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Convergence analysis of a multistep iteration for a finite family of asymptotically quasinonexpansive mappings
Journal of Inequalities and Applications volume 2013, Article number: 423 (2013)
Abstract
This paper is a continuation of the analysis of classical Kuhfittig iteration involving a finite family of asymptotically quasinonexpansive mappings in the general setup of uniformly convex hyperbolic spaces. We establish strong and △convergence results of Kuhfittig iteration, which subsequently help to apply proof mining techniques for the extraction of rates of metastability in the sense of Tao. Additionally, our proposed convergence results extend and improve various results in the current literature.
MSC:47H09, 47H10, 49M05.
1 Introduction
Most of the problems in various disciplines of science are nonlinear in nature, whereas fixed point theory proposed in the setting of normed linear spaces or Banach spaces majorly depends on the corresponding linear structures of those spaces. A nonlinear framework for fixed point theory is a metric space embedded with a ‘convex structure.’ It is remarked that the nonpositively curved spaces play a significant role in many branches of mathematics. The class of hyperbolic spaces  nonlinear in nature  is prominent among nonpositively curved spaces and provides rich geometrical structures for different results with applications in topology, graph theory, multivalued analysis and metric fixed point theory. The study of hyperbolic spaces has been largely motivated and dominated by questions about hyperbolic groups, one of the main objects of study in geometric group theory.
Throughout this paper, we work in the setting of hyperbolic spaces introduced by Kohlenbach [1] which is more restrictive than the hyperbolic type introduced in [2] and more general than the concept of hyperbolic space in [3].
A hyperbolic space is a metric space (X,d) together with a mapping W:{X}^{2}\times [0,1]\to X satisfying

(1)
d(u,W(x,y,\alpha ))\le \alpha d(u,x)+(1\alpha )d(u,y),

(2)
d(W(x,y,\alpha ),W(x,y,\beta ))=\alpha \beta d(x,y),

(3)
W(x,y,\alpha )=W(y,x,(1\alpha )),

(4)
d(W(x,z,\alpha ),W(y,w,\alpha ))\le (1\alpha )d(x,y)+\alpha d(z,w)
for all x,y,z,w\in X and \alpha ,\beta \in [0,1]. A nonempty subset K of a hyperbolic space X is convex if W(x,y,\alpha )\in K for all x,y\in K and \alpha \in [0,1]. The class of hyperbolic spaces contains normed spaces and convex subsets thereof, the Hilbert ball equipped with the hyperbolic metric [4], ℝtrees, Hadamard manifolds as well as CAT(0) spaces in the sense of Gromov (see [5] for a detailed treatment).
The following example accentuates the importance of hyperbolic spaces.
Let {B}_{H} be an open unit ball in a complex Hilbert spaces (H,\u3008\cdot \u3009) w.r.t. the metric
where
Then ({B}_{H},{k}_{{B}_{H}},W) is a hyperbolic space where W(x,y,\alpha ) defines a unique point z in a unique geodesic segment [x,y] for all x,y\in {B}_{H}. The above example is of importance for metric fixed point theory of holomorphic mappings which are {k}_{{B}_{H}}nonexpansive. For a detailed discussion of the topic, we refer to [6].
A hyperbolic space is uniformly convex [7] if for any r>0 and \u03f5\in (0,2], there exists a \delta \in (0,1] such that for all u,x,y\in X, we have
provided d(x,u)\le r, d(y,u)\le r and d(x,y)\ge \u03f5r.
A map \eta :(0,\mathrm{\infty})\times (0,2]\to (0,1], which provides such a \delta =\eta (r,\u03f5) for given r>0 and \u03f5\in (0,2], is known as a modulus of uniform convexity of X. We call η monotone if it decreases with r (for a fixed ϵ), i.e., \mathrm{\forall}\u03f5>0, \mathrm{\forall}{r}_{2}\ge {r}_{1}>0 (\eta ({r}_{2},\u03f5)\le \eta ({r}_{1},\u03f5)).
Let K be a nonempty subset of a metric space (X,d), and let T be a selfmapping on K. Denote by F(T)=\{x\in K:T(x)=x\} the set of fixed points of T. A selfmapping T on K is said to be:

(i)
nonexpansive if d(Tx,Ty)\le d(x,y) for x,y\in K;

(ii)
quasinonexpansive if d(Tx,p)\le d(x,p) for x\in K and for p\in F(T)\ne \mathrm{\varnothing};

(iii)
asymptotically nonexpansive [8] if there exists a sequence {k}_{n}\subset [0,\mathrm{\infty}) and {lim}_{n\to \mathrm{\infty}}{k}_{n}=0 and d({T}^{n}x,{T}^{n}y)\le (1+{k}_{n})d(x,y) for x,y\in K, n\ge 1;

(iv)
asymptotically quasinonexpansive if there exists a sequence {k}_{n}\subset [0,\mathrm{\infty}) and {lim}_{n\to \mathrm{\infty}}{k}_{n}=0 and d({T}^{n}x,p)\le (1+{k}_{n})d(x,p) for x\in K, p\in F(T), n\ge 1;

(v)
uniformly LLipschitzian if there exists a constant L>0 such that d({T}^{n}x,{T}^{n}y)\le Ld(x,y) for x,y\in K and n\ge 1.
It follows from the above definitions that a nonexpansive mapping is quasinonexpansive and that an asymptotically nonexpansive mapping is asymptotically quasinonexpansive. Moreover, an asymptotically nonexpansive mapping is uniformly LLipschitzian. However, the converse of these statements is not true, in general.
The fixed point property (fpp) of various nonlinear mappings has relevant applications in many branches of nonlinear analysis and topology. On the other hand, there are certain situations where it is hard to derive conditions for the (fpp) of certain nonlinear mappings. In such situations, the approximate fixed point property (afpp) is more desirable. Moreover, in a nonlinear domain, the (afpp) of various generalization nonexpansive mappings is still being developed.
The problem of finding a common fixed point of a finite family of nonlinear mappings acting on a nonempty convex domain often arises in applied mathematics, for instance, in convex minimization problems and systems of simultaneous equations. A fundamental result in the construction of common fixed points of a finite family of nonexpansive mappings is essentially due to Kuhfittig [9]. The following iteration is a translation of classical Kuhfittig iteration for a finite family of nonexpansive mappings in hyperbolic spaces.
Let {U}_{0}=I, define
where \delta \le \lambda \le 1\delta for some \delta \in (0,1).
Then the corresponding Kuhfittig iteration in a compact form is defined as follows:
The classical Kuhfittig iteration converges strongly under the compactness condition of K, whereas the weak convergence is established through Opial’s condition. Kuhfittig iteration is comparatively less developed for various nonlinear mappings in a more general setup of spaces with nonpositive sectional curvature such as hyperbolic spaces. To the best of our knowledge, Kuhfittig iteration has never been used as a tool for the approximation of common fixed points of a finite family of asymptotically quasinonexpansive mappings. Moreover, Rhoades [10] mentioned that one can replace λ in the Kuhfittig iteration with a sequence \{{\lambda}_{n}\}. Here a natural question arises:
Question Is Kuhfittig iteration valid for the class of asymptotically quasinonexpansive mappings with a general sequence of control parameters \delta \le {\lambda}_{n}\le 1\delta for some \delta \in (0,1) in the general setup of hyperbolic spaces?
The purpose of this paper is to provide an affirmative answer to the above question. Our convergence results not only can be viewed as an analogue of various existing results but also improve and generalize various results in the current literature; see, for example, [11–25] and the references cited therein.
2 Preliminaries
We start this section with the concept of △convergence which is essentially due to Lim [26] in the general setting of metric spaces. In 2008, Kirk and Panyanak [27] investigated △convergence in CAT(0) spaces and showed that △convergence coincides with the usual weak convergence in Banach spaces. Moreover, both concepts share many useful properties in uniformly convex spaces.
