Convergence theorems for total asymptotically nonexpansive non-self

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1 Wang et al. Journal of Inequalities and Applications 2013, 2013:135 R E S E A R C H Open Access Convergence theorems for total asymptotically nonexpansive non-self mappings in CAT0 spaces Lin Wang 1, Shih-sen Chang 1* and Zhaoli Ma 2 * Correspondence: Changss@yahoo.cn 1 College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan , P.R. China Full list of author information is available at the end of the article Abstract In this paper, we introduce the concept of total asymptotically nonexpansive nonself mappings and prove the demiclosed principle for this kind of mappings in CAT0 spaces. As a consequence, we obtain a -convergence theorem of total asymptotically nonexpansive nonself mappings in CAT0 spaces. Our results extend and improve the corresponding recent results announced by many authors. MSC: 47H09; 47J25 Keywords: total asymptotically nonexpansive nonself mappings; CAT0 space; -convergence; demiclosed principle 1 Introduction AmetricspaceX is a CAT0 space if it is geodesically connected and if every geodesic triangle in X is at least as thin as its comparison triangle in the Euclidean plane. That is to say, let X, d be a metric space, and let x, y X with dx, y =l. Ageodesic path from x to y is an isometry c :[0,l] X such that c0 = x and cl=y. The image of a geodesic path is called a geodesic segment. AmetricspaceX is a uniquely geodesic space if every two points of X are joined by only one geodesic segment. A geodesic triangle x 1, x 2, x 3 in a geodesic space X consists of three points x 1, x 2, x 3 of X and three geodesic segments joining each pair of vertices. A comparison triangle of the geodesic triangle x 1, x 2, x 3 is the triangle x 1, x 2, x 3 := x 1, x 2, x 3 in the Euclidean space R 2 such that dx i, x j =d R 2 x i, x j, i, j =1,2,3. AgeodesicspaceX is a CAT0 space if for each geodesic triangle x 1, x 2, x 3 inx and its comparison triangle := x 1, x 2, x 3 inr 2,theCAT0 inequality dx, y d R 2 x, ȳ 1.1 is satisfied for all x, y and x, ȳ. A thorough discussion on these spaces and their important role in various branches of mathematics is given in [1 5]. In 1976, Lim [6] introducedtheconceptof -convergence in a general metric space. Fixed point theory in a CAT0 space was first studied by Kirk [7, 8]. He showed that every nonexpansive single-valued mapping defined on a bounded closed convex subset of a 2013 Wang et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Wang et al. Journal of Inequalities and Applications 2013, 2013:135 Page 2 of 10 complete CAT0 space always has a fixed point. In 2008, Kirk and Panyanak [9]specialized Lim s concept to CAT0 spaces and proved that it is very similar to the weak convergence in the Banach space setting. So, the fixed point and -convergence theorems for single-valued and multivalued mappings in CAT0 spaces have been rapidly developed and many papers have appeared [10 25]. Let X, d be a metric space. Recall that a mapping T : X X is said to be nonexpansive if dtx, Ty dx, y, x, y X. 1.2 T is said to be asymptotically nonexpansive, if there is a sequence {k n } [1, with k n 1suchthat d T n x, T n y k n dx, y, n 1, x, y