A new iteration process for approximation of fixed points of mean nonexpansive mappings in CAT(0) spaces

Transcription

1 PURE MATHEMATICS RESEARCH ARTICLE A new iteration process for approximation of fixed points of mean nonexpansive mappings in CAT(0) spaces Received: 3 July 07 Accepted: 6 October 07 First Published: 30 October 07 *Corresponding author: A. Abkar Department of Mathemathics, Imam Khomeini International University, Qazvin 3449, Iran abkar@sci.ikiu.ac.ir Reviewing editor: Lishan Liu, Qufu Normal University, China Additional information is available at the end of the article M. Rastgoo and A. Abkar * Abstract: In this paper, we introduce a new iterative algorithm for approximating fixed points of mean nonexpansive mappings in CAT(0) spaces. We prove a Δ-convergence theorem under suitable conditions. The result we obtain improves and extends several recent results stated by many others; they also complement many known recent results in the literature. We then provide some numerical examples to illustrate our main result and to display the efficiency of the proposed algorithm. Subjects: Functional Analysis; Mathematical Analysis; Mathematical Numerical Analysis; Operator Theory; Pure Mathematics Keywords: CAT(0) space; mean nonexpansive mapping; Δ-convergence; iterative algorithm; fixed point AMS subject classifications: 47H09; 47H0; 47J. Introduction Fixed point theory of metric spaces was initiated by the celebrated Banach contraction principle which states that every contraction on a complete metric space has a unique fixed point; moreover, the fixed point can be approximated by Picard s iterates. Perhaps the most influential fixed point theorem in topological fixed point theory is the theorem due to Browder (96) and Gohde (96) independently proving that every nonexpansive self-mapping of a closed, convex, and bounded subset of a uniformly A. Abkar ABOUT THE AUTHOR A. Abkar, a professor of Mathematics, is mainly known for works in Operator Theory of Function Spaces, in particular, Bergman spaces. In January 000, after receiving a PhD from Lund University, Sweden, he was awarded a postdoctoral fellowship by ICTP, Trieste, Italy, as well as the Young Mathematician Prize by IPM in 003. Ali then spent a year on sabbatical at State University of New York in 0. He is a member of editorial board of Bulletin of Iranian Mathematical Society, a member of Iranian Mathematical Society, and American Mathematical Society. He has published more than 60 research papers in high-quality international journals; and works as a reviewer for both MathSciNet and ZbMATH. Recently, his research interest has shifted to Nonlinear Functional Analysis in which he has supervised several PhD students. PUBLIC INTEREST STATAMENT In many applications of mathematics in physical and engineering sciences, a differential equation should be solved. A powerful method for solving these equations is to use the contraction principle in some generalized metric spaces. Several methods have been found to establish the existence of fixed points that are the solutions of some functional or differential equations. Here we have focused not only on the existence of such solutions, but also on the approximation of the solutions. To reach this goal, we have presented a new iterative algorithm that converges to the required solution faster than any other existing algorithm. On the other hand, instead of investigating the class of nonexpansive mappings, we have considered a wider class, i.e. the class of mean nonexpansive mappings. This makes our theorem more powerful as well as more efficient. 07 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license. Page of

