (i.e., Jx∩Jy=∅ for all  x≠y); (3) if E is reflexive, then  J is surjective; (4) if E has Frchet differentiable norm, then  J is uniformly norm-to-norm continuous; (5) if E is uniformly smooth, then  J is uniformly norm-to-norm continuous on each bounded subset of E; (7) if E is a Hilbert space, then  J is the identity operator.As we all know that ifC is a nonempty closed convex subset of a Hilbert space H and PC:H→C is the metric projection of H onto C, then PC is nonexpansive. This fact actually characterizes Hilbert spaces, and, consequently, it is not available in more general Banach spaces. In this connection, Alber [1] has recently introduced a generalized projection operator ΠC in a Banach space E which is an analogue of the metric projection in Hilbert spaces.LetE be a smooth Banach space. Consider the function defined by (1.2)ϕ(x,y)=‖x‖2-2〈x,Jy〉+‖y‖2forx,y∈E.
Observe that, in a Hilbert space H, (1.2) reduces to ϕ(x,y)=∥x-y∥2,x,y∈H.The generalized projectionΠC:E→C is a map that assigns to an arbitrary point x∈E the minimum point of the functional ϕ(x,y), that is, ΠCx=x¯, where x¯ is the solution to the minimization problem (1.3)ϕ(x¯,x)=min⁡y∈Cϕ(y,x),
existence and uniqueness of the operator ΠC follow from the properties of the functional ϕ(x,y) and strict monotonicity of the mapping J (see, e.g., [1, 2]). In Hilbert space, ΠC=PC. It is obvious from the definition of function ϕ that (1.4)(‖y‖-‖x‖)2≤ϕ(y,x)≤(‖y‖+‖x‖)2∀x,y∈E.IfE is a reflexive strictly convex and smooth Banach space, then for x,y∈E, ϕ(x,y)=0 if and only if x=y. It is sufficient to show that if ϕ(x,y)=0, then x=y. From (1.4), we have ∥x∥=∥y∥. This implies that 〈x,Jy〉=∥x∥2=∥Jy∥2. From the definitions of j, we have Jx=Jy. That is, x=y; see [3, 4] for more details.LetC be a closed convex subset of E, and let T be a mapping from C into itself with nonempty set of fixed points. We denote by F(T) the set of fixed points of T. T is called hemi-relatively nonexpansive if ϕ(p,Tx)≤ϕ(p,x) for all x∈C and p∈F(T).A pointp in C is said to be an asymptotic fixed point of T if C contains a sequence {xn} which converges weakly to p such that the strong limn→∞(Txn-xn)=0. The set of asymptotic fixed points of T will be denoted by F̂(T). A hemi-relatively nonexpansive mapping T from C into itself is called relatively nonexpansive if F̂(T)=F(T) (see, [5]).A pointp in C is said to be a strong asymptotic fixed point of T if C contains a sequence {xn} which converges strongly to p such that the strong limn→∞(Txn-xn)=0. The set of strong asymptotic fixed points of T will be denoted by F̃(T). A hemi-relatively nonexpansive mapping T from C into itself is called weak relatively nonexpansive if F̃(T)=F(T) (see, [6]).The following conclusions are obvious: (1) relatively nonexpansive mapping must be weak relatively nonexpansive mapping; (2) weak relatively nonexpansive mapping must be hemi-relatively nonexpansive mapping.In this paper, we will give two examples to show that the inverses of above two conclusions are not hold.In an infinite-dimensional Hilbert space, Mann's iterative algorithm has only weak convergence, in general, even for nonexpansive mappings. Hence in order to have strong convergence, in recent years, the hybrid iteration methods for approximating fixed points of nonlinear mappings has been introduced and studied by various authors.In 2003, Nakajo and Takahashi [7] proposed the following modification of Mann iteration method for a single nonexpansive mapping T in a Hilbert space H: (1.5)x0∈Cchosenonlyarbitrarily,yn=αnxn+(1-αn)Txn,Cn={z∈C:‖yn-z‖≤‖xn-z‖},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qn(x0),
where C is a closed convex subset of H, and PK denotes the metric projection from H onto a closed convex subset K of H. They proved that if the sequence {αn} is bounded above from one, then the sequence {xn} generated by (1.5) converges strongly to PF(T)(x0), where F(T) denote the fixed points set of T.The ideas to generalize the process (1.5) from Hilbert space to Banach space have recently been made. By using available properties on uniformly convex and uniformly smooth Banach space, Matsushita and Takahashi [8] presented their ideas as the following method for a single relatively nonexpansive mapping T in a Banach space E: (1.6)x0∈Cchosenonlyarbitrarily,yn=J-1(αnJx0+(1-αn)JTxn),Cn={z∈C:ϕ(z,yn)≤ϕ(z,xn)},Qn={z∈C:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠCn∩Qn(x0),
where J is the duality mapping on E, and ΠK(·) is the generalized projection from E onto a nonempty closed convex subset K. They proved the following convergence theorem.Theorem MT.
