Some fixed-point theorems for a pair of Reich-Suzuki-type nonexpansive mappings in hyperbolic spaces

Sreya Valiya Valappil; Shaini Pulickakunnel

doi:10.1515/taa-2022-0132

Artikel Open Access

Some fixed-point theorems for a pair of Reich-Suzuki-type nonexpansive mappings in hyperbolic spaces

Sreya Valiya Valappil und Shaini Pulickakunnel

Veröffentlicht/Copyright: 18. Februar 2023

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Aus der Zeitschrift Topological Algebra and its Applications Band 11 Heft 1

Abstract

In this article, we prove some fixed-point results for a pair of Reich-Suzuki-type nonexpansive mappings in uniformly convex W -hyperbolic spaces. We introduce a new iterative scheme and establish its convergence to the fixed points of a pair of Reich-Suzuki-type nonexpansive mappings. We illustrate our main result with an example, and using Matlab code, it is observed that our iteration converges faster than the iteration defined by Garodia et al. for a pair of Reich-Suzuki-type nonexpansive mappings. An application is given to substantiate our main result.

Keywords: hyperbolic space; geodesic space; Reich-Suzuki-type nonexpansive map; fixed point

MSC 2010: 47H09; 47H10; 53C22

1 Introduction

The classical Banach contraction principle (1922) establishes that every contraction map in a complete metric space has a fixed point and the Picard iteration converges to the fixed point. The existence of fixed points for nonexpansive mappings in a bounded closed convex subset of a uniformly convex Banach space was independently proved by Kirk [14], Browder [9], and Gohde [13] in 1965. But in general, the Picard iteration may not converge to the fixed points of a nonexpansive mapping. Therefore, it was necessary to make some new procedures, such as Mann [25], Ishikawa [15], Noor [27], and Abbas and Nazir [1] to deal with nonexpansive-type mappings and get a higher rate of convergence.

The fixed-point results in different metric spaces have been studied extensively [4–7,20,26,29,31,32,36] by various authors. A weaker version of nonexpansive mappings was presented by Suzuki [31] in 2008, and he stated that a map Φ on a subset S of a metric space is Suzuki-type nonexpansive if ρ ( Φ u 1 , Φ u 2 ) ≤ ρ ( u 1 , u 2 ) holds when 1 2 ρ ( u 1 , Φ u 2 ) ≤ ρ ( u 1 , u 2 ) for u 1 , u 2 ∈ S . The class of Suzuki nonexpansive mappings contains the class of nonexpansive mappings. In [29], Pant and Pandey introduced Reich-Suzuki-type nonexpansive mappings, which some other authors referred to as generalized α -nonexpansive mappings [12]. The class of Reich-Suzuki-type nonexpansive mappings contains the class of Suzuki-type nonexpansive mappings making it a more general class of nonexpansive-type mappings. Many authors developed the fixed-point results for these types of mappings [2,3,16,18,19,21,28,33,34].

In [12], Garodia et al. introduced a new iteration process for generalized α -nonexpansive mappings as follows: Let Φ , Ψ be a pair of generalized α -nonexpansive mappings, then

(1) s 0 ∈ S z i = ( 1 − γ i ) s i + γ i Φ s i y i = ( 1 − δ i ) Ψ s i + δ i z i s i + 1 = ( 1 − η i ) Ψ s i + η i Φ y i ,

where { γ i } , { δ i } , { η i } are sequences in ( 0 , 1 ) .

In [35], Ullah and Arshad introduced the following new iteration process known as M ∗ iteration process in Banach spaces as follows: Let S ≠ ∅ be a subset of a Banach space, then

(2) ω 0 ∈ S z i = ( 1 − γ i ) ω i + γ i Φ ω i y i = Φ ( ( 1 − δ i ) ω i + δ i Φ z i ) ω i + 1 = Φ y i ,

where { γ i } , { δ i } , { η i } are sequences in ( 0 , 1 ) .

In this article, we present some fixed-point results for a pair of Reich-Suzuki-type nonexpansive mappings in the setting of UCW-hyperbolic spaces, which include the CAT(0) spaces [24]. We introduce a new iterative scheme and establish its convergence to the fixed points of a pair of Reich-Suzuki-type nonexpansive mappings. An example is given to illustrate our main result, and using Matlab code, it is observed that our iteration converges faster than the iteration defined by Garodia et al. for a pair of Reich-Suzuki-type nonexpansive mappings. An application is given to support our main result.

2 Preliminaries

We recall some basic definitions and results used in this article.

Definition 2.1

[24] A W -hyperbolic space ( X , ρ , W ) is a metric space ( X , ρ ) together with a convexity mapping W : X × X × [ 0 , 1 ] → X such that W ( u 1 , u 2 , τ ) = ( 1 − τ ) u 1 ⊕ τ u 2 satisfying

ρ ( u 3 , W ( u 1 , u 2 , τ ) ) ≤ ( 1 − τ ) ρ ( u 3 , u 1 ) + τ ρ ( u 3 , u 2 ) ,
ρ ( W ( u 1 , u 2 , τ 1 ) , W ( u 1 , u 2 , τ 2 ) ) = ∣ τ 1 − τ 2 ∣ ρ ( u 1 , u 2 ) ,
W ( u 1 , u 2 , τ ) = W ( u 2 , u 1 , 1 − τ ) ,
ρ ( W ( u 1 , u 3 , τ ) , W ( u 2 , u 4 , τ ) ) ≤ ( 1 − τ ) ρ ( u 1 , u 2 ) + τ ρ ( u 3 , u 4 ) .

Definition 2.2

[24] A W -hyperbolic space ( X , ρ , W ) is uniformly convex if for any κ > 0 and any ε ∈ ( 0 , 2 ] there exists θ ∈ ( 0 , 1 ] such that for all u 1 , u 2 ∈ X and t ∈ [ 0 , 1 ] ,

ρ ( u 1 , t ) ≤ κ ρ ( u 2 , t ) ≤ κ ρ ( u 1 , u 2 ) ≥ ε κ ⇒ ρ W u 1 , u 2 , 1 2 , t ≤ ( 1 − θ ) κ .

A mapping ς : ( 0 , ∞ ) × ( 0 , 2 ] → ( 0 , 1 ] providing such a θ ≔ ς ( κ , ε ) for a given κ > 0 and ε ∈ ( 0 , 2 ] is called a modulus of uniform convexity.

Definition 2.3

[23] Let S ≠ ∅ be a subset of W -hyperbolic space ( X , ρ ) and { s i } be a bounded sequence in S . For each s ∈ S , define

Asymptotic radius of { s i } at s as
r ( { s i } , s ) = lim sup i → ∞ ρ ( s i , s ) .
Asymptotic radius of { s i } relative to S as
r ( { s i } , S ) = inf { r ( { s i } , s ) ; s ∈ S } .
Asymptotic center of { s i } relative to S as
A ( { s i } , S ) = { s ∈ S : r ( { s i } , s ) = r ( { s i } , S ) } .

Also, for any S ≠ ∅ , a closed convex subset of X , every bounded sequence { s i } has a unique asymptotic center in X with monotone modulus of uniform convexity.

Definition 2.4

[29] Let S ≠ ∅ be a subset of a metric space. A map Φ : S → S is called Reich-Suzuki-type nonexpansive map if for all u 1 , u 2 ∈ S , there exists some α ∈ [ 0 , 1 ) such that

(3) 1 2 ρ ( u 1 , Φ u 2 ) ≤ ρ ( u 1 , u 2 ) ⇒ ρ ( Φ u 1 , Φ u 2 ) ≤ α ρ ( u 1 , Φ u 1 ) + α ρ ( u 2 , Φ u 2 ) + ( 1 − 2 α ) ρ ( u 1 , u 2 ) .