2 convex Banach space has a fixed point. Fixed point theory in Cartan Alexandrov Toponogov spaces, or briefly in CAT(0) spaces, was first studied by Kirk (00 003, 003). Among other things, he proved that every nonexpansive mapping defined on a bounded closed convex subset of a complete CAT(0) space has a fixed point. Since then the fixed point theorems for various mappings in CAT(0) space have been developed rapidly and numerous papers have appeared (see for example Dhompongsa & Panyanak, 008; Kirk & Panyanak, 008; Leustean, 007; Nanjaras, Panyanak, & Phuengrattana, 00; Shahzad, 009; Shahzad & Markin, 008 and the references therein). As a generalization of nonexpansive mappings, in Zhang (97) introduced the concept of a mean nonexpansive mapping in Banach spaces and proved the existence and uniqueness of fixed points for this type of mappings in Banach spaces with normal structure. The mean nonexpansive mappings were extensively studied by Wu and Zhang (007), and by Yang and Cui (008). In Nakprasit (00) provided an example of a mapping that is mean nonexpansive but not Suzuki-generalized nonexpansive and showed that increasing mean nonexpansiveness implies Suzuki-generalized nonexpansiveness. In Ouahab, Mbarki, Masude, and Rahmoune (0) proved a fixed point theorem for strong semigroups of mean nonexpansive mappings in uniformly convex Banach spaces. In this paper, we shall study mean nonexpansive mappings in the context of CAT(0) spaces. Let (X, d) be a metric space and x, y be two fixed elements in X such that d(x, y) =l. A geodesic path from x to y is an isometry c:[0, l] c([0, ]) X such that c(0) =x, c(l) =y. The image of a geodesic path between two points is called a geodesic segment. A metric space (X, d) is called a geodesic space if every two points of X are joined by a geodesic segment. A geodesic triangle represented by Δ(x, y, z) in a geodesic space consists of three points x, y, z and the three segments joining each pair of the points. A comparison triangle of a geodesic triangle Δ(x, y, z), denoted by Δ(x, y, z) or Δ(x, y, z), is a triangle in the Euclidean space R such that d(x, y) =d R (x, y), d(x, z) =d R (x, z), and d(y, z) =d R (y, z). This is obtainable using the triangle inequality, and it is unique up to isometry on R. Bridson and Haefliger (999) have shown that such a triangle always exists. A geodesic segment joining two points x, y in a geodesic space X is represented by [x, y]. Every point z in the segment is represented by αx ( α)y, where α [0, ], that is, [x, y]: ={αx ( α)y: α [0, ]}. A subset of a metric space X is called convex if for all x, y, [x, y]. A geodesic space is called a CAT(0) space if for every geodesic triangle Δ and its comparison Δ, the following inequality is satisfied: d(x, y) d R (x, y) for all x, y Δ and x, y Δ. Complete CAT(0) spaces are often called Hadamard spaces (see Kirk, 004; Reich & Salinas, 06). Examples of CAT(0) spaces include the R-tree, Hadamard manifold, and Hilbert ball equipped with hyperbolic metric. For more details on these spaces (see for example Abramenko & Brown, 008; Brown, 989; Burago, Burago, & Ivanov, 00). A geodesic space (X, d) is called hyperbolic (see Goebel & Reich, 984; Reich & Shafrir, 990) if, for any x, y, z X, ( d z x, z y ) d(x, y). The class of hyperbolic spaces includes the normed spaces, CAT(0) spaces, and some others. Bashir Ali in (06) presented an example of a hyperbolic space that is not a normed space. Therefore, the class of hyperbolic spaces is more general than the class of normed spaces. Let be a nonempty subset of a CAT(0) spaces (X, d). A self-mapping T: is called nonexpansive if d(tx, Ty) d(x, y) for all x, y. The mapping T is called quasi-nonexpansive if Fix(T) ={x :Tx = x} and d(tx d(x for all x and p Fix(T). In Zhou and Cui in (0) introduced an iterative algorithm to approximate fixed points of mean nonexpansive mappings in CAT(0) spaces; this algorithm is defined in the following way: x, x n+ =( t n t n T(y n ), y n =( s n s n T ), n, Page of

