Mann iteration for monotone nonexpansive mappings in ordered CAT(0) space with an application to integral equations.

In this paper, we establish some convergence results for a monotone nonexpansive mapping in a 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.

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 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.

Preliminaries

To make our paper self-contained, we recall some basic definitions and relevant results.

A metric space X is a 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 endowed with the induced metric is a space. Further, the class of Hilbert spaces are examples of spaces.

The fixed point theory in spaces is gaining attention of researchers, and many results have been obtained for single- and multivalued mappings in a space. For different aspects of fixed point theory in spaces, we refer to [15-24]. The following few results are necessary for our subsequent discussion.

Lemma 2.1

([21])

Let be a space. For and , there exists a unique such that

We use the notation for the unique point h of the lemma.

Lemma 2.2

([21])

Let be a space. For and , we have

Lemma 2.3

([21])

Let X be a space. Then for all and .

Let be a bounded sequence in a complete space X. For , we denote The asymptotic radius is given by and the asymptotic center of is defined as

It is known that in a space, 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 spaces presented a natural framework for Lim’s concept and provided precise analogs of several results in Banach spaces involving weak convergence in space setting.

Definition 2.4

A sequence in X is said to be Δ-convergent to if u is the unique asymptotic center of for every subsequence of . In this case, we write and say that u is the Δ-limit of .

Definition 2.5

A Banach space X is said to satisfy Opial’s condition if for any sequence in X with (⇀ denotes weak convergence), we have for all with .

Examples of Banach spaces satisfying this condition are Hilbert spaces and all spaces (). On the other hand, with fail to satisfy Opial’s condition.

Notice that if given a sequence in X such that Δ-converge to u, then for with , we have So, every space satisfies Opial’s property.

Lemma 2.6

([22])

Every bounded sequence in a complete space admits a Δ-convergent subsequence.

Lemma 2.7

([21])

If G is a closed convex subset of a complete space X and if is a bounded sequence in G, then the asymptotic center of is in G.

Next, we introduce the concept of partial order in the setting of spaces.

Let X be a complete space endowed with partial order “⪯”. An order interval is any of the subsets for any . So, an order interval for all is given by Throughout we will assume that the order intervals are closed and convex subsets of an ordered space .

Definition 2.8

Let G be a nonempty subset of an ordered metric space X. A mapping is said to be:

monotone if for all with ,

monotone nonexpansive if P is monotone and for all with .

Now we present the Mann iteration scheme in the setting of ordered spaces . Let G be a nonempty convex subset of a space X. Then the Mann iteration is as follows: where . In this paper, we prove some Δ-convergence and strong convergence results in spaces.

Some Δ-convergence and strong convergence theorems

We begin with the following important lemma.

Lemma 3.1

Let G be a nonempty closed convex subset of a complete ordered space , and let be a monotone nonexpansive mapping. Fix such that . If is defined by (2.1) with condition , then we have:

for any ,

, provided that Δ-converges to a point .

Proof

(i) We will prove the result by induction on n. Note that if are such that , then for any . This is true because we have assumed that order intervals are convex. Thus we only need to show that for any . We have already assumed that , and hence the inequality holds for . Assume that for . Since for all n, we have that is, . Since P is monotone, we have . By using the transitivity of the order we get . Thus by induction the inequality is true for any .

(ii) Let u be the Δ-limit of . From part (i) we have for all since is increasing and the order interval is closed and convex. Therefore for a fixed ; otherwise, if , then we could construct a subsequence of by leaving the first terms of the sequence , and then the asymptotic center of would not be u, which contradicts the assumption that u is the Δ-limit of the sequence . This completes the proof of part (ii). □

Lemma 3.2

Let G be a nonempty closed convex subset of a complete space , and let be a monotone nonexpansive mapping. Fix such that . If is a sequence described as in (2.1) and with such that , then:

exists, and

.

(i) Since , using part (i) of Lemma 3.1, we have . In particular, for , we have . Using the transitivity of the order, we get . By mathematical induction we have for all . Now we have Since P is a monotone map and for all , we have Thus we have for all . So is a decreasing real sequence bounded below by zero. Hence exists.

(ii) First, consider So exists.

