Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces

Junaid Ahmad; Kifayat Ullah; Muhammad Arshad; Manuel de la Sen; Zhenhua Ma

doi:10.1515/math-2021-0130

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Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces

Junaid Ahmad , Kifayat Ullah , Muhammad Arshad , Manuel de la Sen and Zhenhua Ma

Published/Copyright: December 31, 2021

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$Open Mathematics$

From the journal Open Mathematics Volume 19 Issue 1

Abstract

We get the strong and Δ -convergence of the Picard-Krasnoselskii hybrid iteration scheme to a fixed point of a self-map endowed with the condition ( B γ , μ ) . We use the nonlinear context of CAT(0) spaces for establishing these results. We present a new example of a self-map endowed with ( B γ , μ ) condition and prove that its Picard-Krasnoselskii hybrid iterative process is more effective than the Picard and Krasnoselskii hybrid iterative processes. This improves and extends some recently announced results of the current literature.

Keywords: Picard-Krasnoselskii hybrid iteration; fixed point; (B γ,μ ) condition; Delta and strong convergence; CAT(0) spaces

MSC 2010: 47H09; 47H10

1 Introduction

In 1955, Krasnoselskii [1] noticed a new iterative approach for reckoning fixed points of self-nonexpansive operators. Select a subset K of a Banach space and set P : K → K . One can obtain a sequence { x t } by using the Krasnoselskii iteration in the following way:

(1) x 1 = x ∈ K , x t + 1 = ( 1 − λ ) x t + λ P x t ,

where λ ∈ ( 0 , 1 ) is a real constant in ( 0 , 1 ) .

The iteration process (1) is the extension of the classical Picard iteration [2] process x t + 1 = P x t . As many know, the iterative scheme of Picard is not necessarily convergent in the case of nonexpansive operators. On the other hand, the rate of convergence of the Picard scheme [2] is generally better than the iterative scheme (1). Berinde’s book [3] provides many interesting and fundamental results on these schemes.

Among the other things, in 2017, Okeke and Abbas [4] constructed and studied a hybrid iterative process by combining Picard and Krasnoselskii iterations as follows:

(2) x 1 = x ∈ K , y t + 1 = ( 1 − λ ) x t + λ P x t x t + 1 = P y t . ,

where λ ∈ ( 0 , 1 ) .

Very recently, Abdeljawad et al. [5] studied process (2) for the class of generalized nonexpansive mappings following Patir et al. [6] in the context of Banach spaces. With the help of an example, they noted that the iterative scheme (2) is much better than the both schemes due to Picard and Krasnoselskii. Here, we obtain their results in the general and nonlinear domains of CAT(0) spaces.

2 Preliminaries

Throughout, we will denote by N the set of all natural numbers. Suppose X = ( X , d ) is any given metric space. A self-map P of ∅ ≠ K ⊆ X is regarded as a fixed point for P whenever P v = v . We often represent the set of all fixed points of the self-map P by F P . As many know, the map P is regarded as a quasi-nonexpansive provided that d ( P y , P v ) ≤ d ( y , v ) for every choice of y ∈ K and v ∈ F P . Also, P is said to be endowed with ( C ) condition [7] provided that for y , y ′ ∈ K , the nonexpansive condition d ( P y , P y ′ ) ≤ d ( y , y ′ ) holds whenever the inequality 1 2 d ( y , P y ) ≤ d ( y , y ′ ) holds. We say that the map P is endowed with ( B γ , μ ) condition [6] provided that for y , y ′ ∈ K , the condition d ( P y , P y ′ ) ≤ ( 1 − γ ) d ( y , y ′ ) + μ ( d ( y , P y ′ ) + d ( y ′ , P y ) ) holds whenever the inequality γ d ( y , P y ) ≤ d ( y , y ′ ) + μ d ( y ′ , P y ′ ) holds, where γ ∈ [ 0 , 1 ] and μ ∈ [ 0 , 1 2 ] satisfying 2 μ ≤ γ . For details on these mappings, we refer the reader to [8,9, 10,11].

For more details and literature on CAT(0) spaces, one can refer to the books in [12,13].

We have the following results from [14].

Lemma 2.1

Suppose ( X , d ) is any CAT(0) space.