3 where {s n } n= and {t n } n= are some sequences in (0, ). In this paper, we introduce a new iterative algorithm for approximating fixed points of mean nonexpansive mappings in CAT(0) spaces. Under suitable conditions, we prove a Δ-convergence theorem for our algorithm. The results we obtain improve and extend several recent results in the literature; they also complement many known existing results. We then provide some numerical examples to illustrate our main result, and in this way we display the efficiency of our proposed algorithm.. Preliminaries Throughout this article, (X, d) will stand for a metric space. We denote by N the set of positive integers and by R the set of real numbers. We write x n x to indicate that the sequence {x n } n= converges weakly to x, and x n x to indicate that the sequence {x n } n= converges strongly to x. We start by recalling some basic definitions. Definition. Let be a nonempty subset of (X, d). A mapping T: is said to be nonexpansive if d(tx, Ty) d(x, y), x, y. Definition. Let be a nonempty subset of (X, d). A mapping T: is said to be mean nonexpansive if d(tx, Ty) ad(x, y)+bd(x, Ty), x, y, where a and b are two nonnegative real numbers such that a + b. Obviously, every nonexpansive mapping is a mean nonexpansive mapping (with a = and b = 0). Note that a mean nonexpansive mapping is not necessarily continuous as the following example shows, so that mean nonexpansive mappings are not necessarily nonexpansive. Example Suppose that T:[0, ] [0, ] is a mapping defined by Tx = x + x 6 + ) x 0, ; [ ] x,. Then, T is mean nonexpansive with a =, b =, but not continuous at x =. Thus, T is not a nonexpansive 3 3 mapping. Example Suppose that T:[0, ] [0, ] is a mapping defined by Tx = { x 3 +x x [0, ] is rational; x [0, ] is irrational. Then, T is mean nonexpansive with a = 3, b = 3, but not continuous at any point in [0, ] except x = 4, the fixed point of T. In Suzuki (008) introduced Suzuki-generalized nonexpansive mappings in Banach spaces. Definition.3 Let be a nonempty subset of (X, d). A mapping T: is said to be Suzuki-generalized nonexpansive if d(x, Tx) d(x, y) implies d(tx, Ty) d(x, y) Page 3 of

4 for all x, y. In Nakprasit (00) provided an example of a mapping that is mean nonexpansive but not Suzukigeneralized nonexpansive and showed that increasing mean nonexpansive mappings are Suzukigeneralized nonexpansive. We now turn to some known fact regarding the CAT(0) spaces. Lemma.4 (Nanjaras & Panyanak, 00, Lemma.) Let (X, d) be a CAT(0) space. Then, d(( α)x αy, z) ( α)d(x, z)+αd(y, z) for all α [0, ] and x, y, z X. Lemma. (Nanjaras & Panyanak, 00, Lemma 4.) Let x be a given point in a CAT(0) space (X, d) and {t n } be a sequence in a closed interval [a, b] with 0 < a b < and 0 < a( b). Suppose that {x n } and {y n } are two sequences in X such that () lim sup, x) r, () lim sup d(y n, x) r, (3) lim sup d(( t n t n y n, x) =r for some r 0. Then, lim, y n )=0. Theorem.6 (Zhou & Cui, 0, Theorem 3.) Let be a nonempty bounded closed convex subset of a complete CAT(0) space (X, d) and T: be a mean nonexpansive mapping with b <. Then, T has a fixed point. Theorem.7 (Zhou & Cui, 0, Theorem 