X. 1.3 T is said to be {μ n }, {ν n }, ζ -total asymptotically nonexpansive, if there exist nonnegative sequences {μ n }, {ν n } with μ n 0, ν n 0 and a strictly increasing continuous function ζ :[0, [0, withζ 0 = 0 such that d T n x, T n y dx, y+ν n ζ dx, y + μ n, n 1, x, y X. 1.4 Let X, d be a metric space, and let C be a nonempty and closed subset of X.Recallthat C is said to be a retract of X if there exists a continuous map P : X C such that Px = x, x C.AmapP : X C is said to be a retraction if P 2 = P.IfP is a retraction, then Py = y for all y in the range of P. Definition 1.1 Let X and C be the same as above. A mapping T : C X is said to be {μ n }, {ν n }, ζ -total asymptotically nonexpansive nonself mapping if there exist nonnegative sequences {μ n }, {ν n } with μ n 0, ν n 0 and a strictly increasing continuous function ζ :[0, [0, withζ 0 = 0 such that d TPT n 1 x, TPT n 1 y dx, y+ν n ζ dx, y + μ n, n 1, x, y C, 1.5 where P is a nonexpansive retraction of X onto C. Remark 1.2 From the definitions, it is to know that each nonexpansive nonself mapping is an asymptotically nonexpansive nonself mapping with a sequence {k n =1}, andeach asymptotically nonexpansive mapping is a {μ n }, {ν n }, ζ -total asymptotically nonexpansive mapping with μ n =0,ν n = k n 1, n 1andζ t=t, t 0. Definition 1.3 A nonself mapping T : C X is said to be uniformly L-Lipschitzian if there exists a constant L >0 such that d TPT n 1 x, TPT n 1 y Ldx, y, n 1, x, y C. 1.6 Recently, Chang et al. [26] introduced the following Krasnoselskii-Mann type iteration for finding a fixed point of a total asymptotically nonexpansive mappings in CAT0 spaces. x n+1 =1 α n x n α n T n x n, n

3 Wang et al. Journal of Inequalities and Applications 2013, 2013:135 Page 3 of 10 Under some limit conditions, they proved that the sequence {x n } -converges to a fixed point of T. Inspired and motivated by the recent work of Chang et al. [26], Tang et al. [27], Laowang et al. [19] and so on, the purpose of this paper is to introduce the concept of total asymptotically nonexpansive nonself mappings and prove the demiclosed principle for this kind of mappings in CAT0 spaces. As a consequence, we obtain a -convergence theorem of total asymptotically nonexpansive nonself mappings in CAT0 spaces. The results presented in this paper improve and extend the corresponding recent results in [19, 26, 27]. 2 Preliminaries The following lemma plays an important role in our paper. In this paper, we write 1 tx ty for the unique point z in the geodesic segment joining from x to y such that dz, x =tdx, y, dz, y =1 tdx, y. 2.1 We also denote by [x, y] the geodesic segment joining from x to y,thatis,[x, y]={1 tx ty : t [0, 1]}. AsubsetC of a CAT0 space is convex if [x, y] C for all x, y C. Lemma 2.1 [18] AgeodesicspaceXisaCAT0 space if and only if the following inequality holds: d 2 1 tx ty, z 1 td 2 x, z+td 2 y, z t1 td 2 x, y 2.2 for all x, y, z Xandallt [0, 1]. In particular, if x, y, z are points in a CAT0 space and t [0, 1], then d 1 tx ty, z 1 tdx, z+tdy, z. 2.3 Let {x n } be a bounded sequence in a CAT0 space X.Forx X,weset r x, {x n } = lim sup dx, x n. The asymptotic radius r{x n }of{x n } is given by r {x n } = inf { r x, {x n } : x X }. 2.4 The asymptotic radius r C {x n } of {x n } with respect to C X is given by r C {xn } = inf { r x, {x n } : x C }. 2.5 The asymptotic center A{x n } of {x n } is the set A {x n } = { x X : r x, {x n } = r {x n } }. 2.6

