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On the strong and Δconvergence of new multistep and Siteration processes in a CAT(0) space
Journal of Inequalities and Applications volume 2013, Article number: 482 (2013)
Abstract
In this paper, we introduce a new class of mappings and prove the demiclosedness principle for mappings of this type in a CAT(0) space. Also, we obtain the strong and Δconvergence theorems of new multistep and Siteration processes in a CAT(0) space. Our results extend and improve the corresponding recent results announced by many authors in the literature.
MSC:47H09, 47H10, 54E40, 58C30.
1 Introduction
Contractive mappings and iteration processes are some of the main tools in the study of fixed point theory. There are many contractive mappings and iteration processes that have been introduced and developed by several authors to serve various purposes in the literature (see [1–6]).
Imoru and Olantiwo [7] gave the following contractive definition.
Definition 1 Let T be a selfmapping on a metric space X. The mapping T is called a contractivelike mapping if there exist a constant \delta \in [0,1) and a strictly increasing and continuous function \phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) with \phi (0)=0 such that, for all x,y\in X,
This mapping is more general than those considered by Berinde [8, 9], Harder and Hicks [10], Zamfirescu [11], Osilike and Udomene [12].
By taking \delta =1 in (1.1), we define a new class of mappings as follows.
Definition 2 The mapping T is called a generalized nonexpansive mapping if there exists a nondecreasing and continuous function \phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) with \phi (0)=0 such that, for all x,y\in X,
Remark 1 For x\in F(T) in (1.2), we have
If X is an interval of ℝ, then F(T) is convex. The same is also true in each space with unique geodesic for each pair of points (e.g., metric trees or CAT(0) spaces).
In the case \phi (t)=0 for all t\in [0,\mathrm{\infty}), it is easy to show that every nonexpansive mapping satisfies (1.2), but the inverse is not necessarily true.
Example 1 Let X=[0,2], d(x,y)=xy, \phi (t)=t and define T by
By taking x=2 and y=1.5, we have
but
Therefore T is a generalized nonexpansive mapping, but T is not nonexpansive mapping.
Both a contractivelike mapping and a generalized nonexpansive mapping need not have a fixed point, even if X is complete. For example, let X=[0,\mathrm{\infty}), d(x,y)=xy and define T by
It is proved in Gürsoy et al. [13] that T is a contractivelike mapping. Similarly, one can prove that T is a generalized nonexpansive mapping. But the mapping T has no fixed point.
By using (1.1), it is obvious that if a contractivelike mapping has a fixed point, then it is unique. However, if a generalized nonexpansive mapping has a fixed point, then it need not be unique. For example, let ℝ be the real line with the usual norm \cdot , and let K=[1,1]. Define a mapping T:K\to K by
Now, we show that T is a nonexpansive mapping. In fact, if x,y\in [0,1] or x,y\in [1,0), then we have
If x\in [0,1] and y\in [1,0) or x\in [1,0) and y\in [0,1], then we have
This implies that T is a nonexpansive mapping and so T is a generalized nonexpansive mapping with \phi (t)=0 for all t\in [0,\mathrm{\infty}). But F(T)=\{x\in K;0\le x\le 1\}.
Agarwal et al. [1] introduced the Siteration process which is independent of those of Mann [3] and Ishikawa [2] and converges faster than both of these. We apply this iteration process in a CAT(0) space as
Gürsoy et al. [13] introduced a new multistep iteration process in a Banach space. We modify this iteration process in a CAT(0) space as follows.
For an arbitrary fixed order k\ge 2,
or, in short,
By taking k=3 and k=2 in (1.4), we obtain the SPiteration process of Phuengrattana and Suantai [4] and the twostep iteration process of Thianwan [6], respectively.
In this paper, motivated by the above results, we prove demiclosedness principle for a new class of mappings and the Δconvergence theorems of the new multistep iteration and the Siteration processes for mappings of this type in a CAT(0) space. Also, we present the strong convergence theorems of these iteration processes for contractivelike mappings in a CAT(0) space.
2 Preliminaries on a CAT(0) space
A metric space X is a CAT(0) 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. Fixed point theory in a CAT(0) space was first studied by Kirk (see [14, 15]). He showed that every nonexpansive mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since then the fixed point theory in a CAT(0) space has been rapidly developed and many papers have appeared (see [14–20]). It is worth mentioning that the results in a CAT(0) space can be applied to any CAT(k) space with k\le 0 since any CAT(k) space is a CAT({k}^{\prime}) space for every {k}^{\prime}\ge k (see [[21], p.165]).