3.) Let (X, d) be a complete CAT(0) space and be a nonempty bounded closed convex subset of X. Let T: be a mean nonexpansive mapping with b <, and let {x n } be an approximate fixed point sequence (i.e. lim, Tx n )=0) and {x n } ω. Then, T(ω) =ω. Definition.8 Let {x n } be a bounded sequence in a CAT(0) space (X, d). () The asymptotic radius r({x n }) of {x n } is given by r({x n }): = inf x X {r(x, {x n })}, where r(x, {x n }): = lim sup, x). () The asymptotic center A({x n }) of {x n } is the set A({x n }): ={x X: r(x, {x n }) = r({x n })}. In 006, Dhompongsa et al. proved that for each bounded sequence {x n } in a CAT(0) space, A({x n }) consists of exactly one point (see Proposition 7 in Dhompongsa, Kirk, & Sims, 006). This motivates the following notion of Δ-convergence which is regarded as some sort of weak convergence in CAT(0) spaces; of course this analogy is by no means complete. Definition.9 (Kirk & Panyanak, 008) Let (X, d) be a CAT(0) space. A sequence {x n } in X is said to Δ-converge to x X if and only if x is the unique asymptotic center of all subsequences of {x n }. In this case, we write Δ lim x n = x and x is called the Δ-limit of {x n }. Page 4 of

5 We recall that a bounded sequence {x n } in X is said to be regular if r({x n }) = r({u n }) for every subsequence {u n } of {x n }. It is known that every bounded sequence in a Banach space has a regular subsequence. Proposition.0 (Nanjaras & Panyanak, 00, Proposition 3.). Let {x n } be a bounded sequence in a CAT(0) space (X, d) and let X be a closed convex subset which contains {x n }. Then, (i) Δ lim x n = x implies {x n } x; (ii) if {x n } is regular, then {x n } x implies Δ lim x n = x. Lemma. The following assertions in a CAT(0) space hold: (i) (Kirk & Panyanak, 008). Every bounded sequence in a complete CAT(0) space has a Δ-convergent subsequence. (ii) (Dhompongsa, Kirk, & Panyanak, 007). If {x n } is a bounded sequence in a closed convex subset of a complete CAT(0) space (X, d), then the asymptotic center of {x n } is in. (iii) (Kirk & Panyanak, 008). If {x n } is a bounded sequence in a complete CAT(0) space (X, d) with A({x n }) = {p}, {ν n } is a subsequence of {x n } with A({ν n })={ν}, and the sequence {, ν)} converges, then p = ν. Lemma. (Zhou & Cui, 0, Lemma 4.4) Let be a nonempty closed convex subset of a complete CAT(0) space (X, d) and T: be a mean nonexpansive mapping. If {x n } is a sequence in such that lim, T )) = 0 and Δ lim x n = p, then T(p) =p. Remark.3 By Lemma. and Proposition.0 (ii), if {x n } in Theorem.7 is regular, then the condition b < in Theorem.7 can be removed. 3. A Δ-convergence theorem We begin this section by proving a Δ-convergence theorem for mean nonexpansive mappings in CAT(0) spaces. Indeed, we introduce a new iterative algorithm to approximate the fixed point of our mapping. We shall then compare our algorithm with that of Zhou and Cui (0). Theorem 3. Let (X, d) be a complete CAT(0) space and be a nonempty bounded closed convex subset of (X, d). Let T: be a mean nonexpansive mapping with 0 b <. Let {α n } and {β n= n } be n= sequences in (0, ) such that [c, d],0< c( d). Then, {x n } n= defined by x, z n =( α n α n T ), y n = T(( β n )z n β n T(z n )), x n+ = T(y n ), () is Δ-convergent to some point p Fix(T). Proof steps. Applying Theorem.6, we conclude that Fix(T). Now, we will divide the proof into three Step. First, we will prove that lim exists for each p Fix(T) where {x n } is defined by (). For this purpose, let p Fix(T). Using the fact that {α n } n= (0, ) we obtain d(z n =d(( α n α n T ( α n ) +α n d(t ( α n ) +α n a +α n b ( α n ) +α n () Page of

