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Iterative approximation of attractive points of further generalized hybrid mappings in Hadamard spaces
- Asawathep Cuntavepanit^{1} and
- Withun Phuengrattana^{2}Email author
https://doi.org/10.1186/s13663-019-0653-8
© The Author(s) 2019
- Received: 7 June 2018
- Accepted: 4 January 2019
- Published: 28 January 2019
Abstract
In this paper, we study the class of further generalized hybrid mappings due to Khan (Fixed Point Theory Appl. 2018:8, 2018) in the setting of Hadamard spaces. We prove a demiclosed principle for such mappings in Hadamard spaces. Furthermore, we also prove the Δ-convergence of the sequence generated by the S-iteration process for finding attractive points of further generalized hybrid mappings in Hadamard spaces satisfying the \((\mathbb{S})\) property and the \((\overline{Q_{4}})\) condition. Moreover, we provide a numerical example to illustrate the convergence behavior of the studied iteration and numerically compare the convergence of the studied iteration scheme with the existing schemes. Our results extend some known results which appeared in the literature.
Keywords
- Attractive point
- Further generalized hybrid mappings
- S-iteration
- Δ-convergence
- Hadamard spaces
MSC
- 47H09
- 47H10
1 Introduction
In 2012, Takahashi et al. [3] introduced the class of normally generalized hybrid mappings in a Hilbert space.
Definition 1.1
- (i)
\(\alpha +\beta +\gamma +\delta \geq 0\);
- (ii)
\(\alpha +\beta >0\) or \(\alpha +\gamma >0\); and
- (iii)
\(\alpha \|Tx-Ty\|^{2} +\beta \|x-Ty\|^{2} + \gamma \|Tx-y\|^{2}+ \delta \|x-y\|^{2}\leq 0\), \(\forall x,y\in C\).
They also proved the weak convergence theorem of Mann type for normally generalized hybrid mappings in real Hilbert spaces without convexity assumption on the domain of mappings. To be more precise, they obtained the following result.
Theorem 1.2
In 2015, Kaewkhao et al. [4] extended the results of Takahashi et al. [3] from Hilbert spaces to Hadamard spaces.
In 2018, Khan [1] gave the concept of further generalized mappings (see Definition 1.3 below) which constitutes a generalization of normally generalized hybrid mappings due to Takahashi et al. [3] (see Definition 1.1 above).
Definition 1.3
([1])
- (i)
\(\alpha +\beta +\gamma +\delta \geq 0\), \(\epsilon \geq 0\);
- (ii)
\(\alpha +\beta >0\) or \(\alpha +\gamma >0\); and
- (iii)
\(\alpha \|Tx-Ty\|^{2} +\beta \|x-Ty\|^{2} + \gamma \|Tx-y\|^{2}+ \delta \|x-y\|^{2}+ \epsilon \|x-Tx\|^{2}\leq 0\), \(\forall x,y\in C\).
Obviously, by above definitions, further generalized hybrid is a generalization of normally generalized hybrid when \(\epsilon =0\). It is noteworthy that it contains the class of generalized hybrid, quasi-nonexpansive mappings, quasi-contractive mappings and contractive mappings.
Recently, Khan [1] obtained a weak convergence theorem of Picard–Mann hybrid iterative process [5] for further generalized hybrid mappings in real Hilbert spaces without convexity assumption on the domain of mappings. The iterative process of Khan [1] is faster than Mann, Ishikawa and S-iteration process of Agarwal et al. [6] as shown by him in [5]. However, his results are in a Hilbert space and we want to have some results in Hadamard spaces. Note that no results are available at the moment for further generalized hybrid mappings even for Mann iterative process in Hadamard spaces. We further note that S-iteration process is also faster than Mann and Ishikawa iteration processes (but not Picard–Mann hybrid).
