Kiziltunc and Yolacan Fixed Point Theory and Applications 2013, 2013:90 R E S E A R C H Open Access 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 et al. 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 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 K. A mapping T : K K is called asymptotically nonexpansive if there exists a sequence {k n } [1, withk n 1suchthat T n x T n y k n x y (1.1 for all x, y K and n 1. Goebel and Kirk [1] provedthatifk 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: lim sup n ( sup T n x T n y x y 0. (1.2 x,y K If F(T :={x K : Tx = x} and (1.2 holdsforallx K, y F(T, then T is called asymptotically quasi-nonexpansive in the intermediate sense. Observe that if we define ( a n := sup T n x T n y x y x,y K and σ n = max{0, a n }, (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.
Kiziltunc and Yolacan Fixed Point Theory and Applications 2013, 2013:90 Page 2 of 18 then σ n 0asn and (1.2 is reduced to T n x T n y x y + σn, for all x, y K, n 1. (1.4 The class of mappings which are asymptotically nonexpansive in the intermediate sense was introduced by Bruck et al. [2]. It is known in [3]thatifK 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 {μ n } and {l n }, n 1withμ n, l n 0asn and strictly increasing continuous function φ : R + R + with φ(0 = 0 such that for all x, y K, T n x T n y x y + μn φ ( x y + l n, n 1. (1.5 Remark 1 If φ(λ=λ,then(1.5 is reduced to T n x T n y (1 + μn x y + l n, n 1. (1.6 In addition, if l n =0foralln 1, then total asymptotically nonexpansive mappings coincide with asymptotically nonexpansive mappings. If μ n =0andl n =0foralln 1, we obtain from (1.5 the class of mappings that includes the class of nonexpansive mappings. If μ n =0andl n = σ n = max{0, a n },wherea n := sup x,y K ( T n x T n y x y for all n 1, then (1.5 is reduced to (1.4whichhasbeenstudiedasmappingswhichareasymptotically 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(TintoE, 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. AsubsetK 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 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 P 2 = P. It follows that if a map P is a retraction, then Py = y for all y R(P, the range of P. The concept of asymptotically nonexpansive nonself-mappings was firstly introduced by Chidume et al. [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. LetP : E K be the nonexpansive retraction of E onto K. A nonself mapping T : K E is called asymptotically
Kiziltunc and Yolacan Fixed Point Theory and Applications 2013, 2013:90 Page 3 of 18 nonexpansive if there exists sequence {k n } [1,, k n 1(n suchthat T(PT n 1 x T(PT n 1 y kn x y for all x, y 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. LetP : 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 {μ n } n 1, {l n } n 1 in [0, + withμ n, l n 0as n and a strictly increasing continuous function φ :[0,+ [0,+ withφ(0 =0 such that for all x, y K, T(PT n 1 x T(PT n 1 y x y + μ n φ ( x y + l n, 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 T 1, T 2 : K E be two total nonself asymptotically nonexpansive mappings. Then there exist nonnegative real sequences {μ n } n 1, {l n } n 1 in [0, + with μ n, l n 0 as n and a strictly increasing continuous function φ : R + R + with φ(0 = 0 such that for all x, y K, T i (PT i n 1 x T i (PT i n 1 y x y + μ n φ ( x y + l n, n 1, (1.9 for i =1,2. Proof Since T i : K E is a total nonself asymptotically nonexpansive mappings for i = 1, 2, there exist nonnegative real sequences {μ in }, {l in }, n 1withμ in, l in 0asn and strictly increasing continuous function φ : R + R + with φ i (0 = 0 such that for all x, y K, T i (PT i n 1 x T i (PT i n 1 y x y + μ in φ i ( x y + lin, n 1. Setting μ n = max{μ 1n, μ 2n }, l n = max{l 1n, l 2n }, φ(a=max { φ 1 (a, φ 2 (a } for a 0, then we get nonnegative real sequences {μ n }, {l n }, n 1withμ n, l n 0asn and strictly increasing continuous function φ : R + R + with φ(0 = 0 such that Ti (PT i n 1 x T i (PT i n 1 y x y + μin φ i ( x y + lin x y + μ n φ ( x y + l n, n 1, for all x, y K and each i =1,2.