6 for all n N. This together with the fact that {β n } n= (0, ) yields d(y n =d(t(( β n )z n β n T(z n ) a [ d(( β n )z n β n T(z n ] + b [ d(( β n )z n β n T(z n ] d(( β n )z n β n T(z n ( β n )d(z n +β n d(t(z n ( β n )d(z n +β n ad(z n +β n bd(z n ( β n )d(z n +β n d(z n d(z n (3) for each n N. It now follows that + =d(t(y n ad(y n +bd(y n d(y n. Consequently, we have + for all n. This implies that { } is a decreasing sequence of real numbers. Since this sequence is bounded below, it follows that lim exists. Thus, {x n } is bounded. Step. We will prove that lim, T )) = 0. Without loss of generality, we may assume that (4) r: = lim. () Therefore, lim sup [ d(t lim sup ad(xn +b ] lim sup r. (6) According to (), we also have lim sup d(z n lim sup r. On the other hand, using (3), we can write (7) r = lim sup which implies that + =lim sup d(t(y n lim sup lim sup d(y n lim sup d(z n [ ad(yn +bd(y n ] r = lim sup d(z n. Therefore, r = lim sup d(z n =lim sup d(( α n α n T. (8) (9) Page 6 of

7 Using Lemma. together with (), (6), and (9), we have lim d(x, T(x n n )) = 0. (0) Therefore, we are done. Step 3. Define Ω Δ ): = {ν n } {x n } A({ν n }) Fix(T). We claim that the sequence {x n } is Δ-convergent to a fixed point of T and that Ω Δ ) consists of exactly one point. To this end, we assume that ν Ω Δ ). It follows from the definition of Ω Δ ) that Table. Numerical results corresponding to x = 0.9 and x = 0.9 for 30 steps Numerical results Step Our algorithm for x = 0.9 Our algorithm for x = 0.9 Zhou and Cui algorithm for x = 0.9 Zhou and Cui algorithm for x = Page 7 of

8 Figure. Convergence behaviors corresponding to x = 0.9 and x = 0.9 for 30 steps. Figure. Convergence behaviors ( corresponding ) to x =, for 00 steps. there exists a subsequence {ν n } of {x n } such that A({ν n })={ν}. Now, use the assertion (i) in Lemma. to obtain a subsequence {ρ n } of {ν n } such that Δ lim ρ n = ρ. It now follows from Lemma. that ρ Fix(T). Since the sequence {d(ν n, ρ)} is convergent, it follows from the assertion (ii) in Lemma. that ν = ρ. Therefore, Ω Δ ) Fix(T). Finally, we show that Ω Δ ) consists of exactly one point. Let {ν n } be a subsequence of {x n } such that A({ν n })={ν} and let A({x n }) = {x}. We have already seen that ν = ρ Fix(T). Since {, ρ)} converges, by assertion (iii) in Lemma., we have x = ρ Fix(T), that is, Ω Δ )=x. This completes the proof. 4. Numerical results In the following, we supply a numerical example of a mean nonexpansive mapping satisfying the conditions of Theorem 3., and some numerical experiment results to explain the conclusion of our algorithm. Page 8 of

9 Example 3 Consider X = R with its usual metric, so that X is a complete CAT(0) space. Let = [, ] which is clearly a bounded closed convex subset of X. We consider the mapping T: defined by Tx = x + x 6 + [ x x, [ ],. ), T is discontinuous at x = 0.; consequently, T is neither nonexpansive, nor contractive. Now, we verify that T is mean nonexpansive. [ ) Case : x, y,. By the definition of T, d(t(x), T(y)) = 4 d ( 4 x, 4 y ) = ( 4 d x y + y x, y x + x y ) 4 d(x, y)+ ( 4 d y ), x + ( y ) 4 d, x + ( 4 d x, y ) 4 d(x, y)+ d(x, T(y)) + d(t(x), T(y)). 4 This implies that d(t(x), T(y)) d(x, y)+ d(x, T(y)). 3 3 [ ) [ ] Case : x,, y,. In this case, we have ( x d(t(x), T(y)) = d, y ) ( x = d + T(y) T(y), y + T(x) T(x) ) d(x, T(x)) + d(t(x), T(y)) + d(y, T(y)) d(x, T(y)) + d(t(x), T(y)) + d(t(x), T(y)) + d(x, y)+ d(x, T(y)) = d(x, T(y)) + d(t(x), T(y)) + d(x, y). This implies that d(t(x), T(y)) d(x, y)+ d(x, T(y)). 