Motivated by the above works, we define a class of further generalized hybrid mappings and prove the demiclosed principle for such mapping in Hadamard spaces. Furthermore, we also obtain a Δ-convergence theorem of S-iteration process for further generalized hybrid mappings in Hadamard spaces satisfying the \((\mathbb{S})\) property and the \((\overline{Q_{4}})\) condition. Finally, we provide a numerical example to illustrate the convergence behavior of the S-iteration and numerically compare the convergence of the S-iteration schemes with the existing schemes.
2 Methods
The paper is organized as follows. Section 3 contains the preliminaries, including definitions and lemmas with corresponding references that will be used in the sequel. Section 4 contains the main result of the paper. In Sect. 5, we provide a numerical example to illustrate the convergence behavior of the S-iteration and numerically compare the convergence of the S-iteration schemes with the existing schemes.
3 Preliminaries
Let \((X,d)\) be a metric space. A geodesic from x to y is a map γ from the closed interval \([0,d(x,y)]\subset \mathbb{R}\) to X such that \(\gamma (0)=x\), \(\gamma (d(x,y))=y\) and \(d(\gamma (t_{1}), \gamma (t_{2}))=|t_{1}-t_{2}|\) for all \(t_{1},t_{2}\in [0,d(x,y)]\). The image of γ is called a geodesic (or metric) segment joining x and y. When it is unique, this geodesic segment is denoted by \([x,y]\). The space X is said to be a geodesic metric space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic metric space if there is exactly one geodesic joining x and y for each \(x,y\in X\). A subset C of X is said to be convex, if for any two points \(x,y\in C\), the geodesic joining x and y is contained in C.
Let X be a uniquely geodesic metric space. For each \(x,y\in X\) and for each \(\alpha \in [0,1]\), there exists a unique point \(z\in [x,y]\) such that \(d(x, z) = (1-\alpha ) d(x, y)\) and \(d(y, z) = \alpha d(x, y)\). We denote the unique point z by \(\alpha x \oplus (1-\alpha )y\).
Lemma 3.1
([7])
- (i)
X is a CAT(0) space.
- (ii)X satisfies the (CN) inequality: If \(x,y\in X\) and \(\frac{x\oplus y}{2}\) is the midpoint of x and y, then$$ d \biggl(z,\frac{x\oplus y}{2} \biggr)^{2}\leq \frac{1}{2}d(z,x)^{2}+ \frac{1}{2}d(z,y)^{2} - \frac{1}{4}d(x,y)^{2}, \quad \textit{for all }z\in X. $$
Lemma 3.2
- (i)
\(d(z,\lambda x \oplus (1-\lambda )y) \leq \lambda d(z,x)+ (1- \lambda )d(z,y)\);
- (ii)
\(d(z,\lambda x \oplus (1-\lambda )y)^{2} \leq \lambda d(z,x)^{2}+ (1-\lambda )d(z,y)^{2} -\lambda (1-\lambda ) d(x,y)^{2}\).
A complete CAT(0) space is called an Hadamard space.
It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is an Hadamard space. Other examples include Euclidean spaces \(\mathbb{E}^{2}\), Hilbert spaces, the Hilbert ball [9], hyperbolic spaces [10], R-trees [11], and many others. The fixed point theory in Hadamard spaces was first studied by Kirk [12] in 2003. Since then many authors have published papers on the existence and convergence of fixed points for nonlinear mappings in such spaces (e.g., see [13, 14]).
We now give the definition and collect some basic properties of the Δ-convergence which will be used in the sequel.
Definition 3.3
([16])
A sequence \(\{x_{n}\}\) in an Hadamard space X is said to Δ-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 Δ-\(\lim_{n\to \infty }x_{n}=x\) and call x the Δ-limit of \(\{x_{n}\}\).
Lemma 3.5
([17])
Let C be a nonempty closed convex subset of an Hadamard space X. If \(\{x_{n}\}\) is a bounded sequence in C, then the asymptotic center of \(\{x_{n}\}\) is in C.