Kiziltunc and Yolacan Fixed Point Theory and Applications 2013, 2013:90 Page 4 of 18 In [7], Chidumeet al. study the following iterative sequence: x n+1 = P ( (1 a n x n + a n T(PT n 1 x n, x1 K, n 1, (1.10 to approximatesome fixedpoint of T under suitable conditions. In [13], Wang generalized the iteration process (1.10 as follows: x n+1 = P((1 α n x n + α n T 1 (PT 1 n 1 y n, y n = P((1 α n x n + α n T (1.11 2(PT 2 n 1 x n, x 1 K, n 1, where T 1, T 2 : K E are asymptotically nonexpansive nonself-mappings and {α n }, {α n } 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: x n+1 = P ( (1 a n x n + a n TP [ (1 β n x n + β n Tx n ], x1 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: x 1 K, x n+1 = P((1 α n γ n x n + α n TP((1 β n y n + β n Ty n +γ n u n, y n = P((1 α n γ n x n + α n TP((1 β n x n + β n Tx n+γ n v n, n 1, (1.13 where {α n }, {β n }, {γ n }, {α n }, {β n }, {γ n } are appropriate sequences in [0, 1] and {u n}, {v n } 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 (1.14isdefinedasfollows. Let E be a normed space, K a nonempty convex subset of E, P : E K the nonexpansive retraction of E onto K and T 1, T 2 : K E be two total asymptotically nonexpansive nonself-mappings. Then, for given x 1 K and n 1, we define the sequence {x n } by the iterative scheme: x n+1 = P((1 α n x n + α n T 1 (PT 1 n 1 ((1 β n y n + β n T 1 (PT 1 n 1 y n, y n = P((1 α n x n + α n T 2(PT 2 n 1 ((1 β n x n + β n T (1.14 2(PT 2 n 1 x n, where {α n }, {β n }, {α n }, {β n } areappropriatesequencesin[0,1].clearly,theiterativescheme (1.14 is the generalization of the iterative schemes (1.11, (1.12and(1.13. Under suitable conditions, the sequence {x n } defined by (1.14 can also be generalized to iterative sequence with errors. Thus, all the results proved in this paper can also be
Kiziltunc and Yolacan Fixed Point Theory and Applications 2013, 2013:90 Page 5 of 18 proved for the iterative process with errors. In this case, our main iterative process (1.14 looks like x 1 K, x n+1 = P ( (1 α n γ n x n + α n T 1 (PT 1 n 1 ( (1 β n y n + β n T 1 (PT 1 n 1 y n + γn u n, (1.15 y n = P (( 1 α n γ n xn + α n T 2(PT 2 n 1(( 1 β n xn + β n T 2(PT 2 n 1 x n + γ n v n, where {α n }, {β n }, {γ n }, {α n }, {β n }, {γ n } are appropriate sequences in [0, 1] satisfying α n +β n + γ n =1=α n + β n + γ n and {u n}, {v n } are bounded sequences in K. Observe that the iterative process (1.15 with errors is reduced to the iterative process (1.14whenγ n = γ n =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 δ E : (0, 2] [0, 1] defined by { δ E (ε=inf 1 1 (x + y 2 }. : x = y =1,ε = x y A Banach space E is uniformly convex if and only if δ E (ε > 0 for all ε (0, 2]. The mapping T : K E with F(T is said to satisfy condition (A [17] ifthereisa nondecreasing function f :[0, [0, withf (0 = 0, f (t > 0 for all t (0, such that x Tx f ( d ( x, F(T for all x K,whered(x, F(T = inf{ x p : p F(T}. Two mappings T 1, T 2 : K E are said to satisfy condition (A [18] ifthereisanondecreasing function f :[0, [0, withf (0 = 0, f (t > 0 for all t (0, suchthat 1 ( x T1 x + x T 2 x f ( d(x, F 2 for all x K where d(x, F=inf{ x p : p F = F(T 1 F(T 2 }. Note that condition (A reducestocondition (A when T 1 = T 2 and hence is more general than the demicompactness of T 1 and T 2 [17]. A mapping T : K K is called: (1 demicompact if any bounded sequence {x n } in K such that {x n Tx n } converges has a convergent subsequence; (2 semicompact (or hemicompact if any bounded sequence {x n } in K such that {x n Tx n } 0asn 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 (Aof Senter and Dotson[17].
