a Countable Family of Strict Pseudocontractions in Banach Spaces

Fixed Point Theory and Applications Volume 2010, Article ID 632137,16pages doi:10.1155/2010/632137

Research Article

Weak Convergence Theorems for

a Countable Family of Strict Pseudocontractions in Banach Spaces

Prasit Cholamjiak

and Suthep Suantai

1Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Correspondence should be addressed to Suthep Suantai,[email protected] Received 2 June 2010; Accepted 16 September 2010

Academic Editor: Massimo Furi

Copyrightq2010 P. Cholamjiak and S. Suantai. 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.

We investigate the convergence of Mann-type iterative scheme for a countable family of strict pseudocontractions in a uniformly convex Banach space with the Fr´echet differentiable norm.

Our results improve and extend the results obtained by Marino-Xu, Zhou, Osilike-Udomene, Zhang-Guo and the corresponding results. We also point out that the condition given by Chidume- Shahzad2010is not satisfied in a real Hilbert space. We show that their results still are true under a new condition.

1. Introduction

LetEandE^∗be a real Banach space and the dual space ofE, respectively. LetKbe a nonempty subset ofE. LetJ denote the normalized duality mapping fromEinto 2^E∗ given by Jx {f ∈E^∗: x, fx²f²}, for allx∈E, where·,·denotes the duality pairing between EandE^∗. IfEis smooth orE^∗is strictly convex, thenJis single-valued.

Throughout this paper, we denote the single valued duality mapping byjand denote the set of fixed points of a nonlinear mappingT :K → Eby

FT {x∈K:Txx}. 1.1

Definition 1.1. A mappingTwith domainDTand rangeRTinEis called

ipseudocontractive1if, for allx, y∈DT, there existsjx−y∈Jx−ysuch that Tx−Ty, j

x−y

≤x−y², 1.2

iiλ-strictly pseudocontractive2if for allx, y∈DT, there existλ >0 andjx−y∈ Jx−ysuch that

Tx−Ty, j x−y

≤x−y²−λI−Tx−I−Ty², 1.3

or equivalently

I−Tx−I−Ty, j x−y

≥λI−Tx−I−Ty², 1.4

iiiL-Lipschitzian if, for allx, y∈DT, there exists a constantL >0 such that

Tx−Ty≤Lx−y. 1.5

Remark 1.2. It is obvious by the definition that

1every strictly pseudocontractive mapping is pseudocontractive,

2everyλ-strictly pseudocontractive mapping is1λ/λ-Lipschitzian; see3.

Remark 1.3. LetKbe a nonempty subset of a real Hilbert space andT :K → Ka mapping.

ThenT is said to beκ-strictly pseudocontractive2if, for allx, y ∈ DT, there existsκ ∈ 0,1such that

Tx−Ty² ≤x−y²κI−Tx−I−Ty². 1.6

It is well know that1.6is equivalent to the following:

Tx−Ty, x−y ≤x−y²−1−κ

2 I−Tx−I−Ty². 1.7 It is worth mentioning that the class of strict pseudocontractions includes properly the class of nonexpansive mappings. Moreover, we know from 4 that the class of pseudocontractions also includes properly the class of strict pseudocontractions. A mapping A : E → Eis called accretive if, for allx, y ∈ E, there existsjx−y ∈ Jx−ysuch that Ax−Ay, jx−y ≥ 0. It is also known that A is accretive if and only ifT : I −A is pseudocontractive. Hence, a solution of the equationAu 0 is a solution of the fixed point ofT :I−A. Note that ifT :I−A, thenAisλ-strictly accretive if and only ifTisλ-strictly pseudocontractive.

In 1953, Mann5introduced the iteration as follows: a sequence{xn}defined byx0∈ Kand

xn1αnxn 1−αnTxn, ∀n≥0, 1.8

whereαn∈0,1. IfTis a nonexpansive mapping with a fixed point and the control sequence {α_n}is chosen so that_∞

n0α_n1−α_n ∞, then the sequence{x_n}defined by1.8converges

weakly to a fixed point ofT this is also valid in a uniformly convex Banach space with the Fr´echet differentiable norm6 . However, ifTis a Lipschitzian pseudocontractive mapping, then Mann iteration defined by1.8may fail to converge in a Hilbert space; see4.

