Eric U.OFOEDU Charles E.ONYI

Department of Mathematics，Nnamdi Azikiwe University，Awka，Anambra State，Nigeria

APPROXIMATION OF COMMON FIXED POINT OF FAMILIES OF NONLINEAR MAPPINGS WITH APPLICATIONS?

Eric U.OFOEDU Charles E.ONYI

Department of Mathematics，Nnamdi Azikiwe University，Awka，Anambra State，Nigeria

E-mail：euofoedu@yahoo.com；charles.onyi@gmail.com

It is our purpose in this paper to show that some results obtained in uniformly convex real Banach space with uniformly G?ateaux differentiable norm are extendable to more general reflexive and strictly convex real Banach space with uniformly G?ateaux differentiable norm.Demicompactness condition imposed in such results is dispensed with.Furthermore，Applications of our theorems to approximation of common fixed point of countable infinite family of continuous pseudocontractive mappings and approximation of common solution of countable infinite family of generalized mixed equilibrium problems are also discussed.Our theorems improve，generalize，unify and extend several recently announced results.

nonexpansive mappings，reflexive real Banach spaces；fixed point；uniformly G?ateaux differentiable norm

2010 MR Subject Classification 47H06；47H09；47J05；47J25

1 Introduction

Let E be a real normed space E.A mapping T:D（T）?E→ R（T）?E is called nonexpansive if and only if‖Tx-Ty‖≤‖x-y‖?x，y∈D（T），where D（T）and R（T）denote the domain and the range of the mapping T，respectively.In what follows，we shall require that D（T）?R（T）and denote the fixed point set of an operator T:D（T）→R（T）by Fix（T），that is，F(xiàn)ix（T）:=｛x∈D（T）:Tx=x｝.

Most published results on nonexpansive mappings centered on existence theorems for fixed points of these mappings and iterative approximation of such fixed points.DeMarr［16］in 1963 studied the problem of existence of common fixed point for a family of nonexpansive mappings. He proved the following theorem:

Theorem 1.1（DeMarr［16］） Let E be a real Banach space and let K be a nonempty compact convex subset of E.If ? is a nonempty commuting family of nonexpansive mappings of K into itself，then the family ? has a common fixed point in K.

In 1965，Browder［4］proved the result of DeMarr in a uniformly convex real Banach space E，requiring that K is only bounded，closed，convex and nonempty subset of E.For otherfixed point theorems for families of nonexpansive mappings，the reader may consult any of the following references:Belluce and Kirk［2］，Lim［21］and Bruck［6］.

Considerable research efforts were devoted to developing iterative methods for approximating common fixed points of families of several classes of nonlinear mappings（see e.g.［1，7，11-14，17，18，27］and references there in）.

Maing′e［22］studied the Halpern-type scheme for approximation of a common fixed point of a countable infinite family of nonexpansive mappings in real Hilbert space.Let｛Ti｝i≥1be a countable infinite family of nonexpansive mappings.Define NI:=｛i∈N:Ti/=I｝（I being the identity mapping on a real normed space E）.Maing′e proved the following theorems

Theorem 1.2（Maing′e［22］）Let K be a nonempty closed convex subset of a real Hilbert space H.Let｛Ti｝i≥1be a countable family of nonexpansive self-mappings of K，｛αn｝n≥1and｛σi，n｝n≥1，i∈N be sequences in（0，1）satisfying the following conditions:

Theorem 1.3（Maing′e［22］）Let K be a nonempty closed convex subset of a real Hilbert space H.Let｛Ti｝i≥1be a countable family of nonexpansive self-mappings of K，｛αn｝n≥1and｛σi，n｝n≥1，i∈N be sequences in（0，1）satisfying the following conditions:

converges strongly to a unique fixed point of the contraction PFof，where f:K→K is a strict contraction；and PFis the metric projection from H onto F.

