Home A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
Article Open Access

A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study

  • Fayyaz Ahmad , Kifayat Ullah , Junaid Ahmad , Hasanen A. Hammad EMAIL logo and Reny George
Published/Copyright: May 11, 2024
Download Article (PDF)
Become an author with De Gruyter Brill
Author Information Explore this Subject
Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 57 Issue 1

Abstract

This manuscript is devoted to constructing a novel iterative scheme and reckoning of fixed points for generalized contraction mappings in hyperbolic spaces. Also, we establish Δ and strong convergence results by the considered iteration under the class of mappings satisfying condition (E). Moreover, some qualitative results of the suggested iteration, like weak w 2 -stability and data dependence results, are discussed. Furthermore, to test the efficiency and effectiveness of the proposed iteration, practical experiments are given. To support the theoretical results, illustrative examples are presented. Finally, our results improve and generalize several classical results in the literature of fixed point iterations.

Keywords: iteration process; fixed point; convergence result; data dependence; stability; hyperbolic space
MSC 2010: 47H05; 47H09; 47H10

1 Introduction and preliminaries

Fixed points are considered as the pillar of nonlinear analysis in general and semantic analysis in particular, as they are the bright side that made the turnout for these branches very large by many writers, due to their ease and flexibility. It is common that a self-map J defined on a nonempty subset A of a metric space H is called a Banach-contraction mapping (or a contraction mapping) if for every two elements a , b ∈ A , there exists at least one real number λ ∈ [ 0 , 1 ) such that

(1.1) ϱ ( J a , J b ) ≤ λ ϱ ( a , b ) .

Clearly, if the inequality (1.1) holds for λ = 1 , then J is called a nonexpansive mapping. Once a point a ∗ is available in the domain A with the property J a ∗ = a ∗ , then the point is called a fixed point for J and the relation J a = a is called a fixed point problem. We denote the set of all fixed points of the mapping J by F J . Almost all problems in engineering, mathematical physics, chemistry, and other subjects can be written as fixed-point problems.

In 1922, Banach [1] carried out a basic fixed point theorem for contractions called the Banach contraction principle (BCP). The BCP is important because it suggests the existence and uniqueness of a fixed point and, at the same time, suggests an iterative process called the Picard iteration for calculating this fixed point. This principle is widely applied for finding sought solutions of differential, integral and other types of equations in applied sciences [2–5].

On the other hand, it was an open question until 1965 whether a nonexpansive mapping admits a fixed point in the closed subset of a complete metric space and, if it admits a fixed point, whether the Picard iteration converges to this fixed point. Thus, in the same year, Browder [6], Gohde [7], and Kirk [8] proposed a basic fixed point theorem for nonexpansive mappings in a Banach space setting.

Osilike [9] introduced a generalization of Banach-contraction as follows: The mapping J is called a generalized contraction if for every a , b ∈ A , there exist λ ∈ [ 0 , 1 ) and L ∈ [ 0 , ∞ ) such that

(1.2) ϱ ( J a , J b ) ≤ λ ϱ ( a , b ) + L ϱ ( a , J a ) .

Clearly, the Banach contraction mapping satisfies equation (1.2) with L = 0 , and the class of generalized contractions contains properly the class of the Banach contraction mapping.

In 2008, Suzuki generalized nonexpansive mappings as follows: A mapping J is said to satisfy the condition (C) if, for all a , b ∈ A , one has

1 2 ϱ ( a , J a ) ≤ ϱ ( a , b ) ⇒ ϱ ( J a , J b ) ≤ ϱ ( a , b ) .

Recently, Garcia-Falset et al. [10] generalized the above notion as follows: The mapping J is said to satisfy the condition (E) if, for all a , b ∈ A , there exists μ ≥ 1 such that

ϱ ( a , J b ) ≤ μ ϱ ( J a , J b ) + ϱ ( a , b ) .

The same authors suggested the following properties of mappings, which satisfy condition (E):

Proposition 1.1

Let J : A → A be a given mapping. Then,

  1. if J satisfies condition ( C ) , then J satisfies the condition ( E ) ;

  2. if J satisfies the condition ( E ) , then for all a ∈ A and for all a ∗ ∈ F J , one has ϱ ( J a , J a ∗ ) ≤ ϱ ( a , a ∗ ) ;

  3. if J satisfies the condition ( E ) , then the set F J is essentially closed in A.

The idea of a hyperbolic space introduced by Kohlenbach [11] is as follows:

Definition 1.2

Let ( H , ϱ ) be a metric space, then ( H , ϱ , S ) is called a hyperbolic space if the function S : H × H × [ 0 , 1 ] → H fulfills

  1. ϱ ( r , S ( a , b , α ) ) ≤ α ϱ ( r , a ) + ( 1 − α ) ϱ ( r , b ) ;

  2. ϱ ( S ( a , b , α ) , S ( a , b , β ) ) ≤ ∣ α − β ∣ ϱ ( a , b ) ;

  3. S ( a , b , α ) = S ( b , a , 1 − α ) ;

  4. ϱ ( S ( a , r , α ) , S ( b , s , α ) ) ≤ α ϱ ( a , b ) + ( 1 − α ) ϱ ( r , s )

for all a , b , r , s ∈ H and 0 ≤ α , β ≤ 1 .

A Banach space is an example of a hyperbolic metric space that is linear. Examples of this type of space, that are nonlinear, include Hadamard manifolds and the Hilbert open unit ball. The hyperbolic space H defined above is more general than the hyperbolic spaces in the sense of Takahashi [12], Goebel and Kirk [13], and Reich and Shafrir [14]. Hence, our results will be simultaneously valid in the other hyperbolic space mentioned above. All norm space (and hence all inner product spaces), Hadamarad manifolds, and metric trees are the well-known examples of hyperbolic spaces [15,16]. In this case, for any pair of elements a , b ∈ H and any scalar α ∈ [ 0 , 1 ] , the following holds:

ϱ ( a , S ( a , b , α ) ) = α ϱ ( a , b ) and ϱ ( a , S ( a , b , α ) ) = ( 1 − α ) ϱ ( a , b ) .

It follows that

(1.3) S ( a , b , 1 ) = a and S ( a , b , 0 ) = b .

Definition 1.3

A subset A of a hyperbolic space H is called convex, if and only if S ( a , b , α ) ∈ A for all a , b ∈ A and α ∈ [ 0 , 1 ] .

Definition 1.4

[17] Assume that ( H , ϱ ) is hyperbolic space. We say that H is a uniformly convex if for any 0 < r < ∞ and for each 0 < ε ≤ 2 , there is a 0 < θ < 1 in the way that for any choice of a , u , v ∈ H with ϱ ( u , a ) ≤ r , ϱ ( v , a ) ≤ r , and ρ ( u , v ) ≥ ε r , we have

ϱ S u , v , 1 2 , a ( 1 − θ ) ≤ r .

Definition 1.5

[17] A monotone modulus of uniform convexity (MMUC) is a function Y on ( 0 , ∞ ) × ( 0 , 2 ) to the interval ( 0 , 1 ] that satisfy η = Y ( r , ε ) for any choice of 0 < r < ∞ and 0 < ε < 2 and Y is monotone (i.e., Y decreases with r and ε ).

Recently, Deshmukh et al. [18] proposed two new iterative schemes called JF and Thakur iterations for finding fixed points of a general class of nonexpansive mappings. Now, we assume that α r , β r , γ r ⊂ ( 0 , 1 ) , then by using hypothesis ( S 3 ) and equation (1.3), Mann [19], Khan and Picard-Mann [20], Deshmukh et al. [18], Ullah and Arshad [21], Ali and Ali [22], and Deshmukh et al. [18] iterative processes are, respectively, defined as follows (see Ahmad et al. [23], Sahin [24], and others for details):

(1.4) a 1 ∈ A , a r + 1 = S ( J a r , a r , α r ) , r ≥ 1 ,

(1.5) a 1 ∈ A , b r = S ( J a r , a r , α r ) , a r + 1 = J b r , r ≥ 1 .

(1.6) a 1 ∈ A , c r = S ( J a r , a r , γ r ) , b r = S ( J c r , c r , β r ) , a r + 1 = S ( J b r , J c r , α r ) , r ≥ 1 .

(1.7) a 1 ∈ A , c k = S ( J a r , a r , α r ) , b r = J c r , a r + 1 = J b r , r ≥ 1 .

(1.8) a 1 ∈ A , d r = S ( J a r , a r , α r ) , c r = J d r , b r = J c r , a r + 1 = J b r , r ≥ 1 .

(1.9) a 1 ∈ A , e r = S ( J a r , a r , α r ) , d r = J e r , c r = J d r , b r = S ( J c r , c r , β r ) , a r + 1 = J b r , r ≥ 1 .

Based on iterations (1.7) and (1.8), we introduce the following iteration:

(1.10) a 1 ∈ A , e r = S ( J a r , a r , α r ) , d r = J e r , c r = J d r , b r = J c r , a r + 1 = J b r , r ≥ 1 .