In 1967, Browder-Petryshyn2introduced the class of strict pseudocontractions and proved existence and weak convergence theorems in a real Hilbert setting by using Mann’s iteration1.8with a constant sequence α_n αfor alln. Respectively, Marino-Xu 7and Zhou8extended the results of Browder-Petryshyn2to Mann’s iteration process1.8. To be more precise, they proved the following theorem.

Theorem 1.4see7. LetKbe a closed convex subset of a real Hilbert spaceH. LetT :K → K be aκ-strict pseudocontraction for some 0 ≤ κ < 1, and assume that T admits a fixed point inK.

Let a sequence{x_n}^∞_n0be the sequence generated by Mann’s algorithm1.8. Assume that the control sequence{α_n}^∞_n0is chosen so thatκ < α_n< 1 for allnand_∞

n0α_n−κ1−α_n ∞. Then{x_n} converges weakly to a fixed point ofT.

Meanwhile, Marino, and Xu raised the open question: whetherTheorem 1.4can be extended to Banach spaces which are uniformly convex and have a Fr´echet differentiable norm. Later, Zhou 9 and Zhang-Su 10, respectively, extended the result above to 2- uniformly smooth and q-uniformly smooth Banach spaces which are uniformly convex or satisfy Opial’s condition.

In 2001, Osilike-Udomene11proved the convergence theorems of the Mann5and Ishikawa 12 iteration methods in the framework of q-uniformly smooth and uniformly convex Banach spaces. They also obtained that a sequence{x_n}defined by1.8converges weakly to a fixed point of T under suitable control conditions. However, the sequence {α_n} ⊂ 0,1excluded the canonical choice α_n 1/n, n ≥ 1. This was a motivation for Zhang-Guo13to improve the results in the same space. Observe that the results of Osilike- Udomene11and Zhang-Guo13hold under the assumption that

Cq< q^λ

b^q−1, 1.9

for someb∈0,1andCqis a constant depending on the geometry of the space.

Lemma 1.5 see14–16. Let E be a uniformly smooth real Banach space. Then there exists a nondecreasing continuous functionβ:0,∞ → 0,∞with lim_t→0βt 0 andβct≤cβtfor c≥1 such that, for allx, y∈E, the following inequality holds:

xy² ≤ x²2 y, jx

max{x,1}yβy. 1.10

Recently, Chidume-Shahzad17 extended the results of Osilike-Udomene11and Zhang-Guo13 by using Reich’s inequality 1.10 to the much more general real Banach spaces which are uniformly smooth and uniformly convex. Under the assumption that

βt≤ λt

max{2r,1}, 1.11

for somer >0, they proved the following theorem.

Theorem 1.6 see17. LetE be a uniformly smooth real Banach space which is also uniformly convex andKa nonempty closed convex subset ofE. LetT :K → Kbe aλ-strict pseudocontraction for some 0≤λ < 1 withx^∗∈FT:{x∈K :Txx}/∅. For a fixedx0 ∈K, define a sequence {xn}by

x_n1 1−α_nx_nα_nTx_n, n≥1, 1.12

where{α_n}is a real sequence in0,1satisfying the following conditions:

i_∞

n0α_n∞;

ii_∞

n0α²_n<∞.

Then,{x_n}converges weakly to a fixed point ofT.

However, we would like to point out that the results of Chidume-Shahzad17do not hold in real Hilbert spaces. Indeed, we know from Chidume14that

βt supxty²− x²

t −2

y, jx

:x ≤1,y≤1 . 1.13

IfEis a real Hilbert space, then we have

βt supxty²− x²

t −2

y, x

:x ≤1,y≤1 sup

x²2t x, y

t²y²− x²

t −2

y, x

:x ≤1,y≤1 sup

ty²:y≤1 t.

