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- Open Access
Mann iteration for monotone nonexpansive mappings in ordered CAT(0) space with an application to integral equations
- Izhar Uddin^{1},
- Chanchal Garodia^{1}Email author and
- Juan Jose Nieto^{2}
https://doi.org/10.1186/s13660-018-1925-2
© The Author(s) 2018
- Received: 4 September 2018
- Accepted: 27 November 2018
- Published: 11 December 2018
Abstract
In this paper, we establish some convergence results for a monotone nonexpansive mapping in a \(\operatorname{CAT}(0)\) space. We prove the Δ- and strong convergence of the Mann iteration scheme. Further, we provide a numerical example to illustrate the convergence of our iteration scheme, and also, as an application, we discuss the solution of integral equation. Our results extend some of the relevant results.
Keywords
- \(\operatorname{CAT}(0)\) space
- Fixed point
- Δ-convergence
- Monotone nonexpansive mapping
MSC
- 47H09
- 47H10
1 Introduction
The Banach contraction principle [1] is one of the most fundamental results in fixed point theory and has been utilized widely for proving the existence of solutions of different nonlinear functional equations. In the last few years, many efforts have been made to obtain fixed points in partially ordered sets. In 2004, Ran and Reurings [2] generalized the Banach contraction principle to ordered metric spaces. Later on, in 2005, Nieto and Rodriguez [3] used the same approach to further extend some more results of fixed point theory in partially ordered metric spaces and utilized them to study the existence of solutions of differential equations.
Note that the Banach contraction principle is no longer true for nonexpansive mappings, that is, a nonexpansive mapping need not admit a fixed point on a complete metric space. Also, Picard iteration need not converge for a nonexpansive map in a complete metric space. This led to the beginning of a new era of fixed point theory for nonexpansive mappings by using geometric properties. In 1965, Browder [4], Göhde [5], and Kirk [6] gave three basic existence results for nonexpansive mappings. With a view to locating fixed points of nonexpansive mappings, Mann [7] and Ishikawa [8] introduced two basic iteration schemes.
Now, fixed point theory of monotone nonexpansive mappings is gaining much attention among the researchers. Recently, Bachar and Khamsi [9], Abdullatif et al. [10], and Song et al. [11] proved some existence and convergence results for monotone nonexpansive mappings. Dehaish and Khamsi [12] proved the weak convergence of the Mann iteration for a monotone nonexpansive mapping. In 2016, Song et al. [11] considered the weak convergence of the Mann iteration scheme for a monotone nonexpansive mapping T under some mild different conditions in a Banach space.
The aim of this paper is to study the convergence behavior of the well-known Mann iteration [7] in a \(\operatorname{CAT}(0)\) space for a monotone nonexpansive mapping. Further, we provide a numerical example and application related to solution of an integral equation. Our results generalize and improve several existing results in the literature.
2 Preliminaries
To make our paper self-contained, we recall some basic definitions and relevant results.
A metric space X is a \(\operatorname{CAT}(0)\) 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. For further information about these spaces and the fundamental role they play in various branches of mathematics, we refer to Bridson and Haefliger [13] and Burago et al. [14]. Every convex subset of Euclidean space \(\mathbb{R}^{n}\) endowed with the induced metric is a \(\operatorname{CAT}(0)\) space. Further, the class of Hilbert spaces are examples of \(\operatorname{CAT}(0)\) spaces.
The fixed point theory in \(\operatorname{CAT}(0)\) spaces is gaining attention of researchers, and many results have been obtained for single- and multivalued mappings in a \(\operatorname{CAT}(0)\) space. For different aspects of fixed point theory in \(\operatorname{CAT}(0)\) spaces, we refer to [15–24]. The following few results are necessary for our subsequent discussion.
Lemma 2.1
([21])
We use the notation \((1-z)e\oplus z f\) for the unique point h of the lemma.
Lemma 2.2
([21])
Lemma 2.3
([21])
It is known that in a \(\operatorname{CAT}(0)\) space, \(A(\{u_{n}\})\) consists of exactly one point [25, Proposition 5].
In 1976, Lim [26] introduced the concept of Δ-convergence in a metric space. Later on, Kirk and Panyanak [22] proved that \(\operatorname{CAT}(0)\) spaces presented a natural framework for Lim’s concept and provided precise analogs of several results in Banach spaces involving weak convergence in \(\operatorname{CAT}(0)\) space setting.