LetE be a uniformly convex and uniformly smooth Banach space, let  C be a nonempty closed convex subset of  E, let  T be a relatively nonexpansive mapping from  C into itself, and let  {αn} be a sequence of real numbers such that  0≤αn<1 and   limsupn→∞αn<1. Suppose that  {xn} is given by (1.6), where  J is the duality mapping on  E. If  F(T) is nonempty, then  {xn} converges strongly to  ΠF(T)x0, where  ΠF(T)(·) is the generalized projection from  C onto  F(T).Recently, Plubtieng and Ungchittrakool [9] here proposed the following hybrid iteration method for a countable family of relatively nonexpansive mappings in a Banach space and proved the convergence theorem.Theorem PU.
LetE be a uniformly smooth and uniformly convex Banach space and let  Ĉ and  C be two nonempty closed convex subsets of  E such that  Ĉ⊂C. Let  {Tn} be a sequence of relatively nonexpansive mappings from  C into  E such that  ⋂n=1∞F(Tn) is nonempty, and let  {xn} be a sequence defined as follows:
(1.7)x0∈Ĉ,C1=C,x1=ΠC1x0,yn=J-1(αnJxn+(1-αn)JTnxn),Cn+1={z∈Cn:ϕ(z,yn)≤ϕ(z,xn)},n≥1,xn+1=ΠCn+1x0,
where αn∈[0,1] satisfies either(a)
0≤αn<1 for all  n≥1 and limsupn→∞αn<1 or(b)
liminfn→∞αn(1-αn)>0. Suppose that for any bounded subset B of C there exists an increasing, continuous and convex function hB from R+ into R+ such that hB(0)=0, and
(1.8)lim⁡l,k→∞sup⁡{hB(‖Tlz-Tkz‖):z∈B}=0.
Let T be a mapping from C into E defined by Tx=limn→∞Tnx for all x∈C and suppose that
(1.9)F(T)=⋂n=1∞F(Tn)=⋂n=1∞F̂(Tn)=F̂(T).
Then {xn}, {Tnxn}, and {yn} converge strongly to ΠF(T)x0.In this paper, the authors have obtained the following results: (1) the definition of uniformly closed countable family of nonlinear mappings, (2) strong convergence theorem by the monotone hybrid algorithm for a countable family of hemi-relatively nonexpansive mappings in a Banach space with new method of proof, (3) two examples of uniformly closed countable families of nonlinear mappings and applications, (4) an example which is hemi-relatively nonexpansive mapping but not weak relatively nonexpansive mapping, and (5) an example which is weak relatively nonexpansive mapping but not relatively nonexpansive mapping. Therefore, the results of this paper improve and extend the results of Plubtieng and Ungchittrakool [9] and many others.We need the following definitions and lemmas.Lemma 1.1 (Kamimura and Takahashi [10]).
LetE be a uniformly convex and smooth Banach space, and let  {xn} and  {yn} be two sequences of  E. If  ϕ(xn,yn)→0 and either  {xn} or  {yn} is bounded, then  xn-yn→0.Lemma 1.2 (Alber [1]).
LetC be a nonempty closed convex subset of a smooth Banach space  E and  x∈E. Then,  x0=ΠCx if and only if
(1.10)〈x0-y,Jx-Jx0〉≥0fory∈C.Lemma 1.3 (Alber [1]).
LetE be a reflexive, strictly convex, and smooth Banach space, let  C be a nonempty closed convex subset of  E, and let  x∈E. Then
(1.11)ϕ(y,ΠCx)+ϕ(ΠCx,x)≤ϕ(y,x)∀y∈C.The following lemma is not hard to prove.Lemma 1.4.
LetE be a strictly convex and smooth Banach space, let  C be a closed convex subset of  E, and let  T be a hemi-relatively nonexpansive mapping from  C into itself. Then  F(T) is closed and convex.In this paper, we present the definition of uniformly closed for a sequence of mappings as follows.Definition 1.5.
LetE be a Banach space, C be a closed convex subset of E, let {Tn}n=1∞ be a sequence of mappings of C into E such that ⋂n=1∞F(Tn) is nonempty. We say that {Tn}n=1∞ is uniformly closed if p∈⋂n=1∞F(Tn) whenever {xn}⊂C converges strongly to p and ∥xn-Tnxn∥→0 as n→∞.Lemma 1.6 (see [11, 12]).
LetE be a  p-uniformly convex Banach space with  p≥2. Then, for all  x,y∈E,  j(x)∈Jp(x), and  j(y)∈Jp(y),
(1.12)〈x-y,j(x)-j(y)〉≥cpcp-2p‖x-y‖p,
where Jp is the generalized duality mapping from E into E*, and 1/c is the p-uniformly convexity constant of E.


## 2. Main Results

Definition 2.1.