Let (X,d) be a metric space. A geodesic path joining x\in X to y\in X (or more briefly, a geodesic from x to y) is a map c from a closed interval [0,l]\subset R to X such that c(0)=x, c(l)=y and d(c(t),c({t}^{\prime}))=t{t}^{\prime} for all t,{t}^{\prime}\in [0,l]. In particular, c is an isometry and d(x,y)=l. The image of c is called a geodesic (or metric) segment joining x and y. When it is unique, this geodesic is denoted by [x,y]. The space (X,d) is said to be a geodesic space if every two points of X are joined by a geodesic and X is said to be a uniquely geodesic if there is exactly one geodesic joining x to y for each x,y\in X.
A geodesic triangle \u25b3({x}_{1},{x}_{2},{x}_{3}) in a geodesic metric space (X,d) consists of three points in X (the vertices of △) and a geodesic segment between each pair of vertices (the edges of △). A comparison triangle for the geodesic triangle \u25b3({x}_{1},{x}_{2},{x}_{3}) in (X,d) is a triangle \overline{\u25b3}({x}_{1},{x}_{2},{x}_{3})=\u25b3({\overline{x}}_{1},{\overline{x}}_{2},{\overline{x}}_{3}) in the Euclidean plane {\mathbb{R}}^{2} such that
for i,j\in \{1,2,3\}. Such a triangle always exists (see [21]).
A geodesic metric space is said to be a CAT(0) space [21] if all geodesic triangles of appropriate size satisfy the following comparison axiom.
CAT(0) Let △ be a geodesic triangle in X, and let \overline{\u25b3} be a comparison triangle for △. Then △ is said to satisfy the CAT(0) inequality if for all x,y\in \u25b3 and all comparison points \overline{x},\overline{y}\in \overline{\u25b3},
We observe that if x, {y}_{1}, {y}_{2} are points of a CAT(0) space and if {y}_{0} is the midpoint of the segment [{y}_{1},{y}_{2}], then the CAT(0) inequality implies that
The equality holds for the Euclidean metric. In fact (see [[21], p.163]), a geodesic metric space is a CAT(0) space if and only if it satisfies the inequality (2.1) (which is known as the CN inequality of Bruhat and Tits [22]).
Let x,y\in X, by [[18], Lemma 2.1(iv)] for each t\in [0,1], there exists a unique point z\in [x,y] such that
From now on, we will use the notation (1t)x\oplus ty for the unique point z satisfying (2.2). By using this notation, Dhompongsa and Panyanak [18] obtained the following lemmas which will be used frequently in the proof of our main results.
Lemma 1 Let X be a CAT(0) space. Then
for all t\in [0,1] and x,y,z\in X.
Lemma 2 Let X be a CAT(0) space. Then
for all t\in [0,1] and x,y,z\in X.
3 Demiclosedness principle for a new class of mappings
In 1976 Lim [23] introduced the concept of convergence in a general metric space setting which is called △convergence. Later, Kirk and Panyanak [24] used the concept of △convergence introduced by Lim [23] to prove on a CAT(0) space analogs of some Banach space results which involve weak convergence. Also, Dhompongsa and Panyanak [18] obtained the △convergence theorems for the Picard, Mann and Ishikawa iterations in a CAT(0) space for nonexpansive mappings under some appropriate conditions.
Now, we recall some definitions.
Let \{{x}_{n}\} be a bounded sequence in a CAT(0) space X. For x\in X, we set
The asymptotic radius r(\{{x}_{n}\}) of \{{x}_{n}\} is given by
and the asymptotic radius {r}_{K}(\{{x}_{n}\}) of \{{x}_{n}\} with respect to K\subset X is given by
The asymptotic center A(\{{x}_{n}\}) of \{{x}_{n}\} is the set
and the asymptotic center {A}_{K}(\{{x}_{n}\}) of \{{x}_{n}\} with respect to K\subset X is the set
Proposition 1 ([[25], Proposition 3.2])
Let \{{x}_{n}\} be a bounded sequence in a complete CAT(0) space X, and let K be a closed convex subset of X, then A(\{{x}_{n}\}) and {A}_{K}(\{{x}_{n}\}) are singletons.
Definition 3 ([[24], Definition 3.1])
A sequence \{{x}_{n}\} in a CAT(0) space X is said to be △convergent 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 \u25b3\text{}{lim}_{n}{x}_{n}=x and x is called the △limit of \{{x}_{n}\}.
Lemma 3

(i)
Every bounded sequence in a complete CAT(0) space always has a △convergent subsequence (see [[24], p.3690]).