Remark 2.1

When α = 1 2 , (3) reduces to Kannan nonexpansive mapping, i.e., a self-mapping Φ : X → X is called a Kannan mapping if there exists α ∈ 0 , 1 2 such that

ρ ( Φ u 1 , Φ u 2 ) ≤ α [ ρ ( u 1 , Φ u 1 ) + ρ ( u 2 , Φ u 2 ) ] ,

for all u 1 , u 2 ∈ X . If α = 1 2 , then we say that Φ is a Kannan nonexpansive mapping [11].

Lemma 2.1

[29] Let S ≠ ∅ be a subset of a hyperbolic metric space and Ψ represents a Reich-Suzuki-type nonexpansive mapping on S with a fixed point ζ ∈ S . Then ρ ( Ψ s , ζ ) ≤ ρ ( s , ζ ) for all s ∈ S .

Lemma 2.2

[29] Let S ≠ ∅ be a subset of a hyperbolic space and Φ represents a Reich-Suzuki-type nonexpansive mapping on S. Then for all u 1 , u 2 ∈ S and β < 1 ,

ρ ( u 1 , Φ u 2 ) ≤ 3 + β 1 − β ρ ( u 1 , Φ u 1 ) + ρ ( u 1 , u 2 ) .

Lemma 2.3

[29] Let S ≠ ∅ be a subset of a hyperbolic space and Φ represents a Reich-Suzuki-type nonexpansive mapping on S. Then for all u 1 , u 2 ∈ S ,

ρ ( Ψ u 1 , Ψ 2 u 1 ) ≤ ρ ( u 1 , Ψ u 1 ) ;
Either 1 2 ρ ( u 1 , Ψ u 1 ) ≤ ρ ( u 1 , u 2 ) or 1 2 ρ ( Ψ u 1 , Ψ 2 u 1 ) ≤ ρ ( Ψ u 1 , u 2 ) ;
Either ρ ( Ψ u 1 , Ψ u 2 ) ≤ k ρ ( Ψ u 1 , u 1 ) + k ρ ( u 2 , Ψ u 2 ) + ( 1 − 2 k ) ρ ( u 1 , u 2 ) or ρ ( Ψ 2 u 1 , Ψ u 2 ) ≤ k ρ ( Ψ 2 u 1 , Ψ u 1 ) + k ρ ( Ψ u 2 , u 2 ) + ( 1 − 2 k ) ρ ( Ψ u 1 , u 2 ) .

Theorem 2.1

[22] Let ( X , ρ , W ) be a UCW-hyperbolic space with modulus of uniform convexity ς , and let μ ∈ X . Let ς increase with κ (for a fixed ε ) and let { s i } be a sequence in [ t 1 , t 2 ] for some t 1 , t 2 ∈ ( 0 , 1 ) and { μ i } , { v i } are sequences in X such that lim sup i ρ ( μ i , μ ) ≤ q , lim sup i ρ ( v i , μ ) ≤ q , and lim i ρ ( W ( μ i , v i , s i ) , μ ) = q . Then lim i → ∞ ρ ( μ i , v i ) = 0 .

Throughout this article, we denote a collection of all fixed points of a map Φ by F ( Φ ) .

3 Main results

Motivated by the work in [35], we introduce a new iteration scheme in uniformly convex W-hyperbolic space (UCW-hyperbolic space) for a pair of maps Φ , Ψ : S → S as follows:

Let S ≠ ∅ be a closed convex subset of UCW-hyperbolic space. Let s 0 ∈ S be arbitrary. Then construct the sequence { s i } as follows:

s 0 ∈ S z i = W ( s i , Φ s i , γ i ) , y i = Φ ( W ( s i , Ψ z i , δ i ) ) s i + 1 = Ψ y i , ( ∗ )

where { γ i } , { δ i } are sequences in ( 0 , 1 ) .

Lemma 3.1

Let Φ , Ψ : S → S be a pair of Reich-Suzuki-type nonexpansive mappings, where S is a closed convex subset of a UCW-hyperbolic space X such that F ( Φ ) ∩ F ( Ψ ) ≠ ∅ . Then lim i → ∞ ρ ( s i , ζ ) exists for ζ ∈ F ( Φ ) ∩ F ( Ψ ) , where { s i } is as in ( ∗ ).

Proof

Let ζ ∈ F ( Φ ) ∩ F ( Ψ ) . Then owing to Lemma 2.1 and using iteration ( ∗ ) we have,

(4) ρ ( z i , ζ ) = ρ ( W ( s i , Φ s i , γ i ) , ζ ) ≤ ( 1 − γ i ) ρ ( s i , ζ ) + γ i ρ ( Φ s i , ζ ) ≤ ρ ( s i , ζ )

and

(5) ρ ( y i , ζ ) = ρ ( Φ ( W ( s i , Ψ z i , δ i ) ) , ζ ) ≤ ρ ( W ( s i , Ψ z i , δ i ) , ζ ) ≤ ( 1 − δ i ) ρ ( s i , ζ ) + δ i ρ ( Ψ z i , ζ ) ≤ ρ ( s i , ζ ) .

Now using Lemma 2.1 and (5), we obtain

(6) ρ ( s i + 1 , ζ ) = ρ ( Ψ y i , ζ ) ≤ ρ ( y i , ζ ) ≤ ρ ( s i , ζ ) .

Thus, { ρ ( s i , ζ ) } is decreasing and bounded below. Hence, lim i → ∞ ρ ( s i , ζ ) exists.□

Theorem 3.1

Let S ≠ ∅ be a closed convex subset of a UCW-hyperbolic space X with strictly increasing modulus of uniform convexity. Let Φ , Ψ : S → S be as in Definition 2.4. For s 0 ∈ S , let { s i } be as in ( ∗ ). Then F ( Φ ) ∩ F ( Ψ ) ≠ ∅ iff { s i } is bounded, lim i → ∞ ρ ( Φ s i , s i ) = 0 and lim i → ∞ ρ ( Ψ s i , s i ) = 0 .

Proof

Suppose F ( Φ ) ∩ F ( Ψ ) ≠ ∅ , then from Lemma 3.1, we have lim i → ∞ ρ ( s i , ζ ) exists for ζ ∈ F ( Φ ) ∩ F ( Ψ ) . Let

(7) lim i → ∞ ρ ( s i , ζ ) = q .

From (4) and (5), we obtain

(8) lim sup i → ∞ ρ ( z i , ζ ) ≤ lim sup i → ∞ ρ ( s i , ζ ) = q

and

(9) lim sup i → ∞ ρ ( y i , ζ ) ≤ lim sup i → ∞ ρ ( s i , ζ ) = q .

Since Φ , Ψ are Reich-Suzuki-type nonexpansive, using Lemma 2.1, (8), and (9) we obtain

(10) lim sup i → ∞ ρ ( Φ z i , ζ ) ≤ q ,

(11) lim sup i → ∞ ρ ( Φ y i , ζ ) ≤ q ,

(12) lim sup i → ∞ ρ ( Φ s i , ζ ) ≤ q ,

(13) lim sup i → ∞ ρ ( Ψ z i , ζ ) ≤ q ,

(14) lim sup i → ∞ ρ ( Ψ y i , ζ ) ≤ q ,

and

(15) lim sup i → ∞ ρ ( Ψ s i , ζ ) ≤ q .

Now using (6) and (7), we obtain

(16) lim i → ∞ ρ ( y i , ζ ) = q .