3 3 [ ) [ ] Case 3: y,, x,. The argument is similar to the one in Case. [ ] Case 4: x, y,. The proof is the same as in Case. Hence, T is mean nonexpansive by taking a =, b =. Clearly, 0. is the only fixed point of T. Put 3 3 α n = β n = γ n = n Using MATHEMATICA, we computed the iterates of algorithm () for two different initial points x = 0.9 [, ] and x = 0.9 [, ]. Finally, by the numerical experiments, we compared Zhou and Cui iteration process with our algorithm () (see Table ). Moreover, the convergence behaviors of these algorithms are shown in Figure. We conclude that x n converges to 0.. Example 4 Consider X = R equipped with the Euclidean norm. Let x =(x, x ) R, then the squared distance of x from the origin, O, is Page 9 of

10 x = x + x. Consider = [, ] [, ] which is a bounded, closed, and convex subset of X. We define the mapping K: by ( K(x, x ): = 3 x, ) 3 x K is a nonexpansive mapping. This means that K is a mean nonexpansive mapping with a = and b = 0. Clearly, zero is the only fixed point of the mapping K. In this case, our algorithm is the following: x () =(x (), x () ), (z (n), z (n) )=( α n )(x (n), x (n) )+α n K(x (n), x (n) ), (y (n), y (n) )=K(( β n )(z (n), z (n) )+β n K(z (n), z (n) )), () (x (n+), x (n+) )=K(y (n), y (n) ). Put α n = β n = γ n = n ( Using MATHEMATICA, we have computed the iterates of the algorithm () for x () =, ) for 00 steps. Finally, by the numerical experiments we compared Zhou and Cui iteration process with our algorithm (). The convergence behaviors of these algorithms are shown in Figure. The conclusion is that x n converges to (0, 0). Funding The author received no direct funding for this research. Author details M. Rastgoo m.rastgoo89@gmail.com A. Abkar abkar@sci.ikiu.ac.ir ORCID ID: Department of Mathemathics, Imam Khomeini International University, Qazvin 3449, Iran. Citation information Cite this article as: A new iteration process for approximation of fixed points of mean nonexpansive mappings in CAT(0) spaces, A. Abkar & M. Rastgoo, Cogent Mathematics (07), 4: Correction This article has been republished with minor changes. These changes do not impact the academic content of the article. References Abramenko, P., & Brown, K. S. (008). Buildings: Theory and applications. Graduate texts in mathematics (Vol. 48). New York, NY: Springer. Ali, B. (06). Convergence theorems for finite families of total asymptotically nonexpansive mappings in hyperbolic spaces. Fixed Point Theory and Applications, 06. doi:0.86/s Bridson, M. R., & Haefliger, A. (999). Metric spaces of nonpositive curvature. Berlin: Springer. Browder, F. E. (96). Nonexpansive nonlinear operators in a Banach space. Proceedings of the National Academy of Sciences, 4, Brown, K. S. (989). Buildings. New York, NY: Springer. Burago, D., Burago, Y., & Ivanov, S. (00). A course in metric geometry, Graduate studies in mathematics (Vol. 33). Providence, RI: American Mathematical Society. Dhompongsa, S., Kirk, W. A., & Panyanak, B. (007). Nonexpansive set-valued mappings in metric and Banach spaces. Journal of Nonlinear and Convex Analysis, 8(), 3 4. Dhompongsa, S., Kirk, W. A., & Sims, B. (006). Fixed points of uniformly Lipschitzian mappings. Nonlinear Analysis: Theory, Methods and Applications, 6(4), Dhompongsa, S., & Panyanak, B. (008). On -convergence theorems in CAT(0) spaces. Computers & Mathematics with Applications, 6, Goebel, K., & Reich, S. (984). Uniform convexity, hyperbolic geometry and nonexpansive mappings. New York, NY: Dekker. Gohde, D. (96). Zum Prinzip der kontraktiven Abbidung. Mathematische Nachrichten, 30, 8. In German. Kirk, W. A. (00 003). Seminar of mathematical analysis: Geodesic geometry and fixed point