4 Wang et al. Journal of Inequalities and Applications 2013, 2013:135 Page 4 of 10 And the asymptotic center A C {x n } of {x n } with respect to C X is the set A C {xn } = { x C : r x, {x n } = r C {xn } }. 2.7 Recall that a bounded sequence {x n } in X is said to be regular if r{x n }=r{u n }forevery subsequence {u n } of {x n }. Proposition 2.2 [14] Let X be a complete CAT0 space, let {x n } be a bounded sequence in X, and let C be a closed convex subset of X. Then 1 there exists a unique point u C such that r u, {x n } = inf x C r x, {x n } ; 2 A{x n } and A C {x n } both are singleton. Definition 2.3 [6, 9] LetX be a CAT0 space. A sequence {x n } in X is said to -converge to p X if p is the unique asymptotic center of {u n } for each subsequence {u n } of {x n }.In this case we write -lim x n = p and call pthe -limit of {x n }. Lemma 2.4 [9] Every bounded sequence in a complete CAT0 space always has a -convergent subsequence. Lemma 2.5 [11] Let X be a complete CAT0 space, and let C be a closed convex subset of X. If {x n } is a bounded sequence in C, then the asymptotic center of {x n } is in C. Remark 2.6 Let X be a CAT0 space, and let C beaclosedconvexsubsetofx. Let{x n } be a bounded sequence in C. In what follows, we denote {x n } w w=inf x C x, where x:=lim sup dx n, x. Now we give a connection between the convergence and -convergence. Proposition 2.7 [26] Let X be a CAT0 space, let C be a closed convex subset of X, and let {x n } be a bounded sequence in C. Then -lim x n = pimpliesthat{x n } p. Lemma 2.8 Let C be a closed and convex subset of a complete CAT0 space X, and let T : C X be a uniformly L-Lipschitzian and {μ n }, {ν n }, ζ -total asymptotically nonexpansive nonself mapping. Let {x n } be a bounded sequence in C such that {x n } pand lim dx n, Tx n =0.Then Tp = p. Proof By the definition, {x n } p if and only if A C {x n }={p}. By Lemma 2.5, wehave A{x n }={p}. Since lim dx n, Tx n = 0, by induction we can prove that lim d x n, TPT m x n =0 foreachm

5 Wang et al. Journal of Inequalities and Applications 2013, 2013:135 Page 5 of 10 In fact, it is obvious that the conclusion is true for m = 0. Suppose that the conclusion holds for m 1, now we prove that the conclusion is also true for m +1. Indeed, since T isuniformly L-Lipschitzian, we have d x n, TPT m x n d xn, TPT m 1 x n + d TPT m 1 x n, TPT m x n d x n, TPT m 1 x n + Ldxn, PTx n = d x n, TPT m 1 x n + LdPxn, PTx n d x n, TPT m 1 x n + Ldxn, Tx n 0 asn. 2.8isproved.Henceforeachx C and m 1, from 2.8, we have x:=lim sup dx n, x=lim sup d TPT m 1 x n, x. 2.9 In 2.9takingx = TPT m 1 p, m 1, we have TPT m 1 p = lim sup d TPT m 1 x n, TPT m 1 p lim sup { dxn, p+ν m ζ dx n, p + μ m }. Letting m and taking the superior limit on the both sides, we get that lim sup TPT m 1 p p m Furthermore, for any n, m 1, it follows from inequality 2.2witht = 1 2 that d 2 x n, p TPTm 1 p d2 x n, p+ 1 2 d2 x n, TPT m 1 p 1 4 d2 p, TPT m 1 p Letting n and taking the superior limit on the both sides of the above inequality, for any m 1, we get p TPT m 1 p p TPT m 1 p d2 p, TPT m 1 p Since A{x n }={p}, for any m 1, we have p TPT p 2 m 1 p p TPT m 1 p d2 p, TPT m 1 p, 2.13 which implies that d 2 p, TPT m 1 p 2 TPT m 1 p 2 2 p