1.14

On the other hand, by assumption1.11, we see that

βt≤ λt

max{2r,1} < t, 1.15

which is a contradiction.

It is known that one can extend his result from a single strict pseudocontraction to a finite family of strict pseudocontractions by replacing the convex combination of these mappings in the iteration under suitable conditions. The construction of fixed points for pseudocontractions via the iterative process has been extensively investigated by many authors; see also18–22and the references therein.

Our motivation in this paper is the following:

1to modify the normal Mann iteration process for finding common fixed points of an infinitely countable family of strict pseudocontractions,

2to improve and extend the results of Chidume-Shahzad17from a real uniformly smooth and uniformly convex Banach space to a real uniformly convex Banach space which has the Fr´echet differentiable norm.

Motivated and inspired by Marino-Xu7, Osilike-Udomene11, Zhou8, Zhang- Guo13, and Chidume-Shahzad17, we consider the following Mann-type iteration:x1∈K and

x_n1 1−α_nx_nα_nT_nx_n, n≥1, 1.16

where α_n is a real sequence in 0,1 and {T_n}^∞_n1 is a countable family of strict pseudocontractions on a closed and convex subsetKof a real Banach spaceE.

In this paper, we prove the weak convergence of a Mann-type iteration process1.16 in a uniformly convex Banach space which has the Fr´echet differentiable norm for a countable family of strict pseudocontractions under some appropriate conditions. The results obtained in this paper improve and extend the results of Chidume-Shahzad 17, Marino-Xu 7, Osilike-Udomene11, Zhou8, and Zhang-Guo13in some aspects.

We will use the following notation:

ifor weak convergence and → for strong convergence.

iiωωxn {x: xni x}denotes the weakω-limit set of{xn}.

2. Preliminaries

A Banach spaceEis said to be strictly convex ifxy/2<1 for allx, y∈Ewithxy1 andx /y. A Banach spaceEis called uniformly convex if for each >0 there is aδ >0 such that, forx, y∈Ewithx,y ≤1, andx−y ≥, xy ≤21−δholds. The modulus of convexity ofEis defined by

δE inf

1− 1

2

xy:x,y≤1, x−y≥

, 2.1

for all∈0,2.Eis uniformly convex ifδE0 0, andδE>0 for all 0< ≤2. It is known that every uniformly convex Banach space is strictly convex and reflexive. LetSE {x ∈ E:x1}. Then the norm ofEis said to be Gˆateaux differentiable if

limt→0

xty− x

t 2.2

exists for eachx, y ∈SE. In this caseEis called smooth. The norm ofEis said to be Fr´echet differentiable orE is Fr´echet smooth if, for eachx ∈ SE, the limit is attained uniformly for y∈SE. In other words, there exists a functionε_xswithε_xs → 0 ass → 0 such that

xty− xt

y, jx≤ |t|εx|t| 2.3

for ally∈SE. In this case the norm is Gˆateaux differentiable and

limt→0 sup

y∈SE

1/2xty²−1/2x²

t −

y, jx

0 2.4

for allx∈E. On the other hand, 1

2x² h, jx

≤ 1

2xh²≤ 1

2x²h, jxbh 2.5 for allx, h∈E, wherebis a function defined onR such that lim_t→0bt/t 0. The norm ofEis called uniformly Fr´echet differentiable if the limit is attained uniformly forx, y∈SE.

Letρ_E:0,∞ → 0,∞be the modulus of smoothness ofEdefined by

ρ_Et sup 1

2xyx−y−1 :x∈SE, y≤t

. 2.6

A Banach spaceEis said to be uniformly smooth ifρ_Et/t → 0 ast → 0. Letq > 1, thenE is said to beq-uniformly smooth if there existsc >0 such thatρEt≤ct^q. It is easy to see that ifEisq-uniformly smooth, thenEis uniformly smooth. It is well known thatEis uniformly smooth if and only if the norm ofEis uniformly Fr´echet differentiable, and hence the norm of Eis Fr´echet differentiable, and it is also known that ifEis Fr´echet smooth, thenEis smooth.