In［9］，Chidume et al.proved theorems that extended Theorems 1.2 and 1.3 to ?pspaces，1＜p＜∞.Furthermore，they proved new convergence theorems which are applicable in Lpspaces，1＜p＜∞.Moreover，in their more general setting，some of the conditions on the sequences｛αn｝n≥1and｛σi，n｝n≥1，imposed in Theorem 1.3 were dispensed with or weakened.

Chidume and Chidume［10］proved the following theorems which extended the results obtained by Maing′e［22］and Chidume et al.［9］:

Theorem 1.4（Chidume and Chidume［10］）Let E be a uniformly convex real Banach space.Let K be a closed，convex and nonempty subset of E.Letbe a family of nonexpansive self-mappings of K.Letbe a sequence in（0，1）such thatand=0 for all i∈.Define a family of nonexpansive mappings，where I is is an identity map of K and δ∈（0，1）is fixed.Let｛ztn｝be a sequence satisfying

Theorem 1.5（Chidume and Chidume［10］）Let E be a uniformly convex real Banach space with uniformly G?ateaux differentiable norm.Let K be a closed，convex and nonempty subset of E.Letbe a family of nonexpansive self-mappings of K.For arbitrary fixed δ∈（0，1），define a family of nonexpansive mappingswhere I is is an identity map of K.Assume F:andbe sequences in（0，1）satisfying the following conditions:

Define a sequence｛xn｝iteratively by x1，u∈K，

If at least one of the maps Ti，i=1，2，3，···is demicompact then｛xn｝converges strongly to an element in F

Motivated by the results of Maing′e［22］，Chidume et al.［9］，and Chidume and Chidume［10］，it is our aim in this paper to provide a method of proof which enabled us to obtain the conclusion of Chidume and Chidume［10］in more general reflexive and strictly convex real Banach space with unifromly G?ateaux differentiable norm；and the demicompactness condition imposed in［10］is dispensed with.As applications of our theorems，we obtained strong convergence theorems for approximation of common fixed point of countable infinite family of pseudocontractive mappings in real Hilbert space；in addition，strong convergence theorems for approximation ofcommon solution of countable infinite family of generalized mixed equilibrium problem are also obtained in a real Hilbert space.Our theorems augument，extend，generalize and unify the correponding results of Maing′e［22］，Chidume et al.［9］，and Chidume and Chidume［10］.Our method of proof is of independent interest.

2 Preliminaries

Let E be a real Banach space with dual E?.We denote by J the normalized duality mapping from E to 2E?defined by

where<·，·>denotes the generalized duality pairing between members of E and E?.It is well known that if E?is strictly convex then J is single-valued（see，e.g.，［8，28］）.In the sequel，we shall denote the single-valued normalized duality mapping by j.

Let S:=｛x∈E:‖x‖=1｝.The space E is said to have a G?ateaux differentiable norm if and only if the limit

exists for each x，y∈S，while E is said to have a uniformly G?ateaux differentiable norm if for each y∈S the limit is attained uniformly for x∈S.It is well known that whenever a Banach space has uniformly G?ateaux differentiable norm，then the normalized duality mapping is norm to weak?uniformly continuous on bounded subsets of E.

Let E be a real normed space.The modulus of convexity of E is the function δE:［0，2］→［0，1］defined by

The space E is said to be uniformly convex if and only if δE（?）＞0??∈（0，2］；and the space E is called strictly convex if and only if for all x，y∈E such that‖x‖=‖y‖=1 and for all λ∈（0，1）we have‖λx+（1-λ）y‖＜1.It is well known that every uniformly convex real Banach space is strictly convex and reflexive real Banach space，where we know that a real Banach space E is reflexive if and only if every bounded sequence in E has a subsequence which converges weakly.

A mapping T:D（T）?E→ E is said to be demicompact at h if and only if for any bounded sequence｛xn｝n≥1in D（T）such that（xn-Txn）→ h as n→ ∞，there exists a subsequence say｛xnj｝j≥1of｛xn｝n≥1and x?∈D（T）such that｛xnj｝j≥1converges strongly to x?and x?-Tx?=h.