LetE be a Banach space, let C be a nonempty closed convex subset of E, and let {Tn}n=1∞:C→E, {Sn}n=1∞:C→E be two sequences of mappings. If for any convergence sequence {xn}⊂C, the following holds:
(2.1)lim⁡n→+∞‖Tnxn-Snxn‖=0.
Then {Tn} and {Sn} are said to satisfy the only asymptotically condition.Theorem 2.2.
LetE be a uniformly convex and uniformly smooth Banach space, let  C be a nonempty closed convex subset of  E, let  {Tn}n=1∞:C→E,  {Sn}n=1∞:C→E be two uniformly closed sequences of hemi-relatively nonexpansive mappings satisfying the asymptotically condition such that  F1=⋂n=1∞F(Tn)≠∅,  F2=⋂n=1∞F(Sn)≠∅, and  F=F1⋂F2≠∅. Assume that  {αn}n=0∞,  {βn}n=0∞,  {γn}n=0∞ and  {δn}n=0∞ are four sequences in  [0,1] such that  αn+βn+γn+δn=1,  limn→∞αn=0 and  0<γ≤γn≤1,  δn→0 for some constant  γ∈(0,1). Define a sequence  {xn} in  C by the following algorithm:
(2.2)x0∈Carbitrarily,yn=J-1(αnJx0+βnJxn+γnJTnxn+δnJSnxn),n≥1,Cn={z∈Cn-1:ϕ(z,yn)≤(1-αn)ϕ(z,xn)+αnϕ(z,x0)},n≥1,C0=C,xn+1=ΠCnx0,n≥0.
Then {xn} converges strongly to q=ΠFx0.Proof .
We first show thatCn is closed and convex for all n≥0. From the definitions of Cn, it is obvious that Cn is closed for all n≥0. Next, we prove that Cn is convex for all n≥0. Since
(2.3)ϕ(z,yn)≤(1-αn)ϕ(z,xn)+αnϕ(z,x0)
is equivalent to
(2.4)2〈z,(1-αn)Jxn+αnJx0-Jyn〉≤(1-αn)‖xn‖2+αn‖x0‖2-‖yn‖2,
it is easy to get that Cn is convex for all n≥0.
Next, we show thatF⊂Cn for all n≥1. Indeed, for each p∈F, we have
(2.5)ϕ(p,yn)=ϕ(p,J-1(αnJx0+βnJxn+γnJTnxn+δnJSnxn))=‖p‖2-2〈p,(αnJx0+βnJxn+γnJTnxn+δnJSnxn)〉+‖αnJx0+βnJxn+γnJTnxn+δnJSnxn‖2≤‖p‖2-2αn〈p,Jx0〉-2βn〈p,Jxn〉-2γn〈p,JTnxn〉-2δn〈p,JSnxn〉+αn‖x0‖2+βn‖xn‖+γn‖Tnxn‖2+δn‖Snxn‖2≤αnϕ(p,x0)+βnϕ(p,xn)+γnϕ(p,Tnxn)+δnϕ(p,Snxn)≤αnϕ(p,x0)+(1-αn)ϕ(p,xn).
So, p∈Cn, which implies that F⊂Cn for all n≥1.
Sincexn+1=ΠCnx0 and Cn⊂Cn-1, then we get the following:
(2.6)ϕ(xn,x0)≤ϕ(xn+1,x0),∀n≥0.
Therefore, {ϕ(xn,x0)} is nondecreasing. On the other hand, by Lemma 1.3, we have
(2.7)ϕ(xn,x0)=ϕ(ΠCn-1x0,x0)≤ϕ(p,x0)-ϕ(p,xn)≤ϕ(p,x0),
for all p∈F(T)⊂Cn-1 and for all n≥1. Therefore, ϕ(xn,x0) is also bounded. This together with (3.1) implies that the limit of {ϕ(xn,x0)} exists. Put the following:
(2.8)lim⁡n→∞ϕ(xn,x0)=d.
From Lemma 1.3, we have, for any positive integer m, that
(2.9)ϕ(xn+m,xn+1)=ϕ(xn+m,ΠCnx0)≤ϕ(xn+m,x0)-ϕ(ΠCnx0,x0)=ϕ(xn+m,x0)-ϕ(xn+1,x0),
for all n≥0. This together with (3.6) implies that
(2.10)lim⁡n→∞ϕ(xn+m,xn+1)=0
is, uniformly for all m, holds. By using Lemma 1.1,  we get that
(2.11)lim⁡n→∞‖xn+m-xn+1‖=0
is, uniformly for all m, holds. Then {xn} is a Cauchy sequence; therefore, there exists a point p∈C such that xn→p.
Sincexn+1=ΠCnx0∈Cn, from the definition of Cn, we have the following:
(2.12)ϕ(xn+1,yn)≤(1-αn)ϕ(xn+1,xn)+αnϕ(xn+1,x0).
This together with (2.10) and limn→∞αn=0 implies that
(2.13)lim⁡n→∞ϕ(xn+1,yn)=0.
