Iterative Approximation of a Common Zero of a Countably Infinite Family of

Volume 2008, Article ID 325792,13pages doi:10.1155/2008/325792

Research Article

Iterative Approximation of a Common Zero of a Countably Infinite Family of

m -Accretive Operators in Banach Spaces

E. U. Ofoedu

Department of Mathematics, Nnamdi Azikiwe University, P.M.B. 5025, Awka, 23448 Anambra, Nigeria

Correspondence should be addressed to E. U. Ofoedu,[email protected] Received 1 September 2007; Accepted 4 February 2008

Recommended by Tomas Dominguez Benavides

LetEbe a real reflexive and strictly convex Banach space which has a uniformly Gˆateaux differen- tiable norm and letCbe a closed convex nonempty subset ofE. Strong convergence theorems for approximation of a common zero of a countably infinite family ofm-accretive mappings fromC toEare proved. Consequently, we obtained strong convergence theorems for a countably infinite family of pseudocontractive mappings.

Copyrightq2008 E. U. Ofoedu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

LetEbe a real Banach space with dualE^∗.The normalized duality mapping is the mappingJ : E→2^E∗defined for allx∈Eby

Jx

f^∗∈E^∗: x, f^∗

x²,f^∗x

, 1.1

where·,·denotes the generalized duality pairing between members ofEandE^∗. It is well known that ifE^∗is strictly convex, thenJis single valued. In what follows, the single-valued normalized duality mapping will be denoted byj.

LetE,·be a normed linear space. The norm·is said to be uniformly Gˆateaux differ- entiable if for eachy∈S{x∈E:x1}, the limit

limt→0

xty − x

t 1.2

exists uniformly forx∈S.It is well known thatLpspaces, 1< p <∞,have uniformly Gˆateaux differentiable normsee, e.g., 1. Furthermore, ifEhas a uniformly Gˆateaux differentiable

norm, then the duality mapping is norm-to-weak^∗uniformly continuous on bounded subsets ofE.

LetCbe a nonempty subset of a normed linear spaceE. A mappingT :C→Eis said to be nonexpansive if

Tx−Ty ≤ x−y ∀x, y∈C. 1.3

Most published results on nonexpansive mappings centered on existence theorems for fixed points of these mappings, and iterative approximation of such fixed points.

DeMarr2in 1963 studied the problem of existence of common fixed point for a family of nonlinear nonexpansive mappings. He proved the following theorem.

Theorem 1.1DM. LetEbe a Banach space andCbe a nonempty compact convex subset ofE. IfΩ is a nonempty commuting family of nonexpansive mappings ofCinto itself, then the familyΩhas a common fixed point inC.

In 1965, Browder3proved the result of DeMarr in a uniformly convex Banach space, requiring thatCbe only bounded, closed, convex, and nonempty. For other fixed-point theo- rems for families of nonexpansive mappings, the reader may consult Belluce and Kirk4, Lim 5, and Bruck Jr.6.

In 1973, Bruck Jr.7considered the study of structure of the fixed-point setFT {x∈ C:Txx}of nonexpansive mappingTand established several results.

Kirk8introduced an iterative process given by

xn1α0xnα1Txnα2T²xn· · ·αrT^rxn, 1.4 whereαi ≥ 0, α0 > 0 and_r

i0αi 1,for approximating fixed points of nonexpansive map- pings on convex subset of uniformly convex Banach spaces. Maiti and Saha9worked on and improved the results of Kirk8.

Considerable research efforts have been devoted to develop iterative methods for ap- proximating common fixed pointswhen such fixed points existof families of several classes of nonlinear mappingssee, e.g.,10–18.

LetCbe a nonempty closed and bounded subset of a real Banach spaceE. LetTi :C→ C, i1,2, . . . , rbe a finite family of nonexpansive mappings and let

Sα0Iα1T1α2T2· · ·αrTr, 1.5 where αi ≥ 0, α1 > 0, and_r

i0αi 1.Then the family{Ti}^r_i1 such that the common fixed- point setF:_r

i1FTi/∅is said to satisfy conditionAsee, e.g.,9,19,20if there exists a nondecreasing functionφ:0,∞→0,∞withφ0 0, φε>0 for allε∈0,∞,such thatx−Sx ≥φdx, Ffor allx∈C,wheredx, F inf{x−z:z∈F}.

