WEAK AND STRONG CONVERGENCE OF FINITE FAMILY WITH ERRORS OF NONEXPANSIVE NONSELF-MAPPINGS

WEAK AND STRONG CONVERGENCE OF FINITE FAMILY WITH ERRORS OF NONEXPANSIVE NONSELF-MAPPINGS

S. PLUBTIENG AND K. UNGCHITTRAKOOL

Received 27 September 2005; Revised 5 May 2006; Accepted 8 May 2006

We are concerned with the study of a multistep iterative scheme with errors involving a finite family of nonexpansive nonself-mappings. We approximate the common fixed points of a finite family of nonexpansive nonself-mappings by weak and strong conver- gence of the scheme in a uniformly convex Banach space. Our results extend and improve some recent results, Shahzad (2005) and many others.

Copyright © 2006 S. Plubtieng and K. Ungchittrakool. This is an open access article dis- tributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is prop- erly cited.

1. Introduction

LetKbe a subset of a real normed linear spaceEand letTbe a self-mapping onK.T is said to be nonexpansive providedTx−T yx−yfor allx,y∈K.

Fixed-point iteration process for nonexpansive mappings in Banach spaces includ- ing Mann and Ishikawa iteration processes has been studied extensively by many au- thors to solve the nonlinear operator equations in Hilbert spaces and Banach spaces;

see [3,7,10,11,15,16]. Tan and Xu [15] introduced and studied a modified Ishikawa process to approximate fixed points of nonexpansive mappings defined on nonempty closed convex bounded subsets of a uniformly convex Banach spaceE. Five years later, Xu [18] introduced iterative schemes known as Mann iterative scheme with errors and Ishikawa iterative scheme with errors. Takahashi and Tamura [14] introduced and stud- ied a generalization of Ishikawa iterative schemes for a pair of nonexpansive mappings in Banach spaces. Recently, Khan and Fukhar-ud-din [6] extended their scheme to the modified Ishikawa iterative schemes with errors for two mappings and gave weak and strong convergence theorems. On the other hand, iterative techniques for approximat- ing fixed points of nonexpansive nonself-mappings have been studied by various au- thors; see [4,8,13,19]. Shahzad [12] introduced and studied an iteration scheme for

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2006, Article ID 81493, Pages1–12 DOI10.1155/FPTA/2006/81493

approximating a fixed point of nonexpansive nonself-mappings (when such a fixed point exists) and gave some strong and weak convergence theorems for such mappings.

Inspired and motivated by these facts, we introduce and study a multistep iterative scheme with errors for a finite family of nonexpansive nonself-mappings. Our schemes can be viewed as an extension for two-step iterative schemes of Shahzad [12]. The scheme is defined as follows.

LetK be a nonempty closed convex subset of a uniformly convex Banach space E, which is also a nonexpansive retract ofE. And letT1,T2,. . .,TN:K→Ebe nonexpansive mappings, the following iteration scheme is studied:

x¹_n=Pα¹_nT1x_n+β¹_nx_n+γ¹_nu¹_n, x²_n=Pα²_nT2x_n¹+β²_nxn+γ²_nu²_n,

... ...

x_n+1=x^N_{n =}Pα^N_nT_Nx_n^N₋1 +β^N_nx_n+γ_n^Nu^N_n

(1.1)

withx1∈K,n1, whereP is a nonexpansive retraction with respect to K and{α¹_n}, {α²_n},. . .,{α^N_n},{β¹_n},{β²_n},. . .,{β^N_n},{γ¹_n},{γ²_n},. . .,{γ^N_n}are sequences in [0, 1] withαⁱ_n+ βⁱ_n+γⁱ_n=1 for alli=1, 2,. . .,N, and{u¹_n},{u²_n},. . .,{u^N_n}are bounded sequences inK.

ForN=2,T1=T2≡T,βn=α¹_n,αn=α²_n, andγ¹_n=γ²_n≡0, then (1.1) reduces to the scheme for a mapping defined by Shahzad [12]:

x1=x∈K,

x_n+1=P1−α_nx_n+α_nTP1−β_nx_n+β_nTx_n, (1.2) where{α_n},{β_n}are sequences in [0, 1].

