0 Fixed Point Theory and Applications

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RESEARCH

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An implicit iterative algorithm with errors for two families of generalized asymptotically nonexpansive mappings

Ravi P Agarwal1, Xiaolong Qin2 and Shin Min Kang3*

* Correspondence: smkang@gnu. ac.kr

3Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Korea Full list of author information is available at the end of the article

Abstract

In this paper, an implicit iterative algorithm with errors is considered for two families of generalized asymptotically nonexpansive mappings. Strong and weak convergence theorems of common fixed points are established based on the implicit iterative algorithm.

Mathematics Subject Classification (2000) 47H09 ■ 47H10 ■ 47J25

Keywords: Asymptotically nonexpansive mapping, common fixed point, implicit iterative algorithm, generalized asymptotically nonexpansive mapping

1 Introduction

In nonlinear analysis theory, due to applications to complex real-world problems, a growing number of mathematical models are built up by introducing constraints which can be expressed as subproblems of a more general problem. These constraints can be given by fixed-point problems, see, for example, [1-3]. Study of fixed points of nonlinear mappings and its approximation algorithms constitutes a topic of intensive research efforts. Many well-known problems arising in various branches of science can be studied by using algorithms which are iterative in their nature. The well-known convex feasibility problem which captures applications in various disciplines such as image restoration, computer tomography, and radiation therapy treatment planning is to find a point in the intersection of common fixed point sets of a family of nonexpansive mappings, see, for example, [3-5].

For iterative algorithms, the most oldest and simple one is Picard iterative algorithm. It is known that T enjoys a unique fixed point, and the sequence generated in Picard iterative algorithm can converge to the unique fixed point. However, for more general nonexpansive mappings, Picard iterative algorithm fails to convergence to fixed points of nonexpansive even that it enjoys a fixed point.

Recently, Mann-type iterative algorithm and Ishikawa-type iterative algorithm (implicit and explicit) have been considered for the approximation of common fixed points of nonlinear mappings by many authors, see, for example, [6-24]. A classical convergence theorem of nonexpansive mappings has been established by Xu and Ori [23]. In 2006, Chang et al. [6] considered an implicit iterative algorithm with errors for asymptotically nonexpansive mappings in a Banach space. Strong and weak convergence theorems are

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established. Recently, Cianciaruso et al. [9] considered an Ishikawa-type iterative algorithm for the class of asymptotically nonexpansive mappings. Strong and weak convergence theorems are also established. In this paper, based on the class of generalized asymptotically nonexpansive mappings, an Ishikawa-type implicit iterative algorithm with errors for two families of mappings is considered. Strong and weak convergence theorems of common fixed points are established. The results presented in this paper mainly improve the corresponding results announced in Chang et al. [6], Chidume and Shahzad [7], Cianciaruso et al. [9], Guo and Cho [10], Khan et al. [12], Plubtieng et al. [14], Qin et al. [15], Shzhzad and Zegeye [18], Thakur [21], Thianwan and Suantai [22], Xu and Ori [23], Zhou and Chang [24].

2 Preliminaries

Let C be a nonempty closed convex subset of a Banach space E. Let T : C ® C be a mapping. Throughout this paper, we use F(T) to denote the fixed point set of T.

Recall the following definitions.

T is said to be nonexpansive if

|| Tx — Ty ||<|| x — y ||, Vx, y e C.

T is said to be asymptotically nonexpansive if there exists a positive sequence {hn} c [1, with hn ® 1 as n ® such that

|| Tnx — Tny ||< hn || x — y ||, Vx,y e C,n > 1.

It is easy to see that every nonexpansive mapping is asymptotically nonexpansive with the asymptotical sequence {1}. The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [25] in 1972. It is known that if C is a nonempty bounded closed convex subset of a uniformly convex Banach space E, then every asymptotically nonexpansive mapping on C has a fixed point. Further, the set F(T) of fixed points of T is closed and convex. Since 1972, a host of authors have studied weak and strong convergence problems of implicit iterative processes for such a class of mappings.

T is said to be asymptotically nonexpansive in the intermediate sense if it is continuous and the following inequality holds:

limsup sup(|| Tnx — Tny || - || x — y ||) < 0. (2 1

n^TO x,yeC '

Putting fn = max{0, supx>yiE C(|| Tx - T^yH - ||x - y||)}, we see that £,n ® 0 as n ® Then, (2.1) is reduced to the following:

|| Tnx — Tny ||<|| x — y || +ln, Vx,y e C,n > 1.

The class of asymptotically nonexpansive mappings in the intermediate sense was introduced by Bruck et al. [26] (see also Kirk [27]) as a generalization of the class of asymptotically nonexpansive mappings. It is known that if C is a nonempty closed convex and bounded subset of a real Hilbert space, then every asymptotically nonexpansive self mapping in the intermediate sense has a fixed point; see [28] more details.

