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Strong convergence theorems for two total asymptotically nonexpansive nonself mappings in Banach spaces

Hukmi Kiziltunc* and Esra Yolacan

"Correspondence: hukmu@atauni.edu.tr Department of Mathematics, Faculty of Science, Ataturk University, Erzurum, 25240, Turkey

Abstract

In this paper, we define and study the convergence theorems of a new two-steps iterative scheme for two total asymptotically nonexpansive nonself-mappings in Banach spaces. The results of this paper can be viewed as an improvement and extension of the corresponding results of (Shahzad in Nonlinear Anal. 61:1031-1039, 2005;Thianwan in Thai J. Math. 6:27-38,2008; Ozdemir etal. in Discrete Dyn. Nat. Soc. 2010:307245, 2010) and all the others. MSC: 47H09; 47H10; 46B20

Keywords: total asymptotically nonexpansive mappings; common fixed point; uniformly convex Banach space

ft Spri

ringer

1 Introduction

Let E be a real normed space and K be a nonempty subset of E. A mapping T: K ^ K is called nonexpansive if \\Tx - Ty|| < ||x -y\\ for all x,y e K. A mapping T: K ^ K is called asymptotically nonexpansive if there exists a sequence {kn} c [1, to) with kn ^ 1 such that

|| Tnx - Tny|| < kn\\x - y\\ (1.1)

for all x,y e K and n > 1. Goebel and Kirk [1] proved that if K is a nonempty closed and bounded subset of a uniformly convex Banach space, then every asymptotically nonexpansive self-mapping has a fixed point.

A mapping T is said to be asymptotically nonexpansive in the intermediate sense (see, e.g., [2]) if it is continuous and the following inequality holds:

limsup sup (| Tnx - Tny| - \\x -y\\) < 0. (1.2)

n^TO x,yeK

If F(T) := {x e K : Tx = x} = 0 and (1.2) holds for all x e K, y e F(T), then T is called asymptotically quasi-nonexpansive in the intermediate sense. Observe that if we define

an := sup (|Tnx - Tny| - \\x -y\\) and an = max{0,an}, (1.3)

© 2013 Kiziltunc and Yolacan; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

then an ^ 0 as n and (1.2) is reduced to

Tnx - Tny\ < \\x -y\\ + an, for all x,y e K, n > 1.

The class of mappings which are asymptotically nonexpansive in the intermediate sense was introduced by Bruck et al. [2]. It is known in [3] that if K is a nonempty closed convex bounded subset of a uniformly convex Banach space E and T is a self-mapping of K which is asymptotically nonexpansive in the intermediate sense, then T has a fixed point. It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate sense contains, properly, the class of asymptotically nonexpansive mappings.

Albert et al. [4] introduced a more general class of asymptotically nonexpansive mappings called total asymptotically nonexpansive mappings and studied methods of approximation of fixed points of mappings belonging to this class.

Definition 1 A mapping T: K ^ K is said to be total asymptotically nonexpansive if there exist nonnegative real sequences {¡n} and {ln}, n > 1 with ¡¡n, ln ^ 0 as n ^^ and strictly increasing continuous function 0 : R+ ^ R+ with 0(0) = 0 such that for all x,y e K,

In addition, if ln = 0 for all n > 1, then total asymptotically nonexpansive mappings coincide with asymptotically nonexpansive mappings. If ¡¡n = 0 and ln = 0 for all n > 1, we obtain from (1.5) the class of mappings that includes the class of nonexpansive mappings. If ¡¡n = 0 and ln = an = max{0, an}, where an := supxyeK(\\Tnx - Tny\\ - \\x -y\\) for all n > 1, then (1.5) is reduced to (1.4) which has been studied as mappings which are asymptotically nonexpansive in the intermediate sense.

Iterative techniques for nonexpansive and asymptotically nonexpansive mappings in Banach space including Mann type and Ishikawa type iteration processes have been studied extensively by various authors; see [1,5-11]. However, if the domain of T, D(T), is a proper subset of E (and this is the case in several applications) and T maps D(T) into E, then the iteration processes of Mann type and Ishikawa type have been studied by the authors mentioned above, their modifications introduced may fail to be well defined.

A subset K of E is said to be a retract of E if there exists a continuous map P: E ^ K such that Px = x, for all x e K. Every closed convex subset of a uniformly convex Banach space is a retract. A map P: E ^ K is said to be a retraction if P2 = P. It follows that if a map P is a retraction, then Py = y for all y e R(P), the range of P.

The concept of asymptotically nonexpansive nonself-mappings was firstly introduced by Chidume etal. [7] as the generalization of asymptotically nonexpansive self-mappings. The asymptotically nonexpansive nonself-mapping is defined as follows:

Let K be a nonempty subset of real normed linear space E. Let P: E ^ K be the nonexpansive retraction of E onto K. A nonself mapping T: K ^ E is called asymptotically

Tnx - Tny|| < \\x -y\\ + ¡n0{\\x -y\\) + ln, n > 1.

Remark 1 If $(X) = X, then (1.5) is reduced to

Tnx - Tny| < (1 + ¡n)\\x -y\\ + ln, n > 1.

nonexpansive if there exists sequence {kn} c [1, to), kn — 1 (n — to) such that

|| T(PT)n-1x - T(PT)n-1y\ < kn\\x -y\\ for allx,y e K,n > 1. (1.7)

Chidume et al. [12] introduce a more general class of total asymptotically nonexpansive mappings as the generalization of asymptotically nonexpansive nonself-mappings.