Kiziltunc and Yolacan Fixed Point Theory and Applications 2013, 2013:90 Page 6 of 18 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 {x n } defined in (1.14. In the sequel, we need the following useful known lemmas to prove our main results. Lemma 1 [8] Let {a n }, {b n } and {c n } be sequences of nonnegative real numbers satisfying the inequality a n+1 (1 + b n a n + c n, n 1. If n=1 c n < and n=1 b n <, then (i lim n a n exists; (ii In particular, if {a n } has a subsequence which converges strongly to zero, then lim n a n =0. Lemma 2 [19] Let p >1and R >0be 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, [0, with g(0 = 0 such that λx +(1 λy p λ x p +(1 λ y p W p (λg ( x y, (2.1 for all x, y B R (0 = {x E : x R} and λ [0, 1], where W p (λ=λ(1 λ p + λ p (1 λ. 3 Main results We shall make use of the following lemmas. Lemma 3 Let E be a real Banach space, let K be a nonempty closed convex subset of E which is also a nonexpansive retract of E and T 1, T 2 : K E be two total asymptotically nonexpansive nonself-mappings with sequences {μ n }, {l n } defined by (1.9 such that n=1 μ n <, n=1 l n < and F := F(T 1 F(T 2 ={x K : T 1 x = T 2 x = x}. Assume that there exist M, M >0such that φ(λ M λ for all λ M, i {1, 2}. Starting from an arbitrary x 1 K, define the sequence {x n } by recursion (1.14. Then, the sequence {x n } is bounded and lim n x n p exists, p F. Proof Let p F.Setσ n =(1 β n y n + β n T 1 (PT 1 n 1 y n and δ n =(1 β n x n + β n T 2(PT 2 n 1 x n. Firstly, we note that δ n p = ( 1 β n xn + β n T 2(PT 2 n 1 x n p β n T 2 (PT 2 n 1 x n p + ( 1 β n xn p β n[ xn p + μ n φ ( x n p ] ( + l n + 1 β n xn p x n p + β n μ nφ ( x n p + β n l n. (3.1 Note that φ is an increasing function, it follows that φ(λ φ(m wheneverλ M and (by hypothesis φ(λ M λ if λ M. In either case, we have φ(λ φ(m+m λ (3.2
Kiziltunc and Yolacan Fixed Point Theory and Applications 2013, 2013:90 Page 7 of 18 for some M >0,M >0.Hence,from(3.1and(3.2, we have δ n p x n p + β n μ n[ φ(m+m x n p ] + β n l n ( 1+M μ n xn p + Q 1 (μ n + l n (3.3 for some constant Q 1 >0.From(1.14and(3.3, we have y n p = P (( 1 α n xn + α n T 2(PT 2 n 1 δ n p ( 1 α n xn + α n T 2(PT 2 n 1 δ n p ( 1 α n xn p + α n T2 (PT 2 n 1 δ n p α n[ δn p + μ n φ ( δ n p ] ( + l n + 1 α n xn p α n [( 1+M μ n xn p + Q 1 (μ n + l n ] + α n μ [ n φ(m+m δ n p ] + α n l n + ( 1 α n xn p x n p + M μ n x n p + M μ n δ n p + Q 1 (μ n + l n +μ n φ(m+l n x n p + M ( 2+M μ n μn x n p + M Q 1 μ n (μ n + l n +Q 1 (μ n + l n +μ n φ(m+l n (1 + M 2 μ n x n p + Q 2 (μ n + l n (3.4 for some constant M 2, Q 2 > 0. Similarly, we have σ n p = (1 βn y n + β n T 1 (PT 1 n 1 y n p β T1 n (PT 1 n 1 y n p +(1 βn y n p [ β n yn p + μ n φ ( y n p ] + l n +(1 βn y n p [ y n p + β n μ n φ(m+m y n p ] + β n l n ( 1+M μ n yn p + Q 3 (μ n + l n (3.5 for some constant Q 3 > 0. Substituting (3.4into(3.5 σ n p ( 1+M μ n yn p + Q 3 (μ n + l n ( 1+M [ μ n (1 + M2 μ n x n p + Q 2 (μ n + l n ] + Q 3 (μ n + l n x n p + ( M 2 + M + M μ n M 2 μn x n p + Q 2 (μ n + l n +M Q 2 μ n (μ n + l n + Q 3 (μ n + l n (1 + M 3 μ n x n p + Q 4 (μ n + l n (3.6
Kiziltunc and Yolacan Fixed Point Theory and Applications 2013, 2013:90 Page 8 of 18 for some constant M 3, Q 4 > 0. It follows from (1.14and(3.6that x n+1 p = ( P (1 αn x n + α n T 1 (PT 1 n 1 σ n p (1 αn x n + α n T 1 (PT 1 n 1 σ n p (1 α n x n p +α n T 1 (PT 1 n 1 σ n p [ α n σn p + μ n φ ( σ n p ] + l n +(1 α n x n p [ α n (1 + M3 μ n x n p + Q 4 (μ n + l n ] [ + α n μ n φ(m+m σ n p ] + α n l n +(1 α n x n p x n p + M 3 μ n x n p + Q 4 (μ n + l n + μ n φ(m+m μ n σ n p + l n x n p + ( M 3 + M + M M 3 μ n μn x n p + M Q 4 μ n (μ n + l n +Q 4 (μ n + l n + μ n φ(m+l n (1 + M 4 μ n x n p + Q 5 (μ n + l n (3.7 for some constant M 4, Q 5 >0.Since n=1 μ n <, n=1 l n <, by Lemma 1, weget lim n x n 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 T 1, T 2 : K E be two continuous total asymptotically nonexpansive nonself-mappings with sequences {μ n }, {l n } defined by (1.9 such that n=1 μ n <, n=1 l n < and F := F(T 1 F(T 2 ={x K : T 1 x = T 2 x = x}. Assume that there exist M, M >0such that φ(λ M λ for all λ M, i {1, 2}. Starting from an arbitrary x 1 K, define the sequence {x n } by recursion (1.14. Then, the sequence {x n } converges strongly to a common fixed point of T 1, T 2 if and only if lim inf n d(x n, F=0, where d(x n, F=inf p F x n p, n 1. Proof The necessity is obvious. Indeed, if x n q F (n, then d(x n, F= inf q F d(x n q x n q 0 (n. Now we prove sufficiency. It follows from (3.7thatforx F,wehave x n+1 x (1 + M 4 μ n x n x + Q 5 (μ n + l n = xn x + ξn, (3.8
Kiziltunc and Yolacan Fixed Point Theory and Applications 2013, 2013:90 Page 9 of 18 where ξ n = M 4 μ n x n x + Q 5 (μ n + l n. Since {x n x } is bounded and n=1 μ n <, n=1 l n <,wehave n=1 ξ n <.Hence,(3.8implies inf x n+1 x inf x n x + ξ n, that is x F x F d(x n+1, F d(x n, F+ξ n, (3.9 by Lemma 1(i, it follows from (3.9 that we get lim n d(x n, F exists. Noticing lim inf n d(x n, F = 0, it follows from (3.9 and Lemma 1(ii that we have lim n d(x n, F=0. Now, we prove that {x n } is a Cauchy sequence in E. In fact, from (3.8 that for any n n 0, any m n 1 and any p 1 F,wehavethat x n+m p 1 x n+m 1 p 1 + ξ n+m 1 So by (3.10, we have that x n+m 2 p 1 +(ξ n+m 1 + ξ n+m 2 x n+m 3 p 1 +(ξ n+m 1 + ξ n+m 2 + ξ n+m 3. n+m 1 x n p 1 + ξ k. (3.10 k=n x n+m x n x n+m p 1 + x n p 1 2 x n p 1 + ξ k. (3.11 k=n By the arbitrariness of p 1 F and from (3.11, we have x n+m x n 2d(x n, F+ ξ k, n n 0. (3.12 k=n For any given ε > 0, there exists a positive integer n 1 n 0, such that for any n n 1, d(x n, F< ε 4 and k=n ξ k < ε 2,wehave x n+m x n < ε and so for any m 1 lim x n+m x n = 0. (3.13 n This show that {x n } is a Cauchy sequence in K.SinceK is a closed subset of E and so it is complete. Hence, there exists a p K such that x n p as n. Finally,wehavetoprovethatp F. By contradiction, we assume that p is not in F := F(T 1 F(T 2 ={x K : T 1 x = T 2 x = x}. SinceF is a closed set, d(p, F>0.Thusfor all p F,wehavethat p p 1 p x n + x n p 1. (3.14
Kiziltunc and Yolacan Fixed Point Theory and Applications 2013, 2013:90 Page 10 of 18 This implies that d(p, F p x n + d(x n, F. (3.15 From (3.14and(3.15(n, we have that d(q, F 0. This is a contradiction. Thus p F := F(T 1 F(T 2 ={x K : T 1 x = T 2 x = x}. 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 T 1, T 2 : K E be two continuous total asymptotically nonexpansive nonself-mappings with sequences {μ n }, {l n } defined by (1.9 such that n=1 μ n <, n=1 l n < and F := F(T 1 F(T 2 ={x K : T 1 x = T 2 x = x}. Assume that there exist M, M >0such that φ(λ M λ for all λ M, i {1, 2}. Starting from an arbitrary x 1 K, define the sequence {x n } by recursion (1.15. Suppose that {u n }, {v n } are bounded sequences in K such that n=1 γ n <, n=1 γ n <. Then, the sequence {x n} converges strongly to a common fixed point of T 1, T 2 if and only if lim inf n d(x n, F=0, where d(x n, F=inf p F x n p, n 1. Lemma 4 Let K be a nonempty convex subset of a uniformly convex Banach space E which is also a nonexpansive retract of E and T 1, T 2 : K E be two total asymptotically nonexpansive nonself-mappings with sequences {μ n }, {l n } defined by (1.9 such that n=1 μ n <, n=1 l n < and F := F(T 1 F(T 2 ={x K : T 1 x = T 2 x = x}. Assume that there exist M, M >0such that φ(λ M λ for all λ M, i {1, 2}. Starting from an arbitrary x 1 K, define