T is said to be generalized asymptotically nonexpansive if it is continuous and there exists a positive sequence {hn} c [1, with hn ® 1 as n ® such that

lim sup sup (|| Tnx — Tny || — hn || x — y ||) < 0. (22)

n—x,yeC '

Putting fn = max{0, supx>ye C(| | T"x - Tny|| - hn||x - y||)}, we see that £,n ® 0 as n ® Then, (2.2) is reduced to the following:

|| Tnx — Tny ||< hn || x — y || +fn, Vx,y e C,n > 1.

We remark that if hn = 1, then the class of generalized asymptotically nonexpansive mappings is reduced to the class of asymptotically nonexpansive mappings in the intermediate.

In 2006, Chang et al. [6] considered the following implicit iterative algorithms for a finite family of asymptotically nonexpansive mappings {T1, T2,..., TN} with {an} a real sequence in (0, 1), {wn} a bounded sequence in C and an initial point x0 e C:

x1 = a1x0 + (1 — a1)T1x1 + u1, x2 = a2x1 +(1 — a2)T2x2 + u2,

XN = uNXN—1 + (1 — aN )TNXN + UN, xn+1 = &n+1xN +(1 — an+1)T'nxN+1 + uN+1,

x2N = a2Nx2N—1 + (1 — «2N )T^2N + U2N, x2N+1 = «2N+1x2N + (1 — «2N+1 )T3x2N+1 + U2N+1,

The above table can be rewritten in the following compact form:

xn = anxn—1 + (1 — UnTfyn + Un, Vn > 1,

where for each n > 1 fixed, j(n) - 1 denotes the quotient of the division of n by N and i(n) the rest, i.e., n = (j(n) - 1)N + i(n).

Based on the implicit iterative algorithm, they obtained, under the assumption that C + C c C, weak and strong convergence theorems of common fixed points for a finite family of asymptotically nonexpansive mappings {T1, T2,..., TN}; see [6] for more details.

Recently, Cianciaruso et al. [9] considered a Ishikawa-like iterative algorithm for the class of asymptotically nonexpansive mappings in a Banach space. To be more precise, they introduced and studied the following implicit iterative algorithm with errors.

Iyn = (1 fin $n)xn + finTi(n)xn + $nvn, (2 3)

xn = (1 — an — Kn)xn—1 + anT^y + YnUn, Vn > 1,

where {an}, {bn}, {gn}, and {8n} are real number sequences in [0,1], {wn} and {vn} are bounded sequence in C. Weak and strong convergence theorems are established in a uniformly convex Banach space; see [29] for more details.

In this paper, motivated and inspired by the results announced in Chang et al. [6], Chidume and Shahzad [7], Cianciaruso et al. [9], Guo and Cho [10], Plubtieng et al. [14], Qin et al. [15], Shzhzad and Zegeye [18], Thakur [21], Thianwan and Suantai [22], Xu and Ori [23], Zhou and Chang [24], we consider the following Ishikawa-like implicit iteration algorithm with errors for two finite families of generalized asymptotically nonexpansive mappings {T1, T2,..., TN} and {Si, S2,..., SN}.

x0 e C,

xl = alx0 + в^Т^^а'lxl + в'lS^ + y iVl) + ylUl, x2 = a2xl + eiTiia' 2X2 + в '2 S2x2 + У' 2 V2) + У2 u2,

Xn = aNxn-i + выТы (a'NxN + в 'nSnXn + У 'nvn ) + YnUn , xn+i = аы+i xn + вN+lTN+l(a'N+l xn+i + в 'n+i^n+iXn+i

+ У' N+ivN+i) + Уы+i UN+i,

X2N = a2NX2N-i + в2ыТ2Ы (а'2NX2N + в' 2NS2NX2N + У' 2Nv2N) + Y2NU2N,

X2N+i = a2N+i X2N + в2N+lT2N+l(а' 2N+iX2N+i + в' 2N+iS2N+iX2N+i + У' 2N+l V2N+l) + Y2N+lU2N+l,

where {an}, {bn}, {gn}, and (y^l are sequences in [0,1] such that

an + вп + Уп = a'n + в'п + уП = l for each n > 1. We have rewritten the above table in the following compact form:

Xn = anXn-l + впТ^П)(а'пXn + вПsfnfn + уПVn) + YnUn, n > l,

where for each n > 1 fixed, j(n) - 1 denotes the quotient of the division of n by N and i(n) the rest, i.e., n = (j(n) - 1)N + i(n).

Putting yn = a'nxn + в'пSnxn + уПvn, we have the following composite iterative algorithm:

Iyn = anxn + в 'nSi(ri)Xn + y'nVn, . ,

a -ri(n) ^ (2-4)

Xn = anXn-l + впТ(п)Уп + YnUn, n > l.