For N=2,T1,T2:K →K,T1=T, T2=S, and yn=x¹_n, then (1.1) reduces to the scheme with errors for two mappings defined by

x1=x∈K,

y_n=α¹_nTx_n+β_n¹x_n+γ¹_nu¹_n, xn+1=x²_n=α²_nSyn+β²_nxn+γ²_nu²_n,

(1.3)

where{α¹_n},{α²_n},{β¹_n},{β²_n},{γ_n¹},{γ²_n}are sequences in [0, 1] withα¹_n+β¹_n+γ_n¹=1=α²_n+ β²_n+γ²_nand{u¹_n},{u²_n}are bounded sequences inK.

It is our purpose in this paper to establish several weak and strong convergence theorems of the multistep iterative scheme with errors for a finite family of nonexpansive nonself-mappings. More precisely, we prove weak convergence of these implicit iteration

processes in a uniformly convex Banach space which has the Kadec-Klee property. The results presented in this paper extend and improve the corresponding ones announced by Shahzad [12], and many others.

2. Preliminaries

In this section, we recall the well-known concepts and results.

LetEbe a real Banach space. A subsetKofEis said to be a retract ofEif there exists a continuous mapP:E→Ksuch thatPx=xfor allx∈K. A mapP:E→Eis said to be a retraction ifP²=P. It follows that if a mapPis a retraction, thenP y=yfor all yin the range ofP. A mappingT:K→Eis called demiclosed with respect toy∈Eif for each sequence{xn}inKand eachx∈E,xnxandTxn→yimply thatx∈KandTx=y. A Banach spaceEis said to satisfy Opial’s condition [9] if for any sequence{xn}inE,xnx implies that

lim sup

n→∞

xn−x<lim sup

n→∞

xn−y (2.1)

for ally∈Ewithx=y. A Banach spaceEis said to have the Kadec-Klee property if for every sequence{xn}inE,xnxandxn → xtogether implyxn−x →0. A family {T_i:i=1, 2,. . .,N}ofNnonself-mappings ofK(i.e.,T_i:K→E) withF= ^N_i=1F(T_i)=∅ is said to satisfy condition (B) onKif there is a nondecreasing functionf : [0,∞)→[0,∞) with f(0)=0 andf(r)>0 for allr_∈(0,∞) such that for allx_∈K,

1maxiNx−Tixfd(x,F). (2.2) The family{Ti:i=1, 2,. . .,N} is said to satisfy condition (A^N) if (2.2) is replaced by 1/N^N_i=1x−T_ixf(d(x,F)) for allx∈K. Note that condition (B) reduces to condi- tion (A^N) whenx−T1x = x−T2x = ··· = x−TNx.

A mappingT:K→Eis called semicompact if any sequence{xn}inKsatisfyingxn− Tx_n →0 asn→ ∞has a convergent subsequence.

Lemma 2.1 (Tan and Xu [15]). Let{s_n},{t_n}be two nonnegative sequences satisfying

sn+1sn+tn, ∀n1. (2.3)

If^∞_n=1t_n<∞, then limn→∞s_nexists. Moreover, if there exists a subsequence{s_nj}of{s_n} such thats_nj→0 as j→ ∞, thens_n→0 asn→ ∞.

Lemma 2.2 (Xu [17]). Let p >1 andR >0 be two fixed numbers andEa Banach space.

ThenEis uniformly convex if and only if there exists a continuous, strictly increasing, and convex function g: [0,∞)→[0,∞) withg(0)=0 such thatλx+ (1−λ)y^pλx^p+ (1−λ)y^p−W_p(λ)g(x−y) for allx,y∈B_R(0)= {x∈E:xR}, andλ∈[0, 1], whereWp(λ)=λ(1−λ)^p+λ^p(1−λ).

Lemma 2.3 (Kaczor [5]). LetEbe a real reflexive Banach space such that its dualE^∗has the Kadec-Klee property. Let{xn}be a bounded sequence inEandx^∗,y^∗∈ωw(xn); hereωw(xn)

denote the set of all weak subsequential limits of{xn}. Suppose limn→∞txn+(1−t)x^∗−y^∗ exists for allt∈[0, 1]. Thenx^∗=y^∗.

Lemma 2.4 (Browder [1]). LetEbe a uniformly convex Banach space,Ka nonempty closed convex subset ofE, andT:K→Ea nonexpansive mapping. ThenI−T is demiclosed at zero.

3. Main results

In this section, we prove weak and strong convergence theorems of the iterative scheme given in (1.1) for a finite family of nonexpansive mappings in a Banach space. In order to prove our main results, the following lemmas are needed.