(ii)
Let K be a nonempty closed convex subset of a complete CAT(0) space, and let \{{x}_{n}\} be a bounded sequence in K. Then the asymptotic center of \{{x}_{n}\} is in K (see [[17], Proposition 2.1]).
Lemma 4 ([[18], Lemma 2.8])
If \{{x}_{n}\} is a bounded sequence in a complete CAT(0) space with A(\{{x}_{n}\})=\{x\}, \{{u}_{n}\} is a subsequence of \{{x}_{n}\} with A(\{{u}_{n}\})=\{u\} and the sequence \{d({x}_{n},u)\} converges, then x=u.
Let \{{x}_{n}\} be a bounded sequence in a CAT(0) space X, and let K be a closed convex subset of X which contains \{{x}_{n}\}. We denote the notation
where \mathrm{\Phi}(x)={lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}d({x}_{n},x).
We note that \{{x}_{n}\}\rightharpoonup w if and only if {A}_{K}(\{{x}_{n}\})=\{w\} (see [25]).
Nanjaras and Panyanak [25] gave a connection between the ‘⇀’ convergence and Δconvergence.
Proposition 2 ([[25], Proposition 3.12])
Let \{{x}_{n}\} be a bounded sequence in a CAT(0) space X, and let K be a closed convex subset of X which contains \{{x}_{n}\}. Then \mathrm{\Delta}\text{}{lim}_{n\to \mathrm{\infty}}{x}_{n}=p implies that \{{x}_{n}\}\rightharpoonup p.
By using the convergence defined in (3.1), we obtain the demiclosedness principle for the new class of mappings in a CAT(0) space.
Theorem 1 Let K be a nonempty closed convex subset of a complete CAT(0) space X, and let T:K\to K be a generalized nonexpansive mapping with F(T)\ne \mathrm{\varnothing}. Let \{{x}_{n}\} be a bounded sequence in K such that \{{x}_{n}\}\rightharpoonup w and {lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0. Then Tw=w.
Proof By the hypothesis, \{{x}_{n}\}\rightharpoonup w. Then we have {A}_{K}(\{{x}_{n}\})=\{w\}. By Lemma 3(ii), we obtain A(\{{x}_{n}\})=\{w\}. Since {lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0, then we have
for all x\in K. By taking x=Tw in (3.2), we have
The rest of the proof closely follows the pattern of Proposition 3.14 in Nanjaras and Panyanak [25]. Hence Tw=w as desired. □
Now, we prove the Δconvergence of the new multistep iteration process for the new class of mappings in a CAT(0) space.
Theorem 2 Let K be a nonempty closed convex subset of a complete CAT(0) space X, let T:K\to K be a generalized nonexpansive mapping with F(T)\ne \mathrm{\varnothing}, and let \{{x}_{n}\} be a sequence defined by (1.4) such that \{{\alpha}_{n}\},\{{\beta}_{n}^{i}\}\subset [0,1], i=1,2,\dots ,k2 and \{{\beta}_{n}^{k1}\}\subset [a,b] for some a,b\in (0,1). Then the sequence \{{x}_{n}\} Δconverges to the fixed point of T.
Proof Let p\in F(T). From (1.2), (1.4) and Lemma 1, we have
Also, we obtain
Continuing the above process, we have
This inequality guarantees that the sequence \{d({x}_{n},p)\} is nonincreasing and bounded below, and so {lim}_{n\to \mathrm{\infty}}d({x}_{n},p) exists for all p\in F(T). Let {lim}_{n\to \mathrm{\infty}}d({x}_{n},p)=r. By using (3.3), we get
By Lemma 2, we also have
which implies that
Thus {lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0. To show that the sequence \{{x}_{n}\} △converges to a fixed point of T, we prove that
and {W}_{\u25b3}({x}_{n}) consists of exactly one point. Let u\in {W}_{\u25b3}({x}_{n}). Then there exists a subsequence \{{u}_{n}\} of \{{x}_{n}\} such that A(\{{u}_{n}\})=\{u\}. By Lemma 3, there exists a subsequence \{{v}_{n}\} of \{{u}_{n}\} such that \u25b3\text{}{lim}_{n\to \mathrm{\infty}}{v}_{n}=v\in K. By Proposition 2 and Theorem 1, v\in F(T). By Lemma 4, we have u=v\in F(T). This shows that {W}_{\u25b3}({x}_{n})\subseteq F(T). Now, we prove that {W}_{\u25b3}({x}_{n}) consists of exactly one point. Let \{{u}_{n}\} be a subsequence of \{{x}_{n}\} with A(\{{u}_{n}\})=\{u\}, and let A(\{{x}_{n}\})=\{x\}. We have already seen that u=v and v\in F(T). Finally, since \{d({x}_{n},v)\} converges, by Lemma 4, x=v\in F(T). This shows that {W}_{\u25b3}({x}_{n})=\{x\}. This completes the proof. □
We give the following theorem related to the Δconvergence of the Siteration process for the new class of mappings in a CAT(0) space.