theory (pp. 9 ). Seville, Spain: University of Malaga and Seville, Spain. Kirk, W. A. (003). Geodesic geometry and fixed point theory II. International Conference on Fixed Point Theory and Applications (pp. 3 4). Yokohama Publishers, Yokohama Kirk, W. A. (004). A fixed point theorem in CAT(0) spaces and R-trees. Fixed Point Theory and Applications, 4, Kirk, W. A., & Panyanak, B. (008). A concept of convergence in geodesic spaces. Nonlinear Analysis, 68, Leustean, L. (007). A, Quadratic rate of asymptotic regularity for CAT(0) spaces. Journal of Mathematical Analysis and Applications, 3, Nakprasit, K. (00). Mean nonexpansive mappings and Suzuki-generalized nonexpansive mappings. Journal of Nonlinear Analysis and Optimization,, Nanjaras, B., & Panyanak, B. (00). Demiclosed principle for asymptotically nonexpansive mappings in CAT(0) spaces. Fixed Point Theory and Applications, 00. Article ID Nanjaras, B., Panyanak, B., & Phuengrattana, W. (00). Fixed point theorems and convergence theorems for Suzukigeneralized nonexpansive mappings in CAT(0) spaces. Nonlinear Analysis: Hybrid Systems, 4, 3. Ouahab, A., Mbarki, A., Masude, J., & Rahmoune, M. (0). A fixed point theorem for mean nonexpansive mappings semigroups in uniformly convex Banach spaces. International Journal of Mathematical Analysis, 6, Page 0 of

11 Reich, S., & Salinas, Z. (06). Weak convergence of infinite products of operators in Hadamard spaces. Rendiconti del Circolo Matematico di Palermo, 6, 7. Reich, S., & Shafrir, I. (990). Nonexpansive iterations in hyperbolic spaces. Nonlinear Analysis,, Shahzad, N. (009). Invariant approximations in CAT(0) spaces. Nonlinear Analysis: TMA, 70, Shahzad, N., & Markin, J. (008). Invariant approximations for commuting mappings in CAT(0) and hyperconvex spaces. Journal of Mathematical Analysis and Applications, 337, Suzuki, T. (008). Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. Journal of Mathematical Analysis and Applications, 340, Wu, C., & Zhang, L. J. (007). Fixed points for mean nonexpansive mappings. Acta Mathematica Sinica, English Series, 3, Yang, Y., & Cui, Y. (008). Viscosity approximation methods for mean nonexpansive mappings in Banach spaces. Applied Mathematical Sciences,, Zhang, S. S. (97). About fixed point theorem for mean nonexpansive mapping in Banach spaces. Journal of Sichuan University,, Zhou, J., & Cui, Y. (0). Fixed point theorems for mean nonexpansive mappings in CAT(0) spaces. Numerical Functional Analysis and Optimization, 36, The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license. You are free to: Share copy and redistribute the material in any medium or format Adapt remix, transform, and build upon the material for any purpose, even commercially. The licensor cannot revoke these freedoms as long as you follow the license terms. Under the following terms: Attribution You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. No additional restrictions You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits. Cogent Mathematics (ISSN: 33-83) is published by Cogent OA, part of Taylor & Francis Group. Publishing with Cogent OA ensures: Immediate, universal access to your article on publication High visibility and discoverability via the Cogent OA website as well as Taylor & Francis Online Download and citation statistics for your article Rapid online publication Input from, and dialog with, expert editors and editorial boards Retention of full copyright of your article Guaranteed legacy preservation of your article Discounts and waivers for authors in developing regions Submit your manuscript to a Cogent OA journal at Page of