Therefore, by using  Lemma 1.1, we obtain the following:
(2.14)lim⁡n→∞‖xn+1-yn‖=0.
Since J is uniformly norm-to-norm continuous on bounded sets, then we have the following:
(2.15)lim⁡n→∞‖Jxn+1-Jyn‖=lim⁡n→∞‖Jxn+1-Jxn‖=0.
Noticing that
(2.16)‖Jxn+1-Jyn‖=‖Jxn+1-(αnJx0+βnJxn+γnJTnxn+δnJSnxn)‖=‖αn(Jxn+1-Jx0)+βn(Jxn+1-Jxn)+γn(Jxn+1-JTnxn)+δn(Jxn+1-JTnxn)‖≥γn‖Jxn+1-JTnxn‖-δn‖Jxn+1-JSnxn‖-αn‖Jxn+1-Jx0‖-βn‖Jxn+1-Jxn‖,
which leads to
(2.17)γn‖Jxn+1-JTnxn‖≤‖Jxn+1-Jyn‖+αn‖Jx0-Jxn+1‖+βn‖Jxn+1-Jxn‖+δn‖Jxn+1-JSnxn‖.
From (2.15) andlimn→∞αn=0, limn→∞δn=0, 0<γ≤γn≤1 we obtain that
(2.18)lim⁡n→∞‖Jxn+1-JTnxn‖=0.
Since J-1 is also uniformly norm-to-norm continuous on bounded sets, then we obtain that
(2.19)lim⁡n→∞‖xn+1-Tnxn‖=0.
This together with (2.11) implies that
(2.20)lim⁡n→∞‖xn-Tnxn‖=0.
Sine {Tn} and {Sn} satisfy the asymptotically condition, we also have
(2.21)lim⁡n→∞‖xn-Snxn‖=0.
Since xn→p and {Tn}n=1∞, {Sn}n=1∞ are uniformly closed, we have
(2.22)p∈F=(⋂n=1∞F(Tn))⋂(⋂n=1∞F(Sn)).
Finally, we prove thatp=ΠFx0, from Lemma 1.3, we have
(2.23)ϕ(p,ΠFx0)+ϕ(ΠFx0,x0)≤ϕ(p,x0).
On the other hand, xn+1=ΠCnx0 and F⊂Cn, for all n. Also from Lemma 1.3, we have
(2.24)ϕ(ΠFx0,xn+1)+ϕ(xn+1,x0)≤ϕ(ΠFx0,x0).
By the definition of ϕ(x,y), we know that
(2.25)lim⁡n→∞ϕ(xn+1,x0)=ϕ(p,x0).
Combining (2.24) and (2.25), we know that ϕ(p,x0)=ϕ(ΠFx0,x0). Therefore, it follows from the uniqueness of ΠFx0 that p=ΠFx0. This completes the proof.Whenαn≡0, δn≡0 in Theorem 2.2, we obtain the following result.Theorem 2.3.
LetE be a uniformly convex and uniformly smooth Banach space, let  C be a nonempty closed convex subset of  E, and let  {Tn}n=1∞:C→E be a uniformly closed sequence of hemi-relatively nonexpansive mappings such that  F=⋂n=1∞F(Tn)≠∅. Assume that  {αn}n=0∞ is a sequences in [0,1] such that  0≤αn≤α<1 for some constant  α∈(0,1). Define a sequence  {xn} in  C by the following algorithm:
(2.26)x0∈Carbitrarily,yn=J-1(αnJxn+(1-αn)JTnxn),n≥1,Cn={z∈Cn-1:ϕ(z,yn)≤ϕ(z,xn)},n≥1,C0=C,xn+1=ΠCnx0,n≥0.
Then {xn} converges strongly to q=ΠFx0.


## 3. Applications for Equilibrium Problem

LetE be a real Banach space, and let E* be the dual space of E. Let C be a closed convex subset of E. Let f be a bifunction from C×C to R=(-∞,+∞). The equilibrium problem is to find x∈C such that (3.1)f(x,y)≥0,∀y∈C.
The set of solutions of (1.2) is denoted by EP(f). Given a mapping T:C→E*  let f(x,y)=〈Tx,y-x〉 for all x,y∈C. Then, p∈EP(f) if and only if 〈Tp,y-p〉≥0 for all y∈C, that is, p is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.2). Some methods have been proposed to solve the equilibrium problem in Hilbert spaces, see, for instance, [13–15].For solving the equilibrium problem, let us assume that a bifunctionf satisfies the following conditions:(A1)
f(x,x)=0,forallx∈E,(A2)
f is monotone, that is,  f(x,y)+f(y,x)≤0,forallx,y∈E,(A3)
for allx,y,z∈E, only limsupt↓0f(tz+(1-t)x,y)≤f(x,y),(A4)
for allx∈C, f(x,·) is convex and lower semicontinuous.Lemma 3.1 (Blum and Oettli [13]).