In addition, we establish strong convergence and Δ results for the class of mappings satisfying the condition (E) under considered iteration. Also, some qualitative results of the suggested iteration are examined, including weak w 2 -stability and data dependence results. In addition, real-world tests are conducted to evaluate the effectiveness and efficiency of the suggested iteration, and illustrative examples are provided to corroborate the theoretical results. Ultimately, our results extend and generalize a number of prior results in the literature on fixed-point iterations.

Now, we need some basic definitions and results as follows.

Definition 1.6

Assume that H is a uniformly convex hyperbolic space (UCHS), { a r } is a bounded sequence of H and A is closed convex subset of H . We shall denote the asymptotic radius of { a r } with respect to A by R ( A , { a r } ) and defined by inf { limsup r → ∞ ϱ ( a r , a ) : a ∈ A } . Also, we denote the asymptotic center of { a r } with respect to A by C ( A , { a r } ) and it is defined by { a ∈ A : limsup r → ∞ ϱ ( a r , a ) = R ( A , a r ) } .

Definition 1.7

[25] Let A be a nonempty subset of H . We say that a mapping J : A → A satisfies condition ( I ) if there exists a nondecreasing mapping g : A → A such that g ( c ) = 0 iff c = 0 , g ( c ) > 0 for each c > 0 and ϱ ( a , J a ) ≥ g ( ϱ ( a , F J ) ) .

Lemma 1.8

[26] Let A be a closed convex subset of a UCHS H with an MMUC and { a r } be any bounded sequence in H. Then, the set C ( A , { a r } ) contains a singleton point.

Definition 1.9

Let H be a UCHS. A sequence { a r } ⊂ H is said to be Δ -convergent to a point a ∗ ∈ H , if { u r } is any subsequence of { a r } and a ∗ is a unique asymptotic center of { u r } .

Lemma 1.10

[27] Assume that H is a UCHS with an MMUC. If  0 < i ≤ σ r ≤ j < 1 such that { a r } and { b r } in H satisfy the conditions limsup r → ∞ ϱ ( a r , a ∗ ) ≤ ξ , limsup r → ∞ ϱ ( b r , a ∗ ) ≤ ξ , and lim r → ∞ ϱ ( S ( b r , a r , σ r ) , a ∗ ) = ξ , where ξ ≥ 0 is a real number. Then, lim r → ∞ ϱ ( a r , b r ) = 0 .

2 Convergence results

This section is devoted to approximating fixed points of mappings with condition (E). From here to the end of the article, we shall consider H is a complete UCHS and has an MMUC. We begin this section with the following lemma:

Lemma 2.1

Let A be a closed convex subset of H and J be a self-mapping on A. If J satisfies condition (E) and F J ≠ ∅ , then the sequence { a r } produced by equation (1.10) is essentially Δ -convergent to an element of the set F J .

Proof

We split the proof into the following parts:

Part 1. Assume that a ∗ ∈ F J , we will prove that

(2.1) lim r → ∞ ϱ ( a r , a ∗ )

exists. Since a ∗ ∈ F J , then by Proposition 1.1(ii), we have

(2.2) ϱ ( e r , a ∗ ) = ϱ ( S ( J a r , α r , α r ) , a ∗ ) ≤ α r ϱ ( J a r , a ∗ ) + ( 1 − α r ) ϱ ( a r , a ∗ ) ≤ α r ϱ ( a r , a ∗ ) + ( 1 − α r ) ϱ ( a r , a ∗ ) = ϱ ( a r , a ∗ ) .

Using equation (2.2), we obtain

(2.3) ϱ ( a r + 1 , a ∗ ) = ϱ ( J b r , a ∗ ) ≤ ϱ ( b r , a ∗ ) = ϱ ( J c r , a ∗ ) ≤ ϱ ( c r , a ∗ ) = ϱ ( J d r , a ∗ ) ≤ ϱ ( d r , a ∗ ) = ϱ ( J e r , a ∗ ) ≤ ϱ ( e r , a ∗ ) ≤ ϱ ( a r , a ∗ ) .

Hence,

ϱ ( a r + 1 , a ∗ ) ≤ ϱ ( a r , a ∗ ) .

It follows that the sequence { ϱ ( a r , a ∗ ) } is bounded and nonincreasing, which implies that equation (2.1) is true.

Part 2. We show that

(2.4) lim r → ∞ ϱ ( a r , J a r ) = 0 .

Now, for any a ∗ ∈ F J of F J , put

(2.5) lim r → ∞ ϱ ( a r , a ∗ ) = ξ .

Using equation (2.1) and Proposition 1.1(ii), one has

ϱ ( J a r , a ∗ ) ≤ ϱ ( a r , a ∗ ) .

By equation (2.5), we have

(2.6) limsup r → ∞ ϱ ( J a r , a ∗ ) ≤ ξ .

From equation (2.2), we obtain

ϱ ( e r , a ∗ ) ≤ ϱ ( a r , a ∗ ) ,

Using equation (2.5), we can obtain

(2.7) limsup r → ∞ ϱ ( e r , a ∗ ) ≤ ξ .

Similarly, from equation (2.3), one can write

ϱ ( a r + 1 , a ∗ ) ≤ ϱ ( e r , a ∗ ) ,

From equation (2.5), one has

(2.8) ξ ≤ liminf r → ∞ ϱ ( e r , a ∗ ) .

From equations (2.7) and (2.8), one obtains

(2.9) ξ = lim r → ∞ ϱ ( e r , a ∗ ) .

Since e r = S ( J a r , a r , α r ) , so applying it in equation (2.9), we conclude that

(2.10) lim r → ∞ ϱ ( S ( J a r , a r , α r ) , a ∗ ) = ξ .

Considering equations (2.5), (2.6), and (2.10) with the Lemma 1.10, we have equation (2.4).

Part 3. We prove the Δ convergence of the sequence { a r } toward the fixed point of J . As the sequence { a r } is bounded in A , so according to Lemma 1.8, { a r } admits a unique asymptotic center C ( A , { a r } ) = { a 0 } . Suppose that { g r } is any subsequence of the sequence { a r } such that C ( A , { g r } ) = { g 0 } . Then, by equation (2.4), we have

(2.11) lim r → ∞ ϱ ( g r , J g r ) = 0 .

Now, we claim that the element g 0 is belonging to F J . Since J satisfies condition ( E ) , we obtain

ϱ ( g r , J g 0 ) ≤ μ ϱ ( g r , J g r ) + ϱ ( g r , g 0 ) for some μ ≥ 1 .

Taking limsup r → ∞ in equation (2.11) and in the above inequality, one has

R ( { g r } , J g 0 ) = limsup r → ∞ ϱ ( g r , J g 0 ) ≤ limsup r → ∞ ϱ ( g r , g 0 ) = R ( { g r } , g 0 ) .

Since the asymptotic center contains only one point, we have g 0 = J g 0 , i.e., g 0 is an element of the set F J . Putting g 0 ≠ a 0 . So according to equation (2.1), lim r → ∞ ϱ ( a r , g 0 ) exists. But in our case, the asymptotic center contains only one point. Thus,

limsup r → ∞ ϱ ( g r , g 0 ) < limsup r → ∞ ϱ ( g r , a 0 ) ≤ limsup r → ∞ ϱ ( a r , a 0 ) < limsup r → ∞ ϱ ( a r , g 0 ) = limsup r → ∞ ϱ ( g r , g 0 ) .

Subsequently, we proved that limsup r → ∞ ϱ ( g r , g 0 ) < limsup r → ∞ ϱ ( g r , g 0 ) , which is a clear contradiction. This implies that g 0 is the unique asymptotic center for any sub-sequence of { a r } . This proves the Δ convergence of the sequence { a r } in F J .□

Theorem 2.2

Suppose that A is a compact convex subset of H and J is a self-mapping on A. If J satisfies condition (E) and F J ≠ ∅ , then the sequence { a r } generated by equation (1.10) is essentially strongly convergent to an element of the set F J .

Proof

Since A is convex, then { a r } is completely continuous in A . Applying compactness property of A , we can find a subsequence { a r l } of { a r } that satisfies lim l → ∞ ϱ ( a r l , a ∗ ) = 0 , where a ∗ is some point of A . As J satisfies condition ( E ) , then, we have

(2.12) ϱ ( a r l , J a ∗ ) ≤ μ ϱ ( a r l , J a r l ) + ϱ ( a r l , a ∗ )

for each μ ≥ 1 . In view of equation (2.4), we obtain lim l → ∞ ϱ ( a r l , J a r l ) = 0 .

Using lim l → ∞ ϱ ( a r l , J a r l ) = 0 and keeping lim l → ∞ ϱ ( a r l , y 0 ) = 0 in our mind and applying the limit on both sides of equation (2.12), we obtain lim l → ∞ ϱ ( a r l , J a ∗ ) = 0 .