Definition 2 Let K be a nonempty closed and convex subset of E. Let P: E — K be the nonexpansive retraction of E onto K. A nonself map T: K — E is said to be total asymptotically nonexpansive if there exist sequences {¡n}n>;b {ln}n>1 in [0, +to) with ¡¡n, ln — 0 as n —^ to and a strictly increasing continuous function 0 : [0, +to) — [0, +to) with 0(0) = 0 such that for all x, y e K,

|| T(PT)n-1x - T(PT)n-1y\ < \\x -y\\ + ¡n0(\\x -y\\) + ln, n > 1. (1.8)

Proposition 1 Let K be a nonempty closed and convex subset of E which is also a nonexpansive retraction of E and T1, T2: K — E be two total nonself asymptotically nonexpansive mappings. Then there exist nonnegative real sequences {¡n}n£1, {ln}n>1 in [0, +to) with ¡¡n, ln — 0 asn —to and a strictly increasing continuous function 0 : R+ — R+ with 0(0) = 0 such that for allx, y e K,

\Ti(PTi)n-1x- Ti(PTi)n-1y\ < \\x-y\\ + ¡in0{\\x-y\\) + ln, n > 1, (1.9)

for i = 1,2.

Proof Since Ti: K — E is a total nonself asymptotically nonexpansive mappings for i = 1,2, there exist nonnegative real sequences {¡in}, {lin}, n > 1 with ¡¡in, lin — 0 as n — to and strictly increasing continuous function 0 : R+ — R+ with 0i (0) = 0 such that for all x, y e K,

\Ti(PTi)n-1x- Ti(PTi)n-1y\ < \\x-y\\ + ¡in0i(\\x-y\\) + ln, n > 1. Setting

¡¡n = max{^1n, ¡2n}, ln = max{l1n, l2n}, 0(a) = maxj01(a),02(a)} for a > 0,

then we get nonnegative real sequences {¡n}, {ln}, n > 1 with ¡¡n, ln — 0 as n —to and strictly increasing continuous function 0 : R+ — R+ with 0(0) = 0 such that

\Ti(PTi)n-1x - Ti(PTi)n-1y\ < \\x -y\\ + ¡in0i(\\x -y\\) + lin

< \\x - y\\ + ¡n0(\\x - y\\) + ln, n > 1,

for all x, y e K and each i = 1,2.

In [7], Chidume et al. study the following iterative sequence:

xn+1 = P((1- an)xn + anT(PT)n 1xn), x1 e K,n > 1,

(1.10)

to approximate some fixed point of T under suitable conditions. In [13], Wang generalized the iteration process (1.10) as follows:

where T1, T2: K ^ E are asymptotically nonexpansive nonself-mappings and (an), (an) are sequences in [0,1]. They studied the strong and weak convergence of the iterative scheme (1.11) under proper conditions. Meanwhile, the results of [13] generalized the results of [7].

In [14], Shahzad studied the following iterative sequence:

xn+1 = P((1- an)xn + anTP[(1- ßn)xn + ßnTxn]), x1 e K,n > 1, (1.12)

where T : K ^ E is a nonexpansive nonself-mapping and K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P nonexpansive retraction.

Recently, Thianwan [15] generalized the iteration process (1.12) as follows:

yn = P((1 - an - y'n)xn + anTP((1 - /K + /Txn) + y'Vn), n > 1,

where {an}, {/n}, {yn}, {a'}, {/'}, {y'} are appropriate sequences in [0,1] and {un}, {vn} are bounded sequences in K. He proved weak and strong convergence theorems for nonexpansive nonself-mappings in uniformly convex Banach spaces.

Inspired and motivated by this facts, we define and study the convergence theorems of two steps iterative sequences for total asymptotically nonexpansive nonself-mappings in Banach spaces. The results of this paper can be viewed as an improvement and extension of the corresponding results of [14-16] and others. The scheme (1.14) is defined as follows.

Let E be a normed space, K a nonempty convex subset of E, P: E ^ K the nonexpansive retraction of E onto K and T1, T2: K ^ E be two total asymptotically nonexpansive nonself-mappings. Then, for given x1 e K and n > 1, we define the sequence {xn} by the iterative scheme:

xn+1 = P((1 - an)xn + anT1(PT1)n-1((1 - fin)yn + ^1^1'^)),

(1.14)

yn = P((1 - a')xn + a'T2(PT2)n-1((1 - /3')xn + /3'T2(PT2)n-1xn)),

where {an}, {/n}, {a'}, {/'} are appropriate sequences in [0,1]. Clearly, the iterative scheme (1.14) is the generalization of the iterative schemes (1.11), (1.12) and (1.13).

Under suitable conditions, the sequence {xn} defined by (1.14) can also be generalized to iterative sequence with errors. Thus, all the results proved in this paper can also be

Xn+1 = P((1 - an)xn + anT1(PT1)n 1yn),

yn = P((1 - a'n)xn + an T2(PT2)n-1xn), x1 e K, n > 1,

(1.11)

x1 e K,

xn+1 = P((1 - an - Yn)xn + an TP((1 - ßn)yn + ß n Tyn) + Ynun),

(1.13)

proved for the iterative process with errors. In this case, our main iterative process (1.14) looks like

xi e K,

xn+1 = P((1 - an - Yn)xn + anT1(PT1)n-1

(1.15)

x ((1-/3n)yn + PnT1(PT1)n-1y n + Yn un ,

yn = P((1 - an - Yn)xn + a!nT2PT2Y-1 ((1 - p'n)xn + P'nT2(PT2)n-1xn) + Y,:Vn),

where {an}, {^n}, {y,}, {an}, {fi'n}, {y,} are appropriate sequences in [0,1] satisfying an + jin + Yn = 1 = a'n + ^ + Y, and {un}, {vn} are bounded sequences in K. Observe that the iterative process (1.15) with errors is reduced to the iterative process (1.14) when Yn = Yn = 0.