Lemma 3.1. Let E be a uniformly convex Banach space andK a nonempty closed con- vex subset of E which is also a nonexpansive retract of E. Let T1,T2,. . .,T_N:K→E be nonexpansive mappings. Let {xn} be the sequence defined by (1.1) with ^∞_n=1γⁱ_n<∞ for eachi=1, 2,. . .,N. If ^N_i=1F(Ti)=∅, then limn→∞xn−x^∗exists for allx^∗∈ ^Ni=1F(Ti).

Proof. For eachn1, we note that

x¹_n−x^∗α¹_nT1xn−x^∗+β¹_nxn−x^∗+γ¹_nu¹_n−x^∗ α¹_nxn−x^∗+β¹_nxn−x^∗+γ¹_nu¹_n−x^∗ xn−x^∗+d_n⁰,

(3.1)

whered_n⁰=γ¹_nu¹_n−x^∗. Since^∞_n=1γ¹_n<∞,^∞_n=1d⁰_n<∞. Next, we note that x²_n−x^∗α²_nT2x_n¹−x^∗+β²_nxn−x^∗+γ²_nu²_n−x^∗

α²_nx_n¹−x^∗+β²_nxn−x^∗+γ²_nu²_n−x^∗

=

α²_n+β²_nxn−x^∗+α²_nd_n⁰+γ_n²u²_n−x^∗ xn−x^∗+d_n¹,

(3.2)

whered_n¹=α²_nd_n⁰+γ²_nu²_n−x^∗. Since^∞_n=1d_n⁰<∞and^∞_n=1γ²_n<∞,^∞_n=1d¹_n<∞. Simi- larly, we have

x³_n−x^∗α³_nx_n²−x^∗+β³_nx_n−x^∗+γ³_nu³_n−x^∗ α³_nxn−x^∗+d_n¹+β³_nxn−x^∗+γ³_nu³_n−x^∗ xn−x^∗+α³_nd_n¹+γ³_nu³_n−x^∗=xn−x^∗+d_n²,

(3.3)

whered_n²=α³_nd_n¹+γ³_nu³_n−x^∗, so^∞_n=1d²_n<∞.

By continuing the above method, there exists a nonnegative real sequence{d_nⁱ⁻¹}such that^∞_n=1d_nⁱ⁻¹<∞and

x_nⁱ−x^∗xn−x^∗+d_nⁱ⁻¹, ∀n1,∀i=1, 2,. . .,N. (3.4) Thusxn+1−x^∗ = x^N_n −x^∗xn−x^∗+d_n^N−1for alln∈N. Hence, byLemma 2.1,

limn→∞x_n−x^∗exists. This completes the proof.

Lemma 3.2. LetEbe a uniformly convex Banach space andK a nonempty closed convex subset ofEwhich is also a nonexpansive retract ofE. LetT1,T2,. . .,T_N:K→Ebe nonex- pansive mappings. Let{xn}be the sequence defined by (1.1) with^∞_n=1γ_nⁱ <∞and{αⁱ_n} ⊆ [ε, 1−ε] for alli=1, 2,. . .,N, for someε∈(0, 1). If ^N_i=1F(Ti)=∅, then limn→∞xn− T_ix_n =0 for alli=1, 2,. . .,N.

Proof. Letx^∗∈ ^N_i=1F(T_i). Then, by Lemma 3.1, lim_n→∞x_n−x^∗exists. Let lim_n→∞

xn−x^∗ =r. Ifr=0, then by the continuity of eachTi the conclusion follows. Sup- pose thatr >0. Firstly, we are now to show that limn→∞T_Nx_n−x_n =0. Since{x_n}and {uⁱ_n}are bounded for alli=1, 2,. . .,N, there existsR >0 such thatx_n−x^∗+γⁱ_n(uⁱ_n−x_n), Tix_nⁱ⁻¹−x^∗+γⁱ_n(uⁱ_n−xn)∈BR(0) for alln1 and for alli=1, 2,. . .,N. UsingLemma 2.2, we have

x^N_n −x^∗2α^N_nT_Nx^N_n⁻¹+β^N_nx_n+γ^N_nu^N_n−x^∗2

=α^N_nT_Nx^N_n⁻¹−x^∗+γ^N_nu^N_n−x_n+1−α^N_nx_n−x^∗+γ^N_nu^N_n −x_n² α^N_nT_Nx^N_n⁻¹−x^∗+γ_n^Nu^N_n−x_n²+1−α^N_nx_n−x^∗+γ^N_nu^N_n −x_n²