For any choice of y , y ′ ∈ X and fix η ∈ [ 0 , 1 ] , one can find a unique p 0 ∈ [ y , y ′ ] , where [ y , y ′ ] denotes the line segment joining y and y ′ , such that
(2.1) d ( y , p 0 ) = η d ( y , y 0 ) and d ( y ′ , p 0 ) = ( 1 − η ) d ( y , y ′ ) .
We shall write ( ( 1 − η ) y ⊕ η y ′ ) to represent the unique q 0 satisfying (2.1).
For any choice of y , y ′ , y ″ ∈ X and η ∈ [ 0 , 1 ] , one has
d ( y ″ , η y ⊕ ( 1 − η ) y ′ ) ≤ η d ( y , y ″ ) + ( 1 − η ) d ( y ″ , y ′ ) .

Suppose K is nonempty closed convex subset of a CAT(0) space X . Assume that { x t } is any bounded sequence in X . Choose y ∈ X , then

r ( y , { x t } ) = lim sup t → ∞ d ( x t , y ) .

The asymptotic radius of { x t } relative to K is given by

r ( K , { x t } ) = inf { r ( y , { x t } ) : y ∈ K } ,

and the asymptotic center of { x t } relative to K is the set

A ( K , { x t } ) = { y ∈ K : r ( y , { x t } ) = r ( K , { x t } ) } ,

Notice that, the asymptotic center with respect to the whole space X , we normally represent by A ( { x t } ) .

Remark 2.2

When X is complete CAT(0) space, then the set A ( { x t } ) consists of exactly one element (see [15]).

Definition 2.3

Suppose a bounded sequence { x t } in a CAT(0) space X be given. It is said that { x t } Δ -converges to an element v ∈ X if and only if v is the unique asymptotic center of each subsequence { u t } of { x t } . When v is Δ -limit of { x t } , then we simply write Δ − lim t x t = v and regard v as the Δ -lim of the sequence { x t } .

We also know that, every CAT(0) space always satisfies the opial-type property, i.e., whenever a Δ -convergent sequence { x t } in X with Δ − lim t x t = y , then for each choice of y ′ ∈ X − { y } , we must have

lim sup t → ∞ d ( x t , y ) < lim sup t → ∞ d ( x t , y ′ ) .

Lemma 2.4

[16] If { x t } is any bounded sequence in a CAT(0) space. In this case, { x t } essentially has a Δ -convergent subsequence.

Lemma 2.5

[17] If K is closed convex in a complete CAT(0) space and assume that { x t } ⊆ K { x t } is bounded. Consequently, the asymptotic center of the sequence contained in K .

The following result suggests many examples of mappings having ( B γ , μ ) condition.

Lemma 2.6

Suppose K is any nonempty subset of a CAT(0) space and P : K → K . If P satisfies the condition ( C ) , then P satisfies ( B γ , μ ) condition.

Lemma 2.7

[6] Let K be a nonempty subset of a complete CAT(0) space X and P : K → K satisfies B γ , μ condition. For any choice of v ∈ F P , it follows that each y ∈ K

d ( v , P y ) ≤ d ( v , y ) .

Lemma 2.8

[5] Let K be a nonempty subset of a metric space X . Let P : K → K satisfy the condition B γ , μ . Then, the set F P is closed.

Theorem 2.9

[18] Let K be a nonempty closed convex subset of a complete CAT(0) space X having opial property. Let P : K → K satisfy the B γ , μ condition. If { x t } is sequence in X , such that

{ x t } Δ -converges to v ,
lim t → ∞ d ( x t , P x t ) = 0 , then P v = v .

Proposition 2.10

[18] Let K be a nonempty subset of a Banach space X . If P : K → K satisfies the B γ , μ condition on K . Then, for all y , y ′ ∈ K and c ∈ [ 0 , 1 ] ,

d ( P y , P 2 y ) ≤ d ( P y , P y ′ ) ;
in below, either (a) or (b) true:
c 2 d ( y , P y ) ≤ d ( y , y ′ ) ;
c 2 d ( P y , P 2 y ) ≤ d ( P y , y ′ ) ;
d ( y , P y ′ ) ≤ ( 3 − c + 2 μ ) d ( y , P y ′ ) + 1 − c 2 d ( y , y ′ ) + μ ( d ( y , P y ′ ) + d ( y ′ , P y ) + 2 d ( P y , P 2 y ′ ) ) .

The following facts are taken from [19].

Lemma 2.11

Suppose X is any CAT(0) space and 0 < g ≤ β t ≤ h < 1 for every choice of t ≥ N . Assume that { s t } and { q t } are any two sequences in X with lim sup t → ∞ d ( s t , y ) ≤ κ , lim sup t → ∞ d ( q t , y ) ≤ κ and lim t → ∞ d ( β t s t ⊕ ( 1 − β t ) q t , y ) = κ for some κ ≥ 0 . Then, lim t → ∞ d ( s t , q t ) = 0 .