The purpose of this paper is to define and study the strong convergence theorems of the new iterations for two total asymptotically nonexpansive nonself-mappings in Banach spaces.

2 Preliminaries

Now, we recall the well-known concepts and results.

Let E be a Banach space with dimension E > 2. The modulus of E is the function SE : (0,2] ^ [0,1] defined by

Se(s) = inf 1 -

2(x + y)

= 1, s = ||x - y||

A Banach space E is uniformly convex if and only if SE(s) > 0 for all s e (0,2].

The mapping T : K ^ E with F(T) = 0 is said to satisfy condition (A) [17] if there is a nondecreasing function f : [0,œ) ^ [0, to) with/(0) = 0, f (t) > 0 for all t e (0, to) such that

||x - Tx|| >f(d(x,F(T)))

for all x e K, where d(x, F(T)) = inf{|x -pH :p e F(T)}.

Two mappings T1, T2 : K ^ E are said to satisfy condition (A') [18] if there is a nondecreasing function f : [0, to) ^ [0, to) with/(0) = 0, f (t) > 0 for all t e (0, to) such that

2(\\x - T1x\\ + \\x - T2x\\) >f (d(x,F))

for allx e K where d(x,F) = inf{\x -p\\:p e F = F(T1) n F(T2)}.

Note that condition (A:) reduces to condition (A) when T1 = T2 and hence is more general than the demicompactness of T1 and T2 [17]. A mapping T : K ^ K is called: (1) demicompact if any bounded sequence {xn} in K such that {xn - Txn} converges has a convergent subsequence; (2) semicompact (or hemicompact) if any bounded sequence {xn} in K such that {xn - Txn} ^ 0 as n ^to has a convergent subsequence. Every demicompact mapping is semicompact but the converse is not true in general.

Senter and Dotson [17] have approximated fixed points of a nonexpansive mapping T by Mann iterates, whereas Maiti and Ghosh [18] and Tan and Xu [8] have approximated the fixed points using Ishikawa iterates under the condition (A) of Senter and Dotson [17].

Tan and Xu [8] pointed out that condition (A) is weaker than the compactness of K. We shall use condition (A') instead of compactness of K to study the strong convergence of {xn} defined in (1.14).

In the sequel, we need the following useful known lemmas to prove our main results.

Lemma 1 [8] Let {an}, {bn} and {cn} be sequences of nonnegative real numbers satisfying the inequality

an+1 < (1 + bn)an + cn, n > 1.

f Z^ Cn < to and bn < to, then

(i) limn^TO an exists;

(ii) In particular, if {an} has a subsequence which converges strongly to zero, then limn^TO an = 0.

Lemma 2 [19] Let p >1 and R > 0 be two fixed numbers and E a Banach space. Then E is uniformly convex if and only if there exists a continuous, strictly increasing and convex function g: [0, to) ^ [0, to) with g(0) = 0 such that

||Xx + (1 - X)y||p < X\\xf + (1 - X)\\yf - Wp(X)g(\\x -y\\), (2.1)

for allx,y e Br(0) = {x e E: \\x\\ <R} and X e [0,1], where WP(X) = X(1- X)p + Xp(1- X).

3 Main results

We shall make use of the following lemmas.

Lemma 3 LetE be a real Banach space, letK be a nonempty closed convex subset ofE which is also a nonexpansive retract ofE and T1, T2: K ^ E be two total asymptotically nonexpansive nonself-mappings with sequences {|n}, {ln} defined by (1.9) such that I' < to, ETO=1 ln < to and F := F(T1) n F(T2) = {x e K: T1x = T2x = x} = 0. Assume that there exist M, M* > 0 such that $(X) < M*X for all X > M, i e {1,2}. Starting from an arbitrary x1 e K, define the sequence {xn} by recursion (1.14). Then, the sequence {xn} is bounded and limn^TO \\xn -p\\ exists, p e F.

Proof Letp e F. Set an = (1 - /n)yn + /3n T^PT^n and Sn = (1 - /)xn + / T2(PT2)n-1xn. Firstly, we note that

\\Sn - p\\ = |(1-/n )xn + /3n T2(PT2)n-1xn - p||

< / | T2(PT2)n-1xn -p| + (1 - /) \\xn -p\\

< /n[\xn -p\\ + \\xn -p\\) + ln] + (1-/n)\\xn -p\\

< \\xn -p\\ + /'n\\xn -p\\) + /'nln. (.1)

Note that $ is an increasing function, it follows that $(X) < $(M) whenever X < M and (by hypothesis) $(X) < M*X if X > M. In either case, we have