Moreover, every uniformly smooth Banach space is reflexive. For more details, we refer the reader to14,23. A Banach spaceEis said to satisfy Opial’s condition24ifx∈Eandx_n x;

then

lim sup

n→ ∞ xn−x<lim sup

n→ ∞

xn−y, ∀y∈E, x /y. 2.7

In the sequel, we will need the following lemmas.

Lemma 2.1see23. LetEbe a Banach space andJ:E → 2^E∗the duality mapping. Then one has the following:

ixy²≥ x²2y, jxfor allx, y∈E, wherejx∈Jx;

iixy²≤ x²2y, jxyfor allx, y∈E, wherejxy∈Jxy.

Lemma 2.2see25. LetEbe a real uniformly convex Banach space,K a nonempty, closed, and convex subset of E, and T : K → K a continuous pseudocontractive mapping. Then, I −T is demiclosed at zero, that is, for all sequence{x_n} ⊂Kwithx_n pandx_n−Tx_n → 0 it follows thatpTp.

Lemma 2.3see25. LetEbe a real reflexive Banach space which satisfies Opial’s condition,Ka nonempty, closed and convex subset ofEandT :K → Ka continuous pseudocontractive mapping.

Then,I−Tis demiclosed at zero.

Lemma 2.4see26. LetEbe a real uniformly convex Banach space with a Fr´echet differentiable norm. LetKbe a closed and convex subset ofEand{S_n}^∞_n1a family ofL_n-Lipschitzian self-mappings

onKsuch that_∞

n1Ln−1<∞andF_∞

n1FSn/∅. For arbitraryx1∈K, definexn1 Snxn

for alln≥1. Then for everyp, q∈F, lim_{n→ ∞}x_n, jp−qexists, in particular, for allu, v∈ω_ωx_n andp, q∈F,u−v, jp−q0.

Lemma 2.5 see 17, 27. Let{an},{bn} and {δn}, be sequences of nonnegative real numbers satisfying the inequality

an1 ≤1δnanbn, n≥0. 2.8

If_∞

n0δ_n < ∞and_∞

n0b_n < ∞, then lim_{n→ ∞}a_n exists. If, in addition,{a_n}has a subsequence converging to 0, then lim_{n→ ∞}an0.

To deal with a family of mappings, the following conditions are introduced. Let K be a subset of a real Banach spaceE, and let{T_n}be a family of mappings ofK such that _∞

n1FTn/∅. Then{Tn}is said to satisfy the AKTT-condition28if for each bounded subset BofK,

∞ n1

sup{Tn1z−Tnz:z∈B}<∞. 2.9

Lemma 2.6see28. LetKbe a nonempty and closed subset of a Banach spaceE, and let{Tn}be a family of mappings ofKinto itself which satisfies the AKTT-condition, then the mappingT :K → K defined by

Tx lim

n→ ∞T_nx, ∀x∈K 2.10

satisfies

lim sup

n→ ∞ {Tz−Tnz:z∈B}0 2.11

for each bounded subsetBofK.

So we have the following results proved by Boonchari-Saejung29,30.

Lemma 2.7see29,30. LetKbe a closed and convex subset of a smooth Banach spaceE. Suppose that{T_n}^∞_n1is a family ofλ-strictly pseudocontractive mappings fromKintoEwith_∞

n1FT_n/∅ and{βn}^∞_n1is a real sequence in0,1such that_∞

n1βn1. Then the following conclusions hold:

1G:_∞

n1βnTn:K → Eis aλ-strictly pseudocontractive mapping;

2FG _∞

n1FTn.