LetC be a closed convex subset of a smooth, strictly convex, and reflexive Banach space  E, let  f be a bifunction from  C×C to  R=(-∞,+∞) satisfying (A1)–(A4), and let  r>0 and  x∈E. Then, there exists  z∈C such that
(3.2)f(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C.Lemma 3.2 (Takahashi and Zembayashi [15]).
LetC be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space  E, and let  f be a bifunction from  C×C to  R=(-∞,+∞) satisfying (A1)–(A4). For  r>0, define a mapping  Tr:E→C as follows:
(3.3)Tr(x)={z∈C:f(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C}
for all x∈E. Then, the following hold:(1)
Tr is single-valued;(2)
Tr is a firmly nonexpansive-type mapping, that is, for all x,y∈E,
(3.4)〈Trx-Try,JTrx-JTry〉≤〈Trx-Try,Jx-Jy〉,(3)
F(Tr)=EP(f);(4)
EP(f) is closed and convex;(5)
Tr is also a relatively nonexpansive mapping.Lemma 3.3 (Takahashi and Zembayashi [15]).
LetC be a closed convex subset of a smooth, strictly convex, and reflexive Banach space  E, let  f be a bifunction from  C×C to  R=(-∞,+∞) satisfying (A1)–(A4), let  r>0, and let  x∈E,  q∈F(Tr), then the following holds:
(3.5)ϕ(q,Trx)+ϕ(Trx,x)≤ϕ(q,x).Lemma 3.4.
LetE be a  p-uniformly convex with   p≥2 and uniformly smooth Banach space, and let  C be a nonempty closed convex subset of  E. Let  f be a bifunction from  C×C to  R=(-∞,+∞) satisfying (A1)–(A4). Let  {rn} be a positive real sequence such that  limn→∞rn=r>0. Then the sequence of mappings  {Trn} is uniformly closed.Proof.
(1) Let{xn} be a convergent sequence in C. Let zn=Trnxn for all n, then
(3.6)f(zn,y)+1rn〈y-zn,Jzn-Jxn〉≥0,∀y∈C,(3.7)f(zn+m,y)+1rn+m〈y-zn+m,Jzn+m-Jxn+m〉≥0,∀y∈C.
Putting y=zn+m in (3.6) and y=zn in (3.7), we have
(3.8)f(zn,zn+m)+1rn〈zn+m-zn,Jzn-Jxn〉≥0,∀y∈C,f(zn+m,zn)+1rn+m〈zn-zn+m,Jzn+m-Jxn+m〉≥0,∀y∈C.
So, from (A2), we have
(3.9)〈zn+m-zn,Jzn-Jxnrn-Jzn+m-Jxn+mrn+m〉≥0,
and hence
(3.10)〈zn+m-zn,Jzn-Jxn-rnrn+m(Jzn+m-Jxn+m)〉≥0.
Thus, we have
(3.11)〈zn+m-zn,Jzn-Jzn+m+Jzn+m-Jxn-rnrn+m(Jzn+m-Jxn+m)〉≥0,
which implies that
(3.12)〈zn+m-zn,Jzn+m-Jzn〉≤〈zn+m-zn,Jzn+m-Jxn-rnrn+m(Jzn+m-Jxn+m)〉≥0.
By using Lemma 1.6, we obtain the following:
(3.13)cpcp-2p‖zn+m-zn‖p≤〈zn+m-zn,Jzn+m-Jxn-rnrn+m(Jzn+m-Jxn+m)〉≥0=〈zn+m-zn,(1-rnrn+m)Jzn+m+rnrn+m(Jxn+m-Jxn)〉.
Therefore, we get the following:
(3.14)cpcp-2p‖zn+m-zn‖p-1≤|1-rnrn+m|‖Jzn+m‖+‖rnrn+mJxn+m-Jxn‖.
On the other hand, for any p∈EP(f), from zn=Trnxn, we have
(3.15)‖zn-p‖=‖Trnxn-p‖≤‖xn-p‖,
so that {zn} is bounded. Since limn→∞rn=r>0, this together with (3.14) implies that {zn} is a Cauchy sequence. Hence Trnxn=zn is convergent.
(2) By using Lemma3.2, we know that
(3.16)⋂n=1∞F(Trn)=EP⁡(f)≠∅.
(3) From (1) we know that,limn→∞Trnx exists for all x∈C. So, we can define a mapping T from C into itself by
(3.17)Tx=lim⁡n→∞Trnx,∀x∈C.
It is obvious that T is nonexpansive. It is easy to see that
(3.18)EP⁡(f)=⋂n=1∞F(Trn)⊂F(T).
On the other hand, let w∈F(T) and wn=Trnw, we have
(3.19)f(wn,y)+1rn〈y-wn,Jwn-Jw〉≥0,∀y∈C.
By (A2) we know that
(3.20)1rn〈y-wn,Jwn-Jw〉≥f(y,wn),∀y∈C.