By uniqueness of the limit of a convergent sequence in a metric space, we can write J a ∗ = a ∗ , which implies that a ∗ ∈ F J .

Hence, according to equation (2.1), we conclude that lim r → ∞ ϱ ( a r , a ∗ ) exists. Subsequently, a ∗ is the point of F J and the sequence { a r } is strongly convergent to a ∗ . This completes the proof.□

We now include an example of condition (E) on a hyperbolic space that is not linear.

Example 2.3

Consider H = { ( a 1 , a 2 ) ∈ R 2 : a 1 , a 2 > 0 } . Set a metric ϱ on H as ϱ ( a , b ) = ∣ a 1 − b 1 ∣ + ∣ a 1 a 2 − b 1 b 2 ∣ for each a = ( a 1 , a 2 ) and b = ( b 1 , b 2 ) in H . Then, ϱ form a metric function on H . Now, for any β ∈ [ 0 , 1 ] , we set a function S as follows:

S ( a , b , β ) = ( ( 1 − β ) a 1 + β b 1 , ( 1 − β ) a 1 a 2 + β b 1 b 2 ( 1 − β ) a 1 + β b 1 ) .

Then, ( H , ϱ , β ) is hyperbolic metric space. Assume that A = [ 0.5 , 3 ] × [ 0.5 , 3 ] , which is a compact and closed subset of H , and set a mapping J on A as: J ( a 1 , a 2 ) = ( 1 , 1 ) if ( a 1 , a 2 ) ≠ ( 0.5 , 0.5 ) and J ( a 1 , a 2 ) = ( 1.5 , 1.5 ) if ( a 1 , a 2 ) = ( 0.5 , 0.5 ) . Then J is not Suzuki mapping as 1 2 ϱ ( a , J a ) ≤ ϱ ( a , b ) but ϱ ( J a , J b ) > ϱ ( a , b ) for a = ( 0.5 , 0.5 ) and b = ( 1.1 , 1.1 ) . However, J satisfies the condition ( E ) with μ = 7 . Hence, all the requirements for Theorem 2.2 are satisfied. Hence, the sequence { a r } generated by equation (1.10) converges to the fixed point ( 1 , 1 ) . Note that, we cannot apply directly classical results of the literature to this example because this example is not a linear space and is only a hyperbolic space.

Theorem 2.4

Assume that A is a closed convex subset of H and J is a self-mapping on A. If J satisfies condition (E) and F J ≠ ∅ . Then, the sequence { a r } obtained by equation (1.10) is essentially strongly convergent to an element of the set F J , provided that the condition liminf r → ∞ d ( a r , F J ) = 0 holds.

Proof

For an arbitrary point a ∗ ∈ F J , we proved that lim r → ∞ ϱ ( a r , a ∗ ) exists. From our assumption, one has

lim r → ∞ ϱ ( a r , F J ) = 0 .

Based on Proposition 1.1(iii), we have that F J is closed. The remaining proof follows immediately by [25, Theorem 2].□

A strong convergence under the condition (I) is as follows:

Theorem 2.5

Assume that A is a closed convex subset of H and J is a self-mapping on A. If J satisfies condition (E) and F J ≠ ∅ , then the sequence { a r } obtained by equation (1.10) is essentially strongly convergent to an element of the set, F J provided that the mapping J satisfies condition (I).

Proof

According to equation (2.4), we can write

(2.13) liminf r → ∞ ϱ ( a r , J a r ) = 0 .

But J is enriched with the condition (I), then

(2.14) ϱ ( a r , J a r ) ≥ g ( ϱ ( a r , F J ) ) .

Using equations (2.13) and (2.14), one has

liminf r → ∞ g ( ϱ ( a r , F J ) ) = 0 .

But g ( 0 ) = 0 , so eventually, we obtain

liminf r → ∞ ϱ ( a r , F J ) = 0 .

Therefore, all requirements of Theorem 2.4 are satisfied. Thus, { a r } converges strongly to a point in F J . This finishes the proof.□

Theorem 2.6

Assume that A is a closed subset of H and J is a self-mapping on A. If J satisfies (1.2) with F J = { a ∗ } , then the sequence { a r } generated by equation (1.10) is essentially strongly convergent to an element of the set F J if ∑ r = 1 ∞ α r = ∞ .

Proof

Using ( S 1 ) and equation (1.10), one has

ϱ ( e r , a ∗ ) = ϱ ( S ( J a r , a r , α r ) , a ∗ ) ≤ α r ϱ ( J a r , a ∗ ) + ( 1 − α r ) ( a r , a ∗ ) ≤ ( α r ( λ ϱ ( a r , a ∗ ) + L ϱ ( a ∗ , J a ∗ ) ) ) + ( 1 − α r ) ϱ ( a r , a ∗ ) = ( α r λ + ( 1 − α r ) ) ϱ ( a r , a ∗ ) = ( 1 − α r ( 1 − λ ) ) ϱ ( a r , a ∗ ) .

This implies that

ϱ ( a r + 1 , a ∗ ) = ϱ ( J b r , a ∗ ) ≤ λ ϱ ( b r , a ∗ ) + L ϱ ( a ∗ , J a ∗ ) = λ ϱ ( b r , a ∗ ) = λ ϱ ( J c r , a ∗ ) ≤ λ ( λ ϱ ( c r , a ∗ ) + L ϱ ( c 0 , J a ∗ ) ) = ( λ ) 2 ϱ ( c r , a ∗ ) = ( λ ) 2 ϱ ( J d r , a ∗ ) ≤ ( λ ) 2 ( ( λ ) ϱ ( d r , a ∗ ) + L ϱ ( a ∗ , J a ∗ ) ) = ( λ ) 3 ϱ ( d r , a ∗ ) = ( λ ) 3 ϱ ( J e r , a ∗ ) ≤ ( λ ) 3 ( λ ϱ ( e r , a ∗ ) + L ϱ ( a ∗ , J a ∗ ) ) = ( λ ) 4 ϱ ( e r , a ∗ ) ≤ ( λ ) 4 ( 1 − α r ( 1 − λ ) ) ϱ ( a r , a ∗ ) .

It follows that

ϱ ( a r + 1 , a ∗ ) ≤ ( λ ) 4 ( 1 − α r ( 1 − λ ) ) ϱ ( a r , a ∗ ) ≤ ( λ ) 4 ( 1 − α r ( 1 − λ ) ) ( λ ) 4 ( 1 − α r − 1 ( 1 − λ ) ) ϱ ( a r − 1 , a ∗ ) ⋮ ≤ ( ( λ ) 4 ) r Π m = 1 r ( 1 − α m ( 1 − λ ) ) ϱ ( a 1 , a ∗ ) .

Consequently, we obtain

(2.15) ϱ ( a r + 1 , a ∗ ) ≤ ( ( λ ) 4 ) r Π m = 1 r ( 1 − α m ( 1 − λ ) ) ϱ ( a 1 , a ∗ ) .

Using equation (2.15) and the fact 1 − a ≤ e − a for each a ∈ [ 0 , 1 ] , we have

ϱ ( a r + 1 , a ∗ ) ≤ ( ( λ ) 4 ) r e − ( 1 − λ ) ∑ m = 1 r α r ϱ ( a 1 , a ∗ ) .

Since λ ∈ ( 0 , 1 ) , then by the assumption ∑ r = 1 ∞ α r = ∞ , one can write

lim r → ∞ ϱ ( a r + 1 , a ∗ ) = 0 ,

which implies that

lim r → ∞ a r = a ∗ ∈ F J .

Therefore, the sequence { a r } is strongly convergent to a point in F J .□

Now, to support the above results, we construct the following examples:

Example 2.7

Let A = [ 0 , 1 ] and J : A → A be a mapping described as follows:

J = 0 , if 0 ≤ b < 1 55 , 16 b 17 , if 1 55 ≤ b ≤ 1 .

Now, we illustrate the following:

  1. The set F J is nonempty.

  2. The mapping J is neither Suzuki nonexpansive nor nonexpansive on A ;

  3. The mapping J satisfies condition (E).

For any a ∗ ∈ F J , we obtain a ∗ = 0 , which implies that F J = { 0 } and proves (a). Next, choose a = 1 55 and b = 1 100 , then it is easy to see that (b) is true. Ultimately, for (c), take μ = 17 , then we realized the following cases:

Case (i): Consider a , b ∈ [ 0 , 1 55 ) . It follows that J a = J b = 0 and

ϱ ( a , J b ) = ∣ a − J b ∣ = ∣ a ∣ ≤ 17 ∣ a ∣ = 17 ∣ a − J a ∣ ≤ 17 ∣ a − J a ∣ + ∣ a − b ∣ = 17 ϱ ( a , J a ) + ϱ ( a , b ) = μ ϱ ( a , J a ) + ϱ ( a , b ) .