Lemma 2.8see30. LetK be a closed and convex subset of a smooth Banach spaceE. Suppose that {S_k}^∞_k1 is a countable family of λ-strictly pseudocontractive mappings of K into itself with _∞

k1FSk/∅. For eachn∈N, defineTn:K → Kby T_nxⁿ

k1

β_n^kS_kx, x∈K, 2.12

where{β_n^k}is a family of nonnegative numbers satisfying i_n

k1β_n^k1 for alln∈N;

iiβ^k:lim_{n→ ∞}β^k_n>0 for allk∈N;

iii_∞

n1_n

k1|β^k_n1−β^k_n|<∞.

Then

1eachTnis aλ-strictly pseudocontractive mapping;

2{Tn}satisfies AKTT-condition;

3IfT :K → Kis defined by

Tx^∞

k1

β^kS_kx, x∈K, 2.13

thenTxlim_{n→ ∞}TnxandFT _∞

n1FTn _∞

k1FSk.

For convenience, we will write that {T_n}, T satisfies the AKTT-condition if {T_n} satisfies the AKTT-condition andTis defined byLemma 2.6withFT _∞

n1FT_n.

3. Main Results

Lemma 3.1. LetE be a real Banach space, and letK be a nonempty, closed, and convex subset of E. Let{T_n}^∞_n1 :K → K be a family ofλ-strict pseudocontractions for some 0 < λ < 1 such that F:_∞

n1FT_n/∅. Define a sequence{x_n}byx1∈K,

x_n1 1−α_nx_nα_nT_nx_n, n≥1, 3.1

where{αn} ⊂0,1satisfying_∞

n1αn∞and_∞

n1α²_n<∞. If{Tn}satisfies the AKTT-condition, then

ilim_{n→ ∞}x_n−pexists for allp∈F;

iilim inf_{n→ ∞}xn−Tnxn0.

Proof. Letp∈F, and putL λ1/λ. First, we observe that

xn1−p≤1−αnxn−pαnTnxn−p≤1Lxn−p,

xn1−xnαnTnxn−xn ≤αn1Lxn−p. 3.2

SinceTnis aλ-strict pseudocontraction, there existsjxn1−p∈Jxn1−p. ByLemma 2.1 we have

x_n1−p²x_n−p α_nT_nx_n−x_n²

≤xn−p²2αnTnxn−xn, j

xn1−p x_n−p²2α_nT_nx_n−T_nx_n1, j

x_n1−p 2αnTnxn1−xn1, j

xn1−p

2αnxn1−xn, j

xn1−p

≤x_n−p²2α_nLx_n−x_n1x_n1−p

− 2α_nλT_nx_n1−x_n1²2α_nx_n−x_n1x_n1−p

≤xn−p²2α²_nL1L²xn−p²

− 2α_nλT_nx_n1−x_n1²2α²_n1L²x_n−p²

xn−p²2α²_n1L³xn−p²−2αnλTnxn1−xn1².

3.3

This implies that

x_n1−p²≤

12α²_n1L³x_n−p². 3.4 Hence, by_∞

n1α²_n<∞, we have fromLemma 2.5that lim_{n→ ∞}x_n−pexists; consequently, {xn}is bounded. Moreover, by3.3, we also have

∞ n1

α_nλT_nx_n1−x_n1²≤^∞

n1

x_n−p²−x_n1−p²

21L³M²₁^∞

n1

α²_n<∞, 3.5

whereM1 sup_n≥1{xn−p}. It follows that lim infn→ ∞Tnxn1−xn1 0. Since{xn}is bounded,

x_n1−T_n1x_n1 ≤ x_n1−T_nx_n1T_nx_n1−T_n1x_n1

≤ xn1−T_nx_n1 sup

z∈{xn}T_nz−T_n1z. 3.6 Since {T_n} satisfies the AKTT-condition, it follows that lim inf_{n→ ∞}x_n −T_nx_n 0. This completes the proof ofiandii.

Lemma 3.2. LetEbe a real Banach space with the Fr´echet differentiable norm. Forx∈E, letβ^∗tbe defined for 0< t <∞by

β^∗t sup

y∈SE

xty²− x²

t −2

y, jx

. 3.7

Then, lim_t→0β^∗t 0, and

xh²≤ x²2 h, jx

hβ^∗h 3.8

for allh∈E\ {0}.