6 Wang et al. Journal of Inequalities and Applications 2013, 2013:135 Page 6 of 10 By 2.10and2.14, we have lim m dp, TPT m 1 p=0.hencewehave dtp, p d Tp, TPT m p + d TPT m p, p Ld p, TPT m 1 p + d TPT m p, p 0 asm, i.e., p = Tp,asdesired. The following result can be obtained from Lemma 2.8 immediately. Lemma 2.9 Let C be a closed and convex subset of a complete CAT0 space X, and let T : C X be an asymptotically nonexpansive nonself mapping with a sequence {k n } [1,, k n 1. Let {x n } be a bounded sequence in C such that lim dx n, Tx n =0and -lim x n = p. Then Tp = p. Lemma 2.10 [26] Let X be a CAT0 space, let x Xbeagivenpoint, and let {t n } be a sequence in [b, c] with b, c 0, 1 and 0<b1 c 1 2. Let {x n} and {y n } be any sequences in X such that lim sup dx n, x r, lim d 1 t n x n t n y n, x = r, lim sup dy n, x r and for some r 0. Then lim dx n, y n = Lemma 2.11 [26] Let {a n }, {λ n } and {c n } be the sequences of nonnegative numbers such that a n λ n a n + c n, n 1. If n=1 λ n < and n=1 c n <, then lim a n exists. If there exists a subsequence of {a n } which converges to zero, then lim a n =0. Lemma 2.12 [18] Let X be a complete CAT0 space, and let {x n } be a bounded sequence in X with A{x n }={p}; {u n } is a subsequence of {x n } with A{u n }={u}, and the sequence {dx n, u} converges, then p = u. 3 Main results Theorem 3.1 Let C be a nonempty, closed and convex subset of a complete CAT0 space E. Let T i : C E be a uniformly L-Lipschitzian and total asymptotically nonexpansive nonself mapping with sequences {μ i n } and {υ n i } satisfying lim μ i n =0and lim υ n i =0, and strictly increasing function ξ i :[0, [0, with ξ i 0 = 0, i =1,2.For arbitrarily chosen x 1 K, {x n } is defined as follows: y n = P1 β n x n β n T 2 PT 2 n 1 x n, 3.1 x n+1 = P1 α n x n α n T 1 PT 1 n 1 y n,

7 Wang et al. Journal of Inequalities and Applications 2013, 2013:135 Page 7 of 10 where {μ 1 n }, {μ 2 n }, {υ n 1 }, {υ n 2 1 n=1 μi n <, n=1 υi n <, i =1,2; }, ξ 1, ξ 2, {α n } and {β n } satisfy the following conditions: 2 there exist constants a, b 0, 1 with 0<b1 a 1 2 such that {α n} [a, b] and {β n } [a, b]; 3 there exists a constant M * >0such that ξ i r M * r, r 0, i =1,2. Then the sequence {x n } defined in 3.1 -converges to a common fixed point of T 1 and T 2. Proof We divide the proof into three steps. Step 1. We first show that lim dx n, q exists for each q FT 1 FT 2. Set μ n = max{μ 1 n, μ 2 n } and υ n = {υ n 1, υ n 2 n=1 υi n FT 2, we have }, n = 1,2,...,. Since n=1 μi n <, <, i =1,2,weknowthat n=1 μ n < and n=1 υ n <. For any q FT 1 dx n+1, q =d P 1 α n x n α n T 1 PT 1 n 1 y n, q d 1 α n x n α n T 1 PT 1 n 1 y n, q 1 α n dx n, q+α n d T 1 PT 1 n 1 y n, q 1 α n dx n, q+α n [ dyn, q+υ n ξ 1 dy n, q + μ n ] 1 α n dx n, q+α n [ 1+υn M * dy n, q+μ n ], 3.2 where dy n, q =d P 1 β n x n β n T 2 PT 2 n 1 x n, q d 1 β n x n β n T 2 PT 2 n 1 x n, q 1 β n dx n, q+β n d T 2 PT 2 n 1 x n, q [ 1 β n dx n, q+β n dxn, q+υ n ξ 2 dx n, q ] + μ n 1+β n υ n M * dx n, q+β n μ n. 3.3 Substituting 3.3 into3.2, we have dx n+1, q 1 α n dx n, q+α n [ 