Case (ii): If a , b ∈ [ 1 55 , 1 ] . It follows that J a = 16 a 17 , J b = 16 b 17 , and

ϱ ( a , J b ) = ∣ a − J b ∣ ≤ ∣ a − J a ∣ + ∣ J a − J b ∣ = ∣ a − J a ∣ + ∣ 16 a 17 − 16 b 17 ∣ = ∣ a − J a ∣ + 16 17 ∣ a − b ∣ ≤ ∣ a − J a ∣ + ∣ a − b ∣ ≤ 17 ∣ a − J a ∣ + ∣ a − b ∣ = 17 ϱ ( a , J a ) + ϱ ( a , b ) = μ ϱ ( a , J a ) + ϱ ( a , b ) .

Case (iii): Consider a ∈ [ 0 , 1 55 ) and b ∈ [ 1 55 , 1 ] . It follows that J a = 0 , J b = 16 b 17 , and

ϱ ( a , J b ) = ∣ a − J b ∣ = a − 16 b 17 = 17 a − 16 b 17 = a + 16 a − 16 b 17 ≤ a 17 + 16 a − 16 b 17 = 1 17 ∣ a ∣ + 16 17 ∣ a − b ∣ = ∣ a ∣ + ∣ a − b ∣ ≤ ∣ a − J a ∣ + ∣ a − b ∣ ≤ 17 ∣ a − J a ∣ + ∣ a − b ∣ = 17 ϱ ( a , J a ) + ϱ ( a , b ) = μ ϱ ( a , J a ) + ϱ ( a , b ) .

Case (iv): If a ∈ [ 1 55 , 1 ] and b ∈ [ 0 , 1 55 ] . It follows that J a = 16 a 17 , J b = 0 , and

ϱ ( a , J b ) = ∣ a − J b ∣ = ∣ a ∣ = 17 a 17 = 17 ∣ a − J a ∣ ≤ 17 ∣ a − J a ∣ + ∣ a − b ∣ = 17 ϱ ( a , J a ) + ϱ ( a , b ) = μ ϱ ( a , J a ) + ϱ ( a , b ) .

From the above cases, we conclude that for each a , b ∈ A , ϱ ( a , J b ) ≤ 17 ϱ ( a , J a ) + ϱ ( a , b ) and this proves (c). For numerical results, take α r = β r = γ r = 0.87 , then we obtain the comparison data shown in Table 1, which were produced by iterations (1.10), (1.9), (1.8), (1.7), (1.6), (1.5), and (1.4).

Table 1

Comparison of various iterative processes using J of Example 2.7

r New JF F M Thakur Picard-Mann Picard Mann
1 0.5000000 0.5000000 0.5000000 0.5000000 0.5000000 0.5000000 0.5000000 0.5000000
2 0.3722542 0.375279 0.3955201 0.4202401 0.426625 0.4465051 0.4705882 0.4744117
3 0.2771464 0.281668 0.3128724 0.3532036 0.364018 0.3987337 0.4429065 0.4501330
4 0.2063379 0.211408 0.2474946 0.2968607 0.310599 0.3560734 0.4168532 0.4270968
5 0.1536203 0.158674 0.1957782 0.2495055 0.26501 0.3179772 0.3923324 0.4052395
6 0.1143716 0.119094 0.1548685 0.2097045 0.226127 0.2839569 0.3692540 0.3845007
7 0.0851506 0.089387 0.1225072 0.1762525 0.192943 0.2535765 0.3475332 0.3648233
8 0.0633954 0.0670901 0.0969081 0.1481368 0.164629 0.2264464 0.3270901 0.3461530
9 0.0471984 0.05035 0.0766582 0.1245060 0.140469 0.2022190 0.3078495 0.3284381
10 0.0351396 0.0377943 0.0606397 0.1046449 0.119856 0.1805837 0.2897407 0.3116298
11 0.0261617 0.0283668 0.0479685 0.0879519 0.102267 0.1612631 0.2726971 0.2956817
12 0.0000000 0.021290 0.0379450 0.0739219 0.0872593 0.1440096 0.2566561 0.2805497
13 0.0000000 0.0000000 0.0300160 0.0621299 0.074454 0.1286021 0.2415587 0.2661922
14 0.0000000 0.0000000 0.0237439 0.0522189 0.0635279 0.1148430 0.2273493 0.2525694
15 0.0000000 0.0000000 0.0187823 0.0438890 0.0542052 0.1025560 0.2139759 0.2396438
16 0.0000000 0.0000000 0.0000000 0.0368878 0.0462506 0.0915835 0.2013890 0.2273797
17 0.0000000 0.0000000 0.0000000 0.0310035 0.0394634 0.0817850 0.1895426 0.2157432
18 0.0000000 0.0000000 0.0000000 0.0260578 0.0336721 0.0730349 0.1783930 0.2047022
19 0.0000000 0.0000000 0.0000000 0.0219011 0.0287308 0.0652209 0.1678993 0.1942263
20 0.0000000 0.0000000 0.0000000 0.0184074 0.0245145 0.0582429 0.1580229 0.1842864
21 0.0000000 0.0000000 0.0000000 0.0000000 0.020917 0.1805837 0.1487274 0.1748553
22 0.0000000 0.0000000 0.0000000 0.0000000 0.0178475 0.161263 0.139979 0.165907
23 0.0000000 0.0000000 0.0000000. 0.0000000 0.0000000 0.144011 0.131745 0.157416
. . . . . . . . .
. . . . . . . .
. . . . . . . .
31 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0811151 0.1034030
. . . . . . . .
. . . . . . . .
. . . . . . . .
57 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0263848
. . . . . . . .
. . . . . . . .
. . . . . . . .
71 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

The CPU times in seconds for our new iteration, JF, F, M, Thakur, Picard-Mann, Picard, and Mann iterations shown in Table 1 are, respectively, 0.047, 0.048, 0.048, 0.049, 0.050, 0.050, 0.049, and 0.051. Clearly, our new iteration needs less time for reaching to the fixed point.

Based on Table 1 and Figure 1, it is clear that the suggested iteration (1.10) converges faster than both iterations (1.9), (1.8), (1.7), (1.6), (1.5), and (1.4).

Figure 1 Behaviors of various iterative processes using Example 2.7.
Figure 1

Behaviors of various iterative processes using Example 2.7.

Example 2.8

Let A = [ 0 , 1 ] and J : A → A be a mapping defined as follows:

J = b 3 , if 0 ≤ b < 1 2 , b 4 , if 1 2 ≤ b ≤ 1 .

Here, we illustrate the following:

  1. The set F J contains only one point.

  2. The mapping J is neither Banach-contraction nor nonexpansive on A ;

  3. The mapping J satisfies (1.2).

Now, for any a ∗ ∈ F J , a ∗ = 0 , which shows that F J = { 0 } , and this proves (i). Since the mapping J is discontinuous, it follows that J cannot form a Banach-contraction or nonexpansive on the set A . This proves (ii). For showing (iii), we will take λ = 1 3 and L = 3 4 and discuss the cases below:

P 1 . For any a , b ∈ [ 0 , 1 2 ) , we have

ϱ ( J a , J b ) = ∣ J a − J b ∣ = a 3 − b 3 = 1 3 ∣ a − b ∣ ≤ 1 3 ∣ a − b ∣ + 3 4 ∣ a − J a ∣ = λ ϱ ( a , b ) + L ϱ ( a , J a ) .

P 2 . For any a , b ∈ ( 1 2 , 1 ] , we obtain

ϱ ( J a , J b ) = ∣ J a − J b ∣ = a 4 − b 4 = 1 4 ∣ a − b ∣ ≤ 1 3 ∣ a − b ∣ ≤ 1 3 ∣ a − b ∣ + 3 4 ∣ a − J a ∣ = λ ϱ ( a , b ) + L ϱ ( a , J a ) .

P 3 . For any a ∈ [ 0 , 1 2 ) and b ∈ ( 1 2 , 1 ] , we can write

ϱ ( J a , J b ) = ∣ J a − J b ∣ = a 3 − b 4 = 4 a − 3 b 12 = 3 a − 3 b + a 12 ≤ 3 a − 3 b 12 + a 12 = 1 4 ∣ a − b ∣ + 1 12 ∣ a ∣ ≤ 1 3 ∣ a − b ∣ + 6 12 ∣ a ∣ = 1 3 ∣ a − b ∣ + 3 4 2 a 3 = 1 3 ∣ a − b ∣ + 3 4 ∣ a − J a ∣ = λ ϱ ( a , b ) + L ϱ ( a , J a ) .

P 4 . For any a ∈ ( 1 2 , 1 ] and b ∈ [ 0 , 1 2 ) , we obtain that

ϱ ( J a , J b ) = ∣ J a − J b ∣ = a 4 − b 3 = 3 a − 4 b 12 = 4 a − 4 b − a 12 ≤ 4 a − 4 b 12 + a 12 = 1 3 ∣ a − b ∣ + 1 12 ∣ a ∣ ≤ 1 3 ∣ a − b ∣ + 9 16 ∣ a ∣ = 1 3 ∣ a − b ∣ + 3 4 3 a 4 = 1 3 ∣ a − b ∣ + 3 4 ∣ a − J a ∣ = λ ϱ ( a , b ) + L ϱ ( a , J a ) .