Proof. Letx∈E. SinceEhas the Fr´echet differentiable norm, it follows that

limt→0sup

y∈SE

1/2xty²−1/2x²

t −

y, jx

0. 3.9

Then lim_t→0β^∗t 0, and hence

xty²− x²

t −2

y, jx

≤β^∗t, ∀y∈SE 3.10

which implies that

xty² ≤ x²2t y, jx

tβ^∗t, ∀y∈SE. 3.11

Suppose thath /0. Putyh/handth. By3.11, we have xh²≤ x²2

h, jx

hβ^∗h. 3.12

This completes the proof.

Remark 3.3. In a real Hilbert space, we see thatβ^∗t tfort >0.

In our more general setting, throughout this paper we will assume that

β^∗t≤2t, 3.13

whereβ^∗is a function appearing in3.8.

So we obtain the following results.

Lemma 3.4. Let E be a real Banach space with the Fr´echet differentiable norm, and let K be a nonempty, closed, and convex subset of E. Let {T_n}^∞_n1 : K → K be a family of λ-strict pseudocontractions for some 0 < λ < 1 such thatF : _∞

n1FTn/∅. Define a sequence{xn}by x1∈K,

x_n1 1−α_nx_nα_nT_nx_n, n≥1, 3.14

where {αn} ⊂ 0,1satisfying_∞

n1αn ∞and _∞

n1α²_n < ∞. If{Tn}, Tsatisfies the AKTT- condition, then lim_{n→ ∞}x_n−T_nx_nlim_{n→ ∞}x_n−Tx_n0.

Proof. Letp∈F, and putM2sup_n≥1{xn−Tnxn}>0. Then by3.8and3.13we have

xn1−p²xn−p αnTnxn−xn²

≤x_n−p²2α_nT_nx_n−x_n, j x_n−p

αnTnxn−xnβ^∗αnTnxn−xn

≤x_n−p²−2α_nλx_n−T_nx_n²2α²_nx_n−T_nx_n²

≤xn−p²−2αnλxn−Tnxn²2α²_nM²₂.

3.15

It follows that

∞ n1

αnxn−Tnxn²<∞. 3.16

Observe that

x_n−T_n1x_n1²x_n−T_nx_n T_nx_n−T_n1x_n1²

≤ xn−T_nx_n²2T_nx_n−T_n1x_n1, jx_n−T_n1x_n1 x_n−T_nx_n²2

T_nx_n−T_nx_n1, jx_n−T_n1x_n1 2

Tnxn1−Tn1xn1, jxn−Tn1xn1

≤ xn−Tnxn²2Lxn−xn1xn−Tn1xn1 2T_nx_n1−T_n1x_n1x_n−T_n1x_n1

≤ xn−T_nx_n²2Lx_n−x_n1x_n−T_nx_n 2Lxn−xn1Tnxn−Tnxn1

2Lxn−xn1Tnxn1−Tn1xn1 2T_nx_n1−T_n1x_n1x_n−x_n1 2Tnxn1−Tn1xn1xn1−Tn1xn1

≤ xn−T_nx_n²

2Lα_n2L²α²_n

x_n−T_nx_n² 2LM2α_n2M2α_n2M2T_nx_n1−T_n1x_n1

≤ xn−T_nx_n²2L1Lα_nx_n−T_nx_n² 2M2L2Tnxn1−Tn1xn1.