Liu et al.19introduced the following iteration process:

x0∈C, xn1Sxn, n≥0 1.6

and showed that {xn}_n≥0 defined by1.6 converges to a common fixed point of {Ti}^r_i1 in Banach spaces, provided that{Ti}^r_i1satisfy conditionA. The result of Liu et al.19improves

the corresponding results of Kirk8, Maiti and Saha9, Senter and Dotson20and those of a host of other authors. However, the assumption that the family{Ti}^r_i1satisfies conditionA is strong.

Let Ebe a reflexive and strictly convex Banach space with a uniformly Gˆateaux dif- ferentiable norm. LetTi : E → E, i 1,2, . . . , r be nonexpansive mappings and {xn}_n≥0 a sequence inEdefined iteratively by1.6and suppose thatJ⁻¹ : E^∗ → Eis weakly sequen- tially continuous at 0. IfF : _r

i1FTi/∅, then Jung21in 2002 proved that, under this situation,{xn}_n≥0 converges weakly to a common fixed point of{Ti}^r_i1. In22, Gossez and Lami Dozo proved that for any normed linear space E, the existence of a weakly sequen- tially continuous duality mapping implies that the spaceE satisfies Opial’s conditionthat is, for all sequences{xn}inEsuch that{xn}converges weakly to somex∈E, the inequality lim infn→∞xn−y>lim infn→∞xn−xholds for ally /x, see e.g.,23. It is well known that Lpspaces, 1< p <∞, p /2,do not satisfy Opial’s condition. Consequently, the results of Jung 21are not applicable inLpspaces 1< p <∞, p /2.

Another class of nonlinear mappings now studied is the class of accretive operators. Let Ebe a real normed linear space. A mappingA :DA ⊂ E → Eis said to be accretive if the following inequality holds:

x−y ≤x−ysAx−Ay ∀s >0,∀x, y∈DA, 1.7 whereDAdenotes the domain of the operatorA. It is not difficult to deduce from1.7that the mappingAis accretive if and only ifIsA⁻¹is nonexpansive on the range ofIsA, whereIdenotes the identity operator defined onE. We note that the range,RIsA,ofIsA needs not be all ofE.WhenAis accretive and, in addition, the range ofIsAis all ofE,then Ais called m-accretive.

Our presentation in this paper is primarily motivated by the study of equations of the form

ut Aut f, u0 u0, f∈E. 1.8

It is well known that many physically significant problems can be modeled by equations of the form1.8 whereAis accretive, which is generally called Evolution Equation. Typical exam- ples where such evolution equations occur can be found in the heat, wave, and Schr ¨odinger equationssee, e.g.,24. One of the fundamental results in the theory of accretive operators, due to Browder25, states that ifAis locally Lipschitzian and accretive, thenAism-accretive and this implies that1.8has a solutionu^∗ ∈ DAfor anyf ∈ Ein particular forf 0.

This result was subsequently generalized by Martin26to continuous accretive operators. If in1.8,f0 andutis independent oft, then1.8reduces to

Au0 1.9

whose solutions correspond to the equilibrium points of1.8. There is no known method to obtain a closed form solution of1.9. The general approach for approximating a solution of 1.9is to transform it into a fixed-point problem. DefiningT :I−A,we observe thatx^∗is a solution of1.9if and only ifx^∗is a fixed point ofT i.e.,x^∗∈Tx^∗. Browder25called such an operatorTpseudocontractive.

Consequently, the study of methods of approximating fixed points of pseudocontractive maps, which correspond to equilibrium points of the system1.8, became a flourishing area of research for numerous mathematicianssee, e.g.,27–31.

Remark 1.2. We observe that a mappingA : I−T is accretive if and only if the mappingT is pseudocontractive. It is, therefore, not difficult to seeusing1.7that every nonexpansive mapping is pseudocontractive. The converse, however, does not hold. The following illustrates this fact.

Example 1.3. LetT:0,1→R,|·|be defined by

Tx

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ x−1

2 ifx∈ 0,1

2

, x−1 ifx∈

1 2,1

.