$(X) < $(M) + M*X (3.2)

for some M > 0, M* >0. Hence, from (3.1) and (3.2), we have

Pn -p\\ < \\xn -p\\ + P'n¡ln[4>(M)+M*\\xn -p\\] + P'nln

< (1 + M*^n) \\xn -p\\ + Q^n + ln) (3.3)

for some constant Q1 > 0. From (1.14) and (3.3), we have

\\yn -p\\ = ||P((1 - a'n)xn + anT2(PT2)n-1Sn) -p||

< ||(1- a'n)xn + anT2(PT2)n-1Sn -p|

< (1 - a!n) \ \xn -p\\ + an | T2(PT2)n-1Sn -p |

< an[Pn -p\\ + ¡n0{\\5n -p\\) + ln] + (1 -an)\\xn -p\\

< a'n[(1 +M*^n)\\xn -p\\ + Q^n + ln)]

+ an¡in[0(M)+M*\\8n -p\\] + an ln + (1-a^\xn - p\\

< \\xn -p\\ + M*¡n\xn -p\\ + M*^nPn -p\\

+ Q^V-n + ln) + ¡n0(M) + ln

< \\xn - p\\ + M* (2 + M*^n) ¡n\\xn - p\\

+ M*Q1^n(^n + ln) + QA^-n + ln) + ¡n$(M) + ln

< (1+ M2^n)\\xn -p\\ + Q2(^n + ln) (.4) for some constant M2, Q2 > 0. Similarly, we have

\\an -p\\ = |(1- Pn)yn + PnTx{PT1)n-1yn -p|

< fin || T1{PT1)n-1yn -p| +(1 - fin) \\yn -p\\

< fin [\\yn -p\ + ¡n0{\\yn -p\\) + ln] + (1 - fin)\\yn -p\

< \\yn - p\\ + fin^n \_4>(M) + M* \\yn - p\\] + finln

< (1+M>n)\\yn -p\\ + Q3(^n + ln) (.5) for some constant Q3 > 0. Substituting (3.4) into (3.5)

\\an -p\\ <{ 1+M*^n)\\yn -p\\ + Q3(^n + ln)

< (1+M>n) [(1+ M2^n)\xn -p\\

+ Q2 (¡n + ln)] + Q3(^n + ln)

< \\xn -p\\ + (M2 + M* + M*^nM2)¡¡n\\xn -p\\

+ Q2 (¡n + ln) + M* Q2 ¡¡n (¡n + ln) + Q3(^n + ln)

< (1 + M3^n)\\xn -p\\ + Q4(^n + ln) (.6)

for some constant M3, Q4 > 0. It follows from (1.14) and (3.6) that

\\xn+1-p\\ = |P((1-an)xn + anT1(PT1)n-1a^ -p|

< | (1 - an)xn + anT1(PT1)n-1an-p|

< (1 - an)\xn -py+an|T1(PT1)n-1an -p|

< an[\\an -p\\ + \\an -p\\) + ln] + (1-an)\xn -p\\

< an[(1 + M3In)\\xn -p\\ + Q4(ln + ln)]

+ anin[$(M) + M*\\an -p\\] + anln

+ (1-an)\\xn -p\\

< \\xn -p\\ + M3|n\xn -p\\ + Q4I n + '

+ in$(M) +M*in\\an -p\\ + ln

< \\xn -p\\ + (M3 + M* + M*M3|n)In\\xn -p\\

+ M*Q4ln(ln + ln) + Q4I n + ln)

+ in$(M) + ln

< (1+ M4|n)\\xn -p\ + Qs(l n + ln ) (3.7)

for some constant M4, Q5 > 0. Since ^TO1' < to, ^TO ln < to, by Lemma 1, we get limn^TO \\xn -p\\ exists. This completes the proof. □

Theorem 1 Let K be a nonempty convex subset of a real Banach space E which is also a nonexpansive retract of E and T1, T2 : K ^ E be two continuous total asymptotically nonexpansive nonself-mappings with sequences {|n}, {ln} defined by (1.9) such that STO=1 In < to, ^TO ln < to and F := F(T1) n F(T2) = {x e K: T1x = T2x = x} = 0. Assume that there exist M, M* > 0 such that $(X) < M*X for all X > M, i e {1,2}. Starting from an arbitrary x1 e K, define the sequence {xn} by recursion (1.14). Then, the sequence {xn} converges strongly to a common fixed point of T1, T2 if and only if liminfn^TO d(xn, F) = 0, where d(xn,F) = infpeF \\xn -p\\, n > 1.

Proof The necessity is obvious. Indeed, if xn ^ q e F (n ^ to), then

d(xn,F) = inf d(xn - q) < \\xn - q\\ ^ 0 (n ^ to).

Now we prove sufficiency. It follows from (3.7) that for x* e F, we have

||xn+1-X*| < (1+ M4|n)|Xn - x* | + Q5 (in + ln)

— 11 xn x | + ^n, (3.8)

where H„ = M^n\\x„ - x*|| + Qs(^„ + l„). Since {xn - x*} is bounded and ^TO ¡x„ < to, ln < to, we have ^TO H„ < to. Hence, (3.8) implies

inf ||xn+i -x* | < inf |xn - x* II + H„,

x* gF x* gF

that is

d(xn+1, F) < d(xn, F) + H„, (.9)

by Lemma 1(i), it follows from (3.9) that we get limn—TO d(xn, F) exists. Noticing liminf„—TO d(xn, F) = 0, it follows from (3.9) and Lemma 1(ii) that we have limn—TO d(xn, F ) = 0.