We remark that the implicit iterative algorithm (2.4) is general which includes (2.3) as a special case.

Now, we show that (2.4) can be employed to approximate fixed points of generalized asymptotically nonexpansive mappings which is assumed to be Lipschitz continuous. Let Ti be a L't-Lipschitz generalized asymptotically nonexpansive mapping with a sequence (h'n1 С [l, (x>) such that h'n ^ l as n ® ^ and Si be a LJ-Lipschitz generalized asymptotically nonexpansive mapping with sequences (k'n 1 С [l, (x>) such that k'n ^ l as n ® ^ for each 1 < i < N. Define a mapping Wn : C ® C by

Wn(x) = anXn-l + впТ^П)(а'п X + вП Siln)X + уП Vn) + YnUn, Vn > l.

It follows that

II Wn(x) - Wn(y) II

< вп 11 Tf(n)(a'nx + в'nS™x + y'nVn) - tin)(a'ny + в'nS'l^y + У'nVn) 11

< вnL(a'n || x - y || +в'n II Sf^x - ¿Iffy II)

< вnL(a'n + в'nL) II x - y II, Vx, y e C,

L = max {Lt\ ..., L?, L], ...L?}. (2 . 5)

If pnL(a'n + p^L) < 1 for all n > 1, then Wn is a contraction. By Banach contraction mapping principal, we see that there exists a unique fixed point xn e C such that

Xn = Wn(Xn) = anXn-1 + PnTnfa'nX + P'nSf^x + Y nVn) + YnUn, Vn > 1.

That is, the implicit iterative algorithm (2.4) is well defined.

The purpose of this paper is to establish strong and weak convergence theorem of fixed points of generalized asymptotically nonexpansive mappings based on (2.4).

Next, we recall some well-known concepts.

Let E be a real Banach space and UE = {x e E : ||x|| = 1}. E is said to be uniformly convex if for any e e (0, 2] there exists 8 >0 such that for any x, y e UE,

x — y ||> e implies

< 1 - S.

It is known that a uniformly convex Banach space is reflexive and strictly convex.

Recall that E is said to satisfy Opial's condition [30] if for each sequence {xn} in E, the condition that the sequence xn ® x weakly implies that

lim inf II Xn — X | < lim inf | Xn — y |

n—n—

for all y e E and y * x. It is well known [30] that each lp (1 < p < and Hilbert spaces satisfy Opial's condition. It is also known [29] that any separable Banach space can be equivalently renormed to that it satisfies Opial's condition.

Recall that a mapping T: C ® C is said to be demiclosed at the origin if for each sequence {xn} in C, the condition xn ® x0 weakly and Txn ® 0 strongly implies Tx0 = 0. T is said to be semicompact if any bounded sequence {xn} in C satisfying limn®„||x„ - Txn|| = 0 has a convergent subsequence.

In order to prove our main results, we also need the following lemmas.

Lemma 2.1. [20]Let {an}, {bn} and {cn} be three nonnegative sequences satisfying the following condition:

an+i < (1 + bn)an + Cn, Vn > no,

where n0 is some nonnegative integer. IfYlcn < <x>andYlbn < to, then limn®„ an exists.

Lemma 2.2. [17]Let E be a uniformly convex Banach space and 0<1 < tn < h <1 for all n > 1. Suppose that {xn} and {yn} are sequences of E such that

lim sup || xn ||< r, lim sup || yn ||< r

lim || tnXn + (1 — tn)yn ||= r

hold for some r > 0. Then limn®„ ||xn - yn|| =0.

The following lemma can be obtained from Qin et al. [31] or Sahu et al. [32] immediately.

Lemma 2.3. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C ® C be a Lipschitz generalized asymptotically nonexpansive mapping. Then I - T is demiclosed at origin.

3 Main results

Now, we are ready to give our main results in this paper.

Theorem 3.1. Let E be a real uniformly convex Banach space and C be a nonempty closed convex subset of E. Let Ti : C ® C be a uniformly L't-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence {h,j c [1, to , where h, ^ las n ® ^ and Si : C ® C be a uniformly LJ-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence {fe,j c [1, to , where fej, ^ las n ® ^ for each 1 < i < N. Assume that ^ = p| = F(Ti) p| p|N=1 F(Sj) = 0. Let {un}, {vn} be bounded sequences in C and en = max{hn, kn}, where h, = sup{h, : 1 < i < Njand fe„ = sup{fe, : 1 < i < Nj. Let {an}, {bn}, {gn}, {a,j, {^,jand {y,jbe sequences in [0,1] such that a, + + y, = a, + + y, = lfor each n > 1. Let {xn} be a sequence generated in (2.4). Put = max{0, supxyeC(|| Tj"x - Tj"y || -h, || x - y ||)jand

v, = max{0, supxyeC(|| S,x - S,y || -fe, || x - y ||)j. Let fn = max{^n, vn}, where vn = max{v, : 1 < i < Njand v, = max{v, : 1 < i < Nj. Assume that the following restrictions are satisfied:

(a)L TO =1 y, < TOand n=j y< to;

(b)£^ (e, - 1) < TOand £^ ^ < to;

(c) ^Lfa, + ¡3,L) < 1, where L is defined in (2.5);

(d) there exist constants l, h <= (0, 1) such that l < an, a, < n.

lim || x, - Trx, ||= lim || x, - Srx, ||= 0, Vr e {1,2,...,Nj. Proof. Fixing / e J-, we see that

||y, -/1| = || a^x, + 3'nS1i(n)Xn + y^v, -/ ||

< a\ yx,- / y +3y ^i^f,- / y +yyv,-/ y

< y x, - / y +3Vj(,) y x, - / y +3+ yy v,- /

< ej(u) y x, - / y +3 + yy v, - / y

| x, - / |

= y anxn—1 + ¡nTi(n)yn + / y

< a, || x,-1 - / y +3, y T^)y, - / y +y, y u, - / y

< a, y x,-1 - / y +¡nej(n) y y, - / y +Pnjn) + y, y u, - / || .

< a, y x,-1 - / y +(1 - a^ej, || y, - / || +¡n£j(n) + y, || u, - / || .

Substituting (3.1) into (3.2), we see that

y xn f y

< an y Xn-1 - f y +(1 - an)ej(n)(ej(n) II Xn - f y + fi'nHj(n) + Y'n II vn - f II)

+fin£j(n) + Yn I Un - f I .

< an I Xn-i - f I +(1 - an)e2(n) y Xn - f II +(1 + ej(n))|j(n) +ej(n)Y'n I Vn - f I +Yn I Un - f I .

Notice that J](en - 1) < to. We see from the restrictions (b) and (d) that there exists a positive integer n0 such that

(1 - an)jn) < R < 1, Vn > no, where R = (1 - X)(l + j^)- It follows that

\ 1 - (1 - Ojn) /

(1 + gj(n))fj(n) + ej(n)Y'n II Vn - f II + Yn II Un - f II

1-(1 -an)efin} (33)

(1 +M1)Çj(„) +M1M2Y/n +M3yn + 1-R

where Mi = supB>i(eB}, M2 = supM>i{||v„ - f\\}, and M3 = supB>i{||un -f||}. In view of Lemma 2.1, we see that limn®„ ||xn - f|| exists for each f e F. This implies that the sequence {xn} is bounded. Next, we assume that limn®„ ||xn - f|| = d >0. From (3.1), we see that

y T^Vn - f + Yn(un - Tjjyn) y

Il Tj(n) f II II T'(n) II <y Ti(n) yn - f y +Yn y Un - Ti(n) yn y

< ej(n) y yn - f y +£j(n) + Yn y Un - T^Yn y

< ej(n) y xn - f y +ej(n)?j(n) + ej(n)Y'n y Vn - f y +?j(n) +Y U Tj(n)y .

+ Yn y un Ti(n) yn y . This implies from the restrictions (a) and (b) that limsup y T^Yn - f + Yn(Un - TrnVn) II< d.

Notice that

y Xn-1 - f + Yn(Un - T'lnVn) y<y Xn-1 - f y +Yn y Un - T^Yn y . This shows from the restriction (a) that

limsup y Xn-1 - f + Yn(Un - j))yn) y< d.

On the other hand, we have

d = lim y x, - / y

= „lim y a,(x,-1 - / + Уn(un - T^y,)) + (1 - ^K^y, - / + yn(un - ^jy,» y .

It follows from Lemma 2.2 that

lim y T^k - x,-1 y=0. (3.4)

Notice that

y x, x,-1 y< ft, y Ti(n)yn xn—1 y + y, y u, x,-1 y .

It follows from (3.4) and the restriction (a) that

lim y x, - x,-1 ||= 0. (3.5)

This implies that

lim y x, - x,+i ||= 0, Vl =1,2,..., N. (3.6)

Notice that

y x, -/ + y,(v, - S^x,) ||<|| x, -/ y +y, y v, - S^x, ||

II Sj(,) / ' ( j,) ) II

y Si(n)Xn - J + y n(vn - Si(n)xn) y

<y S^x, - / y +yy v, - S^x, y < ej(,)y x,- / y +j,)+yy v,- sin))Xn

which in turn imply that

lim sup y x, -/ + y,(v, - Siij¡'lrn')Xn) ||< d

lim sup y S^x, - / + y,(v, - S^x,) ||< d.