1.10

Clearly, T is not continuous and thus cannot be nonexpansive. Now, let s > 0, then for x, y∈0,1/2∪1/2,1we obtain that|x−ysI−Tx−I−Ty| ≥ |x−y|.So,Tis pseu- docontrative but not nonexpansive. Thus, the class of pseudocontractive mappings properly contains the class of nonexpansive mappings. Moreover, we see in particular that the operator Ais accretive, if and only if the mappingJA: IA⁻¹is a single-valued nonexpansive map- ping fromRIAtoDAand thatFJA NA,whereNA {x∈DA:Ax0}and FJA {x∈E:JAxx}.see, e.g.,1.

LetCbe a nonempty closed convex subset of a real reflexive and strictly convex Banach spaceEwhich has a uniformly Gˆateaux differentiable norm. LetAi :C→E, i1,2, . . . , rbe a finite family ofm-accretive mappings withN _r

i1NAi/∅.Suppose that every bounded closed convex nonempty subset ofEhas the fixed-point property for nonexpansive mappings;

Zegeye and Shahzad 32 constructed an iterative sequence which converges strongly to a common solution of the equationsAix0, i1,2, . . . , r.

It is our purpose in this paper to construct an iterative algorithm for the approximation of a common zero of a countably infinite family of m-accretive operators in Banach spaces. As a result, we obtain strong convergence theorems for approximation of a common fixed point of a countably inftinite family{Tk}_k∈N of pseudocontractive mappings, provided thatI−Tkism- accretive for allk∈N.Our theorems improve, generalize, and extend the correponding results of Zegeye and Shahzad32 and several other results recently announcedseeRemark 3.18 of this paperfrom a finite family{Ai}^r_i1 ofm-accretive mappings to a countably infinite family {Ak}_k∈Nofm-accretive mappings. Furthermore, our theorems are applicable, in particular in Lpspaces 1< p <∞, and our method of proof is of independent interest.

2. Preliminaries

In the sequel, the following Lemmas and Theorems will be used.

Lemma 2.1see, e.g.,18,27,33. Let{λn}_n≥1be a sequence of nonnegative real numbers satisfying the condition

λn1≤ 1−αn

λnσn, n≥0, 2.1

where {αn}_n≥0 and {σn}_n≥0 are sequences of real numbers such that {αn}_n≥1 ⊂ 0,1,_∞

n1αn

∞. Suppose that σn oαn, n ≥ 0 (i.e., limn→∞σn/αn 0) or _∞

n1|σn| < ∞ or lim sup_n→∞σn/αn≤0, thenλn→0 asn→ ∞.

Lemma 2.2. LetEbe a real normed linear space. Then the following inequality holds: for allx, y∈E, for alljxy∈Jxy,

xy²≤ x²2

y, jxy

. 2.2

Lemma 2.3see7, Lemma 3, page 257. LetCbe a nonempty closed and convex subset of a real strictly convex Banach spaceE. Let{Tk}_k∈Nbe a sequence of nonself nonexpansive mappingsTk:C→ E.Then there exists a nonexpansive mappingT :C→Esuch thatFT _∞

k1FTk.

Proof. If the sequence{Tk}_k∈N does not have a common fixed point, we can assumeT to be translation by any nonzero vector in which caseFT _∞

k1FTk ∅.Otherwise, letx^∗be a common fixed point of{Tk}_k∈N.Let{ξk}_k≥1be any sequence of positive real numbers such that _∞

k1ξk1 and setT :_∞

k1ξkTk.Then the mappingTis well defined, since

Tkx≤Tkx−Tkx^∗Tkx^∗≤x−x^∗x^∗. 2.3 Thus,_∞

k1ξkTkxconverges absolutely for eachx∈C.It is easy to see thatT is nonexpansive and mapsCintoE. Next, we claim thatFT _∞

k1FTk.The inclusion_∞

k1FTk⊂FTis obvious. We prove the reverse inclusion only. Suppose thatTx0x0.Then

x0−x^∗Tx0−x^∗

∞ k1

ξkTkx0−x^∗

∞ k1

ξk

Tkx0−x^∗

≤^∞

k1

ξkTkx0−x^∗.