Letμbe a bounded linear functional defined on ?∞satisfying‖μ‖=1=μ（1，1，···，1，···）. It is known thatμis a mean on N if and only if

for every a= （a1，a2，a3，···）∈ ?∞.In the sequel，we shall use the notationμn（an）instead ofμ（a）.A meanμon N is called a Banach limit ifμn（an）= μn（an+1）for every a=（a1，a2，a3，···）∈?∞.It is well known that ifμis a Banach limit，then

In what follows，we shall need the following lemmas.

Lemma 2.1 Let E be a real normed space，then

for all x，y∈E and for all j（x+y）∈J（x+y）.

Lemma 2.2（Lemma 3 of Bruck［5］） Let K be a nonempty closed convex subset of a strictly convex real Banach space E.Letbe a sequence of nonexpansive mappings from K to E such thatLetbe a sequence of positive numbers such that，then a mapping T on K defined by Txfor all x∈K is well defined， nonexpansive and Fix（T

Lemma 2.3（Xu［27］）Let｛an｝be a sequence of nonnegative real numbers satisfying the following relation:

（ii）limsupσn≤0.

Then，an→0 as n→∞.

Lemma 2.5（Kikkawa and Takahashi［19］）Let Let K be a nonempty closed convex subset of a Banach spaces E with a uniformly G?ateaux differentiable norm，let｛xn｝be a bounded sequence of E and letμbe a mean on N.Let z∈K.Then

3 Implicit Iterative Method for Countable Infnity Family of Nonexpansive Mappings

We begin with the following lemma:

Lemma 3.1（Chidume and Chidume［10］）Let E be a real Banach space.Let Ti:E→E， i=1，2，···，be a countable infinite family of nonexpansive mappings.Leti=1，2，···be sequences in（0，1）such that.Fix a δ∈（0，1）and define afamily of mappings Si:E→E by Six=（1-δ）x+δTix?x∈E，i=1，2，···.For fixed u∈E and for all n∈N，define a map Φn:E→E by Φnx=αnu+σi，nSix，?x∈E，then Φnis a strict contraction on E.Hence，for all n∈N，there is a unique z∈E satisfying

n

Hence，Ψx∈??x∈?，that is，? is invariant under Ψ.Let x?∈Fix（Ψ），then since every closed convex nonempty subset of a reflexive and strictly convex Banach space is a Chebyshev set（see e.g.，［23］，Corollary 5.1.19），there exists a unique u?∈? such that

But x?=Ψx?and Ψu?∈?.Thus，

So，Ψu?=u?.Hence，F(xiàn)ix（Ψ）∩?/=?.This completes the proof. □

In particular，we have that

Now，using（3.1），we have that

So，

Again，taking Banach limit，we obtain

We now show that u?=z?.Suppose for contradiction that u?/=z?，then

But using（3.1），we have that

Thus，

Similarly，we also obtain that≤0 or rather

Adding（3.4）and（3.5），we have that‖z?-u?‖≤0，a contradiction.Thus，z?=u?.Hence，converges strongly toThis completes the proof.

4 Explicit Iterative Method for Countable Infinite Family of Nonexpansive Mappings

For the rest of this paper，｛αn｝∞n=1and｛σi，n｝∞n=1are sequences in（0，1）satisfying the following additional conditions:

Then，

for some M＞0.Thus，

From（4.1），we have that

Using Lemma 2.1，we have that

This implies that

and hence，

Also，since j is norm-to-weak?uniformly continuous on bounded subsets of E，we have that

Moreover，we have that

Using（4.3），（4.4）and（4.5），we obtain from（4.6）that

Finally，using Lemma 2.1 we obtain from（4.1）that

Using（4.7）and Lemma 2.3 in（4.8），we get thatconverges strongly to common fixed point of the familyof nonexpansive mappings. □

5 Application to Approximation of Common Fixed Points of Counably Infnite Family of Continuous Pseudocontractive Mappings

The most important generalization of the class of nonexpansive mappings is，perhaps，the class of pseudocontractive mappings.These mappings are intimately connected with the important class of nonlinear accretive operators.This connection will be made precise in what follows.