Since , using the Lemma 3.1, we have for all . Then, since P is a nonexpansive map and r is a fixed point of P, we have From this we get Since , there exists a subsequence of such that Since exists, it follows that , and this proves the result. □

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 space , and let be a monotone nonexpansive mapping. Fix such that . If is a sequence described as in (2.1), then the conditions and imply that u is a fixed point of P.

Since Δ-, by Lemma 3.1 we get for all . Then from the nonexpansiveness of P and it follows that Thus by the uniqueness of asymptotic center we get , which proves the desired result. □

Theorem 3.4

Let G be a nonempty closed convex subset of a complete space , and let be a monotone nonexpansive mapping with . Fix such that . If is a sequence described as in (2.1), then Δ-converges to a fixed point of P.

From Lemma 3.2 we have that exists for each , so the sequence is bounded, and .

Let , where the union is taken over all subsequences over . To show the Δ-convergence of to a fixed point of P, we will first prove that and thereafter argue that is a singleton set. To show that , let . Then there exists a subsequence of such that . By Lemmas 2.6 and 2.7 there exists a subsequence of such that and . Since and is a subsequence of , we have that . In view of Lemma 3.3, we have , and hence .

Now we wish to show that . If, on the contrary, , then we would have which is a contradiction since X satisfies the Opial condition and hence . Now it remains to show that consists of a single element only. For this, let be a subsequence of . Again, using Lemmas 2.6 and 2.7, we can find a subsequence of such that Δ-. Let and . Previously, we have already proved that ; therefore, it suffices to show that . If , then since , is convergent by Lemma 3.2, By the uniqueness of asymptotic center we have which gives a contradiction. Therefore we must have , which proves that is a singleton set and that a particular element is a fixed point of P. Hence the conclusion follows. □

Theorem 3.5

Let X be a complete space endowed with partial ordering ′⪯′, and let G be a nonempty closed convex subset of X. Let be a monotone nonexpansive mapping such that . Fix such that and . If is a sequence described as in (2.1) such that , then converges to a fixed point of P if and only if .

If the sequence converges to a point , then it is obvious that .

For the converse part, assume that . From Lemma 3.2(i) we have so that Thus forms a decreasing sequence that is bounded below by zero, so exists. As , we have .

Now we prove that is a Cauchy sequence in G. Let ϵ>0 be arbitrary. Since , there exists such that, for all , we have In particular, so there must exist such that Thus, for , we have which shows that is a Cauchy sequence. Since G is a closed subset of a complete metric space X, so G itself is a complete metric space, and therefore must converge in G. Let .

Now P is a monotone nonexpansive mapping, and from Lemma 3.3(i) we have . Also, from the proof of Lemma 3.1 in [12] we can easily deduce that for any . Therefore we have and hence . Thus . □

Numerical example

In this section, we present a numerical example to illustrate the convergence behavior of our iteration scheme (2.1).

Let be a complete metric space with the metric Now, consider the order relation as Let P be defined by Then, clearly, P is not continuous at for , since Also, if , then or for some , and So, P is a monotone nonexpansive map but not a nonexpansive map, and 0 is the unique fixed point of P.

Now, we show the convergence of (2.1) using two different sets of values.

It is evident from the tables (Table 1 and Table 2) and graphs (Fig. 1 and Fig. 2) that our sequence (2.1) converges to 0, which is a fixed point of P.

Table 1

( for all )

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

Table 2

( for all )

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

Figure 1

Graph corresponding to Table 1

Figure 2

Graph corresponding to Table 2

Application to integral equations

In this section, we use our iteration scheme (2.1) to find the solution of following integral equation: where Recall that, for all , we have Next, assume that there exist a nonnegative function and such that for and .

,

is measurable and satisfies the condition for and such that .

Let where ρ is sufficiently large, that is, G is the closed ball of centered at 0 with radius ρ.

Define the operator by Then , and it is a monotone nonexpansive map.

It is worth mentioning that every Hilbert space is a space, and so is . Taking and P as in (5.1) in Theorem 3.4, we get the following result.

Theorem 5.1

Under the above assumptions, the sequence generated by iteration scheme (2.1) converges to a solution of integral equation (IE).