Now

ρ ( s i + 1 , ζ ) = ρ ( Ψ y i , ζ ) ≤ ρ ( y i , ζ ) = ρ ( Φ ( W ( s i , Ψ z i , δ i ) ) , ζ ) ≤ ( 1 − δ i ) ρ ( s i , ζ ) + δ i ρ ( z i , ζ ) .

Thus,

ρ ( s i + 1 , ζ ) − ρ ( s i , ζ ) ≤ δ i ρ ( z i , ζ ) − δ i ρ ( s i , ζ ) ≤ ρ ( z i , ζ ) − ρ ( s i , ζ ) .

Applying limit infimum on both sides, we obtain

(17) q ≤ lim inf i → ∞ ρ ( z i , ζ ) .

Now using (17) and (8), we obtain

lim i → ∞ ρ ( z i , ζ ) = q ,

i.e.,

(18) lim i → ∞ ρ ( W ( s i , Φ s i , γ i ) , ζ ) = q .

Now using Theorem 2.1, (7), (12), and (18), we obtain

(19) lim i → ∞ ρ ( s i , Φ s i ) = 0 .

Now

ρ ( s i , z i ) = ρ ( s i , W ( s i , Φ s i , γ i ) ) ≤ γ i ρ ( Φ s i , s i ) .

Hence,

(20) lim i → ∞ ρ ( s i , z i ) = 0 , using (19) .

Consider

ρ ( y i , ζ ) = ρ ( Φ ( W ( s i , Ψ z i , δ i ) ) , ζ ) ≤ ρ ( W ( s i , Ψ z i , δ i ) , ζ ) using Lemma 2.1 ≤ ( 1 − δ i ) ρ ( s i , ζ ) + δ i ρ ( Ψ z i , ζ ) ≤ ρ ( s i , ζ ) using Lemma 2.1 and (4) ,

which on using (7) and (16) gives

(21) lim i → ∞ ρ ( W ( s i , Ψ z i , δ i ) , ζ ) = q .

Owing to equations (7), (13), (21), and Theorem 2.1, we obtain

(22) lim i → ∞ ρ ( s i , Ψ z i ) = 0 .

Now

ρ ( Ψ s i , s i ) ≤ ρ ( Ψ s i , Ψ z i ) + ρ ( Ψ z i , s i ) ≤ 3 + β 1 − β ρ ( Ψ z i , Ψ 2 z i ) + ρ ( s i , Ψ z i ) + ρ ( s i , Ψ z i ) (using Lemma 2.2) ≤ 3 + β 1 − β ρ ( z i , Ψ z i ) + 2 ρ ( s i , Ψ z i ) (using Lemma 2.3) ≤ 3 + β 1 − β [ ρ ( z i , s i ) + ρ ( s i , Ψ z i ) ] + 2 ρ ( s i , Ψ z i ) .

Now using (20) and (22), we obtain

(23) lim i → ∞ ρ ( Ψ s i , s i ) = 0 .

Hence, lim i → ∞ ρ ( Φ s i , s i ) = 0 and lim i → ∞ ρ ( Ψ s i , s i ) = 0 .

Conversely, let { s i } be bounded, lim i → ∞ ρ ( Φ s i , s i ) = 0 , and lim i → ∞ ρ ( Ψ s i , s i ) = 0 . Let s ∗ ∈ A ( { s i } , S ) . Now,

r ( { s i } , Φ s ∗ ) = lim sup i → ∞ ρ ( { s i } , Φ s ∗ ) ≤ lim sup i → ∞ 3 + β 1 − β ρ ( Φ s i , s i ) + lim sup i → ∞ ρ ( s i , s ∗ ) (using Lemma 2.2) = lim sup i → ∞ ρ ( s i , s ∗ ) = r ( { s i } , s ∗ ) .

Thus, Φ s ∗ ∈ A ( { s i } , S ) . Since A ( { s i } , S ) is singleton, we obtain Φ s ∗ = s ∗ . Similarly, we can prove that Ψ s ∗ = s ∗ . Thus, F ( Φ ) ∩ F ( Ψ ) ≠ ∅ .□

Corollary 3.1

Let S ≠ ∅ be a closed convex subset of a UCW-hyperbolic space X with strictly increasing modulus of uniform convexity. Let Φ , Ψ : S → S be Kannan nonexpansive mappings. For s 0 ∈ S , let { s i } be as in ( ∗ ). Then F ( Φ ) ∩ F ( Ψ ) ≠ ∅ iff { s i } is bounded, lim i → ∞ ρ ( Φ s i , s i ) = 0 , and lim i → ∞ ρ ( Ψ s i , s i ) = 0 .

Proof

Proof follows from Theorem 3.1 and Remark 2.1.□

Theorem 3.2

Let S ≠ ∅ be a closed convex subset of a UCW-hyperbolic space X with strictly increasing modulus of uniform convexity. Let Φ , Ψ : S → S be as in Definition 2.4 with F ( Φ ) ∩ F ( Ψ ) ≠ ∅ . For s 0 ∈ S , let { s i } be as in ( ∗ ). If Φ ( S ) belongs to a compact set, then { s i } → a , for some a ∈ F ( Φ ) ∩ F ( Ψ ) .

Proof

Suppose that Φ ( S ) belongs to a compact set. Then there exists { Φ s i k } of { Φ s i } such that Φ s i k → s as k → ∞ . Now using Theorem 3.1, we obtain

ρ ( s i k , s ) ≤ ρ ( s i k , Φ s i k ) + ρ ( Φ s i k , s ) → 0 ,

as k → ∞ . Also, using Lemma 2.2, we have

ρ ( Φ s i k , Ψ s ) ≤ ρ ( Φ s i k , s i k ) + ρ ( s i k , Ψ s ) ≤ ρ ( Φ s i k , s i k ) + 3 + β 1 − β ρ ( s i k , Ψ s i k ) + ρ ( s i k , s ) , (using Lemma 2.2) → 0 , (using Theorem 3.1)

and

ρ ( Φ s i k , Φ s ) ≤ ρ ( Φ s i k , s i k ) + ρ ( s i k , Φ s ) ≤ ρ ( Φ s i k , s i k ) + 3 + β 1 − β ρ ( s i k , Ψ s i k ) + ρ ( s i k , s ) , using Lemma 2.2 → 0 .

Thus, s i k → s , Φ s i k → Φ s , and Φ s i k → Ψ s . Thus, Φ ( s ) = Ψ ( s ) . Also ρ ( Φ ( s i k ) , s ) → 0 . Thus, Φ ( s ) = Ψ ( s ) = s . Thus, s is a fixed point of Φ and Ψ .

From Lemma 3.1, we have lim i → ∞ ρ ( s i , s ) exists. Thus, lim i → ∞ ρ ( s i , s ) = lim i → ∞ ρ ( s i k , s ) = 0 . Hence, { s i } → s ∈ F ( Φ ) ∩ F ( Ψ ) .□

Corollary 3.2

Let S ≠ ∅ be a closed convex subset of a UCW-hyperbolic space X with strictly increasing modulus of uniform convexity. Let Φ , Ψ : S → S be Kannan nonexpansive mappings with F ( Φ ) ∩ F ( Ψ ) ≠ ∅ . For s 0 ∈ S , let { s i } be as in ( ∗ ). If Φ ( S ) belongs to a compact set, then { s i } → a , for some a ∈ F ( Φ ) ∩ F ( Ψ ) .

Proof

Proof follows from Theorem 3.2 and Remark 2.1.□

CAT(0) space is a special class of UCW-hyperbolic spaces. Hence, the following corollary follows from Theorem 3.2.