A mapping T′with domain D（T′），and range R（T′），in E is called pseudocontractive if and only if for all x，y∈D（T′），the following inequality holds:

for all r＞0.As a consequence of a result of Kato［20］，the pseudocontractive mappings can also be defined in terms of the normalized duality mappings as follows:the mapping T′is calledpseudocontractive if and only if for all x，y∈D（T′），there exists j（x-y）∈J（x-y）such that

It now follows trivially from（5.2）that every nonexpansive mapping is pseudocontractive.We note immediately that the class of pseudocontractive mappings is larger than that of nonexpansive mappings.For examples of pseudocontractive mappings which are not nonexpansive，the reader may see［8］.

To see the connection between the pseudocontractive mappings and the accretive mappings，we introduce the following definition:a mapping A with domain，D（A），and range，R（A），in E is called accretive if and only if for all x，y∈D（A），the following inequality is satisfied:

for all r＞0.Again，as a consequence of Kato［20］，it follows that A is accretive if and only if for all x，y∈D（A），there exists j（x-y）∈J（x-y）such that

It is easy to see from inequalities（5.1）and（5.3）that an operator A is accretive if and only if the mapping T′:=（I-A）is pseudocontractive.Consequently，the fixed point theory for pseudocontractive mappings is intimately connected with the mapping theory of accretive operators.For the importance of accretive operators and their connections with evolution equations，the reader may consult any of the references［8，24］.

Due to the above connection，fixed point theory of pseudocontractive mappings became a flourishing area of intensive research for several authors.It is of interest to note that if E=H is a Hilbert space，accretive operators coincide with the monotone operators，where an operator A with domain，D（A），and range，R（A），in H is called monotone if and only if for all x，y∈D（A），we have that

Recently，Zegeye［30］established the following lemmas.

Lemma 5.1（Zegeye［30］） Let K be a nonempty closed convex subset of a real Hilbert space H.Let T′:K→H be a continuous pseudocontractive mapping，then for all r＞0 and x∈H，there exists z∈K such that

Lemma 5.2（Zegeye［30］）Let K be a nonempty closed convex subset of a real Hilbert space H.Let T′:K→K be a continuous pseudocontractive mapping，then for all r＞0 and x∈H，define a mapping Fr:H→K by

then the following hold:

（1）Fris single-valued；

（2）Fris firmly nonexpansive type mapping，i.e.，for all x，y∈H，

（3）Fix（Fr）is closed and convex；and Fix（Fr）=Fix（T′）for all r＞0.

Remark 5.3 We observe that Lemmas 5.1 and 5.2 hold in particular for r=1.Thus，ifis a family of continuous pseudocontractive mappings and we define

Theorem 5.4 Let H be a real Hilbert space.Let T′

where Six=（1-δ）x+δF（i）1x?x∈H，i=1，2，···.Letbe a sequence in（0，1）such thatand-λi|=0.Let Ψ:=（1-δ）I+δT，where T:=，thenconverges strongly to an element of

6 Application to Approximation of Common Solution of Countably Infinite Generalized Mixed Equilibrium Problems

Let K be a closed convex nonempty subset of a real Hilbert space H with inner product ?·，·?and norm‖·‖.Let f:K×K → R be a bifunction and Φ:K → R∪｛+∞｝be a proper extended real valued function，where R denotes the set of real numbers.Let Θ:K→H be a nonlinear monotone mapping.The generalized mixed equilibrium problem（abbreviated GMEP）for f，Φ and Θ is to find u?∈K such that

The set of solutions for GMEP（6.1）is denoted by

If Φ≡0≡Θ in（6.1），then（6.1）reduces to the classical equilibrium problem（abbreviated EP），that is，the problem of finding u?∈K such that

and GMEP（f，0，0）is denoted by EP（f），where

If f≡0≡Φ in（6.1），then GMEP（6.1）reduces to the classical variational inequality problem and GMEP（0，0，Θ）is denoted by VI（Θ，K），where