2.4

ButTkx^∗x^∗andTkare nonexpansive for allk∈N,soTkx0−x^∗ ≤ x0−x^∗.Since_∞

k1ξk1, 2.4implies that

∞ k1

ξkTkx0−x^∗

x0−x^∗, Tkx0−x^∗x0−x^∗ ∀k∈N.

2.5

SinceE is strictly convex and eachξk > 0 while _∞

k1ξk 1,2.5implies thatTkx0−x^∗ Tmx0−x^∗for allk, m∈N,that is,Tkx0Tmx0for allk, m∈N.Hence,

x0Tx0^∞

k1

ξkTkx0^∞

k1

ξkTmx0Tmx0 ∀m∈N. 2.6 Thus,x0∈_∞

m1FTm.This completes the proof.

Remark 2.4. The proof ofLemma 2.3is as given by Bruck Jr.7. We included it here for com- pleteness of our presentation in this paper.

Theorem 2.5I. (see e.g., [1]). LetAbe a continuous accretive operator defined on a real Banach space EwithDA E.ThenAism-accretive.

Theorem 2.6MJ. (see [34]). Let Cbe a closed convex nonempty subset of a real reflexive Banach spaceEwhich has uniformly Gˆateaux differentiable norm andT : C → Ea nonexpansive mapping withFT/∅.Suppose that every bounded closed convex nonempty subset ofChas the fixed-point property for nonexpansive mappings, then there exists a continuous patht→zt,0< t <1 satisfying zttu 1−tTzt,for arbitrary but fixedu∈C,which converges strongly to a fixed point ofT.

3. Main results

For the rest of this paper,{αn}_n≥1is a real sequence such that{αn}_n≥1 ⊂0,1and satisfiesi limn→∞αn0;ii_∞

n1αn∞and eitheriiilimn→∞|αn−αn−1|/αn0 oriii_∞

n1|αn−αn−1|<

∞.The sequence{ξk}^∞_k1is a sequence of positive real numbers such that_∞

k1ξk1.

We now state and prove our main theorems.

3.1. Strong convergence theorems for a countably infinite family of m-accretive mappings

Theorem 3.1. LetCbe a closed convex nonempty subset of a real reflexive and strictly convex Banach spaceEwhich has a uniformly Gˆateaux differentiable norm. LetAk : C→ E, k ∈ Nbe a countably infinite family ofm-accretive mappings such thatN_∞

k1NAk/∅.Suppose that every bounded closed convex nonempty subset ofChas the fixed point property for nonexpansive mappings. For arbi- traryu, x1∈C,let{xn}n≥1be iteratively generated by

xn1αnu 1−αnSxn, n≥1, 3.1 whereS_∞

k1ξkJAk;JAk IAk⁻¹, k∈N.Then,{xn}_n≥1converges strongly to a common zero of{Ak}_k∈N.

Proof. SinceJAk IAk⁻¹ is nonexpansive for eachk ∈N,we obtain, byLemma 2.3, that S_∞

k1ξkJAk is well defined, nonexpansive, andFS _∞

k1FJAk N.Now, letq∈FS, then we obtain by inductionusing3.1that

xn−q ≤max{x1−q,u−q} 3.2

for alln∈N; hence{xn}_n≥1and{Sxn}_n≥1are bounded. This implies that for someM0>0, xn1−Sxnαnu−Sxn ≤αnM0−→0 asn−→ ∞. 3.3 Moreover, from3.1we obtain that

xn1−xnαnu 1−αn

Sxn−αn−1u−

1−αn−1 Sxn−1 αn−αn−1

u−Sxn−1

1−αn

Sxn−Sxn−1

≤

1−αnxn−xn−1αn−αn−1M0.

3.4

This results in the following two cases.

Case 1. Conditioniiiis satisfied. In this case,xn1−xn ≤ 1−αnxn−xn−1σn,where σnαnβn;βn |αn−αn−1|M0/αn,so thatσnoαn since limn→∞|αn−αn−1|/αn0.

Case 2. Conditioniiiis satisfied. In this case,xn1−xn ≤ 1−αnxn−xn−1σn,where σn|αn−αn−1|M0,so that_∞

n0σn<∞.