1+υn M * 1+β n υ n M * dx n, q+β n μ n + μn ] [ 1+ 1+β n + β n υ n M * α n M * υ n ] dxn, q+1+β n 1+υ n M * α n μ n. 3.4 Since n=1 μ n < and n=1 υ n <, it follows from Lemma 2.11 that lim dx n, q exists for each q FT 1 FT 2. Step 2. We show that lim dx n, T 1 x n =lim dx n, T 2 x n =0. For each q FT 1 FT 2, from the proof of Step 1, we know that lim dx n, qexists. We may assume that lim dx n, q=k.from3.3, we have dy n, q 1+β n υ n M * dx n, q+β n μ n. 3.5 Taking lim sup on both sides in 3.5, we have lim sup dy n, q k. 3.6

8 Wang et al. Journal of Inequalities and Applications 2013, 2013:135 Page 8 of 10 In addition, since d T 1 PT 1 n 1 y n, q dy n, q+υ n ξ 2 dy n, q + μ n 1+υ n M * dy n, q+μ n, we have lim sup d T 1 PT 1 n 1 y n, q k. 3.7 Since lim dx n+1, q=k,itiseasytoprovethat lim d 1 α n x n α n T 1 PT 1 n 1 y n, q = k. 3.8 It follows from Lemma 2.10 that lim d x n, T 1 PT 1 n 1 y n = On the other hand, since dx n, q d x n, T 1 PT 1 n 1 y n + d T1 PT 1 n 1 y n, q d x n, T 1 PT 1 n 1 y n + dyn, q+υ n M * dy n, q+μ n = d x n, T 1 PT 1 n 1 y n + 1+υn M * dy n, q+μ n, we have lim inf dy n, q k.combinedwith3.6, it yields that lim dy n, q=k This implies that lim d 1 β n x n β n T 2 PT 2 n 1 x n, q = k It is easy to show that lim sup d T 2 PT 2 n 1 x n, q k So, it follows from 3.12 and Lemma 2.10 that lim d x n, T 2 PT 2 n 1 x n = Observe that d x n, T 1 PT 1 n 1 x n d xn, T 1 PT 1 n 1 y n + d T1 PT 1 n 1 y n, T 1 PT 1 n 1 x n d x n, T 1 PT 1 n 1 y n + dxn, y n +υ n ξ 1 dx n, y n + μ n = d x n, T 1 PT 1 n 1 y n + 1+υn M * dx n, y n +μ n, 3.14

9 Wang et al. Journal of Inequalities and Applications 2013, 2013:135 Page 9 of 10 where dx n, y n =d P 1 β n x n β n T 2 PT 2 n 1 x n, xn β n d x n, T 2 PT 2 n 1 x n It follows from 3.13that lim dx n, y n = Thus, from 3.9, 3.14and3.16, we have lim d x n, T 1 PT 1 n 1 x n = In addition, since dx n+1, x n =d P 1 α n x n α n T 1 PT 1 n 1 y n, xn d 1 α n x n α n T 1 PT 1 n 1 y n, x n α n d T 1 PT 1 n 1 y n, x n, from 3.9, we have lim dx n+1, x n = Finally, since dx n, T 1 x n dx n, x n+1 +d x n+1, T 1 PT 1 n x n+1 + d T 1 PT 1 n x n+1, T 1 PT 1 n x n + d T1 PT 1 n x n, T 1 x n 1 + Ldx n, x n+1 +d x n+1, T 1 PT 1 n x n+1 + Ld T1 PT 1 n 1 x n, x n, it follows from 3.17 and3.18 thatlim dx n, T 1 x n =0.Similarly,wealsocanshow that lim dx n, T 2 x n =0. Step 3. We show that {x n } -converges to a common fixed point of T 1 and T 2. Let W ω x n = {u n } {x n } A{u n}. Firstly, we show that W ω FT 1 FT 2. Let u W ω, then there exists a subsequence {u n } of {x n } such that A{u n }={u}. By Lemma 2.4 and Lemma 2.5,thereexistsasubsequence{u ni } of {u n } such that -lim i u ni = p K.Since lim dx n, T 1 x n =lim dx n, T 2 x n = 0, it follows from Lemma 2.8 that p FT 1 FT 2. So, lim dx n, p exists. By Lemma 2.12, weknowthatp = u FT 1 FT 2. This implies that W ω x n FT 1 FT 2. Next, let {u n } be a subsequence of {x n } with A{u n }={u} and A{x n } = {v}. Sinceu W ω x n FT 1 FT 2, lim dx n, u exists. By Lemma 2.12, weknowthatv = u. ThisimpliesthatW ω x n contains only one point. Thus, since W ω x n FT 1 FT 2, W ω x n contains only one point and lim dx n, q exists for each q FT 1 FT 2, we know that {x n } -converges to a common fixed point of T 1 and T 2. The proof is completed.