On the other hand, we have

y x, - / y =y anXn-l + ¡nTfl)Уn + УnUn - / y

< a, y x,-1 - / y +3, y Tf,)y, - / y +y, y u, - / ||

< a, y x,-1 - T^y, y + y Tf,)y, -/ y +y, y u, - / y

< a, y x,-1 - T^y, y +ej(,) y y, - / y +j,) + y, y u, - / y,

from which it follows that lim infn®„ ||yn - /|| > d. In view of (3.1), we see that lim supn®„ ||yn - f]| < d. This proves that lim || yn - f ||= d.

n—>W

Notice that

lim || yn - f || = lim || a'n(xn - f + Y' n(vn - S^Xn))

n—W n—W V >

+ (1 - d'nW'fffn - f + Y' n(vn - S'lnfnV H .

This implies from Lemma 2.2 that

lim || Sfnlxn - Xn ||=0. (3.7)

n—w Vn)

On the other hand, we have

II Ti(n)X - y Ti(n)Xn Xn y

<11 Tj(n)X —| Ti(n) n - Tj(n)y Ti(n)yn y + y Tj(n)y Ti(n)yn

<11 Tj(n) X —| Ti(n) n - T'(n)y Ti(n)yn y + y T'(n)y Ti(n)yn

— L | n - yn | + | T'(n)y -X Ti(n)yn Xn-

— n || S}(j)Xn Xn y +Ly n y vn Xn y + y Ti(n)yn Xn-1 y + y Xn-1 Xn y .

This combines with (3.4), (3.5), and (3.7) gives that

lim y TflXn - Xn ||= 0. (3.8)

n—w y J

Since n = (j(n) - 1)N + i(n), where i(n) e {1, 2,..., N}, we see that

Xn Si(n)Xn y — y Xn Si(n)Xn y + y Si(n)Xn Si(n)Xn

j(n) ,, r .. S(n)-1 i(n)Xn y +L y Si(n) Xn - Xn

—| Xn - S —| Xn - S

j(n) || r(|| Sj(n)-1 d(n)-1

i(n)Xn y +L(y Si(n) Xn - Si(n-N)Xn-N

II S(n)-1 II II

+ y Si(n—N)Xn—N Xn-N y + y Xn-N Xn

Notice that

j(n - N) = j(n) - 1 and i(n - N) = i(n).

This in turn implies that

II WW- ciM- II = 11 dM- dM- II

y Si(n) Xn - Si(n-N)Xn-N y =y Si(n) Xn - Si(n) Xn-N y

— L y Xn - Xn-N y

(3.10)

cj(n)-1 Qj(n-N) ,,

Si(n-N)Xn-N - Xn-N y = y Si(n-N)Xn-N - Xn-N y . (3.11)

Substituting (3.10) and (3.11) into (3.9) yields that

II Xn Si(n)Xn II < II Xn Sj(n)Xn II +L(L II Xn Xn-N II

+ II Si(n-N)Xn-N - Xn-N II).

It follows from (3.6) and (3.7) that

Hm II Xn - Si(n)Xn II= 0. (3 . 12)

In particular, we see that

lim—TO II XjN+i - SiXjN+1 II = 0, lim—TO II XjN+2 - S2XjN+2 II = 0,

lim—TO II XjN+N - SNXjN+N II = 0. For any r, s = 1, 2,..., N, we obtain that

II XjN +5 SrXjN+s H

< XjN+s - XjN+r II + II XjN+r - SrXjN+r II + II SrXjN+r - SrXjN+5 I

< (1 + L) II XjN+s - XjN+r II + II XjN+r - SrXjN+r II . Letting j ® we arrive at

lim II XjN+5 - SrXjN+5 II = 0,

j—>TO

which is equivalent to

lim II Xn - SrXn II= 0. (3 . 13)

n—TO v '

Notice that

I Xn Ti(n)Xn I <I Xn Ti(n)Xn II + II Ti(n)Xn Ti(n)Xn I

-II Tj(n) II r II Tj(n)-1 II

^ Xn - Ti(n)Xn II +L II Ti(n) Xn - Xn II

-II jj(n) || r(.. Tj(n)-1 Tj(n)-1

Xn - Ti(n)Xn II +L(I Ti(n) Xn - Ti(n-N)Xn-N

II T,j(n)-1 II II 111

+ II Ti(n-N)Xn-N - Xn-N II + II Xn-N - Xn II).

On the other hand, we have

T'(n)-1 Tj(n)-1 11 = 11 Tj(n)-1 Tj(n)-1

Ti(n) Xn - T i(n-N)Xn-N II =I Ti(n) Xn - Ti(n) X

(3. 14)

(3. 15)

< L II Xn - Xn-N II

II Ti(n-N)Xn-N - Xn-N II = I Ti(n-jv)Xn-N - Xn-N I . (3 . 16)

Substituting (3.15) and (3.16) into (3.14) yields that

I Xn Ti(n)Xn I <I Xn ^fn) Xn II +L(L I Xn Xn-N I

+ II T(n-N)Xn-N - Xn-N II). It follows from (3.6) and (3.8) that

lim II Xn - Ti(n)Xn II= 0. (3 . 17)

M—VlYl v ' v '

In particular, we see that

limfe—«, || XjN+1 - TiXjN+1 ||= 0, limfe—«, || XjN+2 - T2XjN+2 ||= 0,

limfe^TO || XjN+N - TNXjN+N ||= 0.