3 Main results

We now establish a helpful lemma for coming convergence results. From now on, we shall write X instead of complete CAT(0) space X .

Lemma 3.1

Suppose K is any nonempty closed convex subset of X . Let P : K → K be a mapping having B γ , μ condition with F P ≠ ∅ . If { x t } is a sequence of Picard-Krasnoselskii (2) (replacing + by ⊕ ), then lim t → ∞ d ( x t , v ) exists for each v ∈ F P .

Proof

Suppose v ∈ F P . Using Lemma 2.7, we obtain

d ( x t + 1 , v ) = d ( P y t , v ) ≤ d ( y t , v ) = d ( ( 1 − λ ) x t ⊕ λ P x t , v ) ≤ ( 1 − λ ) d ( x t , v ) + λ d ( P x t , v ) ≤ ( 1 − λ ) d ( x t , v ) + λ d ( x t , v ) ≤ d ( x t , v ) .

Thus, { d ( x t , v ) } is bounded as well as nonincreasing. Hence, lim t → ∞ d ( x t , v ) exists for every choice of v ∈ F P .□

Theorem 3.2

Suppose K is any nonempty closed convex subset of X . Let P : K → K be a mapping having B γ , μ condition. If { x t } is a sequence of Picard-Krasnoselskii (2) (replacing + by ⊕ ). Then, F P ≠ ∅ if and only if { x t } is bounded and lim t → ∞ d ( x t , P x t ) = 0 .

Proof

Select an element v ∈ F P . By Lemma 3.1, lim t → ∞ d ( x t , v ) exists. We may suppose that lim t → ∞ d ( x t , v ) = κ . We now need to show that lim t → ∞ d ( x t , v ) = κ . Then, proof of Lemma 3.1 suggests that d ( x t + 1 , v ) ≤ d ( y t , v ) . It follows that

lim inf t → ∞ d ( x t + 1 , v ) ≤ lim inf t → ∞ d ( y t , v ) .

Consequently,

κ ≤ lim inf t → ∞ d ( y t , v ) . □

Again by the proof of Lemma 3.1, d ( y t , v ) ≤ d ( x t , v ) . Hence, lim sup t → ∞ d ( y t , v ) ≤ κ . Therefore, we obtain lim t → ∞ d ( y t , v ) = κ . Also by Lemma 2.7, d ( P x t , v ) ≤ d ( x t , v ) . It follows that lim sup t → ∞ d ( x t , v ) ≤ κ . By Lemma 2.11, we obtain

lim t → ∞ d ( x t , P x t ) = 0 .

Conversely, we suppose { x t } is bounded and lim t → ∞ d ( x t , P x t ) = 0 . We choose v ∈ A ( { x t } ) and show v = P v . By Proposition 2.10(iii), choose γ = c 2 , c ∈ ( 0 , 1 ) , and one has

d ( x t , P v ) ≤ ( 3 − c − 2 μ ) d ( x t , P x t ) + 1 − c 2 d ( x t , v ) + μ ( d ( x t , P v ) + d ( v , P x t ) + 2 d ( P x t , P 2 x t ) ) .

By Lemma 2.7 and Proposition 2.10(i),

d ( x t , P v ) ≤ ( 3 − c + 4 μ ) d ( x t , P x t ) + 1 − c 2 + μ d ( x t , v ) + μ d ( x t , P v ) .

It follows that

d ( x t , P v ) ≤ 3 − c + 4 μ 1 − μ d ( x t , P x t ) + 1 − c 2 + μ 1 − μ d ( x t , v ) .

This implies that

r ( P v , { x t } ) = lim sup t → ∞ d ( x t , P v ) ≤ 1 − c 2 + μ 1 − μ lim sup t → ∞ d ( x t , v ) ≤ lim sup t → ∞ d ( x t , v ) = r ( v , { x t } ) .

Hence, P v ∈ A ( { x t } ) . But this set is singleton, we have P v = v . Hence, F P ≠ ∅ .

Now we suggest the Δ -convergence of the sequence { x t } generated by (2) for the class of mappings having B γ , μ condition in the context of CAT(0) spaces.

Theorem 3.3

Suppose K is any nonempty closed convex subset of X . Let P : K → K be a mapping with B γ , μ condition having F P ≠ ∅ . If { x t } is a sequence of Picard-Krasnoselskii iteration (2) (replacing + by ⊕ ). Then, { x t } Δ -converges to an element of F P .