Definition 2.4
A sequence \(\{u_{n}\}\) in X is said to be Δ-convergent to \(u\in X\) if u is the unique asymptotic center of \(\{v_{n}\}\) for every subsequence \(\{v_{n}\}\) of \(\{u_{n}\}\). In this case, we write \(\Delta \text{-}\lim_{n} {u_{n}}=u\) and say that u is the Δ-limit of \(\{u_{n}\}\).
Definition 2.5
A Banach space X is said to satisfy Opial’s condition if for any sequence \(\{u_{n}\}\) in X with \(u_{n} \rightharpoonup u\) (⇀ denotes weak convergence), we have \(\limsup_{n\to \infty } \|u_{n} - u\| < \limsup_{n\to \infty } \|u_{n} - v\|\) for all \(v\in X\) with \(v \neq u\).
Examples of Banach spaces satisfying this condition are Hilbert spaces and all \(l^{p}\) spaces (\(1 < p < \infty\)). On the other hand, \(L^{p} [0, 2\pi ]\) with \(1 < p \neq 2\) fail to satisfy Opial’s condition.
Lemma 2.6
([22])
Every bounded sequence in a complete \(\operatorname{CAT}(0)\) space admits a Δ-convergent subsequence.
Lemma 2.7
([21])
If G is a closed convex subset of a complete \(\operatorname{CAT}(0)\) space X and if \(\{u_{n}\}\) is a bounded sequence in G, then the asymptotic center of \(\{u_{n}\}\) is in G.
Next, we introduce the concept of partial order in the setting of \(\operatorname{CAT}(0)\) spaces.
Definition 2.8
- (i)
monotone if \(Pu \preceq Pv\) for all \(u, v\in G\) with \(u \preceq v\),
- (ii)monotone nonexpansive if P is monotone andfor all \(u, v\in G\) with \(u\preceq v\).$$\begin{aligned} d(Pu, Pv) \leq d(u, v) \end{aligned}$$
3 Some Δ-convergence and strong convergence theorems
We begin with the following important lemma.
Lemma 3.1
- (i)
\(u_{n} \preceq u_{n+1} \preceq Pu_{n}\) for any \(n\geq 1\),
- (ii)
\(u_{n} \preceq u\), provided that \(\{u_{n}\}\) Δ-converges to a point \(u\in G \).
Proof
(ii) Let u be the Δ-limit of \(\{u_{n}\}\). From part (i) we have \(u_{n} \preceq u_{n+1} \) for all \(n \geq 1\) since \(\{u_{n}\}\) is increasing and the order interval \([u_{m}, \rightarrow)\) is closed and convex. Therefore \(u \in [u_{m}, \rightarrow) \) for a fixed \(m \in \mathbb{N} \); otherwise, if \(u \notin [u_{m}, \rightarrow) \), then we could construct a subsequence \(\{u_{r}\}\) of \(\{u_{n}\}\) by leaving the first \(m-1\) terms of the sequence \(\{u_{n}\}\), and then the asymptotic center of \(\{u_{r}\}\) would not be u, which contradicts the assumption that u is the Δ-limit of the sequence \(\{u_{n}\}\). This completes the proof of part (ii). □
Lemma 3.2
- (i)
\(\lim_{n\to \infty } d(u_{n}, r)\) exists, and
- (ii)
\(\lim_{n\to \infty } d(Pu_{n},u_{n})=0\).
Proof
The following lemma is an analogue of Theorem 3.7 of [22].
Lemma 3.3
Let G be a nonempty closed convex subset of a complete \(\operatorname{CAT}(0)\) space \((X,\preceq)\), and let \(P : G \to G\) be a monotone nonexpansive mapping. Fix \(u_{1}\in G\) such that \(u_{1} \preceq Pu_{1}\). If \(\{u_{n}\}\) is a sequence described as in (2.1), then the conditions \(\Delta \text{-}\lim_{n} {u_{n}}=u\) and \(\lim_{n\to \infty } d(Pu_{n}, u_{n})=0\) imply that u is a fixed point of P.
Proof
Theorem 3.4
Let G be a nonempty closed convex subset of a complete \(\operatorname{CAT}(0)\) space \((X,\preceq)\), and let \(P : G \to G\) be a monotone nonexpansive mapping with \(F(P)\neq \emptyset \). Fix \(u_{1}\in G\) such that \(u_{1} \preceq Pu_{1}\). If \(\{u_{n}\}\) is a sequence described as in (2.1), then \(\{u_{n}\}\) Δ-converges to a fixed point of P.
Proof
From Lemma 3.2 we have that \(\lim_{n\to \infty } d(u_{n}, r)\) exists for each \(r\in F(P)\), so the sequence \(\{u_{n}\}\) is bounded, and \(\lim_{n\to \infty }d(u_{n}, Pu_{n})=0 \).