3.17

By3.17, we have

xn1−Tn1xn1²≤1−αnxn−Tn1xn1²αnTnxn−Tn1xn1²

≤ xn−Tn1xn1²

αnTnxn−Tnxn1Tnxn1−Tn1xn1² xn−Tn1xn1²αnTnxn−Tnxn1²

2α_nT_nx_n−T_nx_n1T_nx_n1−T_n1x_n1 α_nT_nx_n1−T_n1x_n1²

≤ xn−T_n1x_n1²α²_nL²x_n−T_nx_n² 2α²_nLxn−TnxnTnxn1−Tn1xn1 αnTnxn1−Tn1xn1²

≤ xn−Tn1xn1²α²_nL²M₂²

2LM2Tnxn1−Tn1xn1Tnxn1−Tn1xn1²

≤ xn−Tnxn²2L1Lαnxn−Tnxn² 2M2L2Tnxn1−Tn1xn1α²_nL²M²₂ 2LM2Tnxn1−Tn1xn1Tnxn1−Tn1xn1²

≤ xn−Tnxn²2L1Lαnxn−Tnxn² α²_nL²M²₂2M22L2T_nx_n1−T_n1x_n1 T_nx_n1−T_n1x_n1²

≤ xn−T_nx_n²2L1Lα_nx_n−T_nx_n² α²_nL²M²₂2M22L2sup

z∈{xn}Tnz−Tn1z sup

z∈{xn}T_nz−T_n1z².

3.18

Since_∞

n1α_nx_n−T_nx_n² <∞,_∞

n1α²_n <∞, and_∞

n1sup{T_n1z−T_nz:z ∈ {x_n}}< ∞, it follows fromLemma 2.5that lim_{n→ ∞}xn−Tnxnexists. Hence, byLemma 3.1ii, we can conclude that lim_{n→ ∞}xn−Tnxn0. Since

xn−Txn ≤ xn−TnxnTnxn−Txn

≤ xn−T_nx_n sup

z∈{xn}T_nz−Tz, 3.19

it follows fromLemma 2.6that lim_{n→ ∞}x_n−Tx_n0. This completes the proof.

Now, we prove our main result.

Theorem 3.5. LetEbe a real uniformly convex Banach space with the Fr´echet differentiable norm, and letKbe a nonempty, closed, and convex subset ofE. Let{Tn}^∞_n1:K → Kbe a family ofλ-strict pseudocontractions for some 0 < λ < 1 such thatF : _∞

n1FTn/∅. Define a sequence{xn}by x1∈K,

xn1 1−αnxnαnTnxn, n≥1, 3.20

where {αn} ⊂ 0, λ satisfying_∞

n1αn ∞and_∞

n1α²_n < ∞. If{Tn}, Tsatisfies the AKTT- condition, then{x_n}converges weakly to a common fixed point of{T_n}.

Proof. Letp∈F, and defineSn:K → Kby

S_nx 1−α_nxα_nT_nx, x∈K. 3.21

Then_∞

n1FS_n FFT. By3.8, we have for boundedx, y∈Kthat S_nx−S_ny²x−y−α_nx−y−T_nx−T_ny²

≤x−y²−2α_nI−T_nx−I−T_ny, j x−y

α_nx−y−

T_nx−T_nyβ^∗

α_nx−y−

T_nx−T_ny

≤x−y²−2αnλx−y−Tnx−Tny² 2α²_nx−y−Tnx−Tny²

x−y²−2α_nλ−α_nx−y−T_nx−T_ny²

≤x−y².

3.22

This implies thatS_n is nonexpansive. ByLemma 3.1i, we know that{x_n}is bounded. By Lemma 3.4, we also know that lim_{n→ ∞}xn−Txn 0. ApplyingLemma 2.2, we also have ω_ωx_n⊂FT.

Finally, we will show thatω_ωx_nis a singleton. Suppose thatx^∗, y^∗∈ω_ωx_n⊂FT. Hencex^∗, y^∗ ∈_∞

n1FSn. ByLemma 2.4, lim_{n→ ∞}xn, jx^∗−y^∗exists. Suppose that{xnk} and{x_mk}are subsequences of{x_n}such thatx_nk x^∗andx_mk y^∗. Then

x^∗−y^∗2

x^∗−y^∗, j

x^∗−y^∗ lim

k→ ∞

x_nk−x_mk, j

x^∗−y^∗

0. 3.23

Hencex^∗ y^∗; consequently,x_n x^∗ ∈ _∞

n1FS_n F asn → ∞. This completes the proof.