the sequence {x n } by recursion (1.14. Suppose that (i 0<lim inf n α n and 0<lim inf n β n < lim sup n β n <1, and (ii 0<lim inf n α n and 0<lim inf n β n < lim sup n β n <1. Then lim n x n T i (PT i n 1 x n =0for i =1,2. Proof Let p F. Then by Lemma 3, lim n x n p exists. Let lim n x n p = r. If r = 0, then by the continuity of T 1 and T 2 the conclusion follows. Now suppose r >0.Set σ n =(1 β n y n +β n T 1 (PT 1 n 1 y n and δ n =(1 β n x n +β n T 2(PT 2 n 1 x n.since{x n } is bounded, there exists an R >0suchthatx n p, y n p B R (0 for all n 1. Using Lemma 2,wehave, for some constant A 1 >0,that δ n p 2 = ( 1 β n xn + β n T 2(PT 2 n 1 x n p 2 ( 1 β n xn p 2 + β n T 2 (PT 2 n 1 x n p 2 β n ( ( 1 β n g xn T 2 (PT 2 n 1 x n ( 1 β n xn p 2 + β n [ xn p + μ n φ ( x n p ] 2 + l n β n ( ( 1 β n g xn T 2 (PT 2 n 1 x n x n p 2 + A 1 (μ n + l n β n( 1 β n g (xn T 2 (PT 2 n 1 x n. (3.16
Kiziltunc and Yolacan Fixed Point Theory and Applications 2013, 2013:90 Page 11 of 18 It follows from (1.14, Lemma 2,(3.2and(3.16thatforsomeconstantA 2 >0, y n p 2 = (( P 1 α n xn + α n T 2(PT 2 n 1 δ n p 2 ( 1 α n (xn p+α n ( T2 (PT 2 n 1 δ n p 2 ( 1 α n xn p 2 + α n T 2 (PT 2 n 1 δ n p 2 α n ( ( 1 α n g xn T 2 (PT 2 n 1 δ n ( 1 α n xn p 2 + α n [ δn p + μ n φ ( δ n p ] 2 + l n α n ( ( 1 α n g x n T 2 (PT 2 n 1 δ n x n p 2 + A 2 (μ n + l n α n ( ( β n 1 β n g xn T 2 (PT 2 n 1 x n α n( ( 1 α n g xn T 2 (PT 2 n 1 δ n. (3.17 Using Lemma 2 and (3.17, we have, for some constant A 3 >0,that σ n p 2 = (1 β n y n + β n T 1 (PT 1 n 1 y n p 2 ( (1 βn (y n p+β n T1 (PT 1 n 1 y n p 2 (1 β n y n p 2 + β T1 n (PT 1 n 1 y n p 2 β n (1 β n g ( yn T 1 (PT 1 n 1 y n (1 β n y n p 2 [ + β n yn p + μ n φ ( y n p ] 2 + l n β n (1 β n g ( yn T 1 (PT 1 n 1 y n y n p 2 + A 3 (μ n + l n β n (1 β n g ( y n T 1 (PT 1 n 1 y n x n p 2 + A 2 (μ n + l n +A 3 (μ n + l n β n (1 β n g ( yn T 1 (PT 1 n 1 y n α n ( ( 1 α n g xn T 2 (PT 2 n 1 δ n α n ( n( β 1 β n g x n T 2 (PT 2 n 1 x n. (3.18 Similarly, it follows from (1.14, Lemma 2,(3.2and(3.18thatforsomeconstantA 4 >0, x n+1 p 2 = ( P (1 αn x n + α n T 1 (PT 1 n 1 σ n p 2 ( (1 αn (x n p+α n T1 (PT 1 n 1 σ n p 2 (1 α n x n p 2 + α T1 n (PT 1 n 1 σ n p 2 α n (1 α n g ( x n T 1 (PT 1 n 1 σ n (1 α n x n p 2 [ + α n σn p + μ n φ ( σ n p ] 2 + l n α n (1 α n g ( xn T 1 (PT 1 n 1 σ n x n p 2 + A 4 (μ n + l n
Kiziltunc and Yolacan Fixed Point Theory and Applications 2013, 2013:90 Page 12 of 18 α n (1 α n g ( xn T 1 (PT 1 n 1 σ n α n β n (1 β n g ( yn T 1 (PT 1 n 1 y n α n α n ( ( 1 α n g x n T 2 (PT 2 n 1 δ n α n α n ( n( β 1 β n g xn T 2 (PT 2 n 1 x n. (3.19 It follows from (3.19that α n α n ( ( β n 1 β n g x n T 2 (PT 2 n 1 x n xn p 2 x n+1 p 2 + A 4 (μ n + l n, (3.20 and α n α n ( ( 1 α n g x n T 2 (PT 2 n 1 δ n xn p 2 x n+1 p 2 α n β n (1 β n g ( yn T 1 (PT 1 n 1 y n xn p 2 x n+1 p 2 α n (1 α n g (x n T 1 (PT 1 n 1 σ n xn p 2 x n+1 p 2 + A 4 (μ n + l n, (3.21 + A 4 (μ n + l n, (3.22 + A 4 (μ n + l n. (3.23 Since 0 < lim inf n α n,0<lim inf n α n and 0 < lim inf n β n < lim sup n β n <1, there exists n 0 N and n 1, n 2, n 3, n 4 (0, 1 such that 0 < n 1 < α n,0<n 2 < α n and 0 < n 3 < β n < n 4 <1foralln n 0.Thisimpliesby(3.20that n 1 n 2 n 3 (1 n 4 g ( xn T 2 (PT 2 n 1 x n xn p 2 x n+1 p 2 + A 4 (μ n + l n (3.24 for all n n 0. It follows from (3.24thatk n 0,wehave k g (xn T 2 (PT 2 n 1 x n n=n 0 ( k 1 ( xn p 2 x n+1 p 2 k + A 4 (μ n + l n n 1 n 2 n 3 (1 n 4 n=n 0 n=n 0 ( 1 k x n0 p 2 + A 4 (μ n + l n. n 1 n 2 n 3 (1 n 4 n=n 0 Then n=n 0 g( x n T 2 (PT 2 n 1 x n < and therefore lim n g( x n T 2 (PT 2 n 1 x n = 0. Since g is strictly increasing and continuous with g(0 = 0, we have lim xn T 2 (PT 2 n 1 x n =0. (3.25 n