In either case, we obtainbyLemma 2.1that limn→∞xn1−xn 0.This implies that lim_n→∞xn−Sxn0sincexn−Sxn ≤ xn−xn1xn1−Sxn →0 asn→ ∞. For all t∈0,1, define the mappingGt:E→Eby

Gtx:tu 1−tSx, x∈E. 3.5

It is easy to see thatGtis a contraction for eacht∈0,1,and so has for eacht∈0,1a unique fixed pointzt ∈C; using Theorem2.6, we have thatzt→z^∗∈FSast→0.Now,

zt−xnt u−xn

1−t

Szt−xn

. 3.6

So, byLemma 2.2we have that zt−xn²≤1−t²Szt−xn²2t

u−xn, j zt−xn

≤1−t²Szt−SxnSxn−xn²2

zt−xn²

u−zt, j

zt−xn

≤ 1t²

zt−xn²2t

u−zt, j zt−xn

Sxn−xn2zt−xnSxn−xn. 3.7

This implies that

u−zt, j xn−zt

≤ t

2Sxn−xn 2t

M, 3.8

for someM >0.Thus,

lim sup

n→∞

u−zt, j xn−zt

≤ t

2M. 3.9

Moreover, we have that u−zt, j

xn−zt

u−z, j

xn−z

u−z, j xn−zt

−j

xn−z

z−zt, j xn−zt

3.10 Thus, since{xn}n≥1 is bounded, we have thatz^∗−zt, jxn −zt → 0 ast → 0.Also,u− z^∗, jxn−zt−jxn−z^∗ → 0 ast → 0 since the normalized duality mappingj is norm-to- weak^∗unformly continuous on bounded subsets ofE.Thus ast→0, we obtian from3.9and 3.10that

lim sup

n→∞

u−z^∗, j

xn−z^∗

≤0. 3.11

Now, put

μn:max 0,

u−z^∗, j

xn−z^∗

. 3.12

Then, 0≤μnfor alln≥0.It is easy to see thatμn→0 asn→ ∞since by3.11, ifε >0 is given, there existsnε∈Nsuch thatu−z^∗, jxn−z^∗< εfor alln≥nε.Thus, 0≤μn< εfor alln≥nε. So, lim_n→∞μn0.

Next, we obtain from the recursion formula3.1that xn1−z^∗αn

u−z^∗

1−αn

Sxn−z^∗

. 3.13

It follows that

xn1−z^∗2≤ 1−αn

₂Sxn−z^∗22αn

u−z^∗, j

xn1−z^∗

≤

1−αnxn−z^∗22αnμn1

1−αnxn−z^∗γn,

3.14

whereγn2αnμn1.Therefore,γnoαnand byLemma 2.1, we obtain that{xn}_n≥1converges strongly toz^∗ ∈ FS.ButFS _∞

k1FJAk

_∞

k1NAk N.Hence,{xn}_n≥1converges strongly to the common zero of the family{Ak}_k∈Nofm-accretive operators. This completes the proof.

Corollary 3.2. LetCbe a closed convex nonempty subset of a real reflexive and strictly convex Banach spaceEwhich has a uniformly Gˆateaux differentiable norm. LetAk:C→E, k1,2, . . . , rbe a finite family ofm-accretive mappings such thatN _r

k1NAk/∅.Suppose that every bounded closed convex nonempty subset ofChas the fixed point property for nonexpansive mappings. For arbitrary u, x1∈C,let{xn}_n≥1be iteratively generated by

xn1αnu 1−αn

Sxn, n≥1, 3.15

whereS _r

k1αkJAk;JAk IAk⁻¹;{αk}^r_k1is a finite collection of positive real numbers such that_r

k1αk1.Then,{xn}_n≥1converges strongly to a common zero of{Ak}^r_k1. Proof. The mappingS _r

k1αkJAk is clearly nonexpansive. Following the arguement of the proof ofLemma 2.3we get thatFS _r

k1FJAk.The rest follows fromTheorem 3.1. This completes the proof.

Remark 3.3. If, in particular, we consider a singlem-accretive operatorA, the requirement that Ebe strictly convex will be dispensed, in this case, withr1 andSinCorollary 3.2coincides withJA IA⁻¹.

Remark 3.4. We note that ifEis smooth, thenEis reflexive and has a uniformly Gˆateaux differ- entiable norm and with property that every bounded closed convex nonempty subset ofEhas the fixed point property for nonexpansive mappingssee e.g.,1.