Let \(W_{\omega }(\{u_{n}\})=: \bigcup X(\{v_{n}\})\), where the union is taken over all subsequences \(\{v_{n}\}\) over \(\{u_{n}\}\). To show the Δ-convergence of \(\{u_{n}\}\) to a fixed point of P, we will first prove that \(W_{\omega }(\{u_{n}\}) \subset F(P)\) and thereafter argue that \(W_{\omega }(\{u_{n}\})\) is a singleton set. To show that \(W_{\omega }(\{u_{n}\}) \subset F(P)\), let \(y\in W_{\omega }(\{u_{n} \})\). Then there exists a subsequence \(\{y_{n}\}\) of \(\{u_{n}\}\) such that \(X(\{y_{n}\})=y\). By Lemmas 2.6 and 2.7 there exists a subsequence \(\{z_{n}\}\) of \(\{y_{n}\}\) such that \(\Delta \text{-}\lim_{n} z_{n}=z\) and \(z\in G\). Since \(\lim_{n\to \infty } d(Pu _{n}, u_{n})=0\) and \(\{z_{n}\}\) is a subsequence of \(\{u_{n}\}\), we have that \(\lim_{n\to \infty } d(z_{n}, Pz_{n})=0\). In view of Lemma 3.3, we have \(z=Pz\), and hence \(z\in F(P)\).
Theorem 3.5
Let X be a complete \(\operatorname{CAT}(0)\) space endowed with partial ordering ′⪯′, and let G be a nonempty closed convex subset of X. Let \(P : G \to G\) be a monotone nonexpansive mapping such that \(F(P) \neq \emptyset \). Fix \(u_{1}\in G\) such that and \(u_{1} \preceq Pu _{1}\). If \(\{u_{n}\}\) is a sequence described as in (2.1) such that \(\sum_{n=1}^{\infty }\kappa_{n} (1-\kappa_{n})=\infty \), then \(\{u_{n}\}\) converges to a fixed point of P if and only if \(\liminf_{n\to \infty } d(u_{n}, F(P))=0\).
Proof
If the sequence \(\{u_{n}\}\) converges to a point \(u\in F(P)\), then it is obvious that \(\liminf_{n\to \infty } d(u_{n},F(P))=0\).
4 Numerical example
In this section, we present a numerical example to illustrate the convergence behavior of our iteration scheme (2.1).
Now, we show the convergence of (2.1) using two different sets of values.
(\(\kappa_{n} = \frac{2n}{5n+2}\) for all \(n \in \mathbb{N}\))
Step | When \(u_{1} = 0.25\) | \(u_{1} = 0.45\) | \(u_{1} = 0.65\) |
---|---|---|---|
1 | 0.25 | 0.45 | 0.65 |
2 | 0.1607142857142857 | 0.2892857142857142 | 0.4178571428571429 |
3 | 0.1071428571428571 | 0.1928571428571428 | 0.2785714285714286 |
4 | 0.07247899159663865 | 0.1304621848739496 | 0.1884453781512605 |
5 | 0.04941749427043545 | 0.0889514896867838 | 0.1284854851031322 |
6 | 0.03386013496307614 | 0.06094824293353705 | 0.088036350903998 |
7 | 0.02327884278711485 | 0.04190191701680672 | 0.0605249912464986 |
8 | 0.01604352678571429 | 0.02887834821428571 | 0.04171316964285715 |
9 | 0.01107767325680272 | 0.0199398118622449 | 0.02880195046768708 |
10 | 0.007660093209491246 | 0.01378816777708424 | 0.01991624234467724 |
11 | 0.005303141452724708 | 0.00954565461490447 | 0.01378816777708424 |
12 | 0.003674983989168876 | 0.006614971180503976 | 0.00955495837183908 |
13 | 0.002548779218294543 | 0.004587802592930178 | 0.006626825967565813 |
14 | 0.001768928860458153 | 0.003184071948824675 | 0.004599215037191199 |
15 | 0.001228422819762606 | 0.002211161075572691 | 0.003193899331382777 |
16 | 0.000853514556588304 | 0.001536326201858948 | 0.002219137847129592 |
17 | 0.0005932967039699188 | 0.001067934067145854 | 0.00154257143032179 |
18 | 0.0004125798918411505 | 0.0007426438053140709 | 0.001072707718786992 |
19 | 0.0002870120986721047 | 0.0005166217776097884 | 0.0007462314565474726 |
20 | 0.0001997249140244027 | 0.0003595048452439249 | 0.0005192847764634474 |
21 | 0.0001390242048601234 | 0.0002502435687482223 | 0.0003614629326363212 |
22 | 0.0000967972267484037 | 0.0001742350081471267 | 0.0002516727895458498 |