Now, we prove that {xn} is a Cauchy sequence in E. In fact, from (3.8) that for any n > n0, any m > n1 and any p1 g F, we have that

\\xn+m -P1\\ < \\x n+m-1 p1 \\ + H„+m-1

< \\xn+m-2 -p1 \\ + (H„+m-1 + H„+m-2)

< \\xn+m-3 -p1 \\ + (H„+m-1 + Hn+m-2 + H„+m-3)

< \\x„ -p1\ + j2 Hk. (.)

So by (3.10), we have that

\\xn+m - xn \\ < \\x n+m - p1\\ + \\x n p1 \\

< 2\\x„ -pj + J2 Hk. (.11)

By the arbitrariness ofp1 g F and from (3.11), we have

\\xn+m -x„\\<2d(x„, F) + ^^Hk, Vn > n0. (.2)

For any given e >0, there exists a positive integer n1 > n0, such that for any n > n1, d(xn, F)< | and ^TO=n Hk < §,we have \\xn+m - xn \\ < e and so for any m > 1

lim \\xn+m - x„\\ = 0. (3.13)

n—>TO

This show that {xn} is a Cauchy sequence in K. Since K is a closed subset of E and so it is complete. Hence, there exists ap g K such that xn — p as n — to.

Finally, we have to prove that p g F .By contradiction, we assume that p is not in F := F(T1) n F(T2) = {x g K : T1x = T2x = x} = 0. Since F is a closed set, d(p, F) > 0. Thus for all p g F,we have that

\p -p1\\ < \\p - x„\\ + \\x„ -pj. (3!4)

This implies that

d(p,F) < \\p-xn\\ + d(xn,F). (3.15)

From (3.14) and (3.15) (n ^ to), we have that d(q, F) < 0. This is a contradiction. Thus p e F := F(T1) n F(T2) = {x e K: T1x = T2x = x} = 0. This completes the proof. □

On the lines similar to this theorem, we can also prove the following theorem which addresses the error terms.

Theorem 2 Let K be a nonempty convex subset of a real Banach space E which is also a nonexpansive retract of E and T1, T2: K ^ E be two continuous total asymptotically nonexpansive nonself-mappings with sequences {|n}, {ln} defined by (1.9) such that ZTO i' < to, TO l' < to and F := F(T1) n F(T2) = {x e K: T1x = T2x = x} = 0. Assume that there exist M, M* > 0 such that $(X) < M*X for all X > M, i e {1,2}. Starting from an arbitrary x1 e K, define the sequence {xn} by recursion (1.15). Suppose that {un}, {vn} are bounded sequences in K such that ^TO Yn < to, ^TO < to. Then, the sequence {xn} converges strongly to a common fixed point of T1, T2 if and only if liminfn^TO d(xn, F) = 0, where d(xn, F) = infpeF \\xn -p\\, n > 1.

Lemma 4 LetK be a nonempty convex subset of a uniformly convex Banach space E which is also a nonexpansive retract ofE and T1, T2: K ^ E be two total asymptotically nonexpansive nonself-mappings with sequences {|n}, {ln} defined by (1.9) such that XTO In < to, ln < to and F := F(T1) n F(T2) = {x e K: T1x = T2x = x} = 0. Assume that there exist M, M* >0suchthat $(X) < M*Xforall X > M, i e {1,2}. Startingfrom an arbitrary x1 e K, define the sequence {xn} by recursion (1.14). Suppose that

(i) 0 < liminfn^TO an and 0 < liminfn^TO pn < limsupn^TO pn < 1, and

(ii) 0 < liminfn^TO a'n and 0 < liming«, P'n < limsupn^TO p'n < 1. Then limn^TO \\xn - Ti(PTi)n-1xn \\ = 0for i = 1,2.

Proof Let p e F. Then by Lemma 3, limn^«, \\xn -p\\ exists. Let limn^TO \\xn -p\\ = r. If r = 0, then by the continuity of T1 and T2 the conclusion follows. Now suppose r >0. Set an = (1-/n)yn + /nTi(PT1)n-1yn and Sn = (1-/3')xn + P'„T2(PT2)n-1xn.Since {xn} is bounded, there exists an R >0 such that xn -p,yn -p e BR(0) for all n > 1. Using Lemma 2, we have, for some constant A1 > 0, that

\\Sn - p\\2 = ||(1-3n )xn + P'n T2(PT2)n-1xn - p|2

< (1 - P'n) \\xn -p\\2 + & | T2(PT2)n-1xn -p||2

- 3n (1-3')g(|xn - T2(PT2)n-1xn|)

< (1-P'n)\\xn -p\\2 + P'n[\\xn -p\\ + In${\\xn -p\\) + '2

- P'n (1-3')g(|xn - T2(PT2)n-1xn|)

< \\xn -p\2+ A1(|n + ln)

- 3n (1 - 3n)g( |xn - T2 (PT2)n-1xn || ). (3.16)

It follows from (1.14), Lemma 2, (3.2) and (3.16) that for some constant A2 > 0,

\\yn -p\\2 = |P((l-«n)xn + a'nT2(PT2)n-1Sn) -p|2

< | (1 - an)(xn -p)+a'n(T2(PT2)n-1Sn -p) |2

< (1- a'n) \\Xn -p\\2 + an |T2(PT2)n-15n -p|2

- a'n (1- a'n)g(|xn - T2(PT2)n-15n|)

< (1-a^\Xn -p\\2 + a'n[\\&n -p\\ + Pn -p\\) + ln]2

- a'n (1- a'n)g(|xn - T2(PT2)n-15n|)