For any r, s = 1, 2,..., N, we obtain that

H XjN+s - TrXjN +s ||

<| XjN+s - XjN+r || + || XjN+r - TrXjN+r || + || TrXjN+r - TrXjN+s ||

< (1 + L) || XjN+s - XjN+r || + || XjN+r - TrXjN+r || .

Letting j ® we arrive

lim || XjN+s - TrXjN+s ||= 0,

j—>TO

which is equivalent to

lim || Xn - TrXn ||= 0. (3 , 18)

n—TO v '

This completes the proof. □

Now, we are in a position to give weak convergence theorems.

Theorem 3.2. Let Ebe a real Hilbert space and C be a nonempty closed convex subset of E. Let Ti : C ® C be a uniformly L\-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence {h'n} C [1, to , where h'n — las n ® and Si: C ® C be a uniformly L\-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence {fen} C [1, to , where k'n — 1as n ® for each 1 < i < N. Assume that F = p|= F(Ti) p| p|N=1 F(Si) = 0. Let {un}, {vn} be bounded sequences in C and en = max {hn, kn}, where hn = sup{hn : 1 < i < N}and kn = sup{fen : 1 < i < N}. Let {a„}, {bn}, {gn}, {ft}, {P'n}and {yn}be sequences in [0,1] such that an + fin + Yn = a'n + + yn = for each n > 1. Let {xn} be a sequence generated in (2.4). Put v'n = max{0, supX,yeC(|| S^^x - Sny || -fein || X - y ||)}and

v'n = max{0, supX,yeC(|| SnX - Sny || -fen || X - y ||)}. Let £n = max{^n, vn}, where vn = max{vn : 1 < i < N}and vn = max{vn : 1 < i < N}. Assume that the following restrictions are satisfied:

(a) ETO1 Yn < TOcmdJ2Y 'n < TO

(b)£TO=1 (en - 1) < TOandY.^ < to;

(c) finL(a'n + fi'nL) < 1, where L is defined in (2.5);

(d) there exist constants l, h e (0, 1) such that l < an, a'n < n.

Then the sequence {xn} converges weakly to some point in F.

Proof. Since E is a Hilbert space and {xn} is bounded, we can obtain that there exists a subsequence {Xn,} of {xn} converges weakly to x* e C. It follows from (3.13) and (3.18) that

lim || Xn, - TrXn, ||= lim || Xn, - Sx, ||= 0.

Since I - Sr and I - Tr are demiclosed at origin by Lemma 2.3, we see that x* e F. Next, we show that the whole sequence {xn} converges weakly to x*. Suppose the contrary. Then there exists some subsequence {Xn,} of {xn} such that {Xn,} converges weakly to x** e C. In the same way, we can show that x** e F. Notice that we have proved that limn®„ ||xn - f|| exists for each f e F .By virtue of Opial's condition of £, we have

liminf y xn - X* y < liminf || xn - X** ||

p—w p p—w p

= liminf y xn, - X** y

,—w q

< liminf y xn, - X* y .

,—w '

This is a contradiction. Hence, x* = x**. This completes the proof. □

Remark 3.3. Theorem 3.2 which includes the corresponding results announced in Chang et al. [6], Chidume and Shahzad [7], Guo and Cho [10], Plubtieng et al. [14], Qin et al. [15], Thakur [21], Thianwan and Suantai [22], Xu and Ori [23], and Zhou and Chang [24] as special cases mainly improves the results of Cianciaruso et al. [9] in the following aspects.

(1) Extend the mappings from one family of mappings to two families of mappings;

(2) Extend the mappings from the class of asymptotically nonexpansive mappings

to the class of generalized asymptotically nonexpansive mappings.

If Sr = I for each r e {1, 2,..., N} and Y'n = 0, then Theorem 3.2 is reduced to the following.

Corollary 3.4. Let £ be a real Hilbert space and C be a nonempty closed convex subset of £. Let T : C ® C be a uniformly L^-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence (h^ C [1, w , where hln — 1as n ® ^ for each 1 < i < N. Assume that F = p|j=1 F(Ti) = 0. Let {un} be a bounded sequence in C and hn = sup(hn : 1 — i — N}. Let {an}, {bn} and {gn} be sequences in [0,1] such that an + bn + gn = 1 for each n > 1. Let {xn} be a sequence generated in the following process:

X0 e C, Xn = anXn-1 + PnT^Xn + YnUn, n > 1. (3.19)

Put ^ = max{0, supX,yeC(y T'x - Tfy y -hn || X - y ||)}. Let

= max(^n : 1 — i — N}. Assume that the following restrictions are satisfied:

(a) EYn < w;

(b)£^ (hn - 1) < wand £^ ^n < w;

(c) bnL < 1, where L = max(L\ : 1 — i — N};

(d) there exist constants l, h e (0, 1) such that l > an, an — ^

Then the sequence {xn} converges weakly to some point in F.