Theorem 3 Let K be a nonempty closed convex subset of a complete CAT(0) space X, let T:K\to K be a generalized nonexpansive mapping with F(T)\ne \mathrm{\varnothing}, and let \{{x}_{n}\} be a sequence defined by (1.3) such that \{{\alpha}_{n}\},\{{\beta}_{n}\}\subset [a,b] for some a,b\in (0,1). Then the sequence \{{x}_{n}\} Δconverges to the fixed point of T.
Proof Let p\in F(T). Using (1.2), (1.3) and Lemma 1, we have
Also, we obtain
From (3.4) and (3.5), we have
This inequality guarantees that the sequence \{d({x}_{n},p)\} is nonincreasing and bounded below, and so {lim}_{n\to \mathrm{\infty}}d({x}_{n},p) exists for all p\in F(T). Let
Now, we prove that {lim}_{n\to \mathrm{\infty}}d({y}_{n},p)=r. By (3.4), we have
This gives that
or
This gives
By (3.5) and (3.6), we obtain
Then we get
By Lemma 2, we also have
which implies that
Thus {lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0. The rest of the proof follows the pattern of the above theorem and is therefore omitted. □
4 Strong convergence theorems for a contractivelike mapping
Now, we prove the strong convergence of the new multistep iteration process for a contractivelike mapping in a CAT(0) space.
Theorem 4 Let K be a nonempty closed convex subset of a complete CAT(0) space X, let T:K\to K be a contractivelike mapping with F(T)\ne \mathrm{\varnothing}, and let \{{x}_{n}\} be a sequence defined by (1.4) such that \{{\alpha}_{n}\}\subset [0,1), {\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty} and \{{\beta}_{n}^{i}\}\subset [0,1), i=1,2,\dots ,k1. Then the sequence \{{x}_{n}\} converges strongly to the unique fixed point of T.
Proof Let p be the unique fixed point of T. From (1.1), (1.4) and Lemma 1, we have
Also, we obtain
In a similar fashion, we can get
Continuing the above process, we have
In addition, we obtain
From (4.1) and (4.2), we have
Using the fact that 0\le \delta <1, {\alpha}_{j}\in [0,1] and {\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}, we get that
This together with (4.3) implies that
Consequently, {x}_{n}\to p\in F(T) and this completes the proof. □
Remark 2 In Theorem 4 the condition {\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty} may be replaced with {\sum}_{n=0}^{\mathrm{\infty}}{\beta}_{n}^{i}=\mathrm{\infty} for a fixed i=1,2,\dots ,k1.
Finally, we give the strong convergence theorem of the Siteration process for a contractivelike mapping in a CAT(0) space as follows.
Theorem 5 Let K be a nonempty closed convex subset of a complete CAT(0) space X, let T:K\to K be a contractivelike mapping with F(T)\ne \mathrm{\varnothing}, and let \{{x}_{n}\} be a sequence defined by (1.3) such that \{{\alpha}_{n}\},\{{\beta}_{n}\}\subset [0,1]. Then the sequence \{{x}_{n}\} converges strongly to the unique fixed point of T.
Proof Let p be the unique fixed point of T. From (1.1), (1.3) and Lemma 1, we have
Similarly, we obtain
Then, from (4.4) and (4.5), we get that
If \delta \in (0,1), we obtain
Thus we have {x}_{n}\to p\in F(T). If \delta =0, the result is clear. This completes the proof. □
5 Conclusions
The new multistep iteration reduces to the twostep iteration and the SPiteration processes. Also, the class of generalized nonexpansive mappings includes nonexpansive mappings. Then these results presented in this paper extend and generalize some works for a CAT(0) space in the literature.
References
Agarwal RP, O’Regan D, Sahu DR: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal. 2007, 8(1):61–79.
Ishikawa S: Fixed points by a new iteration method. Proc. Am. Math. Soc. 1974, 44: 147–150. 10.1090/S00029939197403364695
Mann WR: Mean value methods in iterations. Proc. Am. Math. Soc. 1953, 4: 506–510. 10.1090/S00029939195300548463
Phuengrattana W, Suantai S: On the rate of convergence of Mann, Ishikawa, Noor and SP iterations for continuous functions on an arbitrary interval. J. Comput. Appl. Math. 2011, 235(9):3006–3014. 10.1016/j.cam.2010.12.022
Rhoades BE: A comparison of various definitions of contractive mappings. Trans. Am. Math. Soc. 1977, 226: 257–290.