According to the above cases, we conclude that ϱ ( J a , J b ) ≤ λ ϱ ( a , b ) + L ϱ ( a , J a ) . Hence, the required proof of (c) is completed.

For numerical results, choose α r = β r = γ r = 0.87 , so we generate some numerical values given in Table 2 from our iteration (1.10) and JF (1.9), F (1.8), M (1.7), Thakur (1.6), Picard-Mann (1.5), and Mann (1.4) iterations. Clearly, once more, our iteration (1.10) gives better approximations in less steps of iterations. Also, the graphical comparison in this case is given in Figure 2.

Table 2

Comparison of various iterative processes using J of Example 2.8

r New JF F M Thakur Picard-Mann Picard Mann
1 0.4900000 0.4900000 0.4900000 0.4900000 0.4900000 0.4900000 0.4900000 0.4900000
2 0.0025407 0.0032013 0.0076222 0.0228666 0.0339844 0.0686000 0.1633333 0.2058000
3 0.0000131 0.0000209 0.0001185 0.0010671 0.0023570 0.0096000 0.0544444 0.0864360
4 0.0000000 0.0000002 0.0000011 0.0000497 0.0001634 0.0013445 0.0181481 0.0363031
5 0.0000000 0.0000000 0.0000000 0.0000021 0.0000113 0.0001882 0.0060493 0.0152473
6 0.0000000 0.0000000 0.0000000 0.0000001 0.0000001 0.0000263 0.0020164 0.0064038
7 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000036 0.0006721 0.0026896
8 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000005 0.0002240 0.0011296
9 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000746 0.0004744
10 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000248 0.0001992
11 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000082 0.0000836
12 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000027 0.0000351
13 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000009 0.0000147
14 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000003 0.0000062
15 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000001 0.0000026
16 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000010
17 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000004
18 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000001
19 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
20 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
21 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
22 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
Figure 2 Behaviors of various iterative processes using Example 2.8.
Figure 2

Behaviors of various iterative processes using Example 2.8.

The CPU times in seconds for our new iteration and JF, F, M, Thakur, Picard-Mann, Picard and Mann iterations in Table 1 are, respectively, 0.047, 0.048, 0.048, and 0.049, 0.050, 0.050, 0.049, and 0.051. Clearly, our new iteration needs less time for reaching to the fixed point.

3 Weak w 2 stability result

In 1987, Harder [28] rigorously examined the idea of stability of an fixed point (FP) iteration process in her PhD thesis as follows:

Definition 3.1

[28] Let J : H → H be a given mapping and a r + 1 = h ( J , a r ) be a fixed point iteration so that { a r } converges to a ∈ F J . For a chosen sequence { q r } in H , define

ε r = ϱ ( q r + 1 , h ( J , q r ) ) for all r ∈ N .

Then, the fixed point iteration method is called J -stable if the assertion below holds

lim r → ∞ ε r = 0 iff lim r → ∞ q r = a .

Several writers have lately examined the idea of stability in Definition 3.1 for various classes of contraction mappings, e.g., see [29–34]. Because the sequence { q r } is arbitrarily chosen, Berinde pointed out in [35] that the concept of stability in Definition 3.1 is not precise. To obtain over this restriction, the same author noted that if { q r } were approximate sequence of { a r } , then the definition would make sense. As a result, any iteration process will be weakly stable if it is stable, but the converse is not true in general.

Definition 3.2

[35] A sequence { q r } ⊂ H is called an approximate sequence of { a r } ⊂ H if, for any b ≥ 1 , there is α = α ( b ) so that

∥ a r − q r ∥ ≤ α for all r ≥ b .

Definition 3.3

[35] Let { a r } be an iterative process defined for a 0 ∈ H , and

(3.1) a r + 1 = h ( J , a r ) , r ≥ 0 ,

where J : H → H be a given mapping. Suppose that { q r } converges to a fixed point a ∗ of J , and for any approximate sequence { q r } ⊂ H of { a r } :

lim r → ∞ ε r = lim r → ∞ ϱ ( q r + 1 , h ( J , q r ) ) = 0 implies lim r → ∞ q r = a ∗ ,

then equation (3.1) is called weakly stable with respect to J or weakly J -stable.

By using the more general concept of the equivalent sequence in place of the approximate sequence in Definition 3.3, Timis [36] studied a new concept of weak stability in 2012 as follows:

Definition 3.4

[37] The sequences { a r } and { q r } are called equivalent if

lim r → ∞ ϱ ( q r , a r ) = 0 .

Definition 3.5

[36] Assume that { a r } is an iterative procedure defined for a 0 ∈ H and

(3.2) a r + 1 = h ( J , a r ) , r ≥ 0 ,

where J : H → H be a self-mapping. Suppose that { a r } converges to a fixed point a ∗ of J , and for any equivalent sequence { q r } ⊂ H of { a r } :

lim r → ∞ ε r = lim r → ∞ ϱ ( q r + 1 , h ( J , q r ) ) = 0 implies lim r → ∞ q r = a ∗ ,

then equation (3.2) is called weakly w 2 -stable with respect to ℑ .

Any analogous sequence is an approximate sequence, as demonstrated with an example in [36], but the converse is not true.

Here, we provide weak w 2 -stability and data dependence theorems for our new iterative process (1.10) in the class of mappings satisfying equation (1.2).

Theorem 3.6

If all requirements of Theorem 2.6  are true. Then, the sequence considered by equation (1.10) is essentially weak w 2 -stable.

Proof

Assume that { a r } is a sequence generated by equation (1.10), { q r } is an equivalent sequence of { a r } , and

ε r = ϱ ( p r + 1 , J q r ) .

Putting q r = J r k , r r = J g r , g r = J t r , and t r = S ( J p r , p r , α r ) . If ε r ⟶ 0 , as r → ∞ , then by equations (1.2) and (1.10), and ( S 4 ), we have

ϱ ( e r , t r ) = ϱ ( S ( J a r , a r , α r ) , S ( J p r , p r , α r ) ) ≤ α r ϱ ( J r , J p r ) + ( 1 − α r ) ϱ ( a r , p r ) ≤ α r ( λ ϱ ( a r , p r ) + L ϱ ( a r , J a r ) ) + ( 1 − α r ) ϱ ( a r , p r ) = ( 1 − α r ( 1 − λ ) ) ϱ ( a r , p r ) + α r L ϱ ( a r , J a r ) .

Also,

ϱ ( d r , g r ) = ϱ ( J e r , J t r ) ≤ λ ϱ ( e r , t r ) + L ϱ ( e r , J e r ) = λ ( ( 1 − α r ( 1 − λ ) ) ϱ ( a r , p r ) + α r L ϱ ( a r , J a r ) ) + L ϱ ( e r , J e r ) .

Now,

ϱ ( b r , q r ) = ϱ ( J c r , J r r ) ≤ λ ϱ ( c r , r r ) + L ϱ ( c r , J c r ) = λ ϱ ( J d r , J g r ) + L ϱ ( c r , J c r ) ≤ λ ( λ ϱ ( d r , g r ) + L ϱ ( d r , J d r ) + L ϱ ( c r , J c r ) ≤ ( λ ) 2 ϱ ( d r , g r ) + λ L ϱ ( d r , J d r ) + L ϱ ( c r , J c r ) ≤ ( λ ) 2 ( λ ( ( 1 − α r ( 1 − λ ) ) ϱ ( a r , p r ) + α r L ϱ ( a r , J a r ) ) ) + L ϱ ( e r , J e r ) ) + λ L ϱ ( d r , J d r ) + L ϱ ( c r , J c r ) = ( λ ) 3 ( 1 − α r ( 1 − λ ) ) ϱ ( a r , p r ) + ( λ ) 3 α r L ϱ ( a r , J a r ) + ( λ ) 2 L ϱ ( e r , J e r ) + λ L ϱ ( d r , J d r ) + L ϱ ( c r , J c r ) .

This implies that

ϱ ( p r + 1 , a ∗ ) ≤ ϱ ( p r + 1 , a r + 1 ) + ϱ ( a r + 1 , a ∗ ) ≤ ϱ ( p r + 1 , J q r ) + ϱ ( J q r , J b r ) + ϱ ( a r + 1 , a ∗ ) ≤ ε r + ϱ ( J q r , J b r ) + ϱ ( a r + 1 , a ∗ ) ≤ ε r + λ ϱ ( q r , b r ) + L ϱ ( b r , J b r ) + ϱ ( a r + 1 , a ∗ ) ≤ ε r + λ [ ( λ ) 3 ( 1 − α r ( 1 − λ ) ) ϱ ( a r , p r ) + ( λ ) 3 α r L ϱ ( a r , J a r ) + ( λ ) 2 L ϱ ( e r , J e r ) + λ L ϱ ( d r , J d r ) + L ϱ ( c r , J c r ) ] + L ϱ ( b r , J b r ) + ϱ ( a r + 1 , a ∗ ) = ε r + ( λ ) 4 ( 1 − α r ( 1 − λ ) ) ϱ ( a r , p r ) + ( λ ) 4 α r L ϱ ( a r , J a r ) + ( λ ) 3 L ϱ ( e r , J e r ) + ( λ ) 2 L ϱ ( d r , J d r ) + λ L ϱ ( c r , J c r ) + L ϱ ( b r , J b r ) + ϱ ( a r + 1 , a ∗ ) .