Lemma 3.6
([8])
Let \(\{x_{n}\}\) be a sequence in an Hadamard space X with \(A(\{x_{n}\}) = \{x\}\). If \(\{u_{n}\}\) is a subsequence of \(\{x_{n}\}\) with \(A(\{u_{n}\}) = \{u\}\) and \(\{d(x_{n},u)\}\) converges, then \(x = u\).
In 2008, Berg and Nikolaev [18] introduced the concept of quasilinearization in an Hadamard space X as follows:
In 2013, Kakavandi [20] introduced the concept of \((\mathbb{S})\) property for an Hadamard space as follows.
Definition 3.7
An Hadamard space X satisfies the \((\mathbb{S})\) property if for any \((x,y)\in X\times X\) there exists a point \(y_{x}\in X\) such that \([\overrightarrow{xy}]=[\overrightarrow{y_{x}x}]\).
Moreover, Kakavandi also proved the characterization of Δ-convergence for Hadamard spaces satisfying the \((\mathbb{S})\) property as follows.
Lemma 3.8
Let X be an Hadamard space, \(\{x_{n}\}\) be a bounded sequence in X and \(x\in X\). If X satisfies the \((\mathbb{S})\) property, then Δ-\(\lim_{n\rightarrow \infty }x_{n}=x\) if and only if \(\lim_{n\rightarrow \infty }\langle \overrightarrow{xx_{n}}, \overrightarrow{xy}\rangle =0\) for all \(y\in X\).
In 2008, Kirk and Panyanak [16] introduced a geometric condition on Hadamard spaces called the \((Q_{4})\) condition as follows.
Definition 3.9
In 2013, Kakavandi [20] modified the \((Q_{4})\) condition as follows.
Definition 3.10
We can see that Hilbert spaces and every Hadamard space of constant curvature satisfy the \((\overline{Q_{4}})\) condition. Anyway, since \((\overline{Q_{4}})\) implies \((Q_{4})\), there are some Hadamard spaces that do not satisfy such a condition. The following results were obtained by Kaewkhao et al. [4].
Lemma 3.11
Let X be an Hadamard space satisfying the \((\overline{Q_{4}})\) condition. Let C be a nonempty subset of X. Then, for any mapping \(T:C\rightarrow X\), \(A(T)\) is closed and convex.
Lemma 3.12
Let X be an Hadamard space and C be a closed convex subset of X. Let \(x\in X\) and \(y\in C\). Then \(y=P_{C}x\) if and only if \(\langle \overrightarrow{xy},\overrightarrow{yz}\rangle \geq 0\) for all \(z\in C\).
We also need the following lemmas due to Kaewkhao et al. [4].
Lemma 3.13
- (i)
the sequences \(\{d(x_{n},y)\}\) and \(\{d(Tx_{n},y)\}\) are bounded for all \(y\in C\);
- (ii)
\(\mu _{n} d(x_{n},y)=\mu _{n}d(Tx_{n},y)\) for any Banach limit \(\mu _{n}\) on \(l^{\infty }\).
Lemma 3.14
Let X be an Hadamard space and C be a closed convex subset of X. Let \(\{x_{n}\}\) be a bounded sequence in X. If \(d(x_{n+1},z) \leq d(x_{n},z)\) for all \(z\in C\), then \(\lim_{n\to \infty }P_{C}x _{n}= z_{0}\in C\), where \(P_{C}\) is the metric projection from X onto C.
4 Results and discussion
Moreover, in Hadamard spaces, a further generalized hybrid mapping is defined analogously to Definition 1.3 as follows.
Definition 4.1
- (i)
\(\alpha +\beta +\gamma +\delta \geq 0\), \(\epsilon \geq 0\);
- (ii)
\(\alpha +\beta >0\) or \(\alpha +\gamma >0\); and
- (iii)
\(\alpha d(Tx,Ty)^{2}+\beta d(x,Ty)^{2}+\gamma d(Tx,y)^{2}+\delta d(x,y)^{2}+\epsilon d(x,Tx)^{2}\leq 0\), \(\forall x,y\in C\).