Corollary 3.3

Let S ≠ ∅ be a closed convex subset of a CAT(0) space. Let Φ , Ψ : S → S be as in Definition 2.4 with F ( Φ ) ∩ F ( Ψ ) ≠ ∅ . For s 0 ∈ S , let { s i } be as in ( ∗ ). If Φ ( S ) belongs to a compact set, then { s i } → a , for some a ∈ F ( Φ ) ∩ F ( Ψ ) .

Corollary 3.4

Let S ≠ ∅ be a closed convex subset of a UCW-hyperbolic space X with strictly increasing modulus of uniform convexity. Let Φ , Ψ : S → S be Suzuki nonexpansive with F ( Φ ) ∩ F ( Ψ ) ≠ ∅ . For s 0 ∈ S , let { s i } be as in ( ∗ ). If Φ ( S ) belongs to a compact set, then { s i } → a , for some a ∈ F ( Φ ) ∩ F ( Ψ ) .

Proof

Every Suzuki-type nonexpansive mapping is Reich-Suzuki-type. Hence, Theorem 3.2 provides the proof.□

Theorem 3.3

Let S ≠ ∅ be a closed convex subset of a complete UCW-hyperbolic space X with strictly increasing modulus of uniform convexity. Let Φ , Ψ : S → S be as in Definition 2.4 with F ( Φ ) ∩ F ( Ψ ) ≠ ∅ . For s 0 ∈ S , let { s i } be as in ( ∗ ). Then { s i } → s ∈ F ( Φ ) ∩ F ( Ψ ) iff lim inf i → ∞ ρ ( s i , F ( Φ ) ∩ F ( Ψ ) ) = 0 .

Proof

If { s i } → s ∈ F ( Φ ) ∩ F ( Ψ ) , then we have ρ ( s i , s ) → 0 . Thus, lim inf i → ∞ ρ ( s i , F ( Φ ) ∩ F ( Ψ ) ) = 0 .

Conversely assume that lim inf i → ∞ ρ ( s i , F ( Φ ) ∩ F ( Ψ ) ) = 0 . Using (6), for any s ∈ F ( Φ ) ∩ F ( Ψ ) we have

ρ ( s i + 1 , s ) ≤ ρ ( s i , s ) .

Thus,

(24) ρ ( s i + 1 , F ( Φ ) ∩ F ( Ψ ) ) ≤ ρ ( s i , F ( Φ ) ∩ F ( Ψ ) ) .

Hence, { ρ ( s i + 1 , F ( Φ ) ∩ F ( Ψ ) ) } is decreasing and bounded below. Thus, lim i → ∞ ρ ( s i + 1 , F ( Φ ) ∩ F ( Ψ ) ) exists. But we assumed that lim inf i → ∞ ρ ( s i + 1 , F ( Φ ) ∩ F ( Ψ ) ) = 0 . Hence, lim i → ∞ ( ρ ( s i + 1 , F ( Φ ) ∩ F ( Ψ ) ) ) = 0 .

Now we prove that { s i } is Cauchy. Let ε > 0 . Since lim i → ∞ ρ ( s i + 1 , F ( Φ ) ∩ F ( Ψ ) ) = 0 , there exists i 0 such that for all i ≥ i 0 , we obtain

ρ ( s i , F ( Φ ) ∩ F ( Ψ ) ) < ε 4 .

In particular,

inf { ρ ( s i 0 , q ) : q ∈ F ( Φ ) ∩ F ( Ψ ) } < ε 4 .

Thus, there exists some q 0 ∈ F ( Φ ) ∩ F ( Ψ ) such that ρ ( s i 0 , q 0 ) < ε 2 .

For j , k ≥ i 0 , we have

ρ ( s j + k , s j ) ≤ ρ ( s j + k , q 0 ) + ρ ( q 0 , s j ) ≤ ρ ( s i 0 , q 0 ) + ρ ( s i 0 , q 0 ) < ε 2 + ε 2 = ε .

Hence, { s i } is Cauchy. Since S is closed, we have { s i } → g for some g ∈ S .

Now using Lemma 2.2 and Theorem 3.1, we have

ρ ( g , Φ g ) ≤ ρ ( g , s i ) + ρ ( s i , Φ g ) ≤ ρ ( g , s i ) + 3 + β 1 − β ρ ( s i , Φ s i ) + ρ ( s i , g ) → 0 .

Thus, Φ g = g . Similarly, we can prove Ψ g = g . Thus, g ∈ F ( Φ ) ∩ F ( Ψ ) .□

Corollary 3.5

Let S ≠ ∅ be a closed convex subset of a complete UCW-hyperbolic space X with strictly increasing modulus of uniform convexity. Let Φ , Ψ : S → S be Kannan nonexpansive mappings with F ( Φ ) ∩ F ( Ψ ) ≠ ∅ . For s 0 ∈ S , let { s i } be as in ( ∗ ). Then { s i } → g ∈ F ( Φ ) ∩ F ( Ψ ) iff lim inf i → ∞ ρ ( s i , F ( Φ ) ∩ F ( Ψ ) ) = 0 .

Proof

Proof follows from Theorem 3.3 and Remark 2.1.□

Every complete CAT(0) space is complete UCW-hyperbolic space with strictly increasing modulus of uniform convexity. Hence, the following corollary follows from Theorem 3.3.

Corollary 3.6

Let S ≠ ∅ be a closed convex subset of a complete CAT(0) space X with strictly increasing modulus of uniform convexity. Let Φ , Ψ : S → S be as in Definition 2.4 with F ( Φ ) ∩ F ( Ψ ) ≠ ∅ . For s 0 ∈ S , let { s i } be as in ( ∗ ). Then { s i } → s ∈ F ( Φ ) ∩ F ( Ψ ) iff lim inf i → ∞ ρ ( s i , F ( Φ ) ∩ F ( Ψ ) ) = 0 .

Corollary 3.7

Let S ≠ ∅ be a closed convex subset of a UCW-hyperbolic space X with strictly increasing modulus of uniform convexity. Let Φ , Ψ : S → S be a pair of nonexpansive mappings with F ( Φ ) ∩ F ( Ψ ) ≠ ∅ . For s 0 ∈ S , let { s i } be as in ( ∗ ). Then { s i } → s ∈ F ( Φ ) ∩ F ( Ψ ) iff lim inf i → ∞ ρ ( s i , F ( Φ ) ∩ F ( Ψ ) ) = 0 .

Proof

Every nonexpansive mapping is Reich-Suzuki-type nonexpansive. Hence, the result follows from Theorem 3.3.□

Two mappings Φ , Ψ : S → S said to satisfy condition (A) [30], if there exists a nondecreasing function h : [ 0 , ∞ ) → [ 0 , ∞ ) with h ( 0 ) = 0 and h ( r ) > 0 for r > 0 such that

ρ ( a , Φ a ) ≥ h ( ρ ( a , F ( Φ ) ∩ F ( Ψ ) ) )

ρ ( a , Ψ a ) ≥ h ( ρ ( a , F ( Φ ) ∩ F ( Ψ ) ) ) ,

for all a ∈ S .

Theorem 3.4

Let S ≠ ∅ be a closed convex subset of a complete UCW-hyperbolic space X with strictly increasing modulus of uniform convexity. Let Φ , Ψ : S → S be as in Definition 2.4 with F ( Φ ) ∩ F ( Ψ ) ≠ ∅ . For s 0 ∈ S , let { s i } be as in ( ∗ ). If Φ , Ψ : S → S satisfies condition (A), then { s i } → s for some s ∈ F ( Φ ) ∩ F ( Ψ ) .