< \\xn -p\2 + A2(^n + ln)

- an P'n (1-P'n )g (|xn - T2(PT2)n-1xn|)

- an(1 - a'n)g(|xn - T2(PT2)n-1Sn |). (3.17) Using Lemma 2 and (3.17), we have, for some constant A3 > 0, that

\K -p\\2 = |(1-Pn)yn + pnTx{PT1)n-1yn -p|2

< | (1 - fin)(jn -p) + Pn(T1(PT1)n-1yn -p) |2

< (1- fin)\\yn - p\2 + PnWT1(PT1)n-1yn - p|2

- fin (1- A,)g(|yn - T1(PT1)n-1yn|)

< (1-fin)\\yn - p\2+ fin [\\yn - p\\ + \\yn - p\\) + l^2

- fin (1- fin)g(|yn - T1(PT1)n-1yn|)

< \\yn -p\\2 + A3(^n + ln)

- fin (1- fin)g(|yn - T1(PT1)n-1yn|)

< \\xn -p\2+ A2(^n + ln)+A3(^n + ln)

- fin (1- fin)g(|yn - T1(PT1)n-1yn|)

- a'n(1- a'n)g(|xn - T2(PT2)n-15n|)

- anfi'n (1 - fi'n)g( |xn - T2(PT2)n-1xn |). (3.18) Similarly, it follows from (1.14), Lemma 2, (3.2) and (3.18) that for some constant A4 > 0,

\\xn+1-p\\2 = |P((1- an)xn + anT1(PT1)n-1an) -p|2

< |(1-an)(xn -p)+an(T1(PT1)n-1an -p)|2

< (1 - an)\\xn -p\\2 + an | T1(PT1)n-1an -p|2

- an(1 - an)g(|xn - T1(PT1)n-1an |)

< (1- an)\xn -p\\2 + an [\\On -p\\ + \K -p\\) + ln]2

- an(1 - an)g(|xn - T1(PT1)n-1an |)

< \\xn -p\2+ A4(^n + ln)

- an(1 - an)g(|xn - T1(PT1)n-1an |)

- an3n(1 - 3n)g(|yn - T1(PT1)n-1yn |)

- ana' (1-a')g(|xn - T2(PT2)n-1Sn |)

- ana' fa(1 - 3n)g(|xn - T2(PT2)n-1xn |). (3.19) It follows from (3.19) that

ana'n 3n (1-3' )g (|xn - T2(PT2)n-1xn|) < \\xn - p\\2- \\xn+1- p\2

+ A4(|n + ln), (3.20)

ana'n(1 - a')g(|xn - T2(PT2)n-1Sn |) < \\xn -p\\2 - \\xn+1 -p\\2

+ A4(|n + ln), (3.21)

an3n(1- 3n)g(|yn - T1(PT1)n-1yn|) < \\xn -p\2- \\xn+1-p\2

+ A4(|n + ln), (3.22)

an(1 - an)g(|xn - T1(PT1)n-1an|) < \\xn -p\\2 - \\xn+1 -p\\2

+ A4(|n + ln). (3.23)

Since 0 < liminfn^TO an, 0 < liming«, a'n and 0 < liming«, P'n < limsupn^TO3n < 1, there exists n0 e N and n1, n2, n3, n4 e (0,1) such that 0 < n1 < an, 0 < n2 < a'n and 0 < n3 < f3'n < n4 < 1 for all n > no. This implies by (3.20) that

n1n2'3(1- '4)g( |xn - T2(PT2)n-1Xn|) < \\xn - p\\2- \ix„+1 - p\\2

+ A4(|n + ln) (3.24)

for all n > n0. It follows from (3.24) that k > n0, we have

J2g{ |xn - T2(PT2)n-1xn|)

1 / k k \

< n n n (1 - n ) I ^ ^\xn -p\2 - \\xn+1 -+ A^J2(in + ln) I

< -\-r 1 \\xn0 -p\2 + A^y~](ln + ln)\.

n1n2n3(1 - n4^ nt;0 /

Then £TO=n0 g(\xn - T2(PT2)n-1xn\\) < to and therefore lim«^g(\\xn - T2(PT2)n-1xn\\) = 0. Since g is strictly increasing and continuous withg(0) = 0, we have

lim ||xn - T2(PT2)n-1xn | = 0. (3.25)

By a similar method, together with (3.21), (3.22) and (3.23), it can be show that

(3.26)

lim \\xn - T2(PT2)n-1Sn || = 0, lim \\yn - T^PT^n || = 0,

1_llVi" " M_liVi" "

lim \\xn - Ti(PTi) On \\ =0.

It follows from (1.14) that

\\yn - xn\\ = \P((l-«n )xn + a'n T2(PT2)n-1Sn) - Pxn\ < \T2(PT2)n-1&n-xn\\. This together with (3.26) implies that

lim \\yn -xn\\ =0. (3.27)

It follows from (3.26) and (3.27) that

\\ T1(PT1)n-1xn - xn \ < \\ T1(PT1)n-1xn - T1(PT1)n-1yn \

+ \T1(PT1)n-1yn - yn \ + \\yn - xn\\

< \\yn - xn\\ + \\yn - xj\) + ln

+ \T1(PT1)n-1yn - yn \ + \\yn - xn\\ ^ 0, as n ^TO. (.28)