Since wn→Tw=w and from (A4), we have f(y,w)≤0, for all y∈C. Then, for t∈(0,1] and y∈C,
(3.21)0=f(ty+(1-t)w,ty+(1-t)w)≤tf(ty+(1-t)w,y)+(1-t)f(ty+(1-t)w,w)≤tf(ty+(1-t)w,y).
Therefore, we have
(3.22)f(ty+(1-t)w,y)≥0.
Letting t↓0 and using (A3), we get the following:
(3.23)f(w,y)≥0,∀y∈C,
and hence w∈EP(f). From above two respects, we know that F(T)=⋂n=0∞F(Trn).
Next we show{Trn} is uniformly closed. Assume xn→x and ∥xn-Trnxn∥→0, from above results, we know that Tx=limn→∞Trnx. On the other hand, from ∥xn-Trnxn∥→0, we also get limn→∞Trnx=x, so that x∈F(T)=⋂n=1∞F(Trn). That is, the sequence of mappings {Trn} is uniformly closed. This completes the proof.Theorem 3.5.
LetE be a  p-uniformly convex with  p≥2 and uniformly smooth Banach space, and let  C be a nonempty closed convex subset of  E. Let  f and  g be two bifunctions from  C×C to  R=(-∞,+∞) satisfying (A1)–(A4). Assume that  {αn}n=0∞,  {βn}n=0∞,  {γn}n=0∞, and  {δn}n=1∞ are four sequences in  [0,1] such that  αn+βn+γn+δn=1,  limn→∞αn=0,limn→∞δn=0, and  0<γ≤γn≤1 for some constant  γ∈(0,1). Let  {xn} be a sequence generated by
(3.24)x0∈Carbitrarily,yn=J-1(αnJx0+βnJxn+γnJTrnxn+δnJSrnxn),n≥1,Cn={z∈Cn-1:ϕ(z,yn)≤(1-αn)ϕ(z,xn)+αnϕ(z,x0)},n≥1,C0=C,xn+1=ΠCnx0,n≥0,
where
(3.25)Tr(x)={z∈C:f(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C},∀x∈E,Sr(x)={z∈C:g(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C},∀x∈E,
and limn→∞rn=r>0. Assume that the mappings Tn and Sn satisfy the asymptotically condition. Then {xn} converges strongly to q=ΠEP(f)∩∩EP(g)x0.Proof.
By Lemma3.4, {Trn}n=1∞, {Srn}n=1∞ are uniformly closed; therefore, by using Theorem 2.2 and Lemma 3.2, we can obtain the conclusion of Theorem 3.5. This completes the proof.Whenαn≡0, δn≡0 in the Theorem 3.5, we obtain the following result.Theorem 3.6.
LetE be a  p-uniformly convex with  p≥2 and uniformly smooth Banach space, and let  C be a nonempty closed convex subset of  E. Let  f be a bifunction from  C×C to  R=(-∞,+∞) satisfying (A1)–(A4). Assume that  {αn}n=0∞ is a sequences in  [0,1] such that  0≤αn≤α<1 for some constant  α∈(0,1). Define a sequence  {xn} in  C by the following algorithm:
(3.26)x0∈Carbitrarily,yn=J-1(αnJxn+(1-αn)JTrnxn),n≥1,Cn={z∈Cn-1:ϕ(z,yn)≤ϕ(z,xn)},n≥1,C0=C,xn+1=ΠCnx0,n≥0,
where limn→∞rn=r>0. Then {xn} converges strongly to q=ΠEP(f)x0.


## 4. Applications for Maximal Monotone Operators

In this section, we apply our above results to prove some strong convergence theorem concerning maximal monotone operators in a Banach spaceE.LetA be a multivalued operator from E to E* with domain D(A)={z∈E:Az≠∅} and range R(A)={z∈E:z∈D(A)}. An operator A is said to be monotone if (4.1)〈x1-x2,y1-y2〉≥0
for each x1,x2∈D(A) and y1∈Ax1, y2∈Ax2. A monotone operator A is said to be maximal if it's graph G(A)={(x,y):y∈Ax} is not properly contained in the graph of any other monotone operator. We know that if A is a maximal monotone operator, then A-10 is closed and convex. The following result is also wellknown.Theorem 4.1.
LetE be a reflexive, strictly convex, and smooth Banach space, and let  A be a monotone operator from  E to  E*. Then  A is maximal if and only if  R(J+rA)=E*. for all  r>0.LetE be a reflexive, strictly convex, and smooth Banach space, and let A be a maximal monotone operator from E to E*. Using Theorem 4.1 and strict convexity of E, we obtain that for every r>0 and x∈E, there exists a unique xr such that (4.2)Jx∈Jxr+rAxr.Then we can define a single-valued mappingJr:E→D(A) by Jr=(J+rA)-1J and such a Jr is called the resolvent of A, we know that A-10=F(Jr) for all r>0.Theorem 4.2.