Proof

From inequality (24), we have lim i → ∞ ρ ( s i , F ( Φ ) ∩ F ( Ψ ) ) exists and Theorem 3.1 gives

lim i → ∞ ρ ( s i , Φ s i ) = 0 = lim i → ∞ ρ ( s i , Ψ s i ) .

Now using condition (A), we have

lim i → ∞ h ( ρ ( s i , F ( Φ ) ∩ F ( Ψ ) ) ) ≤ lim i → ∞ ρ ( s i , Φ s i ) = 0

lim i → ∞ h ( ρ ( s i , F ( Φ ) ∩ F ( Ψ ) ) ) ≤ lim i → ∞ ρ ( s i , Ψ s i ) = 0 .

Thus, lim i → ∞ h ( ρ ( s i , F ( Φ ) ∩ F ( Ψ ) ) ) = 0 .

Since h is a nondecreasing function satisfying h ( 0 ) = 0 and h ( r ) > 0 for r ∈ ( 0 , ∞ ) , we have lim i → ∞ ρ ( s i , F ( Φ ) ∩ F ( Ψ ) ) = 0 . Thus from Theorem 3.3, the sequence { s i } → s ∈ F ( Φ ) ∩ F ( Ψ ) .□

Corollary 3.8

Let S ≠ ∅ be a closed convex subset of a complete UCW-hyperbolic space X with strictly increasing modulus of uniform convexity. Let Φ , Ψ : S → S be Kannan nonexpansive mappings with F ( Φ ) ∩ F ( Ψ ) ≠ ∅ . For s 0 ∈ S , let { s i } be as in ( ∗ ). If Φ , Ψ : S → S satisfies condition (A), then { s i } → s for some s ∈ F ( Φ ) ∩ F ( Ψ ) .

Proof

Proof follows from Theorem 3.4 and Remark 2.1.□

Corollary 3.9

Let S ≠ ∅ be a closed convex subset of a complete CAT(0) space X with strictly increasing modulus of uniform convexity. Let Φ , Ψ : S → S be as in Definition 2.4 satisfying condition (A) and F ( Φ ) ∩ F ( Ψ ) ≠ ∅ . For s 0 ∈ S , let { s i } be as in ( ∗ ). Then { s i } → s for some s ∈ F ( Φ ) ∩ F ( Ψ ) .

Proof

Every complete CAT(0) space is UCW-hyperbolic space with strictly increasing modulus of uniform convexity. Hence, the proof follows from Theorem 3.4.□

Corollary 3.10

Let S ≠ ∅ be a closed convex subset of a complete UCW-hyperbolic space X with strictly increasing modulus of uniform convexity. Let Φ , Ψ : S → S be Suzuki-type nonexpansive mapping satisfying condition (A) and F ( Φ ) ∩ F ( Ψ ) ≠ ∅ . For s 0 ∈ S , let { s i } be as in ( ∗ ). Then { s i } → s for some s ∈ F ( Φ ) ∩ F ( Ψ ) .

Proof

Every nonexpansive mapping is Reich-Suzuki-type nonexpansive. Hence, the result follows from Theorem 3.4.□

4 Numerical example

Example 4.1

The following is an example for a Reich-Suzuki-type nonexpansive mapping. Let X = R 2 and consider the metric ρ on X defined by

(25) ρ ( ( u 1 , v 1 ) , ( u 2 , v 2 ) ) = ∣ v 1 − v 2 ∣ , if u 1 = u 2 ∣ u 1 − u 2 ∣ + ∣ v 1 ∣ + ∣ v 2 ∣ , if u 1 ≠ u 2 .

We can see that ( R 2 , ρ ) is a complete R -tree [10]. Hence, it is a CAT ( 0 ) space [8]. Consider the sets,

(26) S = ( 0 , y ) : 0 ≤ y ≤ 1 2 and T = 1 2 , y : 0 ≤ y ≤ 1 2 .

Define the function, ξ : S ∪ T → S ∪ T as follows:

(27) ξ ( u , v ) = 1 2 , v 2 , if ( u , v ) ∈ S 0 , v 2 , if ( u , v ) ∈ T .

Now, we prove that ξ is a Reich-Suzuki-type nonexpansive mapping.

Consider the following cases.

Using (25), (26), and (27), we have

Case 1: ( u 1 , v 1 ) , ( u 2 , v 2 ) ∈ S , where u 1 = u 2 = 0 and 0 ≤ v 1 , v 2 ≤ 1 2 .

ρ ( ξ ( u 1 , v 1 ) , ξ ( u 2 , v 2 ) ) = ρ 1 2 , v 1 2 , 1 2 , v 2 2 = ∣ v 1 2 − v 2 2 ∣ ≤ 1 4 + 1 4 ≤ 1 2 1 2 + ∣ v 1 ∣ + ∣ v 1 ∣ 2 + 1 2 1 2 + ∣ v 2 ∣ + ∣ v 2 ∣ 2 = 1 2 ρ ( 0 , v 1 ) , 1 2 , v 1 2 + 1 2 ρ ( 0 , v 2 ) , 1 2 , v 2 2 = 1 2 ρ ( ( u 1 , v 1 ) , ξ ( u 1 , v 1 ) ) + 1 2 ρ ( ( u 2 , v 2 ) , ξ ( u 2 , v 2 ) ) .

Case 2: ( u 1 , v 1 ) , ( u 2 , v 2 ) ∈ T , where u 1 = u 2 = 1 2 and 0 ≤ v 1 , v 2 ≤ 1 2 .

ρ ( ξ ( u 1 , v 1 ) , ξ ( u 2 , v 2 ) ) = ρ 0 , v 1 2 , 0 , v 2 2 = v 1 2 − v 2 2 ≤ 1 2 ∣ v 1 ∣ + 1 2 ∣ v 2 ∣ ≤ 1 2 1 2 + ∣ v 1 ∣ + v 1 2 + 1 2 1 2 + ∣ v 2 ∣ + v 2 2 = 1 2 ρ 1 2 , v 1 , 0 , v 1 2 + 1 2 ρ 1 2 , v 2 , 0 , v 2 2 = 1 2 ρ ( ( u 1 , v 1 ) , ξ ( u 1 , v 1 ) ) + 1 2 ρ ( ( u 2 , v 2 ) , ξ ( u 2 , v 2 ) ) .

Case 3: ( u 1 , v 1 ) ∈ S , ( u 2 , v 2 ) ∈ T , where u 1 = 0 , u 2 = 1 2 , and 0 ≤ v 1 , v 2 ≤ 1 2 .

ρ ( ξ ( u 1 , v 1 ) , ξ ( u 2 , v 2 ) ) = ρ 1 2 , v 1 2 , 0 , v 2 2 = 1 2 + ∣ v 1 ∣ 2 + v 2 2 ≤ 1 2 + 1 2 ∣ v 1 ∣ 2 + 1 2 ∣ v 1 ∣ 2 + 1 2 v 2 2 + v 2 2 ≤ 1 4 + 1 2 ∣ v 1 ∣ + 1 2 ∣ v 1 ∣ 2 + 1 4 + 1 2 v 2 2 + 1 2 ∣ v 2 ∣ = 1 2 1 2 + ∣ v 1 ∣ + ∣ v 1 ∣ 2 + 1 2 1 2 + ∣ v 2 ∣ + v 2 2 = 1 2 ρ ( 0 , v 1 ) , 1 2 , v 1 2 + 1 2 ρ 1 2 , v 2 , 0 , v 2 2 = 1 2 [ ρ ( ( u 1 , v 1 ) , ξ ( u 1 , v 1 ) ) ] + 1 2 [ ρ ( ( u 2 , v 2 ) , ξ ( u 2 , v 2 ) ) ] .