The following lemma is a demiclosedness principle for a further generalized hybrid mapping in an Hadamard space.
Lemma 4.2
Let X be an Hadamard space X satisfying the \((\mathbb{S})\) property. Let C be a nonempty subset of X and let \(T:C\rightarrow C\) be a further generalized hybrid mapping. Let \(\{x_{n}\}\) be a bounded sequence in C such that \(\lim_{n\rightarrow \infty }d(x_{n},Tx_{n})=0\) and Δ-\(\lim_{n\rightarrow \infty }x_{n}=z\). Then \(z\in A(T)\).
Proof
In what follows we get a Δ-convergence theorem for a further generalized hybrid mapping in an Hadamard space.
Theorem 4.3
Proof
Step 1. We will show that \(\lim_{n\rightarrow \infty }d(u,x_{n})\) exists for all \(u\in A(T)\).
Step 2. We will show that \(\lim_{n\rightarrow \infty }d(x_{n},Tx_{n})= 0\).
Remark 4.4
Theorem 4.3 extends and improves the results of Kaewkhao et al. [4] from a normally generalized hybrid mapping to a further generalized hybrid mapping. In fact, we present the S-iteration process for solving the attractive point problem of further generalized hybrid mappings in Hadamard spaces.
It is known that a Hilbert space satisfies both the \((\mathbb{S})\) property and the \((\overline{Q_{4}})\) condition. Furthermore, Δ-convergence and weak convergence are the same in a Hilbert space. Thus, we have the following theorem.
Theorem 4.5
Moreover, the following example shows that there is an Hadamard space satisfying both the \((\mathbb{S})\) property and the \(( \overline{Q_{4}})\) condition, which is not a Hilbert space.
Example 4.6
([4])
Remark 4.7
Theorem 4.5 extends and improves the results of Takahashi et al. [3] from a normally generalized hybrid mapping to a further generalized hybrid mapping. In fact, we present the S-iteration process for solving the attractive point problem of further generalized hybrid mappings in Hilbert spaces.
5 Numerical example for the main result
In this section, we give a numerical example supporting our main results and compare the convergence of the studied method (4) with the Mann and Ishikawa iterations.
Example 5.1
Iterates of S-iteration, Mann iteration, and Ishikawa iteration for \(x_{1} =u_{1}=z_{1}= -0.5\)
n | S-iteration | Mann iteration | Ishikawa iteration | |||
---|---|---|---|---|---|---|
\(x_{n}\) | \(|x_{n}-x_{n-1}|\) | \(u_{n}\) | \(|u_{n}-u_{n-1}|\) | \(z_{n}\) | \(|z_{n}-z_{n-1}|\) | |
1 | −0.5000000 | – | −0.5000000 | – | −0.5000000 | – |
2 | 0.6742424 | 1.1742e + 00 | −0.0833333 | 4.1667e − 01 | −0.1590909 | 3.4091e − 01 |
3 | 0.8462461 | 1.7200e − 01 | 0.2107843 | 2.9412e − 01 | 0.1550166 | 3.1411e − 01 |
4 | 0.9271811 | 8.0935e − 02 | 0.4260250 | 2.1524e − 01 | 0.4077682 | 2.5275e − 01 |
5 | 0.9654324 | 3.8251e − 02 | 0.5960916 | 1.7007e − 01 | 0.5982240 | 1.9046e − 01 |