LetE be a uniformly convex and uniformly smooth Banach space, let  A be a maximal monotone operator from  E to  E*, and let  Jr be a resolvent of  A for  r>0. Then for any sequence  {rn}n=1∞ such that  liminfn→∞rn>0,  {Jrn}n=1∞ is a uniformly closed sequence of hemi-relatively nonespansive mappings.Proof.
Firstly, we show that{Jrn}n=1∞ is uniformly closed. Let {zn}⊂E be a sequence such that zn→p and limn→∞∥zn-Jrnzn∥=0. Since J is uniformly norm-to-norm continuous on bounded sets, we obtain that
(4.3)1rn(Jzn-JJrnzn)⟶0.
It follows from
(4.4)1rn(Jzn-JJrnzn)∈AJrnzn
and the monotonicity of A that
(4.5)〈w-Jrnzn,w*-1rn(Jzn-JJrnzn)〉≥0
for all w∈D(A) and w*∈Aw. Letting n→∞, we have 〈w-p,w*〉≥0 for all w∈D(A) and w*∈Aw. Therefore, from the maximality of A, we obtain that p∈A-10=F(Jrn) for all n≥1, that is, p∈⋂n=1∞F(Jrn).
Next we show thatJrn is a hemi-relatively nonexpansive mapping for all n≥1. For any w∈E and p∈F(Jrn)=A-10, from the monotonicity of A, we have
(4.6)ϕ(p,Jrnw)=‖p‖2-2〈p,JJrnw〉+‖Jrnw‖2=‖p‖2+2〈p,Jw-JJrnw-Jw〉+‖Jrnw‖2=‖p‖2+2〈p,Jw-JJrnw〉-2〈p,Jw〉+‖Jrnw‖2=‖p‖2-2〈Jrw-p-Jrnw,Jw-JJrnw〉-2〈p,Jw〉+‖Jrnw‖2=‖p‖2-2〈Jrnw-p,Jw-JJrnw〉+2〈Jrnw,Jw-JJrnw〉-2〈p,Jw〉+‖Jrnw‖2≤‖p‖2+2〈Jrnw,Jw-JJrnw〉-2〈p,Jw〉+‖Jrnw‖2=‖p‖2-2〈p,Jw〉+‖w‖2-‖Jrnw‖2+2〈Jrnw,Jw〉-‖w‖2=ϕ(p,w)-ϕ(Jrnw,w)≤ϕ(p,w).
This implies that Jrn is a hemi-relatively nonexpansive mapping for all n≥1. This completes the proof.Theorem 4.3.
LetE be a uniformly convex and uniformly smooth Banach space, let  A and  B be two maximal monotone operators from  E to  E* with nonempty common zero point set  A-1(0)∩B-1(0), let  JrA be a resolvent of  A and  JrB a resolvent of  B for  r>0. Assume that  {αn}n=0∞,  {βn}n=0∞,  {γn}n=0∞, and  {δn}n=1∞ are four sequences in  [0,1] such that  αn+βn+γn=1,  limn→∞αn=0,  limn→∞δn=0, and  0<γ≤γn≤1 for some constant  γ∈(0,1). Let  {xn} be a sequence generated by
(4.7)x0∈Carbitrarily,yn=J-1(αnJx0+βnJxn+γnJJrnAxn+δnJJrnBxn),n≥1,Cn={z∈Cn-1:ϕ(z,yn)≤(1-αn)ϕ(z,xn)+αnϕ(z,x0)},n≥1,C0=C,xn+1=ΠCnx0,n≥0,
where liminfn→∞rn>0. Assume that the mappings JrnA and JrnB satisfy the asymptotically condition. Then {xn} converges strongly to q=ΠA-1(0)∩B-1(0)x0.Proof.
From Theorem4.2, {Jrn}n=1∞ is uniformly closed countable family of hemi-relatively nonexpansive mappings, on the other hand, A-1(0)=⋂n=1∞F(Jrn), by using Theorem 2.2, we can obtain the conclusion of Theorem 4.3.Whenαn≡0,δn≡0 in Theorem 4.3, we obtain the following result.Theorem 4.4.
LetE be a uniformly convex and uniformly smooth Banach space, let  A be a maximal monotone operator from  E to  E* with nonempty zero point set  A-1(0), and let  Jr be a resolvent of  A for  r>0. Assume that  {αn}n=0∞ is a sequence in  [0,1] such that  0≤αn≤α<1 for some constant  α∈(0,1). Define a sequence  {xn} in  C by the following algorithm:
(4.8)x0∈Carbitrarily,yn=J-1(αnJxn+(1-αn)JJrnxn),n≥1,Cn={z∈Cn-1:ϕ(z,yn)≤ϕ(z,xn)},n≥1,C0=C,xn+1=ΠCnx0,n≥0,
where liminfn→∞rn>0. Then {xn} converges strongly to q=ΠA-1(0)x0.