Consequently, we obtain

(3.3) ϱ ( p r + 1 , a ∗ ) ≤ ε r + ( λ ) 4 ( 1 − α r ( 1 − λ ) ) ϱ ( a r , p r ) + ( λ ) 4 α r L ϱ ( a r , J a r ) + ( λ ) 3 L ϱ ( e r , J e r ) + ( λ ) 2 L ϱ ( d r , J d r ) + λ L ϱ ( c r , J c r ) + L ϱ ( b r , J b r ) + ϱ ( a r + 1 , a ∗ ) .

Since lim r → ∞ ϱ ( a r , J a r ) = lim r → ∞ ϱ ( b r , J b r ) = lim r → ∞ ϱ ( d r , J d r ) = lim r → ∞ ϱ ( e r , J e r ) = 0 and lim r → ∞ ϱ ( a r , a ∗ ) = 0 , then, one has

ϱ ( a r , J a r ) ≤ ϱ ( a r , a ∗ ) + ϱ ( J a r , a ∗ ) ≤ ϱ ( a r , a ∗ ) + λ ϱ ( a r , a ∗ ) + L ϱ ( a ∗ , J a ∗ ) = ( 1 + λ ) ϱ ( a r , a ∗ ) ⟶ 0 .

By the proof of Theorem 2.6, ϱ ( e r , J a ∗ ) ≤ ( 1 − α r ( 1 − λ ) ) ϱ ( a r , a ∗ ) , we obtain

ϱ ( e r , J e r ) ≤ ϱ ( e r , a ∗ ) + ϱ ( J e r , a ∗ ) ≤ ϱ ( e r , a ∗ ) + λ ϱ ( e r , a ∗ ) + L ϱ ( a ∗ , J a ∗ ) = ( 1 + λ ) ϱ ( e r , a ∗ ) ≤ ( 1 + λ ) [ ( 1 − α r ( 1 − λ ) ) ] ϱ ( a r , a ∗ ) ≤ ( 1 + λ ) ϱ ( a r , a ∗ ) ⟶ 0 .

Also,

ϱ ( d r , J d r ) ≤ ϱ ( d r , a ∗ ) + ϱ ( J d r , a ∗ ) ≤ ϱ ( d r , a ∗ ) + λ ϱ ( d r , a ∗ ) + L ϱ ( a ∗ , J a ∗ ) = ( 1 + λ ) ϱ ( d r , a ∗ ) ≤ ( 1 + λ ) [ ( 1 − α r ( 1 − λ ) ) ] ϱ ( a r , a ∗ ) ≤ ( 1 + λ ) ϱ ( a r , a ∗ ) ⟶ 0 .

Similarly,

ϱ ( b r , J b r ) ≤ ϱ ( b r , a ∗ ) + ϱ ( J b r , a ∗ ) ≤ ϱ ( b r , a ∗ ) + λ ϱ ( b r a ∗ ) + L ϱ ( a ∗ , J a ∗ ) = ( 1 + λ ) ϱ ( b r , a ∗ ) = ( 1 + λ ) ϱ ( J c r , a ∗ ) ≤ ( 1 + λ ) ( λ ϱ ( c r , a ∗ ) + L ϱ ( a ∗ , J a ∗ ) = ( 1 + λ ) λ ϱ ( c r , a ∗ ) = ( 1 + λ ) λ ϱ ( J d r , a ∗ ) ≤ ( 1 + λ ) λ ( λ ϱ ( d r , a ∗ ) + L ϱ ( a ∗ , J a ∗ ) = ( 1 + λ ) ( λ ) 2 ϱ ( d r , a ∗ ) = ( 1 + λ ) ( λ ) 2 ϱ ( J e r , a ∗ ) ≤ ( 1 + λ ) ( λ ) 2 ( λ ϱ ( e r , a ∗ ) + L ϱ ( a ∗ , J a ∗ ) ) = ( 1 + λ ) ( λ ) 3 ϱ ( e r , a ∗ ) ≤ ( 1 + λ ) ( λ ) 3 [ ( 1 − α r ( 1 − λ ) ) ] ϱ ( a r , a ∗ ) ≤ ( 1 + λ ) ( λ ) 3 ϱ ( a r , a ∗ ) ⟶ 0 .

Since { p r } and { a r } are equivalent, we have lim r → ∞ ϱ ( a r , p r ) = 0 . Using lim r → ∞ ϱ ( a r , J a r ) = lim r → ∞ ϱ ( b r , J b r ) = lim r → ∞ ϱ ( d r , J d r ) = lim r → ∞ ϱ ( e r , J e r ) = 0 = lim r → ∞ ε r . Hence, equation (3.3) can be written as follows:

lim r → ∞ ϱ ( p r + 1 , a ∗ ) = 0 .

This implies that { a r } is weak w 2 -stable with respect to J .□

4 Data dependence result

When we compute the fixed point using an iterative process, we may use an approximate mapping of the given operator. In this case, we may have a certain estimate between the original fixed point and the approximate fixed point. This approach is known as data dependence and is widely studied by authors [38,39]. Now, we carry out a data dependency result for our iterative process.

Definition 4.1

[35] A mapping J ˜ is said to be approximate mapping for J if and only if, for any choice of a , b ∈ H and any ε > 0 , we have ϱ ( J a , J ˜ b ) ≤ ε .

Lemma 4.2

[40]. Let { a r } , { b r } , and { c r } be three sequences of nonnegative real numbers such that { b r } ∈ ( 0 , 1 ) satisfying ∑ r = 1 ∞ b r = ∞ and c r ≥ 0 for any r ∈ N . If r 1 ∈ N such that r ≥ r 1 , and

a r + 1 ≤ ( 1 − b r ) a r + b r c r ,

then,

0 ≤ limsup r → ∞ a r ≤ limsup r → ∞ c r .

Now, we introduce our main result in this section.

Theorem 4.3

Assume that H, A, and J are defined as in Theorem 2.6. Let J ˜ : A → A be an approximate mapping of J for some ε . Suppose that the sequences { a r } and { a ˜ r } are defined by (1.10) and

(4.1) a ˜ 1 ∈ A , e ˜ r = S ( J ˜ a ˜ r , a ˜ r , α r ) , d ˜ r = J ˜ e ˜ r , c ˜ r = J ˜ d ˜ r , b ˜ r = J ˜ c ˜ r , a ˜ r + 1 = J ˜ b ˜ r , r ≥ 1 .

respectively, where α r ∈ ( 0 , 1 ) satisfying the condition ∑ r = 1 ∞ α r = ∞ . Then, if a ∗ = J a ∗ and a ∗ ˜ = J ˜ a ∗ ˜ , we have

ϱ ( a ∗ , a ∗ ˜ ) ≤ ( λ 4 + λ 3 + λ 2 + λ + 1 ) ε 1 − λ 4 ,

where λ ∈ [ 0 , 1 ) .

Proof

It follows from equations (1.2), (1.10), and (4.1) that

ϱ ( e r , e ˜ r ) = ϱ ( S ( J a r , a r , α r ) , S ( J ˜ a ˜ r , a ˜ r , α r ) ) ≤ α r ϱ ( J a r , J ˜ a ˜ r ) + ( 1 − α r ) ϱ ( a r , a ˜ r ) ) ≤ α r ϱ ( J a r , J a ˜ r ) + α r ϱ ( J a ˜ r , J ˜ a ˜ r ) + ( 1 − α r ) ϱ ( a r , a ˜ r ) ≤ α r ϱ ( J a r , J a ˜ r ) + α r ε + ( 1 − α r ) ϱ ( a r , a ˜ r ) ≤ α r ( λ ϱ ( a r , a ˜ r ) + L ϱ ( a r , J a r ) ) + α r ε + ( 1 − α r ) ϱ ( a r , a ˜ r ) = ( 1 − α r ( 1 − λ ) ) ϱ ( a r , a ˜ r ) + α r L ϱ ( a r , J a r ) + α r ε .