If f≡0≡Θ，then GMEP（6.1）reduces to the following minimization problem:

and GMEP（0，Φ，0）is denoted by Argmin（Φ），where

If Θ≡0，then（6.1）becomes the mixed equilibrium problem（abbreviated MEP）and GMEP（f，Φ，0）is denoted by MEP（f，Φ），where

If Φ≡0，then（6.1）reduces to the generalized equilibrium problem（abbreviated，GEP）and GMEP（f，0，Θ）is denoted by GEP（f，Θ），where

If f≡0，then GMEP（6.1）reduces to the generalized variational inequality problem（abbreviated GVIP）and GMEP（0，Φ，Θ）is denoted by GVI（Φ，Θ，K），where

The generalized mixed equilibrium problem（GMEP）includes as special cases the monotone inclusion problems，saddle point problems，variational inequality problems，minimization problems，optimization problems，vector equilibrium problems，Nash equilibria in noncooperative games.Furthermore，there are several other problems，for example，the complementarity problems and fixed point problems，which can also be written in the form of the generalized mixed equilibrium problem.In other words，the generalized mixed equilibrium problem is a unifying model for several problems arising from engineering，physics，statistics，computer science，optimization theory，operations research，economics and countless other fields.For the past 20 years or so，many existence results have been published for various equilibrium problems（see e.g.［3，25，29］）.

In the sequel，we shall require that the bifunction f:K×K→R satisfies the following conditions:

（A1）f（x，x）=0?x∈K；

（A2）f is monotone，in the sense that f（x，y）+f（y，x）≤0 for all x，y∈K；

t→0

（A4）the function y→f（x，y）is convex and lower semicontinuous for all x∈K.

Lemma 6.1（Compare with Lemma 2.4 of［25］）Let C be a closed convex nonempty subset of a real Hilbert space H.Let f:K×K → R be a bifunction satisfying conditions（A1）-（A4）；Θ:K→H a continuous monotone mapping and Φ:K→R∪｛+∞｝a proper lower semicontinuous convex function.Then，for all r＞0 and x∈H there exists u∈K such that

Moreover，if for all x∈H we define a mapping Gr:H→2Kby

then the following hold:

（1）Gris single-valued for all r＞0；

（2）Gris firmly nonexpansive，that is，for all x，z∈H，

（3）Fix（Gr）=GMEP（f，Φ，Θ）for all r＞0；

（4）GMEP（f，Φ，Θ）is closed and convex.

Remark 6.2 We observe that Lemmas 6.1 holds in particular for r=1.Thus，if we define

where Six=（1-δ）x+?x∈H，i=1，2，···.Letbe a sequence in（0，1）such that=1 andLet Ψ:=（1-δ）I+δT，where T:，then｛zn｝converges strongly to an element of

where Six=（1-δ）x+?x∈H，i=1，2，···.Letbe a sequence in（0，1）such that=1 and-λi|=0.Let Ψ:=（1-δ）I+δT，where T:=，then｛xn｝converges strongly to an element ofGMEP（fi，Φi，Θi）.

Remark 6.5 Prototypes for our iteration parameters are:

Remark 6.6 It is well known that every real Hilbert is a reflexive and strictly convex real Banach space with uniformly G?ateaux differentiable norm；thus Theorems 5.4，5.5，6.3 and 6.4 hold.

Remark 6.7 The addition of bounded error terms in any of our recursion formulas leads to no further generalization.

Remark 6.8 If f:K → K is a contraction map and we replace u by f（xn）in the recursion formulas of our theorems，we obtain what some authors now call viscosity iteration process.We observe that all our theorems in this paper carry over trivially to the so-called viscosity process.One simply replaces u by f（xn），repeats the argument of this paper，using the fact that f is a contraction map.Furthermore，we must note that method of proof of Theorems 3.4 and 4.1 easily carries over to the so-called nonself nonexpansive mappings.

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?Received February 20，2013；revised March 13，2015.