Let \{{x}_{n}\} be a bounded sequence in a hyperbolic space X. For x\in X, define a continuous functional r(\cdot ,\{{x}_{n}\}):X\to [0,\mathrm{\infty}) by
The asymptotic radius r(\{{x}_{n}\}) of \{{x}_{n}\} is given by
The asymptotic center of a bounded sequence \{{x}_{n}\} with respect to a subset K of X is defined as follows:
This is the set of minimizers of the functional r(\cdot ,\{{x}_{n}\}). If the asymptotic center is taken with respect to X, then it is simply denoted by A(\{{x}_{n}\}). It is known that uniformly convex Banach spaces and even CAT(0) spaces enjoy the property that ‘bounded sequences have unique asymptotic centers with respect to closed convex subsets.’ The following lemma is due to Leustean [28] and ensures that this property also holds in a complete uniformly convex hyperbolic space.
Lemma 2.1 [28]
Let (X,d,W) be a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity. Then every bounded sequence \{{x}_{n}\} in X has a unique asymptotic center with respect to any nonempty closed convex subset K of X.
Recall that a sequence \{{x}_{n}\} in X is said to △converge to x\in X if x is the unique asymptotic center of \{{u}_{n}\} for every subsequence \{{u}_{n}\} of \{{x}_{n}\}. In this case, we write \mathrm{\u25b3}{lim}_{n}{x}_{n}=x and call x the △limit of \{{x}_{n}\}. A sequence \{{x}_{n}\} is said to be quasiFejér monotone w.r.t. a set K if d({x}_{n+1},x)\le d({x}_{n},x)+{\u03f5}_{n} for all x\in K and {\u03f5}_{n}\ge 0 for all n\ge 0. This concept generalizes the classical concept of Fejér monotone sequence in a sense that it satisfies the standard Fejér monotonicity property within an additional error term {\u03f5}_{n}. A mapping T:K\to K is semicompact if every bounded sequence \{{x}_{n}\}\subset K satisfying d({x}_{n},T{x}_{n})\to 0 has a convergent subsequence.
In the sequel, we need the following useful results.
Lemma 2.2 [29]
Let (X,d,W) be a uniformly convex hyperbolic space with monotone modulus of uniform convexity η. Let x\in X and \{{\alpha}_{n}\} be a sequence in [a,b] for some a,b\in (0,1). If \{{x}_{n}\} and \{{y}_{n}\} are sequences in X such that {lim\hspace{0.17em}sup}_{n\u27f6\mathrm{\infty}}d({x}_{n},x)\le c, {lim\hspace{0.17em}sup}_{n\u27f6\mathrm{\infty}}d({y}_{n},x)\le c and {lim}_{n\u27f6\mathrm{\infty}}d(W({x}_{n},{y}_{n},{\alpha}_{n}),x)=c for some c\ge 0, then {lim}_{n\to \mathrm{\infty}}d({x}_{n},{y}_{n})=0.
Lemma 2.3 [29]
Let K be a nonempty closed convex subset of a uniformly convex hyperbolic space, and let \{{x}_{n}\} be a bounded sequence in K such that A(\{{x}_{n}\})=\{y\} and r(\{{x}_{n}\})=\rho. If \{{y}_{m}\} is another sequence in K such that {lim}_{m\to \mathrm{\infty}}r({y}_{m},\{{x}_{n}\})=\rho, then {lim}_{m\to \mathrm{\infty}}{y}_{m}=y.
Lemma 2.4 [11]
Let \{{a}_{n}\}and \{{b}_{n}\} be two sequences of nonnegative real numbers such that {\sum}_{n=1}^{\mathrm{\infty}}{b}_{n}<\mathrm{\infty}. If {a}_{n+1}\le (1+{b}_{n}){a}_{n}, n\ge 1, then {lim}_{n\to \mathrm{\infty}}{a}_{n} exists.
3 Main results
Throughout this section, we assume that the mappings {\{{U}_{n(i)}\}}_{i=1}^{k} are nonexpansive and satisfy F:=({\bigcap}_{i=1}^{k}F({T}_{i}))\cap ({\bigcap}_{i=1}^{k}F({U}_{n(i)}))\ne \mathrm{\varnothing}. We are now in a position to prove our main convergence results.
Theorem 3.1 Let K be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η, and let {\{{T}_{i}\}}_{i=1}^{k} be a finite family of uniformly LLipschitzian asymptotically quasinonexpansive selfmappings of K with a sequence \{{t}_{n}\}\subset [1,\mathrm{\infty}) such that {lim}_{n\to \mathrm{\infty}}{t}_{n}=1 and {\sum}_{n=1}^{\mathrm{\infty}}({t}_{n}1)<\mathrm{\infty}. Assume that F\ne \mathrm{\varnothing}, then the sequence \{{x}_{n}\} defined as