Kiziltunc and Yolacan Fixed Point Theory and Applications 2013, 2013:90 Page 13 of 18 By a similar method, together with (3.21, (3.22and(3.23, it can be show that lim xn T 2 (PT 2 n 1 δ n =0, lim yn T 1 (PT 1 n 1 y n =0, n n (3.26 lim xn T 1 (PT 1 n 1 σ n =0. n It follows from (1.14that y n x n = P (( 1 α n xn + α n T 2(PT 2 n 1 δ n Pxn T 2 (PT 2 n 1 δ n x n. This together with (3.26impliesthat lim n y n x n =0. (3.27 It follows from (3.26and(3.27that T1 (PT 1 n 1 x n x n T1 (PT 1 n 1 x n T 1 (PT 1 n 1 y n + T1 (PT 1 n 1 y n y n + yn x n y n x n + μ n φ ( y n x n + l n + T1 (PT 1 n 1 y n y n + yn x n 0, as n. (3.28 That is lim n T 1 (PT 1 n 1 x n x n = 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 T 1, T 2 : K E be two continuous total asymptotically nonexpansive nonself-mappings with sequences {μ n }, {l n } defined by (1.9 such that n=1 μ n <, n=1 l n < and F := F(T 1 F(T 2 ={x K : T 1 x = T 2 x = x}. Assume that there exist M, M >0such that φ(λ M λ for all λ M, i {1, 2}; and that one of T 1, T 2 is demicompact (without loss of generality, we assume T 1 is demicompact. Starting from an arbitrary x 1 K, define the sequence {x n } by recursion (1.14. Suppose that (i 0<lim inf n α n and 0<lim inf n β n < lim sup n β n <1, and (ii 0<lim inf n α n and 0<lim inf n β n < lim sup n β n <1. Then the sequence {x n } converges strongly to some common fixed points of T 1 and T 2. Proof It follows from (1.14and(3.26that x n+1 x n = P ( (1 α n x n + α n T 1 (PT 1 n 1 σ n Pxn T 1 (PT 1 n 1 σ n x n 0, as n. (3.29 It follows Lemma 4 and (3.29that x n T i (PT i n 2 x n x n x n 1 + x n 1 T i (PT i n 2 x n 1 + Ti (PT i n 2 x n 1 T i (PT i n 2 x n
Kiziltunc and Yolacan Fixed Point Theory and Applications 2013, 2013:90 Page 14 of 18 2 x n x n 1 + xn 1 T i (PT i n 2 x n 1 + μ n 1 φ ( x n x n 1 + l n 1 0, as n,fori =1,2. (3.30 Since T i is continuous and P is nonexpansive retraction, it follows from (3.30 thatfor i =1,2 T i (PT i n 1 x n T i x n = T i P ( T i (PT i n 2 x n T i Px n 0, as n. (3.31 Hence, by Lemma 4 and (3.31, we have x n T i x n xn T i (PT i n 1 x n + Ti (PT i n 1 x n T i x n 0, as n,fori =1,2. (3.32 Since T 1 is demicompact, from the fact that lim n x n T 1 x n =0and{x n } is bounded, there exists a subsequence {x nk } of {x n } that converges strongly to some q K as k. Hence, it follows from (3.32 thatt 1 x nk q, T 2 x nk q as k and it follows from (3.31andT i is continuous that Ti (PT i n k 1 x nk T i q Ti (PT i n k 1 x nk T i x nk + Ti x nk T i q = T i PT i (PT i n k 2 x nk T i Px nk + T i x nk T i q 0, as n,fori =1,2. (3.33 Observe that q T 1 q q x nk + xnk T 1 (PT 1 n k 1 x nk + T1 (PT 1 n k 1 x nk T 1 q. Taking limit as k and using the fact that Lemma 4 and (3.33wehavethatT 1 q = q and so q F(T 1. Also we get q T 2 q q x nk + x nk T 2 (PT 2 n k 1 x nk + T 2 (PT 2 n k 1 x nk T 2 q. Taking limit as k and using the fact that Lemma 4 and (3.33wehavethatT 2 q = q and so q F(T 2. Therefore, we obtain that q F. It follows from (3.7, Lemma 1 and lim k x nk = q that {x n } converges strongly to q 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 T 1, T 2 : K E be two continuous total asymptotically nonexpansive nonself-mappings with sequences {μ n }, {l n } defined by (1.9 such that n=1 μ n <, n=1 l n < and satisfying the condition (A. Assume that there exist M, M >0such that φ(λ M λ for all λ M, i {1, 2} and F := F(T 1 F(T 2 ={x K : T 1 x = T 2 x = x}. Starting from an arbitrary x 1 K, define the sequence {x n } by recursion (1.14. Suppose that