10 Wang et al. Journal of Inequalities and Applications 2013, 2013:135 Page 10 of 10 Competing interests The authors declare that they have no competing interests. Authors contributions All authors read and approved the final manuscript. Author details 1 College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan , P.R. China. 2 School of Information Engineering, The College of Arts and Sciences, Yunnan Normal University, Kunming, Yunnan , P.R. China. Acknowledgements The authors would like to express their thanks to the referees for their helpful comments and suggestions. This work was supported by the Natural Scientific Research Foundation of Yunnan Province Grant No. 2011FB074. Received: 18 December 2012 Accepted: 10 March 2013 Published: 29 March 2013 References 1. Bridson, MR, Haefliger, A: Metric Spaces of Non-Positive Curvature. Grundlehren der Mathematischen Wissenschaften, vol Springer, Berlin Brown, KS: Buildings. Springer, New York Goebel, K, Reich, S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Monographs and Textbooks in Pure and Applied Mathematics, vol. 83. Dekker, New York Burago, D, Burago, Y, Ivanov, S: A Course in Metric Geometry. Graduate Studies in Mathematics, vol. 33. Am. Math. Soc., Providence Gromov, M: Metric Structures for Riemannian and Non-Riemannian Spaces. Progress in Mathematics, vol Birkhäuser, Boston Lim, TC: Remarks on some fixed point theorems. Proc. Am. Math. Soc. 60, Kirk, WA: Geodesic geometry and fixed point theory. In: Seminar of Mathematical Analysis Malaga/Seville, 2002/2003. Colecc. Abierta, vol. 64, pp Seville University Publications, Seville Kirk, WA: Geodesic geometry and fixed point theory II. In: International Conference on Fixed Point Theory and Applications, pp Yokohama Publications, Yokohama Kirk, WA, Panyanak, B: A concept of convergence in geodesic spaces. Nonlinear Anal., Theory Methods Appl. 6812, Dhompongsa, S, Kaewkhao, A, Panyanak, B: Lim s theorems for multivalued mappings in CAT0 spaces. J. Math. Anal. Appl. 3122, Dhompongsa, S, Kirk, WA, Panyanak, B: Nonexpansive set-valued mappings in metric and Banach spaces. J. Nonlinear Convex Anal. 81, Kirk, WA: Fixed point theorems in CAT0 spaces and R-trees. Fixed Point Theory Appl , Shahzad, N, Markin, J: Invariant approximations for commuting mappings in CAT0 and hyperconvex spaces. J. Math. Anal. Appl. 3372, Dhompongsa, S, Kirk, WA, Sims, B: Fixed points of uniformly Lipschitzian mappings. Nonlinear Anal., Theory Methods Appl. 654, Laowang, W, Panyanak, B: Strong and -convergence theorems for multivalued mappings in CAT0 spaces. J. Inequal. Appl. 2009, Article ID Shahzad, N: Invariant approximations in CAT0 spaces. Nonlinear Anal., Theory Methods Appl. 7012, Nanjaras, B, Panyanak, B, Phuengrattana, W: Fixed point theorems and convergence theorems for Suzuki-generalized nonexpansive mappings in CAT0 spaces. Nonlinear Anal. Hybrid Syst. 41, Dhompongsa, S, Panyanak, B: On -convergence theorems in CAT0 spaces. Comput. Math. Appl. 5610, Laowang, W, Panyanak, B: Approximating fixed points of nonexpansive nonself mappings in CAT0 spaces. Fixed Point Theory Appl. 2010, ArticleID doi: /2010/ Leustean, L: A quadratic rate of asymptotic regularity for CAT0-spaces. J. Math. Anal. Appl. 3251, Saejung, S: Halpern s iteration in CAT0 spaces. Fixed Point Theory Appl. 2010, Article ID doi: /2010/ Cho, YJ, Ciric, L, Wang, SH: Convergence theorems for nonexpansive semigroups in CAT0 spaces. Nonlinear Anal doi: /j.na Abkar, A, Eslamian, M: Fixed point and convergence theorems for different classes of generalized nonexpansive mappings in CAT0 spaces. Comput. Math. Appl doi: /j.camwa Nanjaras, B, Panyanak, B: Demiclosed principle for asymptotically nonexpansive mappings in CAT0 spaces. Fixed Point Theory Appl. 2010, ArticleID doi: /2010/ Shahzad, N: Fixed point results for multimaps in CAT0 spaces. Topol. Appl. 1565, Chang, SS, Wang, L, Lee, HWJ, Chan, CK, Yang, L: Total asymptotically nonexpansive mappings in CAT0 space demiclosed principle and -convergence theorems for total asymptotically nonexpansive mappings in CAT0 spaces.appl.math.comput.219, Tang, JF, Chang, SS, Lee, HWJ, Chan, CK: Iterative algorithm and -convergence theorems for total asymptotically nonexpansive mappings in CAT0 spaces. Abstr. Appl. Anal. 2012,Article ID doi: /2012/ doi: / x Cite this article as: Wang et al.: Convergence theorems for total asymptotically nonexpansive non-self mappings in CAT0 spaces. Journal of Inequalities and Applications :135.