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |
16 | 0.9999899 | 1.1023e − 05 | 0.9962619 | 2.1566e − 03 | 0.9974837 | 1.5930e − 03 |
17 | 0.9999952 | 5.2699e − 06 | 0.9976368 | 1.3749e − 03 | 0.9984638 | 9.8015e − 04 |
18 | 0.9999977 | 2.5201e − 06 | 0.9985102 | 8.7334e − 04 | 0.9990648 | 6.0091e − 04 |
19 | 0.9999989 | 1.2054e − 06 | 0.9990631 | 5.5292e − 04 | 0.9994320 | 3.6724e − 04 |
20 | 0.9999995 | 5.7667e − 07 | 0.9994121 | 3.4904e − 04 | 0.9996558 | 2.2380e − 04 |
Iterates of S-iteration, Mann iteration, and Ishikawa iteration for \(x_{1} =u_{1}=z_{1}= 0.4\)
n | S-iteration | Mann iteration | Ishikawa iteration | |||
---|---|---|---|---|---|---|
\(x_{n}\) | \(|x_{n}-x_{n-1}|\) | \(u_{n}\) | \(|u_{n}-u_{n-1}|\) | \(z_{n}\) | \(|z_{n}-z_{n-1}|\) | |
1 | 0.4000000 | – | 0.4000000 | – | 0.4000000 | – |
2 | 0.7181818 | 3.1818e − 01 | 0.5000000 | 1.0000e − 01 | 0.5181818 | 1.1818e − 01 |
3 | 0.8669850 | 1.4880e − 01 | 0.6176471 | 1.1765e − 01 | 0.6450471 | 1.2687e − 01 |
4 | 0.9370031 | 7.0018e − 02 | 0.7219251 | 1.0428e − 01 | 0.7512207 | 1.0617e − 01 |
5 | 0.9700950 | 3.3092e − 02 | 0.8043177 | 8.2393e − 02 | 0.8312256 | 8.0005e − 02 |
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |
16 | 0.9999913 | 9.5358e − 06 | 0.9981890 | 1.0448e − 03 | 0.9989430 | 6.6918e − 04 |
17 | 0.9999958 | 4.5591e − 06 | 0.9988551 | 6.6612e − 04 | 0.9993547 | 4.1173e − 04 |
18 | 0.9999980 | 2.1802e − 06 | 0.9992782 | 4.2311e − 04 | 0.9996071 | 2.5242e − 04 |
19 | 0.9999990 | 1.0428e − 06 | 0.9995461 | 2.6788e − 04 | 0.9997614 | 1.5427e − 04 |
20 | 0.9999995 | 4.9889e − 07 | 0.9997152 | 1.6910e − 04 | 0.9998554 | 9.4011e − 05 |
From Tables 1 and 2, we see that both \(\{x_{n}\}\), \(\{u_{n}\}\) and \(\{z_{n}\}\) converge to \(1\in A(T)\) and observe that \(|x_{n}-1|\leq |u _{n}-1|\) and \(|x_{n}-1|\leq |z_{n}-1|\), so the sequence \(\{x_{n}\}\) generated by S-iteration converges faster than both \(\{u_{n}\}\) generated by Mann iteration and \(\{z_{n}\}\) generated by Ishikawa iteration.
6 Conclusions
The results presented in this paper modify, extend and improve the corresponding results of Takahashi et al. [3] and Kaewkhao et al. [4], and others. The main aim of this paper is to prove the demiclosed principle for further generalized hybrid mapping and the Δ-convergence of the sequence generated by the S-iteration process for finding attractive points of such mappings in Hadamard spaces satisfying the \((\mathbb{S})\) property and the \((\overline{Q_{4}})\) condition. We also provide a numerical example to illustrate and support our results at the end.
Declarations
Acknowledgements
The authors appreciate the support of their institutes.
Availability of data and materials
Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.