23 | 0.00006741235434263828 | 0.0001213422378167489 | 0.0001752721212908597 |
24 | 0.0000469581784523506 | 0.0000845247212142311 | 0.0001220912639761116 |
25 | 0.00003271676367581804 | 0.0000588901746164725 | 0.000085063585557127 |
26 | 0.00002279868964810942 | 0.00004103764136659699 | 0.00005927659308508453 |
27 | 0.000015889995815349 | 0.00002860199246762819 | 0.00004131398911990741 |
28 | 0.00001107660292237831 | 0.00001993788526028096 | 0.00002879916759818363 |
29 | 7.722420347291922 × 10^{−6} | 0.00001390035662512546 | 0.00002007829290295901 |
30 | 5.384680854404231 × 10^{−6} | 9.69242553792 × 10^{−6} | 0.00001400017022145 |
31 | 3.755106385308214 × 10^{−6} | 6.759191493554787 × 10^{−6} | 9.76327660180 × 10^{−6} |
(\(\kappa_{n} = {\sqrt{\frac{n}{2n+3}}}\) for all \(n \in \mathbb{N} \))
Step | When \(u_{1} = 1.5\) | \(u_{1} = 2.5\) | \(u_{1} = 3.5\) |
---|---|---|---|
1 | 1.5 | 2.5 | 3.5 |
50 | 1.000070736246516 | 2.000070736246516 | 3.000070736246516 |
100 | 1.000000020810691 | 2.000000020810692 | 3.000000020810691 |
150 | 1.000000000006688 | 2.000000000006688 | 3.000000000006689 |
200 | 0.1721565830342946 | 1.090775274459172 | 2.056164187502003 |
250 | 0.00005853496173392133 | 1.00003086450209 | 2.000019096386024 |
300 | 2.015550211966978 × 10^{−8} | 1.000000010627658 | 2.000000006575511 |
350 | 7.001008916808291 × 10^{−12} | 1.000000000003692 | 2.000000000002284 |
400 | 2.447383506364153 × 10^{−15} | 0.0918987936870247 | 1.030161439675001 |
450 | 8.59732107577053 × 10^{−19} | 0.00003228278010981194 | 1.000010595298216 |
500 | 3.031773634861876 × 10^{−22} | 1.13842533894431 × 10^{−8} | 1.000000003736344 |
550 | 1.072466692421632 × 10^{−25} | 4.027092404879379 × 10^{−12} | 1.000000000001322 |
600 | 3.803545320138658 × 10^{−29} | 1.42822416570892 × 10^{−15} | 0.03556653915576083 |
650 | 1.351861194143804 × 10^{−32} | 5.076213541974867 × 10^{−19} | 0.00001264110719020337 |
5 Application to integral equations
- (i)
\(h\in L^{2} ([0, 1], \mathbb{R})\),
- (ii)\(B: [0,1] \times [0, 1] \times L^{2} ([0, 1], \mathbb{R}) \rightarrow \mathbb{R}\) is measurable and satisfies the conditionfor \(t, v \in [0, 1]\) and \(u, w \in L^{2} ([0, 1], \mathbb{R})\) such that \(u \leq w\).$$\begin{aligned} 0 \leq \bigl\vert B(t, v, u) - B(t, v, w)\bigr\vert \leq \Vert u - w \Vert \end{aligned}$$
It is worth mentioning that every Hilbert space is a \(\operatorname{CAT}(0)\) space, and so is \(L^{2}([0, 1], \mathbb{R})\). Taking \(X = L^{2}([0, 1], \mathbb{R})\) and P as in (5.1) in Theorem 3.4, we get the following result.
Declarations
Acknowledgements
Authors are grateful to the anonymous referees for their valuable suggestions and pointing out an omission in example.
Funding
The first author is thankful to University Grants Commission, India for the financial assistance in form of BSR-Startup reserach grant. The second author is grateful to University Grants Commission, India, for providing financial assistance in the form of the Junior Research Fellowship. The research of J.J. Nieto has been partially supported by AEI of the Ministerio de Economia y Competitividad of Spain under Grant MTM2016-75140-P and cofinanced by European Community fund FEDER and XUNTA de Galicia under grants GRC2015-004 and R2016/022.
Authors’ contributions
The authors have contributed in this work on an equal basis. 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
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