This implies that

ϱ ( a r + 1 , a ˜ r + 1 ) = ϱ ( J b r , J ˜ b ˜ r ) ≤ ϱ ( J b r , J b ˜ r ) + ϱ ( J b ˜ r , J ˜ b ˜ r ) ≤ ϱ ( J b r , J b ˜ r ) + ε ≤ λ ϱ ( b r , b ˜ r ) + L ϱ ( b r , J r ) + ε ≤ λ [ ϱ ( J c r , J ˜ c ˜ r ) ] + L ϱ ( b r , J b r ) + ε ≤ λ [ ϱ ( J c r , J c ˜ r ) + ϱ ( J c ˜ r , J ˜ c ˜ r ) ] + L ϱ ( b r , J b r ) + ε ≤ λ [ ϱ ( J c r , J c ˜ r ) + ε ] + L ϱ ( b r , J b r ) + ε ≤ λ [ λ ϱ ( c r , c ˜ r ) + L ϱ ( c r , J c r ) + ε ] + L ϱ ( b r , J b r ) + ε = ( λ ) 2 ϱ ( c r , c ˜ r ) + λ L ϱ ( c r , J c r ) + λ ε + L ϱ ( b r , J b r ) + ε ≤ ( λ ) 2 [ ϱ ( J d r , J ˜ d ˜ r ) ] + λ L ϱ ( c r , J c r ) + λ ε + L ϱ ( b r , J b r ) + ε ≤ ( λ ) 2 [ ϱ ( J d r , J d ˜ r ) + ϱ ( J d ˜ r , J ˜ d ˜ r ) ] + λ L ϱ ( c r , J c r ) + λ ε + L ϱ ( b r , J b r ) + ε ≤ ( λ ) 2 [ ϱ ( J d r , J d ˜ r ) + ε ] + λ L ϱ ( c r , J c r ) + λ ε + L ϱ ( b r , J b r ) + ε ≤ ( λ ) 2 [ λ ϱ ( d r , d ˜ r ) + L ϱ ( d r , J d r ) + ε ] + λ L ϱ ( c r , J c r ) + λ ε + L ϱ ( b r , J b r ) + ε = ( λ ) 3 ϱ ( d r , d ˜ r ) + ( λ ) 2 L ϱ ( d r , J d r ) + ( λ ) 2 ε + λ L ϱ ( c r , J c r ) + λ ε + L ϱ ( b r , J b r ) + ε = ( λ ) 3 ϱ ( J e r , J ˜ e ˜ r ) + ( λ ) 2 L ϱ ( d r , J d r ) + ( λ ) 2 ε + λ L ϱ ( c r , J c r ) + λ ε + L ϱ ( b r , J b r ) + ε ≤ ( λ ) 3 ( ϱ ( J e r , J e ˜ r ) + ϱ ( J e ˜ r , J ˜ e ˜ r ) ) + ( λ ) 2 L ϱ ( d r , J d r ) + ( λ ) 2 ε + λ L ϱ ( c r , J c r ) + λ ε + L ϱ ( b r , J b r ) + ε ≤ ( λ ) 3 ( ϱ ( J e r , J e ˜ r ) + ε ) + ( λ ) 2 L ϱ ( d r , J d r ) + ( λ ) 2 ε + λ L ϱ ( c r , J c r ) + λ ε + L ϱ ( b r , J b r ) + ε ≤ ( λ ) 3 ( λ ϱ ( e r , e ˜ r ) + L ϱ ( e r , J e r ) + ε ) + ( λ ) 2 L ϱ ( d r , J d r ) + ( λ ) 2 ε + λ L ϱ ( c r , J c r ) + λ ε + L ϱ ( b r , J b r ) + ε ≤ ( λ ) 4 ϱ ( e r , e ˜ r ) + ( λ ) 3 L ϱ ( e r , J e r ) + ( λ ) 3 ε + ( λ ) 2 L ϱ ( d r , J d r ) + ( λ ) 2 ε + λ L ϱ ( c r , J c r ) + λ ε + L ϱ ( b r , J b r ) + ε ≤ ( λ ) 4 [ ( 1 − α r ( 1 − λ ) ) ϱ ( a r , a ˜ r ) + α r L ϱ ( a r , J a r ) + α r ε ] + ( λ ) 3 L ϱ ( e r , J e r ) + ( λ ) 3 ε + ( λ ) 2 L ϱ ( d r , J d r ) + ( λ ) 2 ε + λ L ϱ ( c r , J c r ) + λ ε + L ϱ ( b r , J b r ) + ε = ( λ ) 4 [ ( 1 − α r ( 1 − λ ) ) ϱ ( a r , a ˜ r ) ] + ( λ ) 4 α r L ϱ ( a r , J a r ) + ( λ ) 4 α r ε + ( λ ) 3 L ϱ ( e r , J e r ) + ( λ ) 3 ε + ( λ ) 2 L ϱ ( d r , J d r ) + ( λ ) 2 ε + λ L ϱ ( c r , J c r ) + λ ε + L ϱ ( b r , J b r ) + ε

Consequently, we obtain

(4.2) ϱ ( a r + 1 , a ˜ r + 1 ) ≤ ( λ ) 4 [ ( 1 − α r ( 1 − λ ) ) ϱ ( a r , a ˜ r ) ] + ( λ ) 4 α r L ϱ ( a r , J a r ) + ( λ ) 4 α r ε + ( λ ) 3 L ϱ ( e r , J e r ) + ( λ ) 3 ε + ( λ ) 2 L ϱ ( d r , J d r ) + ( λ ) 2 ε + λ L ϱ ( c r , J c r ) + λ ε + L ϱ ( b r , J b r ) + ε .

Since λ ∈ [ 0 , 1 ) , we have ( λ ) 4 ∈ ( 0 , 1 ) . Thus, we can find a constant f ∈ ( 0 , 1 ) such that

(4.3) ( λ ) 4 = 1 − f .

From equation (4.3) and the assumption α r ≤ 1 , we obtain 1 − α r ( 1 − λ ) ≤ 1 for all r ∈ N . Hence, (4.2) can be written as follows:

(4.4) ϱ ( a r + 1 , a ˜ r + 1 ) ≤ ( 1 − f ) ϱ ( a r , a ˜ r ) + f ( λ ) 4 L ϱ ( a r , J a r ) + ( λ ) 3 L ϱ ( e r , J e r ) + ( λ ) 2 L ϱ ( d r , J d r ) + λ L ϱ ( c r , J c r ) + L ϱ ( b r , J b r ) f . + f λ 4 ε + λ 3 ε + λ 2 ε + λ ε + ε f .

Now, set

A r = ϱ ( a r + 1 , a ˜ r + 1 ) , B r = f , C k = ( ( λ ) 4 L ϱ ( a r , J a r ) + ( λ ) 3 L ϱ ( e r , J e r ) + ( λ ) 2 L ϱ ( d r , J d r ) + λ L ϱ ( c r , J c r ) + L ϱ ( b r , J b r ) + ( λ ) 4 ε + ( λ ) 3 ε + ( λ ) 2 ε + λ ε + ε ) ∕ ( 1 − ( λ ) 4 ) .

Considering equation (4.4) with Lemma 4.2 and using lim r → ∞ ϱ ( a r , J a r ) = lim r → ∞ ϱ ( c r , J c r ) = lim r → ∞ ϱ ( b r , J b r ) = lim r → ∞ ϱ ( d r , J d r ) = lim r → ∞ ϱ ( e r , J e r ) = 0 , we have

ϱ ( a ∗ , a ∗ ˜ ) ≤ ( λ 4 + λ 3 + λ 2 + λ + 1 ) ε 1 − λ 4 .

This completes the proof.□

5 Conclusion and future work

A new general fixed-point iterative process is constructed in the hyperbolic space setting for finding fixed points of mappings with condition (E) and generalized contractions. We provided a Δ -convergence theorem for mappings with condition (E). Also, some strong convergence theorems are obtained for our new iterative process in the class of mappings with condition (E). All these results are supported by a numerical example. It is seen that our new iterative process has a leading rate of convergence over the classical iterative processes of the literature. Second, we analyzed our new iteration with a class of generalized contractions and obtained a strong convergence theorem. Eventually, weak w 2 -stability and data dependence results are obtained. Our results improve the corresponding results in [24] and [23]. In future, the authors have a plan to extend the main results of this article to the setting of multivalued mappings and to the setting of common fixed points. The authors also made a plan to prove the main results of this article in the setting of modular metric spaces.

Acknowledgements

The authors extend their appreciation to the Prince Sattam bin Abdulaziz University for funding this research work through the project number PSAU/2024/R/1445.

  1. Funding information: This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2024/R/1445).

  2. Author contributions: All authors contributed equally to the writing of this article. All authors have accepted responsibility for entire content of the manuscript and approved its submission.

  3. Conflict of interest: All authors confirm that they have no conflict of interest.

  4. Ethical approval: The conducted research is not related to either human or animal use.

  5. Informed consent: Not applicable.

  6. Data availability statement: Data sharing is not applicable to the article as no datasets were generated or analyzed during the current study.