Proof

By Theorem 3.2, we have { x t } is bounded. Thus, we can write A ( { x t } ) = { a } for some a ∈ X . We prove that A ( { x t s } ) = { a } for any choice of subsequence { x t s } of { x t } . Assume that { x t s } is any subsequence of { x t } having A ( { x t s } ) = { b } . But { x t s } is bounded, so we may choose a subsequence { x t k } of { x t s } , such that { x t k } Δ -converges to z for some z ∈ X . Applying Theorems 3.2 and 2.9, we obtain z ∈ F P , and so lim t → ∞ d ( x t , z ) exists. If z ≠ b , then it follows from the uniqueness of asymptotic center that

lim t → ∞ d ( x t , z ) = lim sup k → ∞ d ( x t k , z ) < lim sup k → ∞ d ( x t k , b ) ≤ lim sup s → ∞ d ( x t s , b ) < lim sup s → ∞ d ( x t s , z ) = lim t → ∞ d ( x t , z ) ,

which is clearly a contradiction. Thus, we conclude that b = z ∈ F P . Now we assume that a ≠ b . Then,

lim t → ∞ d ( x t , b ) = lim sup s → ∞ d ( x t s , b ) ≤ lim sup s → ∞ d ( x t k , a ) ≤ lim sup t → ∞ d ( x t , a ) < lim sup t → ∞ d ( x t , b ) = lim t → ∞ d ( x t , b ) .

Consequently, we get a contradiction and we agree to write a = b ∈ F P . Thus, we have reached to the fact that { x t } is Δ -convergent in the set F P .□

Theorem 3.4

Suppose K is any nonempty closed convex subset of X . Let P : K → K be a mapping with B γ , μ condition having F P ≠ ∅ . Let { x t } be a sequence of Picard-Krasnoselskii iteration (2) (replacing + by ⊕ ). If lim inf t → ∞ d ( x t , F P ) = 0 , then { x t } strongly converges to an element of F P .

Proof

For every choice of v ∈ F P , we have from Lemma 3.1, lim t → ∞ d ( x t , v ) exists. It follows that lim t → ∞ d ( x t , F P ) exists, and so we have

lim t → ∞ d ( x t , F P ) = 0 .

From the aforementioned limit, we may construct a sequence { x t s } in { x t } and { p s } in F P , such that d ( x t s , p s ) ≤ 1 2 s , s ∈ N . Moreover, the proof of Lemma 3.1 provides that { x t } is nonincreasing. Thus,

d ( x t s + 1 , p s ) ≤ d ( x t s , p s ) ≤ 1 2 s .

We shall prove that { p s } is a Cauchy sequence in F P .

d ( p s + 1 , p s ) ≤ d ( p s + 1 , x t s + 1 ) + d ( x t s + 1 , p s ) ≤ 1 2 s + 1 + 1 2 s ≤ 1 2 s − 1 → 0 , as s → ∞ .

From the aforementioned equation, we may observe that the sequence { p s } ⊆ F P is Cauchy. But Lemma 2.8 tells that the set F P is closed in K . So, p s → p 0 for some p 0 ∈ F P . Now Lemma 3.1 suggests that lim t → ∞ d ( x t , p 0 ) exists. This completes the proof.□

The following is the CAT(0) version of the condition I .

Definition 3.5

Recall that a self-mapping P on K subset of X is said to satisfy the condition ( I ) [20] if and only if there exists ρ : [ 0 , ∞ ) → [ 0 , ∞ ) satisfying ρ ( 0 ) = 0 and ρ ( u ) > 0 for every u > 0 , such that

d ( y , P y ) ≥ ρ ( d ( y , F P ) ) for every choice of y ∈ K .

Theorem 3.6

Suppose K is any nonempty closed convex subset of X . Let P : K → K be a mapping with B γ , μ condition having F P ≠ ∅ . Let { x t } be a sequence of Picard-Krasnoselskii iteration (2) (replacing + by ⊕ ). If P has condition ( I ) , then { x t } strongly converges to an element of F P .

Proof

In the view of Theorem 3.2, we can write

lim inf t → ∞ d ( P x t , x t ) = 0 .

Applying condition ( I ) of P , we have

lim inf t → ∞ d ( x t , F P ) = 0 .

The conclusion now clearly follows from Theorem 3.4.□

4 Numerical example

In this section, we are interested in the rate of convergence. We first construct a new example of a mapping P as follows, which is Patir mapping but not Suzuki.

Example 1

Define an operator P : [ 4 , 7 ] → [ 4 , 7 ] as follows:

P y = y + 4 2 if y ≠ 7 , 4 if y = 7 .