△converges to a common fixed point of {\{{T}_{i}\}}_{i=1}^{k}.
Proof Let p\in F, then observe that
Since {\sum}_{n=1}^{\mathrm{\infty}}({t}_{n}^{k1}1)<\mathrm{\infty}, therefore Lemma 2.4 implies that {\{d({x}_{n},p)\}}_{n=1}^{\mathrm{\infty}} is convergent. Consequently, this fact asserts that the sequence {\{d({x}_{n},p)\}}_{n=1}^{\mathrm{\infty}} is bounded. Let M\in \mathbb{N} be a bound of the sequence {\{d({x}_{n},p)\}}_{n=1}^{\mathrm{\infty}}such that d({x}_{n},p)\le M for all n\ge 1. Let {t}_{n}:=1+{r}_{n}, then observe the following variant of estimate (3.2):
where {\theta}_{k1}=\left(\genfrac{}{}{0ex}{}{k1}{1}\right)+\left(\genfrac{}{}{0ex}{}{k1}{2}\right)+\left(\genfrac{}{}{0ex}{}{k1}{3}\right)+\cdots +\left(\genfrac{}{}{0ex}{}{k1}{k1}\right) and {\theta}_{k1}{r}_{n}M is finite.
Hence, estimate (3.3) implies that {\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}} is quasiFejér monotone w.r.t. F. Therefore, {\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}} is bounded, and hence Lemma 2.1 implies that \{{x}_{n}\} has a unique asymptotic center {A}_{K}(\{{x}_{n}\})=\{x\}.
For the △convergence of \{{x}_{n}\}, we first show that the sequence \{{x}_{n}\} is asymptotic regular w.r.t. the k thmapping {S}_{k}, that is, {lim}_{n\to \mathrm{\infty}}d({x}_{n},{S}_{k}{x}_{n})=0. For this, we reason as follows.
Since {\{d({x}_{n},p)\}}_{n=1}^{\mathrm{\infty}} is convergent, therefore, without loss of any generality, we can assume that
where {S}_{k}^{n}:={T}_{k}^{n}{U}_{n(k1)}. The case c=0 is trivial. Moreover, observe that
It follows from estimates (3.4)(3.5) and Lemma 2.2 that
Note that d({x}_{n},{x}_{n+1})={\lambda}_{n}d({x}_{n},{S}_{k}^{n}{x}_{n}), therefore letting n\to \mathrm{\infty} and using (3.6), we have
Now observe that
Taking the lim sup on both sides of the above estimate and using (3.6)(3.7), we get the required asymptotic regularity of the k thmapping {S}_{k}, that is,
Let \{{u}_{n}\} be any subsequence of \{{x}_{n}\} with {A}_{K}(\{{u}_{n}\})=\{u\}, then
Next, we show that u\in F({S}_{k}). For this, we define a sequence \{{z}_{n}\} in K by {z}_{i}={S}_{k}^{i}u.
So, we calculate
Since {S}_{k} is uniformly LLipschitzian with the Lipschitz constant {L}_{K}, therefore, the above estimate yields
Taking limsup on both sides of the above estimate and using (3.8), we have
This implies that r({z}_{i},\{{u}_{n}\})r(u,\{{u}_{n}\})\to 0 as i\to \mathrm{\infty}. It follows from Lemma 2.3 that {lim}_{i\to \mathrm{\infty}}{S}_{k}^{i}u=u. As {S}_{k} is uniformly continuous, so we get that {S}_{k}(u)={S}_{k}({lim}_{i\to \mathrm{\infty}}{S}_{k}^{i}v)={lim}_{i\to \mathrm{\infty}}{S}_{k}^{i+1}u=u. That is, u\in F({S}_{k}) and hence u is the common fixed point of {T}_{k} and {U}_{k1}. Reasoning as above  by utilizing the uniqueness of asymptotic centers  we get that x=u. This infers that u is the unique asymptotic center of \{{x}_{n}\} for every subsequence \{{u}_{n}\} of \{{x}_{n}\}.
To proceed further, we show that
where {S}_{k1}^{n}={T}_{k1}^{n}{U}_{n(k2)}.
For this, we reason as follows.
Observe that estimate (3.2) implies that
Applying liminf on both sides of the above estimate and utilizing the fact that \delta \le {\lambda}_{n}\le 1\delta and {lim}_{n\to \mathrm{\infty}}{t}_{n}=1, we get that
On simplification, we have
On the other hand,
Taking limsup on both sides of the above estimate, we have
Estimates (3.9)(3.10) collectively imply that
Further, observe that
Appealing to Lemma 2.2 and utilizing estimates (3.11)(3.12), we have
Reasoning as above, we can show that:

(i)
{lim}_{n\to \mathrm{\infty}}d({x}_{n},{S}_{k1}{x}_{n})=0;

(ii)
u is the common fixed point of {T}_{k1} and {U}_{k2}.
Continuing in a similar fashion, we can show that u is the common fixed point of {S}_{k2}:={T}_{k2}{U}_{k3}, {S}_{k3}:={T}_{k3}{U}_{k4}, …, {S}_{1}:={T}_{1}{U}_{0}. Hence u\in F:={\bigcap}_{i=1}^{k}F({T}_{i}). This completes the proof. □
The strong convergence of iteration (3.1) can easily be established under compactness condition of K or T(K). Next, we give a necessary and sufficient condition for the strong convergence of iteration (3.1).
Theorem 3.2 Let K be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η, and let {\{{T}_{i}\}}_{i=1}^{k} be a finite family of uniformly Lipschitzian asymptotically quasinonexpansive selfmappings of K with a sequence \{{t}_{n}\}\subset [1,\mathrm{\infty}) such that {lim}_{n\to \mathrm{\infty}}{t}_{n}=1 and {\sum}_{n=1}^{\mathrm{\infty}}({t}_{n}1)<\mathrm{\infty}. Assume that F\ne \mathrm{\varnothing}, then the sequence \{{x}_{n}\} defined in (3.1) converges strongly to a common fixed point of {\{{T}_{i}\}}_{i=1}^{k} if and only if lim{inf}_{n\to \mathrm{\infty}}d({x}_{n},F)=0.
Proof The necessity of the conditions is obvious. Thus, we only prove the sufficiency. It follows from estimate (3.2) that {\{d({x}_{n},p)\}}_{n=1}^{\mathrm{\infty}} converges. Moreover, lim{inf}_{n\to \mathrm{\infty}}d({x}_{n},F)=0 implies that {lim}_{n\to \mathrm{\infty}}d({x}_{n},F)=0. This completes the proof. □
Since the class of asymptotically nonexpansive mappings is properly contained in the class of asymptotically quasinonexpansive mappings, therefore, we now list the following useful corollaries of Theorems (3.1)(3.2).
Corollary 3.3 Let K be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η, and let {\{{T}_{i}\}}_{i=1}^{k} be a finite family of uniformly Lipschitzian asymptotically nonexpansive selfmappings of K with a sequence \{{t}_{n}\}\subset [1,\mathrm{\infty}) such that {lim}_{n\to \mathrm{\infty}}{t}_{n}=1 and {\sum}_{n=1}^{\mathrm{\infty}}({t}_{n}1)<\mathrm{\infty}. Assume that F\ne \mathrm{\varnothing}, then the sequence \{{x}_{n}\} defined in (3.1) △converges to a common fixed point of {\{{T}_{i}\}}_{i=1}^{k}.
Corollary 3.4 Let K be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η, and let {\{{T}_{i}\}}_{i=1}^{k} be a finite family of uniformly Lipschitzian asymptotically nonexpansive selfmappings of K with a sequence \{{t}_{n}\}\subset [1,\mathrm{\infty}) such that {lim}_{n\to \mathrm{\infty}}{t}_{n}=1 and {\sum}_{n=1}^{\mathrm{\infty}}({t}_{n}1)<\mathrm{\infty}. Assume that F\ne \mathrm{\varnothing}, then the sequence \{{x}_{n}\} defined in (3.1) converges strongly to a common fixed point of {\{{T}_{i}\}}_{i=1}^{k} if and only if lim{inf}_{n\to \mathrm{\infty}}d({x}_{n},F)=0.
Concluding remarks (i) Following the line of action of the results proved so far, we can prove these results with suitable changes for the following classes of nonlinear mappings:

(a)
generalized asymptoticallyquasinonexpansive mappings (i.e., \parallel {T}^{n}xp\parallel \le {u}_{n}\parallel xp\parallel +{\delta}_{n}, where {lim}_{n\to \mathrm{\infty}}{u}_{n}=1 and {lim}_{n\to \mathrm{\infty}}{\delta}_{n}=0);

(b)
asymptotically nonexpansive mappings in the intermediate sense [30]\{\mathit{\text{i.e.}},\phantom{\rule{0.25em}{0ex}}{lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{sup}_{x,y\in C}(\parallel {T}^{n}x{T}^{n}y\parallel \parallel xy\parallel )\le 0\}.
Moreover, these proofs even hold for asymptotically weaklyquasinonexpansive mappings [31].

(ii)
It is worth mentioning that Kuhfittig iteration for a finite family of nonexpansive mappings is analyzed in the general setup of uniformly convex hyperbolic spaces resulting in explicit and uniform rates of asymptotical regularity [32]; whereas for iteration (3.1), there does not seem to exist a computable rate of asymptotic regularity, let alone a rate of metastability (in the sense of Tao [33]) in cases where strong convergence holds.
Future work We intend to extract explicit and effective rates of metastability of Kuhfittig iteration involving a finite family of asymptotically quasinonexpansive mappings in the general setup of uniformly convex hyperbolic spaces.
References
Kohlenbach U: Some logical metatheorems with applications in functional analysis. Trans. Am. Math. Soc. 2005, 357: 89–128. 10.1090/S0002994704035159
Goebel K, Kirk WA: Iteration processes for nonexpansive mappings. Contemp. Math. 21. In Topological Methods in Nonlinear Functional Analysis. Edited by: Singh SP, Thomeier S, Watson B. Am. Math. Soc., Providence; 1983:115–123.
Reich S, Shafrir I: Nonexpansive iterations in hyperbolic spaces. Nonlinear Anal. 1990, 15: 537–558. 10.1016/0362546X(90)90058O
Goebel K, Reich S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Dekker, New York; 1984.
Bridson M, Haefliger A: Metric Spaces of NonPositive Curvature. Springer, Berlin; 1999.
Kohlenbach U Springer Monogr. Math. In Applied Proof Theory: Proof Interpretations and Their Use in Mathematics. Springer, Berlin; 2008.
Leustean L:A quadratic rate of asymptotic regularity for CAT(0)spaces. J. Math. Anal. Appl. 2007, 325: 386–399. 10.1016/j.jmaa.2006.01.081
Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 1972, 35: 171–174. 10.1090/S00029939197202985003
Kuhfittig PKF: Common fixed points of nonexpansive mappings by iteration. Pac. J. Math. 1981, 97(1):137–139. 10.2140/pjm.1981.97.137
Rhoades BE: Finding common fixed point of nonexpansive mappings by iteration. Bull. Aust. Math. Soc. 2000, 62: 307–310. 10.1017/S0004972700018785
Chang SS, Cho YJ, Zhou H: Demiclosed principal and weak convergence problems for asymptotically nonexpansive mappings. J. Korean Math. Soc. 2001, 38(6):1245–1260.
Chidume CE, Ali B: Convergence theorems for finite families of asymptotically quasinonexpansive mappings. J. Inequal. Appl. 2007., 2007: Article ID 68616 10.1155/2007/68616
Fukharuddin H, Khan AR, Khan MAA: A new implicit algorithm of asymptotically quasinonexpansive mappings in uniformly convex Banach spaces. IAENG Int. J. Appl. Math. 2012., 42: Article ID 3
Fukharuddin H, Khan AR: Approximating common fixed points of asymptotically nonexpansive maps in uniformly convex Banach spaces. Comput. Math. Appl. 2007, 53: 1349–1360. 10.1016/j.camwa.2007.01.008
Gu F, Fu Q: Strong convergence theorems for common fixed points of multistep iterations with errors in Banach spaces. J. Inequal. Appl. 2009., 2009: Article ID 819036 10.1155/2009/819036