Kiziltunc and Yolacan Fixed Point Theory and Applications 2013, 2013:90 Page 15 of 18 (i 0<lim inf n α n and 0<lim inf n β n < lim sup n β n <1, and (ii 0<lim inf n α n and 0<lim inf n β n < lim sup n β n <1. Then the sequence {x n } converges strongly to some common fixed points of T 1 and T 2. Proof By Lemma 3, weseethatlim n x n p and so, lim n d(x n, F exists for all p F. Also,by(3.32, lim n x n T i x n =0fori = 1, 2. It follows from condition (A that lim f ( d(x n, F ( 1 ( lim x T1 x + x T 2 x =0. (3.34 n n 2 That is, lim n f ( d(x n, F =0. (3.35 Since f :[0, [0, is a nondecreasing function satisfying f (0 = 0, f (t > 0 for all t (0,, therefore, we have lim n d(x n, F=0. (3.36 Now we can take a subsequence {x nk } of {x n } and sequence {y k } F such that x nk y k <2 k for all integers k 1. Using the proof method of Tan and Xu [8], we have x nk+1 y k x nk y k <2 k, (3.37 and hence y k+1 y k y k+1 x nk+1 + x nk+1 y k 2 (k+1 +2 k <2 k+1. (3.38 We get that {y k } is a Cauchy sequence in F and so it converges. Let y k y. SinceF is closed, therefore, y F and then x nk y. Aslim n x n p exists, x n y 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 T 1, T 2 : K E be two continuous total asymptotically nonexpansive nonself-mappings with sequences {μ n }, {l n } defined by (1.9 such that n=1 μ n <, n=1 l n < and F := F(T 1 F(T 2 ={x K : T 1 x = T 2 x = x}. Assume that there exist M, M >0such that φ(λ M λ for all λ M, i {1, 2}; and that one of T 1, T 2 is demicompact (without loss of generality, we assume T 1 is demicompact. Starting from an arbitrary x 1 K, define the sequence {x n } by recursion (1.15. Suppose that {u n }, {v n } are bounded sequences in K such that n=1 γ n <, n=1 γ n <. Suppose that (i 0<lim inf n α n and 0<lim inf n β n < lim sup n β n <1, and (ii 0<lim inf n α n and 0<lim inf n β n < lim sup n β n <1. Then the sequence {x n } converges strongly to some common fixed points of T 1 and T 2.
Kiziltunc and Yolacan Fixed Point Theory and Applications 2013, 2013:90 Page 16 of 18 Theorem 6 Let K be a nonempty convex subset of a real Banach space E which is also a nonexpansive retract of E and T 1, T 2 : K E be two continuous total asymptotically nonexpansive nonself-mappings with sequences {μ n }, {l n } defined by (1.9 such that n=1 μ n <, n=1 l n < and satisfying the condition (A. Assume that there exist M, M >0such that φ(λ M λ for all λ M, i {1, 2} and F := F(T 1 F(T 2 ={x K : T 1 x = T 2 x = x}. Starting from an arbitrary x 1 K, define the sequence {x n } by recursion (1.15. Suppose that {u n }, {v n } are bounded sequences in K such that n=1 γ n <, n=1 γ n <. Suppose that (i 0<lim inf n α n and 0<lim inf n β n < lim sup n β n <1, and (ii 0<lim inf n α n and 0<lim inf n β n < lim sup n β n <1. Then the sequence {x n } converges strongly to some common fixed points of T 1 and T 2. Remark 2 If T 1 and T 2 are asymptotically nonexpansive mappings, then l n =0and φ(λ=λ so that the assumption that there exist M, M >0suchthatφ(λ M λ for all λ M, i {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 isthereallinewiththeusualnorm, K =[0, andp be the identity mapping. Assume that T 1 x = x and T 2 x = sin x for x K. Letφ be a strictly increasing continuous function such that φ : R + R + with φ(0 = 0. Let {μ n } n 1 and {l n } n 1 be two nonnegative real sequences defined by μ n = 1 and l n 2 n = 1, for all n 1(lim n 3 n μ n =0 and lim n l n =0.SinceT 1 x = x for x K,wehave T n 1 x T n 1 y x y. For all x, y K,weobtain T n 1 x T n 1 y x y μ n φ ( x y l n x y x y μ n φ ( x y l n 0 for all n =1,2,...,{μ n } n 1 and {l n } n 1 with μ n, l n 0asn and so T 1 is a total asymptotically nonexpansive mapping. Also, T 2 x = sin x for x K,wehave T n 1 x T n 1 y x y. For all x, y K,weobtain T n 2 x T n 2 y x y μ n φ ( x y l n x y x y μ n φ ( x y l n 0