Funding
The first author was supported by Mahidol University Kanchanaburi Campus, Kanchanaburi, Thailand.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Khan, S.H.: Iterative approximation of common attractive points of further generalized hybrid mappings. Fixed Point Theory Appl. 2018, 8 (2018) MathSciNetView ArticleGoogle Scholar
- Takahashi, W., Takeuchi, Y.: Nonlinear ergodic theorem without convexity for generalized hybrid mappings in a Hilbert space. J. Nonlinear Convex Anal. 12, 399–406 (2011) MathSciNetMATHGoogle Scholar
- Takahashi, W., Wong, N.-C., Yao, J.-C.: Attractive point and weak convergence theorems for new generalized hybrid mappings in Hilbert spaces. J. Nonlinear Convex Anal. 13, 745–757 (2012) MathSciNetMATHGoogle Scholar
- Kaewkhao, A., Inthakon, W., Kunwai, K.: Attractive points and convergence theorems for normally generalized hybrid mappings in CAT(0) spaces. Fixed Point Theory Appl. 2015, 96 (2015) MathSciNetView ArticleGoogle Scholar
- Khan, S.H.: A Picard–Mann hybrid iterative process. Fixed Point Theory Appl. 2013, 69 (2013) MathSciNetView ArticleGoogle Scholar
- Agarwal, R.P., O’Regan, D., Sahu, D.R.: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal. 8(1), 61–79 (2007) MathSciNetMATHGoogle Scholar
- Bridson, M., Haefliger, A.: Metric Spaces of Non-Positive Curvature. Springer, Berlin (1999) View ArticleGoogle Scholar
- Dhompongsa, S., Panyanak, B.: On Δ-convergence theorems in CAT(0) spaces. Comput. Math. Appl. 56, 2572–2579 (2008) MathSciNetView ArticleGoogle Scholar
- Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Dekker, New York (1984) MATHGoogle Scholar
- Kohlenbach, U.: Some logical metatheorems with applications in functional analysis. Trans. Am. Math. Soc. 357, 89–128 (2015) MathSciNetView ArticleGoogle Scholar
- Tits, J.: A Theorem of Lie-Kolchin for Trees, Contributions to Algebra: A Collection of Papers Dedicated to Ellis Kolchin. Academic Press, New York (1977) MATHGoogle Scholar
- Kirk, W.A.: Geodesic geometry and fixed point theory. In: Seminar of Mathematical Analysis, Malaga/Seville, 2002/2003. Colecc. Abierta, vol. 64, pp. 195–225. Univ. Sevilla Secr. Publ., Seville (2003) Google Scholar
- Reich, S., Salinas, Z.: Weak convergence of infinite products of operators in Hadamard spaces. Rend. Circ. Mat. Palermo 65, 55–71 (2016) MathSciNetView ArticleGoogle Scholar
- Reich, S., Shafrir, I.: Nonexpansive iterations in hyperbolic spaces. Nonlinear Anal. 15, 537–558 (1990) MathSciNetView ArticleGoogle Scholar
- Dhompongsa, S., Kirk, W.A., Sims, B.: Fixed points of uniformly Lipschitzian mappings. Nonlinear Anal. 65, 762–772 (2006) MathSciNetView ArticleGoogle Scholar
- Kirk, W.A., Panyanak, B.: A concept of convergence in geodesic spaces. Nonlinear Anal. 68, 3689–3696 (2008) MathSciNetView ArticleGoogle Scholar
- Dhompongsa, S., Kirk, W.A., Panyanak, B.: Nonexpansive set-valued mappings in metric and Banach spaces. J. Nonlinear Convex Anal. 8, 35–45 (2007) MathSciNetMATHGoogle Scholar
- Berg, I.D., Nikolaev, I.G.: Quasilinearization and curvature of Alexandrov spaces. Geom. Dedic. 133, 195–218 (2008) View ArticleGoogle Scholar
- Kakavandi, B.A., Amini, M.: Duality and subdifferential for convex functions on complete CAT(0) metric spaces. Nonlinear Anal. 73, 3450–3455 (2010) MathSciNetView ArticleGoogle Scholar
- Kakavandi, B.A.: Weak topologies in complete CAT(0) metric spaces. Proc. Am. Math. Soc. 141, 1029–1039 (2013) MathSciNetView ArticleGoogle Scholar
- Takahashi, W.: Nonlinear Function Analysis. Yokahama Publishers, Yokahama (2000) Google Scholar
- Mann, W.R.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–510 (1953) MathSciNetView ArticleGoogle Scholar
- Ishikawa, S.: Fixed points by a new iteration method. Proc. Am. Math. Soc. 44, 147–150 (1974) MathSciNetView ArticleGoogle Scholar