To show that P is not Suzuki mapping, let y = 6.2 , y ′ = 7 . We see that 1 2 d ( y , P y ) < d ( y , y ′ ) , but d ( T y , T y ′ ) > d ( y , y ′ ) . Thus, P does not satisfy condition ( C ) . On the other hand, P has B 1 , 1 2 condition.

C 1 : If we take y , y ′ ∈ [ 4 , 7 ) , then

( 1 − γ ) d ( y , y ) + μ ( d ( y , P y ′ ) + d ( y ′ , P y ) ) = 1 2 ( ∣ y − P y ′ ∣ + ∣ y ′ − P y ∣ ) = 1 2 y − y ′ + 4 2 + y ′ − y + 4 2 ≥ 1 2 3 y 2 − 3 y ′ 2 = 3 4 ∣ y − y ′ ∣ ≥ 1 2 ∣ y − y ′ ∣ = d ( P y , P y ′ ) .

C 2 : Now we select y ∈ [ 4 , 7 ) and y ′ = 7 . Then,

( 1 − γ ) d ( y , y ′ ) + μ ( d ( x , P y ′ ) + d ( y ′ , P y ) ) = 1 2 ( ∣ y − P y ′ ∣ + ∣ y ′ − P y ∣ ) = 1 2 ∣ y − 4 ∣ + y ′ − y + 4 2 = 1 2 ∣ y − 4 ∣ + 1 2 y ′ − y + 4 2 ≥ 1 2 ∣ y − 4 ∣ = d ( P y , P y ′ ) .

C 3 : If we take y = y ′ = 7 , then we have

( 1 − γ ) d ( y , y ′ ) + μ ( d ( y , P y ′ ) + d ( y ′ , P y ) ) ≥ 0 = d ( P y , P y ′ ) .

Let the value of λ be 1 2 ∈ ( 0 , 1 ) . Then, the strong convergence of the Picard, Krasnoselskii and Picard-Krasnoselskii hybrid iterative process to the fixed point v = 4 can be easily observed in Table 1.

Table 1

Numerical results obtained from Picard-Krasnoselskii hybrid, Picard and Krasnoselskii iterative schemes to an element v = 4 ∈ F P

t	Picard-Krasnoselskii	Picard	Krasnoselskii
1	4.2	4.2	4.2
2	4.075000000	4.100000000	4.150000000
3	4.028125000	4.028125000	4.112500000
4	4.010546875	4.025000000	4.084375000
5	4.003955078	4.012500000	4.063281250
6	4.001483154	4.006250000	4.047460937
.	.	.	.
.	.	.	.
.	.	.	.
22	4	4.000000095	4.000475681
.	.	.	.
.	.	.	.
.	.	.	.
30	4	4	0.005932617
.	.	.	0.005932617
.	.	.	0.005932617
.	.	.	0.005932617
35	4	4	4

Remark 4.1

It is seen in Table 1 and Figure 1 that Picard-Krasnoselskii hybrid iterative scheme has a much better converge rate vs the iterative schemes due to Picard and Krasnoselskii in the larger setting of maps having B γ , μ condition.

$Figure 1 Graphical representation of the values obtained in Table 1.$

Figure 1

Graphical representation of the values obtained in Table 1.

5 Conclusion

The extension of fixed point results from the context of the linear domain to the general context of nonlinear domain has its own significance. In [21], Takahashi suggested the notion of convexity in metric spaces and studied some fixed point results in this context. This discovery then began different convexity structures in the context of metric spaces. In [5], Abdeljawad et al. established some convergence results for mappings having ( B γ , μ ) condition in the linear context of Banach spaces. In this paper, we have extended their results to the nonlinear context of CAT(0) spaces. We have also suggested a new example of mapping with ( B γ , μ ) condition and showed that its Picard-Krasnoselskii hybrid iterative process is more effective than the Picard and Krasnoselskii iterative processes.

Funding information: 1. Research project of basic scientific research business expenses of provincial colleges and universities in Hebei Province: 2021QNJS11; 2. Innovation and improvement project of academic team of Hebei University of Architecture (Mathematics and Applied Mathematics) No. TD202006; and 3. The Major Project of Education Department in Hebei (No. ZD2021039).
Author contributions: All authors provided equal contributions to this article.
Conflict of interest: The authors declare that they have no conflicts of interest.
Data availability statement: No data were used to support this study.

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Received: 2021-06-07

Revised: 2021-10-19

Accepted: 2021-11-12

Published Online: 2021-12-31

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2021-0130

Keywords for this article

Picard-Krasnoselskii hybrid iteration; fixed point; (B γ,μ ) condition; Delta and strong convergence; CAT(0) spaces

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