That is limn^TO \\ T1(PT1)n-1xn - xn\\ = 0. The proof is completed. □

Theorem 3 Let K be a nonempty convex subset of a real Banach space E which is also a nonexpansive retract of E and T1, T2: K ^ E be two continuous total asymptotically nonexpansive nonself-mappings with sequences {¡n}, {ln} defined by (1.9) such that STOd lxn < <»,£TO=1 ln < to and F := F(T1) n F(T2) = {x e K: T1x = T2x = x} = 0. Assume that there exist M,M* >0 such that $(k) < M*k for all k > M, i e {1,2}; and that one of T1, T2 is demicompact (without loss of generality, we assume T1 is demicompact). Starting from an arbitrary x1 e K, define the sequence {xn} by recursion (1.14). Suppose that

(i) 0 < liminfn^TO an and 0 < liminfn^TO jin < limsupn^TO jin < 1, and

(ii) 0 < liminfn^TO a'n and 0 < liming«, P'n < limsupn^TO p'n < 1.

Then the sequence {xn} converges strongly to some common fixed points ofT1 and T2.

Proof It follows from (1.14) and (3.26) that

\\xn+1-xn\\ = \\P((1-an)xn + anT1(PT1)n-1On) - Pxn\< \\ T1(PTi)n-1On-xn \

^ 0, as n ^to. (.9)

It follows Lemma 4 and (3.29) that

\xn - Ti(PTi) xn\ < \xn - xn-1 \\ + \xn-1 - Ti(PTi) xn-1 \\

+ \Ti(PTi)n-2xn-1- Ti(PTi)n-2xn\

— 2\\xn xn-1 \\ + |xn-1 Ti(PTi) xn-1|

+ !n-10(\\xn - xn-1^ + l n-1 ^ 0, as n ^TO, for i = 1,2. (3.30)

Since Ti is continuous and P is nonexpansive retraction, it follows from (3.30) that for i = 1,2

|| Ti(PTi)n-1xn - Tixn | = || TiP(Ti(PTi)n-2)xn - TiPxn |

^ 0, as n ^TO. (.31)

Hence, by Lemma 4 and (3.31), we have

\\xn - Tix n\\ — \\xn - Ti(PTi)n-1xn\ + mPTi)n- xn Tixn|

^ 0, as n ^TO, for i = 1,2. (3.32)

Since T1 is demicompact, from the fact that limn^TO \\xn - T1xn \\ = 0 and {xn} is bounded, there exists a subsequence {xnk} of {xn} that converges strongly to some q e K as k ^ to. Hence, it follows from (3.32) that T1 x„k ^ q, T2x„k ^ q as k ^to and it follows from (3.31) and Ti is continuous that

|| Ti(PTi)nk-1xnk - Tiq| — || Ti(PTi)nk-1xnk - Tixnk | + HTx^ - Tiq\\

= | TiPTi (PTi)nk-2xnk - TiPxnk | + mxnk - Tiq\ ^ 0, as n ^ to, for i = 1,2. (3.33)

Observe that

\\q - T1 q\\ — \\q - xnk \\ + ||x„k - T1(PT1)'k-1xnk | + || T1(PT1)'k-1xnk - T1q|.

Taking limit as k ^to and using the fact that Lemma 4 and (3.33) we have that T1q = q and so q e F(T1). Also we get

\\q - T2q\\ — \\q - xnk \\ + W - T2(PT2)"k-1xnk | + | T2(PT2)nk-1xnk - T2q|.

Taking limit as k ^to and using the fact that Lemma 4 and (3.33) we have that T2q = q and so q e F(T2). Therefore, we obtain that q e F. It follows from (3.7), Lemma 1 and limk^TO x„k = q that {xn} converges strongly to q e F. This completes the proof. □

Theorem 4 Let K be a nonempty convex subset of a real Banach space E which is also a nonexpansive retract of E and T1, T2 : K ^ E be two continuous total asymptotically nonexpansive nonself-mappings with sequences {|n}, {ln} defined by (1.9) such in < to, to=1 ln < to and satisfying the condition (A'). Assume that there exist M,M* >0 such that 0(X) — M*X for all X > M, i e {1,2} and F := F(T1) n F(T2) = {x e K: T1x = T2x = x} = 0. Startingfrom an arbitrary x1 e K, define the sequence {xn} by recursion (1.14). Suppose that

(i) 0 < liminfn—« an and 0 < liminfn—« fn < limsupn—« fn < 1, and

(ii) 0 < liminfn—« a'n and 0 < liming«, fn < limsupn—« fn < 1.

Then the sequence {xn} converges strongly to some common fixed points ofT1 and T2.

Proof By Lemma 3, we see that limn—« \\xn -p\\ and so, limn—«d(xn,F) exists for all p e F. Also, by (3.32), limn—« \\xn - Tixn \\ = 0 for i = 1,2. It follows from condition (A') that

lim f (d(xn, F)) < lim ( - (\\x - Tix\ + \\x - T2x\\) | =0. (3.34)

n—« v ' n—«\ 2 !