References

[1] B. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fund. Math. 3 (1922), 133–181. 10.4064/fm-3-1-133-181Search in Google Scholar

[2] O. Bouftouth, S. Kabbaj, T. Abdeljawad, and A. Khan, Quasi controlled K-metric spaces over C∗-algebras with an application to stochastic integral equations, Comput. Modeling Eng. Sci. 135 (2023), 2649–2663. 10.32604/cmes.2023.023496Search in Google Scholar

[3] K. Gopalan, S. T. Zubair, and T. Abdeljawad, New fixed point theorems in operator valued extended hexagonal b-like metric spaces, Palestine J. Math. 11 (2022), 48–56. 10.3390/axioms11010019Search in Google Scholar

[4] H. A. Hammad, P. Agarwal, and L. G. J. Guirao, Applications to boundary value problems and homotopy theory via tripled fixed point techniques in partially metric spaces, Mathematics 9 (2021), no. 16, 2012. 10.3390/math9162012Search in Google Scholar

[5] H. A. Hammad, H. Aydi, and Y. U. Gaba, Exciting fixed point results on a novel space with supportive applications, J. Function Spaces 2021, (2021), Article ID 6613774, 12 pages. 10.1155/2021/6613774Search in Google Scholar

[6] F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. USA. 54 (1965), 1041–1044. 10.1073/pnas.54.4.1041Search in Google Scholar PubMed PubMed Central

[7] D. Gohde, Zum Prinzip der Kontraktiven Abbildung, Math. Nachr. 30 (1965), 251–258. 10.1002/mana.19650300312Search in Google Scholar

[8] W. A. Kirk, A fixed point theorem for mappings which do not increase distance, Amer. Math. Monthly 72 (1965), 1004–1006. 10.2307/2313345Search in Google Scholar

[9] M. O. Osilike, Stability results for the Ishikawa fixed point iteration procedure, Indian J. Pure Appl. Math. 26 (1995), 937–945. Search in Google Scholar

[10] J. Garcia-Falset, E. Llorens-Fuster, and T. Suzuki, Fixed point theory for a class of generalized nonexpansive mappings, J. Math. Anal. Appl. 375 (2011), 185–195. 10.1016/j.jmaa.2010.08.069Search in Google Scholar

[11] U. Kohlenbach, Some logical metatheorems with applications in functional analysis, Trans. Amer. Math. Soc. 357 (2004), 89–128. 10.1090/S0002-9947-04-03515-9Search in Google Scholar

[12] W. Takahashi, A convexity in metric spaces and nonexpansive mappings, Kodai Math. Seminar Rep. 22 (1970), 142–149. 10.2996/kmj/1138846111Search in Google Scholar

[13] K. Goebel and W. A. Kirk, Iteration processes for nonexpansive mappings, in: S. P. Singh, S. Thomeier, B. Watson, (Eds), Topological Methods Nonlinear Functional Analysis, Contemporary Mathematics, vol. 21, 1983, pp. 115–123. 10.1090/conm/021/729507Search in Google Scholar

[14] S. Reich and I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal. 15 (1990), 537–558. 10.1016/0362-546X(90)90058-OSearch in Google Scholar

[15] M. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer, Berlin, 1999. 10.1007/978-3-662-12494-9Search in Google Scholar

[16] K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings, Marcel Dekker, New York, 1984. Search in Google Scholar

[17] T. Shimizu, and W. Takahashi, Fixed points of multivalued mappings in certain convex metric spaces, Topo. Methods Nonlinear Anal. 8 (1996), 197–203. 10.12775/TMNA.1996.028Search in Google Scholar

[18] A. Deshmukh, D. Gopal, and V. Rakocević, Two new iterative schemes to approximate the fixed points for mappings, Int. J. Nonlinear Sci. Numer. Simulat. 24 (2023), 1265–1309. 10.1515/ijnsns-2021-0141Search in Google Scholar

[19] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506–510. 10.1090/S0002-9939-1953-0054846-3Search in Google Scholar

[20] S. H. Khan and A Picard-Mann hybrid iterative process, Fixed Point Theory Appl. 2013 (2013), 1–10. 10.1186/1687-1812-2013-69Search in Google Scholar

[21] K. Ullah and M. Arshad, Numerical reckoning fixed points for Suzuki’s generalized nonexpansive mappings via new iteration process, Filomat. 32 (2018), 187–196. 10.2298/FIL1801187USearch in Google Scholar

[22] J. Ali and F. Ali, A new iterative scheme to approximating fixed points and the solution of a delay differential equation, J. Nonlinear Convex Anal. 21 (2020), 2151–2163. Search in Google Scholar

[23] J. Ahmad, K. Ullah, and M. Arshad, Convergence, weak w2 stability, and data dependence results for the F iterative scheme in hyperbolic spaces, Numerical Algor. 91 (2022), 1755–1778. 10.1007/s11075-022-01321-ySearch in Google Scholar

[24] A. Sahin, Some new results of M-iteration process in hyperbolic spaces, Carpathian J. Math. 35 (2019), 221–232. 10.37193/CJM.2019.02.10Search in Google Scholar

[25] H. F. Senter and W. G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44 (1974), 375–380. 10.1090/S0002-9939-1974-0346608-8Search in Google Scholar

[26] 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, 2010, pp. 193–209. 10.1090/conm/513/10084Search in Google Scholar

[27] A. R. Khan, H. Fukhar-ud-din, and M. A. A. Khan, An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces, Fixed Point Theory Appl. 54 (2012), 1–12. 10.1186/1687-1812-2012-54Search in Google Scholar

[28] A. M. Harder, Fixed point theory and stability results for fixed-point iteration procedures, Ph.D. thesis, University of Missouri-Rolla, Missouri, 1987. Search in Google Scholar

[29] S. Hassan, M. De la Sen, P. Agarwal, Q. Ali, and A. Hussain, A new faster iterative scheme for numerical fixed points estimation of Suzuki’s generalized nonexpansive mappings, Math. Probl. Eng. 2020, (2020), 3863819. 10.1155/2020/3863819Search in Google Scholar

[30] G. A. Okeke, M. Abbas, and M. De la Sen, Approximation of the fixed point of multivalued quasi-nonexpansive mappings via a faster iterative process with applications, Discrete Dyn. Nat. Soc. 2020, (2020), 8634050. 10.1155/2020/8634050Search in Google Scholar

[31] H. A. Hammad, H. U. Rehman, and M. Zayed, Applying faster algorithm for obtaining convergence, stability, and data dependence results with application to functional-integral equations, AIMS Math. 7 (2022), no. 10, 19026–19056. 10.3934/math.20221046Search in Google Scholar

[32] H. A. Hammad, H. U. Rehman, and M. De la Sen, A novel four-step iterative scheme for approximating the fixed point with a supportive application, Inf. Sci. Lett. 10(2021), no. 2, 333–339. 10.18576/isl/100214Search in Google Scholar

[33] H. A. Hammad, H. U. Rehman, and M. De la Sen, A New four-step iterative procedure for approximating fixed points with application to 2D Volterra integral equations, Mathematics. 10 (2022), 4257. 10.3390/math10224257Search in Google Scholar

[34] T. M. Tuyen and H. A. Hammad, Effect of shrinking projection and CQ-methods on two inertial forward-backward algorithms for solving variational inclusion problems, Rendiconti del Circolo Matematico di Palermo 70 (2021), no. 3, 1669–1683. 10.1007/s12215-020-00581-8Search in Google Scholar

[35] V. Berinde, Iterative Approximation of Fixed Points, Springer, Berlin, 2007. 10.1109/SYNASC.2007.49Search in Google Scholar

[36] I. Timis, On the weak stability of Picard iteration for some contractive type mappings, Annals Uni. Craiova Math. Computer Sci. Series, 37, (2010), 106–114. Search in Google Scholar

[37] T. Cardinali and P. Rubbioni, A generalization of the Caristi fixed point theorem in metric spaces, Fixed Point Theory. 11 (2010), 3–10. Search in Google Scholar

[38] F. Gursoy, V. Karakaya, and B. E. Rhoades, Data dependence results of new multi-step and S-iterative schemes for contractive-like operators, Fixed Point Theory Appl. 2013 (2013), 1–12. 10.1186/1687-1812-2013-76Search in Google Scholar

[39] F. Gursoy, A. R. Khan, M. Erturk, and V. Karakaya, Weak w2-stability and data dependence of Mann iteration method in Hilbert spaces, RACSAM 2017 (2017), 1–10. DOI: https://doi.org/10.1007/s13398-017-0447-y. 10.1007/s13398-017-0447-ySearch in Google Scholar

[40] S. M. Soltuz and T. Grosan, Data dependence for Ishikawa iteration when dealing with contractive like operators, Fixed Point Theory Appl. 2008 (2008), 1–7. 10.1155/2008/242916Search in Google Scholar
Received: 2023-03-17
Revised: 2023-11-14
Accepted: 2023-12-12
Published Online: 2024-05-11

© 2024 the author(s), published by De Gruyter

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

Articles in the same Issue

  1. Regular Articles
  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
https://doi.org/10.1515/dema-2023-0151
Keywords for this article
iteration process; fixed point; convergence result; data dependence; stability; hyperbolic space
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 18.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/dema-2023-0151/html
Scroll to top button