## 5. Examples

Firstly, we give an example which is hemi-relatively nonexpansive mapping but not weak relatively nonexpansive mapping.Example 5.1.
LetE=Rn and x0≠0 be a any element of E. We define a mapping T:E→E as follows:
(5.1)T(x)={(12+12n+1)x0ifx=(12+12n)x0,-xifx≠(12+12n)x0,
for n=1,2,3,…. Next we show that T is a hemi-relatively nonexpansive mapping but no weak relatively nonexpansive mapping. First, it is obvious that F(T)={0}. In addition, it is easy to see that
(5.2)‖Tx‖≤‖x‖,∀x∈E.
This implies that
(5.3)‖Tx‖2-‖x‖2≤2〈0,JTx-Jx〉=2〈p,JTx-Jx〉
for all p∈F(T). It follows from above inequality that
(5.4)‖p‖2-2〈p,JTx〉+‖Tx‖2≤‖p‖2-2〈p,Jx〉+‖x‖2,
for all p∈F(T) and x∈E. That is
(5.5)ϕ(p,Tx)≤ϕ(p,x),
for all p∈F(T) and x∈E; hence T is a hemi-relatively nonexpansive mapping. Finally, we show that T is not weak relatively nonexpansive mapping. In fact that, letting
(5.6)xn=(12+12n)x0,n=1,2,3,…
from the definition of T, we have
(5.7)Txn=(12+12n+1)x0,n=1,2,3,…
which implies that ∥xn-Txn∥→0 and xn→x0 (xn⇀x0) as n→∞. That is x0∈F̃(T) but x0∈¯F(T).
Next, we give an example which is weak relatively nonexpansive mapping but not relatively nonexpansive mapping.Example 5.2.
LetE=l2, where
(5.8)l2={ξ=(ξ1,ξ2,ξ3,…,ξn,…):∑n=1∞|ξn|2<∞},‖ξ‖=(∑n=1∞|ξn|2)1/2,∀ξ∈l2,〈ξ,η〉=∑n=1∞ξnηn,∀ξ=(ξ1,ξ2,ξ3,…,ξn,…),η=(η1,η2,η3,…,ηn,….)∈l2.
It is well known that l2 is a Hilbert space, so that (l2)*=l2. Let {xn}⊂E be a sequence defined by
(5.9)x0=(1,0,0,0,…),x1=(1,1,0,0,…),x2=(1,0,1,0,0,…),x3=(1,0,0,1,0,0,…),⋮xn=(ξn,1,ξn,2,ξn,3,…,ξn,k,…),
where
(5.10)ξn,k={1ifk=1,n+1,0ifk≠1,k≠n+1,
for all n≥1. Define a mapping T:E→E as follows:
(5.11)T(x)={nn+1xnifx=xn(∃n≥1),-xifx≠xn(∀n≥1).Conclusion 1.
{xn} converges weakly to  x0.Proof.
For anyf=(ζ1,ζ2,ζ3,…,ζk,…)∈l2=(l2)*, we have
(5.12)f(xn-x0)=〈f,xn-x0〉=∑k=2∞ζkξn,k=ζn+1⟶0,
as n→∞. That is, {xn} converges weakly to x0.Conclusion 2.
{xn} is not a Cauchy sequence, so that, it does not converge strongly to any element of  l2.Proof.
In fact, we have∥xn-xm∥=2 for any n≠m. Then {xn} is not a Cauchy sequence.Conclusion 3.
T has a unique fixed point  0, that is,  F(T)={0}.Proof .
The conclusion is obvious.Conclusion 4.
x0 is an asymptotic fixed point of  T.Proof.
Since{xn} converges weakly to x0 and
(5.13)‖Txn-xn‖=‖nn+1xn-xn‖=1n+1‖xn‖⟶0
as n→∞, so that, x0 is an asymptotic fixed point of T.Conclusion 5.
T has a unique strong asymptotic fixed point  0, so that  F(T)=F̃(T).Proof .
In fact that, for any strong convergent sequence{zn}⊂E such that zn→z0 and ∥zn-Tzn∥→0 as n→∞, from Conclusion 2, there exists sufficiently large nature number N such that zn≠xm, for any n,m>N. Then Tzn=-zn for n>N, and it follows from ∥zn-Tzn∥→0 that 2zn→0 and hence zn→z0=0.Conclusion 6.
T is a weak relatively nonexpansive mapping.Proof.
SinceE=l2 is a Hilbert space, we have
(5.14)ϕ(0,Tx)=‖0-Tx‖2=‖Tx‖2≤‖x‖2=‖x-0‖2=ϕ(0,x),∀x∈E.
From Conclusion 2, we have F(T)=F̃(T), then T is a weak relatively nonexpansive mapping.Conclusion 7.
T is not a relatively nonexpansive mapping.Proof.
From Conclusions3 and 4, we have F(T)≠F̂(T), so that T is not a relatively nonexpansive mapping.



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*Source: 401960-2012-02-28.xml*