That is,

lim f {d(xn, F)) = 0. (3.35)

n—>« v '

Since f: [0,«) — [0,«) is a nondecreasing function satisfying f (0) = 0, f (t) > 0 for all t e (0,«), therefore, we have

lim d(xn, F) = 0. (3.36)

n—«

Now we can take a subsequence {xnk} of {xn} and sequence {yk} c F such that \\xnk -yk\\ < 2-k for all integers k > 1. Using the proof method of Tan and Xu [8], we have

\\xnk+i- yk\\<\\xnk - yk \\ < 2-k, (3.37)

and hence

\\yk+i - yk \\ < \\yk+i - xnk+1 \\ + \xnk+i - yk \\ < 2-(k+1) + 2-k < 2-k+1. (3.38)

We get that {yk} is a Cauchy sequence in F and so it converges. Let yk — y. Since F is closed, therefore, y e F and then xnk — y. As limn—« \\xn -p\\ exists, xn — y e F. This completes the proof. □

In a way similar to the above, we can also prove the results involving error terms as follows.

Theorem 5 Let K be a nonempty convex subset of a real Banach space E which is also a nonexpansive retract of E and T1, T2: K ^ E be two continuous total asymptotically nonexpansive nonself-mappings with sequences {¡n}, {ln} defined by (1.9) such that 5ZTO lxn < to, £ TO ln < to and F := F (T1) n F (T2) = {x e K: T1x = T2x = x} = 0. Assume that there exist M,M* >0 such that $(k) < M*k for all k > M, i e {1,2}; and that one of T1, T2 is demicompact (without loss of generality, we assume T1 is demicompact). Starting from an arbitrary x1 e K, define the sequence {xn} by recursion (1.15). Suppose that {un}, {vn} are bounded sequences in K such that^TO=\ Yn < to, £ TO y!n < to. Suppose that

(i) 0 < liminfn^TO an and 0 < liminfn^TO ßn < limsupn^TO ßn < 1, and

(ii) 0 < liminfn^TO a'n and 0 < liming«, ß'n < limsupn^TO ß'n < 1.

Then the sequence {xn} converges strongly to some common fixed points ofT1 and T2.

Theorem 6 Let K be a nonempty convex subset of a real Banach space E which is also a nonexpansive retract of E and T1, T2 : K ^ E be two continuous total asymptotically nonexpansive nonself-mappings with sequences {fn}, {ln} defined by (1.9) such that^TO fn < to,£TO l' < to and satisfying the condition (A'). Assume that there exist M,M* > 0 such that 4>(X) — M*X for all X > M, i e {1,2} and F := F(T1) n F(T2) = {x e K: T1x = T2x = x} = 0. Starting from an arbitrary x1 e K, define the sequence {xn} by recursion (1.15). Suppose that {un}, {vn} are bounded sequences in K such that J^TO Yn < to, y' < to. Suppose that

(i) 0 < liminfn^TO an and 0 < liminfn^TO (3n < limsupn^TO (3n < 1, and

(ii) 0 < liminfn^TO a'n and 0 < liming«, 3n < limsupn^TO 3n < 1.

Then the sequence {xn} converges strongly to some common fixed points ofT1 and T2.

Remark 2 If T1 and T2 are asymptotically nonexpansive mappings, then ln = 0 and ф(Х) = X so that the assumption that there exist M,M* >0 such that ф(Х) < M*X for all X > M, i e{1, 2} in the above theorems is no longer needed. Hence, the results in the above theorems also hold for asymptotically nonexpansive mappings. Thus, the results in this paper improvement and extension the corresponding results of [14,15] and [16] from asymptotically nonexpansive (or nonexpansive) mappings to total asymptotically nonexpansive nonself-mappings under general conditions.

Example 1 Let E is the real line with the usual norm | -|, K =[0, то) and P be the identity mapping. Assume that T1x = x and T2x = sin x for x e K. Let ф be a strictly increasing continuous function such that ф : R+ ^ R+ with ф(0) = 0. Let {¡n}n>1 and {ln}n>1 be two nonnegative real sequences defined by ¡¡n = n2 and ln = 4r, for all n > 1 (limn^œ ¡¡n = 0 and limn^œ ln = 0). Since T1x = x for x e K, we have

|I> - Tfy] < |x -y\. For all x, y e K, we obtain

\T"x - Tfy) - |x -y| - 1г„Ф{\x -y|) - ln

< |x - y | - |x - y| - ¡.1пф( |x - y|) - ln

for all n = 1,2,..., {¡n}n>1 and {ln}n>1 with ¡¡n, ln ^ 0 as n ^то and so T1 is a total asymptotically nonexpansive mapping. Also, T2x = sin x for x e K, we have

l^x - Tfyl < |x -y|. For all x, y e K, we obtain

\ T2x - T£y\ - |x -y| - ¡пФ{^ -y|) - ln < |x - y | - |x - y| - ¡¡пФ( |x - y|) - ln <0

for all n = 1,2,..., {¡n}n>1 and {ln}n>1 with ¡xn, ln — 0 as n — « and so T2 is a total asymptotically nonexpansive mapping. Clearly, F := F(T1) n F(T2) = {0}. Set

a„ = an =

ß'n = ßn =

Yn = Yn =

n is even, n is odd

and vn = un =

for n > 1. Thus, the conditions of Theorem 2 are fulfilled. Therefore, we can invoke Theorem 2 to demonstrate that the iterative sequence {xn} defined by (1.15) converges strongly to 0.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

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

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The authors are very gratefulto the referees for their carefulreading of manuscript, valuable comments and suggestions.

Received: 1 December 2012 Accepted: 25 March 2013 Published: 10 April 2013

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doi:10.1186/1687-1812-2013-90

Cite this article as: Kiziltunc and Yolacan: Strong convergence theorems for two total asymptotically nonexpansive nonself mappings in Banach spaces. Fixed Point Theory and Applications 2013 2013:90.

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