Next, we are in a position to state strong convergence theorems in a Banach space.

Theorem 3.5. Let £ be a real uniformly convex Banach space and C be a nonempty closed convex subset of £. Let Ti : C ® C be a uniformly LJ-Lipschitz and generalized

asymptotically nonexpansive mapping with a sequence {h'} C [1, w , where hn — 1as n ® ^ and Si : C ® C be a uniformly LJ-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence {fen} C [1, w , where fen — 1as n ® ^ for each 1 < i < N. Assume that F = p| = F(Ti) p| pN F(S^ = 0. Let {un}, {vn} be bounded sequences in C and en = max{hn, kn}, where hn = sup{hn : 1 — i — N}and fen = sup{fen : 1 — i — N}. Let {an}, {bn}, {gn}, {an}, {^Jand {Y'}be sequences in [0,1] such that an + fin + Yn = a'n + ¡i'n + y' = for each n > 1. Let {xn} be a sequence generated in (2.4). Put ^n = max{0, supX,yeC(|| T'x - T'y y -hin || x - y ||)}and

v'n = max{0, supX,yeC(|| S'x - S'y || -fein || X - y ||)}. Let fn = max{^n, vn}, where vn = maxiv' : 1 — i — N}and vn = maxiv' : 1 — i — N}. Assume that the following restrictions are satisfied:

(a) nw

=1 Yn < wand n=1 y 'n < w;

(b)£w=1 («' - 1) < wand £w=1 In < w;

(c) PnL(a'n + p'nL) < 1, where L is defined in (2.5);

(d) there exist constants l h e (0, 1) such that l > an, a' — ty

If one of mappings in {T1, T2,..., TN} or one of mappings in {Si, S2,..., SN} are semicom-pact, then the sequence {xn} converges strongly to some point in F.

Proof. Without loss of generality, we may assume that S1 are semicompact. It follows from (3.13) that

lim y xn - S1xn ||= 0.

By the semicompactness of S1, we see that there exists a subsequence {Xnp} of {xn} such that Xnp — w e C strongly. From (3.13) and (3.18), we have

y w - Srw y — y w - x„f y + y x„f - SrXnp y + || SrX„f - SrW y

y w - TrW y — y w - Xnp y + y Xnp - TrXnp y + y TrX'p - TrW y

Since Sr and Tr are Lipshcitz continuous, we obtain that w e F. From Theorem 3.1, we know that limn®„||xn -f|| exists for each f e F. That is, limn®„||xn - w|| exists. From Xnp — w, we have

lim y xn - w ||= 0.

This completes the proof of Theorem 3.5. □

If Sr = I for each r e {1, 2,..., N} and y' = 0, then Theorem 3.5 is reduced to the following.

Corollary 3.6. Let £ be a real uniformly convex Banach space and C be a nonempty closed convex subset of £. Let Ti : C ® C be a uniformly LJ-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence {h'} C [1, w , where h' — 1as « ® ^ for each 1 < i < N. Assume that F = p| = F(Ti) = 0. Let {un} be a bounded sequence in C and hn = sup{h' : 1 — i — N}. Let {an}, {bn} and {gn} be sequences i« [0,1] such that an + bn + gn = 1 for each n > 1. Let {xn} be a sequence generated i«

(3.19). Put t^n = max{0,supxyeC(|| T^x - Tny || -hhn || x - y ||)}. Let

/xn = max{^n : 1 < i < N}. Assume that the following restrictions are satisfied:

(a) E^=1 Yn < TO

(b) E^=1 (hn - 1) < TOand Y,^=1 tn < to;

(c) bnL < 1, where L = max{Lj : 1 < i < N};

(d) there exist constants l, h e (0, 1) such that l > an, a'n < n.

If one of mappings in [Tlt T2,..., TN} is semicompact, then the sequence {xn} converges strongly to some point in F.