Khan AR, Domlo AA, Fukharuddin H: Common fixed point Noor iteration for a finite family of asymptotically quasinonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 2008, 341: 1–11. 10.1016/j.jmaa.2007.06.051
Khan AR, Khamsi MA, Fukharuddin H:Strong convergence of a general iteration scheme in CAT(0)spaces. Nonlinear Anal. 2011, 74: 783–791. 10.1016/j.na.2010.09.029
Fukharuddin, H, Khan, MAA: Convergence analysis of a general iteration schema of nonlinear mappings in hyperbolic spaces. Fixed Point Theory Appl. (in press)
Osilike MO, Aniagbosor SC: Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings. Math. Comput. Model. 2000, 32: 1181–1191. 10.1016/S08957177(00)001990
Sahin A, Basarır M: On the strong convergence of a modified S iteration process for asymptotically quasinonexpansive mappings in a CAT(0) space. Fixed Point Theory Appl. 2013., 2013: Article ID 12 10.1186/16871812201312
Sahin A, Basarır M: On the strong convergence and △convergence of SP iteration on CAT(0) space. J. Inequal. Appl. 2013., 2013: Article ID 311 10.1186/1029242X2013311
Schu J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 1991, 43: 153–159. 10.1017/S0004972700028884
Schu J: Iterative construction of fixed points of asymptotically nonexpansive mappings. J. Math. Anal. Appl. 1991, 158: 407–413. 10.1016/0022247X(91)90245U
Tan KK, Xu HK: Fixed point iteration process for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 1994, 122(3):733–739. 10.1090/S00029939199412039935
Yao Y, Liou YC: New iterative schemes for asymptotically quasinonexpansive mappings. J. Inequal. Appl. 2010., 2010: Article ID 934692 10.1155/2010/934692
Lim TC: Remarks on some fixed point theorems. Proc. Am. Math. Soc. 1976, 60: 179–182. 10.1090/S0002993919760423139X
Kirk W, Panyanak B: A concept of convergence in geodesic spaces. Nonlinear Anal. 2008, 68: 3689–3696. 10.1016/j.na.2007.04.011
Leustean L: Nonexpansive iterations in uniformly convex W hyperbolic spaces. Contemporary Mathematics 513. In Nonlinear Analysis and Optimization I: Nonlinear Analysis. Edited by: Leizarowitz A, Mordukhovich BS, Shafrir I, Zaslavski A. Am. Math. Soc., Providence; 2010:193–209.
Khan AR, Fukharuddin H, Khan MAA: An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 54 10.1186/16871812201254
Bruck R, Kuczumow T, Reich S: Convergence of iterates of asymptotically nonexpansive mappings in Banach space with uniform Opial property. Colloq. Math. 1993, 65(2):169–179.
Kohlenbach U, Lambov B, et al.: Bounds on iterations of asymptotically quasinonexpansive mappings. In International Conference on Fixed Point Theory and Applications Valencia, 2003. Edited by: Garcia Falset J. Yokohama Publishers, Yokohama; 2004:143–172.
Khan MAA, Kohlenbach U: Bounds on Kuhfittig’s iteration schema in uniformly convex hyperbolic spaces. J. Math. Anal. Appl. 2013, 403: 633–642. 10.1016/j.jmaa.2013.02.058
Tao T: Soft analysis, hard analysis, and the finite convergence principle. 298. In Structure and Randomness: Pages from Year One of a Mathematical Blog. Am. Math. Soc., Providence; 2008.
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Khan, M.A.A. Convergence analysis of a multistep iteration for a finite family of asymptotically quasinonexpansive mappings. J Inequal Appl 2013, 423 (2013). https://doi.org/10.1186/1029242X2013423
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DOI: https://doi.org/10.1186/1029242X2013423