−W2

α^N_ngTNx^N_n⁻¹−xn

α^N_nx^N_n⁻¹−x^∗+γ_n^Nu^N_n −xn²+1−α^N_nxn−x^∗+γ^N_nu^N_n −xn²

−W2

α^N_ngT_Nx^N_n⁻¹−x_n

α^N_nxn−x^∗+d^N_n⁻²+γ^N_nu^N_n −xn² +1−α^N_nxn−x^∗+d_n^N−2+γ_n^Nu^N_n −xn²

−W2

α^N_ngTNx^N_n⁻¹−xn

=x_n−x^∗+λ^N_n⁻²²−W2

α^N_ngT_Nx_n^N−1−x_n,

(3.5) whereλ^N_n⁻²:=d^N_n⁻²+γ^N_nu^N_n −x^∗. Observe thatε³W2(α^N_n) now (3.5) implies that ε³g(T_Nx^N_n⁻¹−x_n)x_n−x^∗2− x_n+1−x^∗2+ρ_n^N−2, where ρ^N_n⁻²:=2λ^N_n⁻²x_n− x^∗2+ (λ^N_n⁻²)². Since^∞_n=1d^N_n⁻²<∞and^∞_n=1γ^N_n⁻²<∞, we get^∞_n=1ρ^N_n⁻²<∞. This implies that limn→∞g(TNx^N_n⁻¹−xn)=0. Sinceg is strictly increasing and continuous

at 0, it follows that limn→∞TNx^N_n⁻¹−xn =0. Note that

x_n−x^∗x_n−T_Nx^N_n⁻¹+T_Nx^N_n⁻¹₋x^∗ x_n−T_Nx^N_n⁻¹+x^N_n⁻¹−x^∗,

(3.6)

for alln1. Thusr=limn→∞xn−x^∗lim infn→∞x^N_n⁻¹−x^∗lim sup_n→∞x_n^N−1− x^∗rand therefore limn→∞x^N_n⁻¹−x^∗ =r. Using the same argument in the proof above, we have

x^N_n⁻¹−x^∗2α^N_n⁻¹x^N_n⁻²−x^∗+γ^N_n⁻¹u^N_n⁻¹−x^∗2 +1−α^N_n⁻¹xn−x^∗+γ^N_n⁻¹u^N_n⁻¹−x^∗2

−W2

α^N_n⁻¹gTN−1x_n^N−2−xn

α^N_n⁻¹x_n−x^∗+d_n^N−3+γ_n^N−1u^N_n⁻¹−x^∗2 +1−α^N_n⁻¹x_n−x^∗+d_n^N−3+γ^N_n⁻¹u^N_n⁻¹−x^∗2

−W2

α^N_n⁻¹gT_N−1x_n^N−2−x_n xn−x^∗2+ρ^N_n⁻³−W2

α^N_n⁻¹gTN−1x_n^N−2−xn.

(3.7)

This implies thatε³g(T_N−1x^N_n⁻²−x_n)x_n−x^∗2− x^N_n⁻¹−x^∗2+ρ^N_n⁻³and there- fore limn→∞TN−1x_n^N−2−xn =0. Thus, we have

x_n−T_Nx_nx_n−T_Nx_n^N−1+T_Nx_n^N−1−T_Nx_n x_n−T_Nx_n^N−1+x^N_n⁻¹−x_n

=xn−TNx^N_n⁻¹+Pα^N_n⁻¹TN−1x^N_n⁻²+β^N_n⁻¹xn+γ_n^N−1u^N_n⁻¹−Pxn xn−TNx_n^N−1+α^N_n⁻¹xn−TN−1x^N_n⁻²+γ_n^N−1u^N_n⁻¹−xn.

(3.8) Since lim_n→∞x_n−T_Nx_n^N−1 =0, lim_n→∞x_n−T_N−1x^N_n⁻² =0, and^∞_n=1γ^N_n⁻¹<∞, it follows that limn→∞xn−TNxn =0. Similarly, by using the same argument as in the proof above, we have limn→∞xn−TN−2x^N_n⁻³ =limn→∞xn−TN−3x^N_n⁻⁴ =,. . .,= lim_n→∞x_n−T2x¹_n=0. This implies that lim_n→∞x_n−T_N−1x_n=lim_n→∞x_n−T_N−2x_n = ,. . .,=limn→∞xn−T3xn =0. It remains to show that

lim_n→∞x_n−T1x_n=0, lim_n→∞x_n−T2x_n=0. (3.9)