Thus, in all the cases we have

ρ ( ξ ( u 1 , v 1 ) , ξ ( u 2 , v 2 ) ) ≤ 1 2 [ ρ ( ( u 1 , v 1 ) , ξ ( u 1 , v 1 ) ) ] + 1 2 [ ρ ( ( u 2 , v 2 ) , ξ ( u 2 , v 2 ) ) ] ,

i.e.,

ρ ( ξ ( u 1 , v 1 ) , ξ ( u 2 , v 2 ) ) ≤ 1 2 [ ρ ( ( u 1 , v 1 ) , ξ ( u 1 , v 1 ) ) ] + 1 2 [ ρ ( ( u 2 , v 2 ) , ξ ( u 2 , v 2 ) ) ] + ( 1 − 2 α ) [ ρ ( ( u 1 , v 1 ) , ( u 2 , v 2 ) ) ] , for α = 1 2 .

Hence, ξ is Reich-Suzuki-type nonexpansive for α = 1 2 .

Now we illustrate Theorem 3.1 using two Reich-Suzuki-type nonexpansive mappings.

Example 4.2

Consider the function

(28) Φ ( s ) = − s 3 , if s ∈ [ − 2 , 0 ) ⧹ − 1 4 0 , if s = − 1 4 − s 5 , if s ∈ [ 0 , 2 ] .

Clearly, 0 is the fixed point of Φ , and Φ is not continuous at s = − 1 4 . Hence, Φ is not nonexpansive. Now we prove that Φ is Reich-Suzuki-type nonexpansive with constant α = 1 2 . Consider the following cases.

Case-1: s , w ∈ [ − 2 , 0 ) ⧹ − 1 4 . Consider

ρ ( Φ s , Φ w ) = 1 3 ∣ s − w ∣ ≤ 1 3 ∣ s ∣ + 1 3 ∣ w ∣ ≤ 2 3 ∣ s ∣ + 2 3 ∣ w ∣ = 1 2 s + s 3 + 1 2 w + w 3 = 1 2 ρ ( s , Φ s ) + 1 2 ρ ( w , Φ w ) .

Case-2: s , w ∈ [ 0 , 2 ] .

Consider

ρ ( Φ s , Φ w ) = 1 5 ∣ s − w ∣ ≤ 1 5 ∣ s ∣ + 1 5 ∣ w ∣ ≤ 3 5 ∣ s ∣ + 3 5 ∣ w ∣ = 1 2 s + s 5 + 1 2 w + w 5 = 1 2 ρ ( s , Φ s ) + 1 2 ρ ( w , Φ w ) .

Case-3: s ∈ [ − 2 , 0 ) , w ∈ [ 0 , 2 ] .

Consider

ρ ( Φ s , Φ w ) = 1 3 ∣ s ∣ + 1 5 ∣ w ∣ ≤ 2 3 ∣ s ∣ + 3 5 ∣ w ∣ ≤ 1 2 s + s 3 + 1 2 w + w 5 = 1 2 ρ ( s , Φ s ) + 1 2 ρ ( w , Φ w ) .

Case-4: s ∈ [ − 2 , 0 ) , w = − 1 4 .

Consider

ρ ( Φ s , Φ w ) = − s 3 − 0 ≤ 4 6 ∣ s ∣ + 1 8 = 1 2 s + s 3 + 1 2 − 1 4 + 0 = 1 2 ρ ( s , Φ s ) + 1 2 ρ ( w , Φ w ) .

Case-5: s ∈ [ 0 , 2 ] , w = − 1 4 .

Consider

ρ ( Φ s , Φ w ) = − s 5 + 0 ≤ 3 5 ∣ s ∣ + 1 8 = 1 2 s + s 5 + 1 2 1 4 + 0 = 1 2 ρ ( s , Φ s ) + 1 2 ρ ( w , Φ w ) .

Thus, in all the cases we have ρ ( Φ s , Φ w ) ≤ 1 2 ρ ( s , Φ s ) + 1 2 ρ ( w , Φ w ) , i.e.,

ρ ( Φ s , Φ w ) ≤ α ρ ( s , Φ s ) + α ρ ( w , Φ w ) + ( 1 − 2 α ) ρ ( s , w ) for α = 1 2 .

Hence, Φ is Reich-Suzuki-type nonexpansive for α = 1 2 .

Now consider the function

(29) Ψ ( s ) = − s 4 , if s ∈ [ − 2 , 0 ) ⧹ − 1 4 0 , if s = − 1 4 − s 6 , if s ∈ [ 0 , 2 ] .

Now we prove that Ψ is Reich-Suzuki-type nonexpansive with constant α = 1 2 . Consider the following cases.

Case-1: s , w ∈ [ − 2 , 0 ) ⧹ − 1 4 .

Consider

ρ ( Ψ s , Ψ w ) = 1 4 ∣ s − w ∣ ≤ 1 4 ∣ s ∣ + 1 4 ∣ w ∣ ≤ 5 8 ∣ s ∣ + 5 8 ∣ w ∣ = 1 2 s + s 4 + 1 2 w + w 4 = 1 2 ρ ( s , Ψ s ) + 1 2 ρ ( w , Ψ w ) .

Case-2: s , w ∈ [ 0 , 2 ] .

Consider

ρ ( Ψ s , Ψ w ) = 1 6 ∣ s − w ∣ ≤ 1 6 ∣ s ∣ + 1 6 ∣ w ∣ ≤ 7 12 ∣ s ∣ + 7 12 ∣ w ∣ = 1 2 s + s 6 + 1 2 w + w 6 = 1 2 ρ ( s , Ψ s ) + 1 2 ρ ( w , Ψ w ) .

Case-3: s ∈ [ − 2 , 0 ) ⧹ − 1 4 , w ∈ [ 0 , 2 ] .

Consider

ρ ( Ψ s , Ψ w ) = − s 4 + w 6 ≤ 1 4 ∣ s ∣ + 1 6 ∣ w ∣ ≤ 5 8 ∣ s ∣ + 7 12 ∣ w ∣ = 1 2 s + s 4 + 1 2 w + w 6 = 1 2 ρ ( s , Ψ s ) + 1 2 ρ ( w , Ψ w ) .

Case-4: s ∈ [ − 2 , 0 ) ⧹ − 1 4 , w = − 1 4 .

Consider

ρ ( Ψ s , Ψ w ) = − s 4 ≤ 5 8 ∣ s ∣ + 1 8 = 1 2 s + s 4 + 1 2 1 4 = 1 2 ρ ( s , Ψ s ) + 1 2 ρ ( w , Ψ w ) .

Case-5: s ∈ [ 0 , 2 ] , w = − 1 4 .

Consider

ρ ( Ψ s , Ψ w ) = − s 6 + 0 ≤ 7 12 ∣ s ∣ + 1 8 = 1 2 s + s 6 + 1 2 − 1 4 + 0 = 1 2 ρ ( s , Ψ s ) + 1 2 ρ ( w , Ψ w ) .

Thus, in all the cases we have ρ ( Ψ s , Ψ w ) ≤ 1 2 ρ ( s , Ψ s ) + 1 2 ρ ( w , Ψ w ) , i.e.,

ρ ( Ψ s , Ψ w ) ≤ α ρ ( s , Ψ s ) + α ρ ( w , Ψ w ) + ( 1 − 2 α ) ρ ( s , w ) , for α = 1 2 .