Kiziltunc and Yolacan Fixed Point Theory and Applications 2013, 2013:90 Page 17 of 18 for all n =1,2,...,{μ n } n 1 and {l n } n 1 with μ n, l n 0asn and so T 2 is a total asymptotically nonexpansive mapping. Clearly, F := F(T 1 F(T 2 ={0}.Set α n = α n = n n +1, γ n = γ n = 1 n, 2 1 β n = β, n is even, 2 1 n = 1 3, n is odd and v n = u n = n +1 for n 1. Thus, the conditions of Theorem 2 are fulfilled. Therefore, we can invoke Theorem 2 to demonstrate that the iterative sequence {x n } defined by (1.15convergesstrongly to 0. Competing interests The authors declare that they have no competing interests. Authors contributions All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript. Acknowledgements Dedicated to Professor Hari M Srivastava. The authors are very grateful to the referees for their careful reading of manuscript, valuable comments and suggestions. Received: 1 December 2012 Accepted: 25 March 2013 Published: 10 April 2013 References 1. Goebel, K, Kirk, WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 35, 171-174 (1972 2. Bruck, RE, Kuczumow, T, Reich, S: Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property. Colloq. Math. 65, 169-179 (1993 3. Kirk, WA: Fixed point theorems for non-lipschitzian mappings of asymptotically nonexpansive type. Isr. J. Math. 17, 339-346 (1974 4. Albert, YI, Chidume, CE, Zegeye, H: Approximating fixed points of total asymptotically nonexpansive mappings. Fixed Point Theory Appl. 2006, ArticleID10673 (2006 5. Schu, J: Weak and strong convergence of a fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 43, 153-159 (1991 6. Osilike, MO, Aniagbosar, SC: Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings. Math. Comput. Model. 32, 1181-1191 (2000 7. Chidume, CE, Ofoedu, EU, Zegeye, H: Strong and weak convergence theorems for fixed points of asymptotically nonexpansive mappings. J. Math. Anal. Appl. 280, 364-374 (2003 8. Tan, KK, Xu, HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 178, 301-308 (1993 9. Khan, SH, Takahashi, W: Approximanting common fixed points of two asymptotically nonexpansive mappings. Sci. Math. Jpn. 53(1, 143-148 (2001 10. Khan, SH, Hussain, N: Convergence theorems for nonself asymptotically nonexpansive mappings. Comput. Math. Appl. 55, 2544-2553 (2008 11. Qin, X, Su, Y, Shang, M: Approximating common fixed points of non-self asymptotically nonexpansive mappings in Banach spaces. J. Appl. Math. Comput. 26, 233-246 (2008 12. Chidume, CE, Ofoedu, EU: Approximation of common fixed points for finite families of total asymptotically nonexpansive mappings. J. Math. Anal. Appl. 333, 128-141 (2007 13. Wang, L: Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings. J. Math. Anal. Appl. 323, 550-557 (2006 14. Shahzad, N: Approximanting fixed points of non-self nonexpansive mappings in Banach spaces. Nonlinear Anal. 61, 1031-1039 (2005 15. Thianwan, S: Weak and strong convergence theorems for new iterations with errors for nonexpansive nonself-mapping. Thai J. Math. 6, 27-38 (2008. Special Issue (Annual Meeting in Mathematics, 2008 16. Ozdemir, M, Akbulut, S, Kiziltunc, H: Convergence theorems on a new iteration process for two asymptotically nonexpansive nonself-mappings with errors in Banach spaces. Discrete Dyn. Nat. Soc. 2010, Article ID307245 (2010. doi:10.1155/2010/307-245 17. Senter, HF, Dotson, WG: Approximating fixed points of nonexpansive mappings. Proc. Am. Math. Soc. 44, 375-380 (1974 18. Maiti, N, Ghosh, MK: Approximating fixed points by Ishikawa iterates. Bull. Aust. Math. Soc. 40, 113-117 (1989 19. Xu, HK: Inequalities in Banach spaces with applications. Nonlinear Anal. 16, 1127-1138 (1991
Kiziltunc and Yolacan Fixed Point Theory and Applications 2013, 2013:90 Page 18 of 18 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.