Theorem 3.7. Let E be a real uniformly convex Banach space and C be a nonempty closed convex subset of E. Let Ti : C ® C be a uniformly L\-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence [h'n} c [1, to , where h'n ^ las n ® and Si : C ® C be a uniformly L's-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence [fen} c [1, to , where k'n ^ las n ® for each 1 < i < N. Assume that F = p| = F(T^ p| p|N=1 F(Sj) = 0. Let {un}, {vn} be bounded sequences in C and en = max{hn, kn}, where hn = sup[hn : 1 < i < N}and kn = sup[kn : 1 < i < N}. Let {an}, {bn}, {gn}, [a'n}, [ß'n}and [Y^be sequences in [0,1] such that an + ßn + Yn = a'n + ß'n + Yn = 1/or each n > 1. Let {xn} be a sequence generated in (2.4). Put i-iln = max[0, supxyeC(|| Tfx - Tfy || -hn || x - y ||)}and

v'n = max[0, supxyeC(|| Snx - Sny || —kn || x - y ||)}. Let £n = max{^n, vn}, where vn = max[vn : 1 < i < N}and vn = max[vn : 1 < i < N}. Assume that the following restrictions are satisfied:

(a) ETO1 Yn < TOand YTO1 Y 'n < to;

(b)E(en - 1) < TOand Y< to;

(c) finL(a'n + fi'nL) < 1, where L is defined in (2.5);

(d) there exist constants l, h e (0, 1) such that l > an,.a'n < n.

If there exists a nondecreasing function g : [0, ® [0, with g(0) = 0 and g(m) > 0 for all m e (0, such that

max {|| x - Srx ||} + max {|| x - Trx ||} > g(dist(x, F)), Vx e C,

1<r<N 1<r<N

then the sequence {xn} converges strongly to some point in F.

Proof. In view of (3.13) and (3.18) that g(dist(xn, F)) — 0, which implies dist(xn, F) — 0. Next, we show that the sequence {xn} is Cauchy. In view of (3.3), we obtain by putting

(1 + M1)(ej(n) - 1) , (1+ M1)£j(n)+ M1M2Y'n + M3Yn

a, 1 =-—- and on =-—-

n 1 - R n 1 - R

|| Xn - f ||< (1 + an) || Xn-1 - f || +bn.

It follows, for any positive integers m, n, where m >n >n0, that

y Xm - p y— B y Xn - p y +B£bi + bm,

where B = exp{^w=1 an}. It follows that

y Xn Xm y — y Xn f y + y Xm f y

— (1 + B) y Xn - f y +B J2 bi + bm.

Taking the infimum over all f e F, we arrive at

y Xn Xm y— (1 + B)dist(Xn, + + bm.

In view of w=1 bn < w and dist(Xn, F) — 0, we see that {x«} is a Cauchy sequence in C and so {xn} converges strongly to some x* e C. Since Tr and Sr are Lipschitz for each r e {1, 2,..., N}, we see that F is closed. This in turn implies that x* e F. This completes the proof. □

If Sr = I for each r e {1, 2,..., N} and y' = 0, then Theorem 3.7 is reduced to the following.

Corollary 3.8. Let £ be a real uniformly convex Banach space and C be a nonempty closed convex subset of £. Let Ti : C ® C be a uniformly LJ-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence {h'} C [1, w , where h' — 1as ft ® ^ for each 1 < i < N. Assume that F = p| = F(Ti) = 0. Let {un} be a bounded sequence in C and hn = sup{h' : 1 — i — N}. Let {an}, {b«} and {gn} be sequences i« [0,1] such that an + b« + gn = 1 for each n > 1. Let {xn} be a sequence generated i« (3.19). Put < = max{0,supX,yeC(y T^x - T^y | -h' || X - y ||)}. Let /xn = max{^in : 1 — i — N}. Assume that the following restrictions are satisfied:

(a) Lw=1 Yn < w;

(b)Ew (hn - 1) < wan^^ ^n < w;

(c) b«L < 1, where L = max{L\ : 1 — i — N};

(d) there exist constants l, h e (0, 1) such that, l > an, a' — n

If there exists a nondecreasingfunction g : [0, ® [0, with g(0) = 0 and g(m) > 0 for all m e (0, such that

max {|| x - Trx ||} > g(dist(x, F)), Vx e C,

1—r—N

then the sequence {xn} converges strongly to some point in F.

Acknowledgements

The authors are indebted to the referees for their helpfulcomments. The work was supported partially by Natural Science Foundation of Zhejiang Province (Y6110270).

Author details

department of Mathematics, Texas A&M University - Kingsville, Kingsville, TX 78363-8202, USA 2Schoolof

Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou

450011, China ^Department of Mathematics and RINS, Gyeongsang NationalUniversity, Jinju 660-701, Korea

Authors' contributions

Allauthors contributed equally and significantly in writing this paper. Allauthors read and approved the final

manuscript.

Competing interests

The authors declare that they have no competing interests.

Received: 13 April 2011 Accepted: 27 September 2011 Published: 27 September 2011

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doi:10.1186/1687-1812-2011-58

Cite this article as: Agarwal et al.: An implicit iterative algorithm with errors for two families of generalized asymptotically nonexpansive mappings. Fixed Point Theory and Applications 2011 2011:58.

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