Thus, we have the following corollary.

Corollary 3.5. LetCbe a closed convex nonempty subset of a real uniformly smooth Banach spaceE.

LetA :C→ Ebe anm-accretive operator withNA/∅.For arbitraryu, x1 ∈C,let the sequence {xn}_n≥1be iteratively generated by

xn1αnu 1−αn

JAxn, n≥1, 3.16

whereJA: IA⁻¹.Then{xn}_n≥1converges strongly to somex^∗∈NA.

Remark 3.6. If inTheorem 3.1we considerCE,then the condition thatAkism-accretive for eachk∈Ncould be replaced with the continuity of eachAk.

Thus, we have the following theorem.

Theorem 3.7. LetEbe a real reflexive and strictly convex Banach space which has a uniformly Gˆateaux differentiable norm. Let Ak : E → E, k ∈ Nbe a countably infinite family of continuous accretive operators such thatN_∞

k1NAk/∅.Suppose that every bounded closed convex nonempty subset ofEhas the fixed point property for nonexpansive mappings. For arbitraryu, x1 ∈E,let{xn}n≥1be iteratively generated by

xn1αnu 1−αn

Sxn, n≥1, 3.17

where S _∞

k1ξkJAk;JAk I Ak⁻¹. Then,{xn}_n≥1 converges strongly to a common zero of {Ak}_k∈N.

Proof. By Theorem2.5, we have thatAkism-accretive for eachk ∈ N.The rest follows from Theorem 3.1.

Corollary 3.8. LetEbe a real reflexive and strictly convex Banach space which has a uniformly Gˆateaux differentiable norm. LetAk:E→E, k1,2, . . . , rbe a finite family of continuous accretive operators such thatN_r

k1NAk/∅.Suppose that every bounded closed convex nonempty subset ofEhas the fixed point property for nonexpansive mappings. For arbitraryu, x1 ∈E,let{xn}_n≥1be iteratively generated by

xn1αnu 1−αn

Sxn, n≥1, 3.18

whereS_r

k1αkJAk;JAk IAk⁻¹,where{αk}^r_k1is a finite collection of positive real numbers such that_r

k1αk1.Then,{xn}_n≥1converges strongly to a common zero of{Ak}^r_k1. 3.2. Strong convergence theorem for countably infinite family of

pseudocontractive mappings

Theorem 3.9. LetCbe a closed convex nonempty subset of a real reflexive and strictly convex Banach spaceEwhich has a uniformly Gˆateaux differentiable norm. LetTk : C → E, k ∈ Nbe a countably infinite family of pseudocontractive mappings such that for eachk ∈N,I−Tkism-accretive onC andF_∞

k1FTk/∅.LetJTk I I−Tk⁻¹ 2I−Tk⁻¹for eachk ∈N.Suppose that every

bounded closed convex nonempty subset ofChas the fixed-point property for nonexpansive mappings.

For arbitraryu, x1∈C,let{xn}n≥1be iteratively generated by xn1αnu

1−αn

Txn, n≥1, 3.19

whereT _∞

k1ξkJTk.Then,{xn}_n≥1converges strongly to a common fixed point of{Tk}_k∈N.

Proof. PutAk : I −Tkfor eachk ∈ N.It is then obvious thatNAk FTkand hence _∞

k1NAk F _∞

k1FTk.Besides, Ak ism-accretive for each k ∈ N.Thus, the proof follows fromTheorem 3.1.

Corollary 3.10. LetCbe a closed convex nonempty subset of a real reflexive and strictly convex Banach spaceEwhich has a uniformly Gˆateaux differentiable norm. LetTk:C→E, k1,2, . . . , rbe a finite family of pseudocontractive mappings such that for eachk 1,2, . . . , r,I−Tkism-accretive onC andF _r

k1FTk/∅.LetJTk I I−Tk⁻¹ 2I−Tk⁻¹for eachk 1,2, . . . , r.Suppose that every nonempty bounded closed convex subset ofChas the fixed-point property for nonexpansive mappings. For arbitraryu, x1∈C,let{xn}_n≥1be iteratively generated by

xn1αnu 1−αn

Txn, n≥1, 3.20

whereT _r

k1αkJTk and{αk}^r_k1 is a finite collection of positive numbers such that_r

k1αk 1.