As a direct consequence of Theorem 3.5, Lemmas 2.7 and 2.8 we also obtain the following results.

Theorem 3.6. LetEbe a real uniformly convex Banach space with the Fr´echet differentiable norm, and let K be a nonempty, closed, and convex subset of E. Let {S_k}^∞_k1 be a sequence of λ_k-strict pseudocontractions ofKinto itself such that_∞

k1FSk/∅and inf{λk :k∈N}λ > 0. Define a sequence{xn}byx1∈K,

xn1 1−αnxnαn

n k1

β^k_nSkxn, n≥1, 3.24

where{α_n} ⊂0, λsatisfying_∞

n1α_n ∞and_∞

n1α²_n<∞and{β^k_n}satisfies conditions (i)–(iii) ofLemma 2.8. Then,{xn}converges weakly to a common fixed point of{Sk}^∞_k1.

Remark 3.7. iTheorems3.5and3.6extend and improve Theorems 3.3 and 3.4 of Chidume- Shahzad17in the following senses:

ifrom real uniformly smooth and uniformly convex Banach spaces to real uniformly convex Banach spaces with Fr´echet differentiable norms;

iifrom finite strict pseudocontractions to infinite strict pseudocontractions.

Using Opial’s condition, we also obtain the following results in a real reflexive Banach space.

Theorem 3.8. Let E be a real Fr´echet smooth and reflexive Banach space which satisfies Opial’s condition, and let K be a nonempty, closed, and convex subset ofE. Let{Tn}^∞_n1 be a family ofλ- strict pseudocontractions for some 0< λ <1 such thatF :_∞

n1FT_n/∅. Define a sequence{x_n} byx1∈K,

xn1 1−αnxnαnTnxn, n≥1, 3.25

where {α_n} ⊂ 0, λ satisfying_∞

n1α_n ∞and_∞

n1α²_n < ∞. If{T_n}, Tsatisfies the AKTT- condition, then{x_n}converges weakly to a common fixed point of{T_n}.

Proof. Let p ∈ F. ByLemma 3.1i, we know that limn→ ∞xn −p exists. Since E has the Fr´echet differentiable norm, byLemma 3.4, we know that lim_{n→ ∞}x_n−Tx_n 0. It follows fromLemma 2.3thatω_ωx_n ⊂ FT F. Finally, we show thatω_ωx_nis a singleton. Let x^∗, y^∗∈ωωxn, and let{xnk}and{xmk}be subsequences of{xn}chosen so thatxnk x^∗and x_mk y^∗. Ifx^∗/y^∗, then Opial’s condition ofEimplies that

n→ ∞limxn−x^∗ lim

k→ ∞xnk−x^∗< lim

k→ ∞xnk−y^∗ lim

k→ ∞xmk−y^∗

< lim

k→ ∞xmk−x^∗ lim

n→ ∞xn−x^∗. 3.26

This is a contradiction, and thus the proof is complete.

Theorem 3.9. Let E be a real Fr´echet smooth and reflexive Banach space which satisfies Opial’s condition, and letK be a nonempty, closed, and convex subset ofE. Let {S_k}^∞_k1 be a sequence of

λk-strict pseudocontractions ofKinto itself such that_∞

k1FSk/∅and inf{λk:k ∈N}λ >0.

Define a sequence{x_n}byx1∈K,

xn1 1−αnxnαn

n k1

β^k_nSkxn, n≥1, 3.27

where{αn} ⊂0, λsatisfying_∞

n1αn ∞and_∞

n1α²_n<∞and{β^k_n}satisfies conditions (i)–(iii) ofLemma 2.8. Then,{x_n}converges weakly to a common fixed point of{S_k}^∞_k1.

Acknowledgments

The authors would like to thank the referees for valuable suggestions. This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, and the Thailand Research Fund, Thailand. The first author is supported by the Royal Golden Jubilee Grant PHD/0261/2551 and by the Graduate School, Chiang Mai University, Thailand.

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