Hence, Ψ is Reich-Suzuki-type nonexpansive with α = 1 2 .

Clearly, 0 is the common fixed point of Φ and Ψ . Table 1 shows that our new iteration ( ∗ ) converges faster than iteration (1) to the fixed point 0. Matlab code is used to show the convergence with two initial values s 0 = − 1.2 and s 0 = 0.8 and set γ i = δ i = η i = 1 4 (Figure 1).

Table 1

Convergence of iterations

i	Initial value s 0 = − 1.2		Initial value s 0 = 0.8
	s i + 1 for iteration ( ∗ )	s i + 1 for iteration (1)	s i + 1 for iteration ( ∗ )	s i + 1 for iteration (1)
1	− 0.047222222222222	0.148750000000000	− 0.028833333333333	− 0.072000000000000
2	− 0.001858281893004	− 0.018438802083333	0.001039201388889	0.006480000000000
3	− 0.000073126833752	0.002285643174913	− 0.000037454550058	− 0.000583200000000
4	− 0.000002877676328	− 0.000283324518557	0.000001349924408	0.000052488000000
5	− 0.000000113241893	0.000035120435113	− 0.000000048653526	− 0.000004723920000
6	− 0.000000004456278	− 0.000004353470603	0.000000001753554	0.000000425152800
7	− 0.000000000175363	0.000000539648960	− 0.000000000063201	− 0.000000038263752
8	− 0.000000000006901	− 0.000000066893986	0.000000000002278	0.000000003443738
9	− 0.000000000000272	0.000000008292067	− 0.000000000000082	− 0.000000000309936
10	− 0.000000000000011	− 0.000000001027871	0.000000000000003	0.000000000027894
11	0.000000000000000	0.000000000127413	0.000000000000000	− 0.000000000002510
⋮		⋮		⋮
14		−0.000000000000243		0.000000000000002
15		0.000000000000030		0.000000000000000
16		− 0.000000000000004
17		0.000000000000000

Figure 1

Convergence of iterations.

5 Application

In this section, we analyze how our main result can be applied to the nonlinear integral equation of the form:

(30) q ( α ) = h ( α ) + γ ∫ 0 1 χ ( α , β ) g ( ( α , β , q ( β ) ) ) d β .

Let L 2 ( [ 0 , 1 ] ) denote the set of all square integrable functions on [0,1] with innerproduct ⟨ q , s ⟩ = ∫ 0 1 q ( α ) ¯ s ( α ) d α for all q , s ∈ L 2 ( [ 0 , 1 ] ) .

Theorem 5.1

Let S ≠ ∅ be a closed convex subset of the Hilbert space L 2 ( [ 0 , 1 ] ) and Λ : S → S defined by

(31) Λ q ( α ) = h ( α ) + γ ∫ 0 1 χ ( α , β ) g ( ( α , β , q ( β ) ) ) d β , α , β ∈ [ 0 , 1 ] , γ ≥ 0 .

Assume that the following conditions hold:

h : [ 0 , 1 ] → R and g : [ 0 , 1 ] × [ 0 , 1 ] × R → R are continuous.
There exists a constant k ≥ 0 such that ∣ g ( α , β , q ) − g ( α , β , s ) ∣ ≤ k ∣ q ( α ) − s ( α ) ∣ for all α , β ∈ [ 0 , 1 ] .
χ : [ 0 , 1 ] × [ 0 , 1 ] → R is continuous with ∣ χ ( α , β ) ∣ ≤ λ for some λ ≥ 0 .
γ λ k ≤ 1 .

Then equation (30) has a solution iff ‖ Λ s i − s i ‖ → 0 , where { s i } is as in ( ∗ ). If Λ ( S ) is contained in a compact set, then { s i } converges to the fixed point of Λ .

Proof

Let q , s ∈ L 2 ( [ 0 , 1 ] ) , then

∣ Λ q ( α ) − Λ s ( α ) ∣ 2 = h ( α ) + γ ∫ 0 1 χ ( α , β ) g ( α , β , q ( β ) ) d β − h ( α ) − γ ∫ 0 1 χ ( α , β ) g ( α , β , s ( β ) ) d β 2 ≤ γ 2 ∫ 0 1 ∣ χ ( α , β ) ∣ 2 ∣ g ( α , β , q ( β ) ) − g ( α , β , s ( β ) ) ∣ 2 d β ≤ γ 2 λ 2 k 2 ∫ 0 1 ∣ q ( β ) − s ( β ) ∣ 2 d β .

Since γ λ k ≤ 1 , we obtain

‖ Λ q − Λ s ‖ ≤ ‖ q − s ‖ .

Now take α = 0 and Ψ = Λ in Definition 2.4, then we obtain Λ is Reich-Suzuki-type nonexpansive. Now take Φ = Ψ = Λ in ( ∗ ). Thus, all conditions in Theorem 3.1 are satisfied. Hence, equation (30) has a solution in S ⊆ L 2 ( [ 0 , 1 ] ) iff ‖ Λ s i − s i ‖ → 0 . If Λ ( S ) is contained in a compact set, then using Corollary 3.3, we obtain { s i } converges to the fixed point of Λ .□

6 Conclusion

In this work, we presented some fixed-point results for a pair of Reich-Suzuki-type nonexpansive mappings in UCW-hyperbolic spaces. We introduced a new iterative scheme and established its convergence to the fixed points of a pair of Reich-Suzuki-type nonexpansive mappings and illustrated our main result with an example. We also observed the convergence of our iteration to fixed points of Reich-Suzuki-type nonexpansive mappings and compared the convergence with iteration (1) for a pair of Reich-Suzuki-type nonexpansive mappings using Matlab code and found that our new iteration converges faster than iteration (1). An application is given to substantiate our main result.

sreya.v.v@gmail.com

Acknowledgements

Sreya Valiya Valappil is highly grateful to University Grant Commission, India, for providing financial support in the form of Junior/Senior Research fellowship.

Conflict of interest: The authors declare no conflict of interest.

References

[1] M. Abbas and T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Matematicki Vesnik 66 (2014), no. 2, 223–234. Suche in Google Scholar

[2] O. Alqahtani, V. M. Himabindu, and E. Karapınar, On Pata-Suzuki-type contractions, Mathematics, 7 (2019), no. 8, 720. 10.3390/math7080720Suche in Google Scholar

[3] H. H. Alsulami, E. Karapınar, H. Piri, Fixed points of generalized-Suzuki-type contraction in complete-metric spaces, Disc. Dyn. Nature Soc. 2015 (2015), 1–8.10.1155/2015/969726Suche in Google Scholar

[4] M. Asadi, E. Karapınar, and A. Kumar, alpha-psi Geraghty contractions on generalized metric spaces, J. Inequalit. Appl. 2014 (2014), no. 1, 1–21, DOI: https://doi.org/10.1186/1029-242X-2014-423. 10.1186/1029-242X-2014-423Suche in Google Scholar

[5] M. Asadi, On Ekeland’s variational principle in M-metric spaces, J. Nonlinear Convex Anal. 17 (2016), no. 6, 1151–1158, http://www.yokohamapublishers.jp/online2/opjnca/vol17/p1151.html. Suche in Google Scholar

[6] M. Asadi, Discontinuity of control function in the (F,Φ,θ)-contraction in metric spaces, Filomat 31 (2017), no. 17, 5427–5433, DOI: https://doi.org/10.2298/FIL1717427A. 10.2298/FIL1717427ASuche in Google Scholar