Thianwan S: Common fixed points of new iterations for two asymptotically nonexpansive nonselfmappings in a Banach space. J. Comput. Appl. Math. 2009, 224: 688–695. 10.1016/j.cam.2008.05.051
Imoru CO, Olantiwo MO: On the stability of Picard and Mann iteration processes. Carpath. J. Math. 2003, 19(2):155–160.
Berinde V: On the stability of some fixed point procedures. Bul. ştiinţ.  Univ. Baia Mare, Ser. B Fasc. Mat.Inform. 2002, 18(1):7–14.
Berinde V: On the convergence of the Ishikawa iteration in the class of quasi contractive operators. Acta Math. Univ. Comen. 2004, 73(1):119–126.
Harder AM, Hicks TL: Stability results for fixed point iteration procedures. Math. Jpn. 1988, 33(5):693–706.
Zamfirescu T: Fix point theorems in metric spaces. Arch. Math. 1972, 23(1):292–298. 10.1007/BF01304884
Osilike MO, Udomene A: Short proofs of stability results for fixed point iteration procedures for a class of contractivetype mappings. Indian J. Pure Appl. Math. 1999, 30(12):1229–1234.
Gürsoy F, Karakaya V, Rhoades BE: Data dependence results of new multistep and S iterative schemes for contractivelike operators. Fixed Point Theory Appl. 2013., 2013: Article ID 76 10.1186/16871812201376
Kirk WA: Geodesic geometry and fixed point theory. Colecc. Abierta 64. In Seminar of Mathematical Analysis. Universidad de Sevilla Secr. Publ., Sevilla; 2003:195–225.
Kirk WA: Geodesic geometry and fixed point theory II. In International Conference on Fixed Point Theory and Applications. Yokohama Publ., Yokohama; 2004:113–142.
Dhompongsa S, Kirk WA, Sims B: Fixed points of uniformly Lipschitzian mappings. Nonlinear Anal., Theory Methods Appl. 2006, 65(4):762–772. 10.1016/j.na.2005.09.044
Dhompongsa S, Kirk WA, Panyanak B: Nonexpansive setvalued mappings in metric and Banach spaces. J. Nonlinear Convex Anal. 2007, 8(1):35–45.
Dhompongsa S, Panyanak B:On △convergence theorems in CAT(0) spaces. Comput. Math. Appl. 2008, 56(10):2572–2579. 10.1016/j.camwa.2008.05.036
Şahin A, Başarır 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
Şahin A, Başarır M: On the strong and △convergence of SP iteration on aCAT(0)space. J. Inequal. Appl. 2013., 2013: Article ID 311 10.1186/1029242X2013311
Bridson M, Haefliger A Grundlehren der Mathematischen Wissenschaften 319. In Metric Spaces of NonPositive Curvature. Springer, Berlin; 1999.
Bruhat F, Tits J: Groupes réductifs sur un corps local. Publ. Math. Inst. Hautes Études Sci. 1972, 41: 5–251. 10.1007/BF02715544
Lim TC: Remarks on some fixed point theorems. Proc. Am. Math. Soc. 1976, 60: 179–182. 10.1090/S0002993919760423139X
Kirk WA, Panyanak B: A concept of convergence in geodesic spaces. Nonlinear Anal., Theory Methods Appl. 2008, 68(12):3689–3696. 10.1016/j.na.2007.04.011
Nanjaras B, Panyanak B:Demiclosed principle for asymptotically nonexpansive mappings in CAT(0) spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 268780 10.1155/2010/268780
Acknowledgements
The authors are grateful to the referee for his/her careful reading and valuable comments and suggestions which led to the present form of the paper. This paper has been presented in International Conference ‘Anatolian Communications in Nonlinear Analysis (ANCNA 2013)’ in Abant Izzet Baysal University, Bolu, Turkey, July 0306, 2013. This paper was supported by Sakarya University Scientific Research Foundation (Project number: 20130200003).
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Başarır, M., Şahin, A. On the strong and Δconvergence of new multistep and Siteration processes in a CAT(0) space. J Inequal Appl 2013, 482 (2013). https://doi.org/10.1186/1029242X2013482
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DOI: https://doi.org/10.1186/1029242X2013482
Keywords
 CAT(0) space
 contractivelike mapping
 strong convergence
 Δconvergence
 new multistep iteration
 Siteration
 fixed point