Then,{xn}n≥1converges strongly to a common fixed point of{Tk}^r_k1.

Corollary 3.11. LetCbe a nonempty closed convex subset of a real uniformly smooth Banach spaceE.

LetT :C→Ebe pseudocontractive mappings such thatI−Tism-accretive onCandFT/∅.Let JT I I−T⁻¹ 2I−T⁻¹. For arbitraryu, x1∈C,let{xn}_n≥1be iteratively generated by

xn1αnu 1−αn

JTxn, n≥1. 3.21

Then,{xn}_n≥1converges strongly to a fixed point ofT.

Theorem 3.12. LetEbe a real reflexive and strictly convex Banach spaceEwhich has a uniformly Gˆateaux differentiable norm. LetTk : E → E, k ∈ Nbe a countably infinite family of continuous pseudocontractive mappings such thatF_∞

k1FTk/∅.LetJTk I I−Tk⁻¹ 2I−Tk⁻¹ for eachk ∈ N.Suppose that every bounded closed convex nonempty subset ofChas the fixed point property for nonexpansive mappings. For arbitraryu, x1∈C,let{xn}_n≥1be iteratively generated by

xn1αnu 1−αn

Txn, n≥1, 3.22

whereT _∞

k1ξkJTk.Then,{xn}_n≥1converges strongly to a common fixed point of{Tk}_k∈N. Proof. The proof follows fromTheorem 3.9.

Corollary 3.13. LetEbe a real reflexive and strictly convex Banach spaceEwhich has a uniformly Gˆateaux differentiable norm. LetTk : E → E, k 1,2, . . . , r be a finite family of continuous pseu- docontractive mappings such thatF _r

k1FTk/∅.LetJTk I I−Tk⁻¹ 2I−Tk⁻¹for eachk1,2, . . . , r.Suppose that every bounded closed convex nonempty subset ofEhas the fixed-point property for nonexpansive mappings. For arbitraryu, x1∈E,let{xn}_n≥1be iteratively generated by

xn1αnu 1−αn

Txn, n≥1, 3.23

whereT _r

k1αkJTk;JTk ITk⁻¹,where{αk}^r_k1is a finite collection of positive numbers such that_r

k1αk1.Then,{xn}_n≥1converges strongly to a common fixed point of{Tk}^r_k1.

Corollary 3.14. LetEbe a real uniformly smooth Banach space. LetT :E→Ebe continuous pseudo- contractive mappings such thatFT/∅.LetJT II−T⁻¹ 2I−T⁻¹. For arbitraryu, x1∈E, let{xn}_n≥1be iteratively generated by

xn1αnu 1−αn

JTxn, n≥1. 3.24

Then,{xn}_n≥1converges strongly to fixed point ofT.

Remark 3.15. A prototype for the sequence{αn}_n≥1satisfying the conditions on our iteration parameter is the sequence{1/n1}_n≥1.We note that conditionsiiiandiiiare not compa- rable, sincee.g.the sequence{βn}_n≥1given by

βn

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

√1

n, ifnis odd

√ 1

n−1, ifnis even

3.25

satisfiesiiibut does not satisfyiiisee e.g.,33.

Remark 3.16. The addition of bounded error terms to our recursion formulas leads to no further generalization.

Remark 3.17. Iff:K→Kis a contraction mapping and we replaceubyfxnin 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 replacesubyfxn, repeats the argument of this paper, using the fact that fis a contraction map.

Remark 3.18. Our theorems improve, extend, and generalize the corresponding results of Zeg- eye and Shahzad32and that of a host of other authors from approximation of a common zero common fixed pointof a finite family of accretive (pseudocontractive) operators to approximation of a common zerocommon fixed pointof a countably infinite family of accretive (pseudocon- tractive) operators. Furthermore,Theorem 3.12extends the corresponding results of Liu et al.

19, Maiti and Saha9, Senter and Dotson20, Jung17from approximation of a common fixed point of a finite family of nonexpansive mappings to the approximation of common fixed points of a countably infinite family of continuous psedocontractive mappings, without as- suming that our operators satisfy the so-called condition A. Our theorems are applicable, in particular, inLpspaces, 1< p <∞.

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