[7] M. Asadi, M. Azhini, E. Karapinar, and H. Monfared, Simulation functions over M-metric spaces, East Asian Math. J. 33 (2017), no. 5, 559–570, DOI: https://doi.org/10.7858/eamj.2017.039. Suche in Google Scholar

[8] M. Bridson and A. Haefliger, Metric Spaces of Nonpositive Curvature, Springer-Verlag, Berlin, 1999. 10.1007/978-3-662-12494-9Suche in Google Scholar

[9] F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. National Academy Sci. 54 (1965), no. 4, 1041–1044. 10.1073/pnas.54.4.1041Suche in Google Scholar PubMed PubMed Central

[10] R. Espinola and P. Lorenzo, Metric fixed point theory on hyperconvex spaces: Recent progress, Arab J. Math. (Springer) 1 (2012), 439–463. 10.1007/s40065-012-0044-zSuche in Google Scholar

[11] M. Gabeleh and N. Shahzad, Best proximity points, cyclic Kannan maps and geodesic metric spaces, J. Fixed Point Theory Appl. 18 (2016), no. 1, 167–188. 10.1007/s11784-015-0272-xSuche in Google Scholar

[12] C. Garodia, I. Uddin, and D. Baleanu, On constrained minimization, variational inequality and split feasibility problem via new iteration scheme in Banach spaces, Bullet. Iranian Math. Soc. 2021 (2021), 1–20. 10.1007/s41980-021-00596-6Suche in Google Scholar

[13] D. Göhde, Zum prinzip der kontraktiven abbildung, Math. Nachrichten 30 (1965), no. 3–4, 251–258. 10.1002/mana.19650300312Suche in Google Scholar

[14] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Am. Math. Monthly 72 (1965), no. 9, 1004–1006. 10.2307/2313345Suche in Google Scholar

[15] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), no. 1, 147. 10.1090/S0002-9939-1974-0336469-5Suche in Google Scholar

[16] E. Karapinar, Remarks on Suzuki (C)-condition. In: Dynamical Systems and Methods, Springer, New York, NY, 2012, pp. 227–243. 10.1007/978-1-4614-0454-5_12Suche in Google Scholar

[17] E. Karapınar, H. Salahifard, and S. Mansour Vaezpour, Demiclosedness principle for total asymptotically nonexpansive mappings in spaces, J. Appl. Math. 2014 (2014), 1–11. 10.1155/2014/738150Suche in Google Scholar

[18] E. Karapınar, A. Fulga, and S. S. Yesilkaya, New results on Perov-interpolative contractions of Suzuki-type mappings, J. Funct. Spaces 2021 (2021), 1–7. 10.1155/2021/9587604Suche in Google Scholar

[19] S. Khatoon, I. Uddin, and D. Baleanu, Approximation of fixed point and its application to fractional differential equation, J. App. Math. Comp. 66 (2021), no. 1, 507–525. 10.1007/s12190-020-01445-1Suche in Google Scholar

[20] M. A. Kutbi, E. Karapınar, J. Ahmad, and A. Azam, Some fixed point results for multi-valued mappings in b-metric spaces, J. Ineq. App. 2014 (2014), no. 1, 1–11. 10.1186/1029-242X-2014-126Suche in Google Scholar

[21] T. Laokul, B. Panyanak, N. Phudolsitthiphat, and S. Suantai, Common endpoints of generalized Suzuki-Kannan-Ćiriç type mappings in hyperbolic spaces, Carpathian J. Math., 38 (2022), no. 2, 445–457. 10.37193/CJM.2022.02.14Suche in Google Scholar

[22] W. Laowang and B. Panyanak, Approximating fixed points of nonexpansive nonself mappings in CAT (0) spaces, Fixed Point Theory Appl. 2010 (2009), 1–11. 10.1155/2010/367274Suche in Google Scholar

[23] L. Leustean, Nonexpansive iterations in uniformly convex W-hyperbolic spaces, in: A. Leizarowitz, B. S. Mordukhovich, I. Shafrir, and A. Zaslavski, (eds.), Nonlinear Analysis and Optimization I: Nonlinear Analysis, Contemporary Mathematics, vol. 513, American Mathematical Society, Providence, 2010, pp. 193–209. 10.1090/conm/513/10084Suche in Google Scholar

[24] L. Leustean, A quadratic rate of asymptotically regularity for CAT (0)-spaces, J. Math. Anal. Appl. 325 (2007), 386–399. 10.1016/j.jmaa.2006.01.081Suche in Google Scholar

[25] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), no. 3, 506–510. 10.1090/S0002-9939-1953-0054846-3Suche in Google Scholar

[26] H. Monfared, M. Asadi, M. Azhini, and D. O’Regan, F(ψ,φ)-Contractions for α-admissible mappings on M-metric spaces, Fixed Point Theory Appl. 2018 (2018), no. 1, 1–17, https://link.springer.com/article/10.1186/s13663-018-0647-y. 10.1186/s13663-018-0647-ySuche in Google Scholar

[27] M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000), no. 1, 217–229. 10.1006/jmaa.2000.7042Suche in Google Scholar

[28] A. E. Ofem, H. Işık, F. Ali, and J. Ahmad, A new iterative approximation scheme for Reich-Suzuki-type nonexpansive operators with an application, J. Inequalit. Appl. 2022 (2022), no. 1, 1–26. 10.1186/s13660-022-02762-8Suche in Google Scholar

[29] R. Pant, and R. Pandey, Existence and convergence results for a class of nonexpansive-type mappings in hyperbolic spaces, Appl. General Topol. 20 (2019), no. 1, 281–295. 10.4995/agt.2019.11057Suche in Google Scholar

[30] H. F. Senter and W. G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44 (1974), no. 2, 375–380. 10.1090/S0002-9939-1974-0346608-8Suche in Google Scholar

[31] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, Journal Math. Anal. Appl. 340 (2008), no. 2, 1088–1095. 10.1016/j.jmaa.2007.09.023Suche in Google Scholar

[32] W. Takahashi, A convexity in metric space and nonexpansive mappings, I, In: Kodai Mathematical Seminar Reports, vol. 22, no. 2, Department of Mathematics, Tokyo Institute of Technology, 1970, pp. 142–149. 10.2996/kmj/1138846111Suche in Google Scholar

[33] I. Uddin, S. Aggarwal, and A. A. Abdou, Approximation of endpoints for α-Reich-Suzuki nonexpansive mappings in hyperbolic metric spaces, Mathematics 9 (2021), no. 1, 1692. 10.3390/math9141692Suche in Google Scholar

[34] K. Ullah, J. Ahmad, M. De La Sen, and M. N. De La Sen, Approximating fixed points of Reich-Suzuki-type nonexpansive mappings in hyperbolic spaces, J. Math. 2020 (2020), 1–6. 10.1155/2020/2169652Suche in Google Scholar

[35] K. Ullah and M. Arshad, New iteration process and numerical reckoning fixed point in Banach spaces, University Politehnica Bucharest Scientific Bulletin Series A, 79 (2017), no. 4, 113–122. Suche in Google Scholar

[36] M. Younis, D. Singh, M. Asadi, and V. Joshi, Results on contractions of Reich type in graphical b-metric spaces with applications, Filomat, 33 (2019), no. 17, 5723–5735, DOI: https://doi.org/10.2298/FIL1917723Y. 10.2298/FIL1917723YSuche in Google Scholar

Received: 2022-07-30

Revised: 2022-11-22

Accepted: 2022-12-22

Published Online: 2023-02-18

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hyperbolic space; geodesic space; Reich-Suzuki-type nonexpansive map; fixed point

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