Fixed point and multidimensional fixed point theorems with applications to nonlinear matrix equations in terms of weak altering distance functions

Definition 1.1

([16]). A function ψ : [0, ∞) → [0, ∞) is said to be an altering distance function if it satisfies the following conditions:

1. ψ is continuous and nondecreasing;

2. ψ(t)= 0 if and only if t = 0.

Example 1.2

Define ψ₁, ψ₂, ψ₃, ψ₄:[0, ∞) → [0, ∞) by ψ₁(t) = t, ψ₂(t) = t², ψ₃(t) = te^t, ψ₄(t) = ln(l+t) for all t ≥ 0. We see that ψ₁, ψ₂, ψ₃, and ψ₄ are altering distance functions because ψ₁, ψ₂, ψ₃, and ψ₄ are continuous and nondecreasing.

Moreover, ψ_i (t) = 0 if and only if t = 0 for all i = 1,2,3, 4. (The graphs of functions ψ₁, ψ₂, ψ₃, and ψ₄ show in Figure 1).

Fig. 1

Graphs ψ₁, ψ₂, ψ₃, ψ₄ in Example 1.2.

In 2012, Yan et al. [17] discussed some results on existence and uniqueness of a fixed point in partially ordered metric spaces by using the concept of an altering distance function as follows.

Theorem 1.3

([17]). Let (X, ⪯) be a partially ordered set and suppose that there exists a metric d in X such that (X, d) is a complete metric space. Suppose that Τ : X → X is a continuous and nondecreasing mapping such that

where ψ is an altering distance function and φ : [0, ∞) → [0, ∞) is a continuous function with the condition: ψ(t) > φ(t) for all t > 0. If there exists x₀∈ X such that x₀⪯ T_x0, then Τ has a fixed point.

On the other hand, the notion of a coupled fixed point was firstly investigated by Guo and Lakshmikantham in [18]. In 2006, Bhaskar and Lakshmikantham [19] were the first to introduce the notion of mixed monotone property in partially ordered metric spaces. They also established the classical coupled fixed point theorems for mappings by using this property under contractive type conditions. Due to the important role of such results for the investigation of solutions of nonlinear differential and integral equations, several authors have studied various generalizations of these results. In this continuation, several authors introduced concepts of a tripled fixed point, quadruple fixed point and multidimensional fixed point. They also proved new results on the existence and uniqueness of multidimensional fixed points that were presented in partially ordered metric spaces and another spaces. For instance, Rus [20] obtained the existence and uniqueness results for a multidimensional fixed point of nonlinear mappings satisfying the contractive condition with a control function in the following class:

together with an approximating iterative scheme, in the setting of partially ordered metric spaces.

In 2015, Soleimani Rad et al. [21] compared relation between multidimensional fixed point results and fixed point theorems concerning various contractive conditions which are depended on contractive constants in abstract metric spaces and metric-like spaces. Also, they claimed that these results are true for another spaces with several contractive conditions. Recently, Su et al. [22] have discussed the existence and uniqueness results for multidimensional fixed point of contraction mappings involving some control functions in complete metric spaces.

In this paper, we present a new concept of weak altering distance function and establish fixed point theorems for generalized contraction mappings in complete metric spaces endowed with a transitive relation by using the idea of a weak altering distance function. Also, we provide an example that supports our main result where previous results in literature are not applicable. Our results generalize and improve the main result of Yan et al. [17] and several well-known results given by some authors in partially metric spaces. Moreover, we present a new extension of multidimensional fixed point theorems in metric spaces endowed with a transitive relation. Furthermore, we apply our results to prove the existence and uniqueness of a solution of nonlinear matrix equations. Finally, we use some numerical examples to show the iterative method is feasible to confirm the existence and uniqueness of positive definite solution of a nonlinear matrix equation.

Definition 2.1

([23]). Let X be a nonempty set and Τ : X^N → X be a given mapping. An element (x₁, x₂, . . ., x_n)∈ X^N is said to be a fixed point of Ν-order of the mapping Τ if

Definition 2.2

([24]). Let X be a nonempty set. A subset ℜ of X² is called a binary relation on X. Notice that for each pair x, y ∈ X, one of the following conditions holds:

1. (x, y)∈ ℜ, which amounts to saying that “x is ℜ-related to y” or “x relates to y under ℜ.” Sometimes, we write xℜy instead of (x, y)∈ ℜ;

2. (x, y)∈ℜ which means that “x is not ℜ-related to y” or “x does not relate to y under ℜ.”

Definition 2.3

A binary relation ℜ defined on a nonempty set X is called transitive if (x, y)∈ ℜ and (y, z)∈ ℜ implies (x, z) ∈ ℜ.

Definition 2.4

([25]). Let ℜ be a binary relation defined on a nonempty set X and x, y ∈ X. We say that x and y are ℜ-comparative if either (x, y)∈ ℜ or (y, x) ∈ ℜ. We denote it by [x, y]∈ ℜ.

Definition 2.5

([25]). Let X be a nonempty set and Τ be a self-mapping on X. A binary relation ℜ defined on X is called Τ-closed if for any x, y ∈ X, (x, y)∈ ℜ ⇒ (Tx, Ty)∈ ℜ.

Definition 2.6

([25]). Let X be a nonempty set and ℜ be a binary relation on X. A sequence {x_n}⊂ X is called ℜ-preserving if

Definition 2.7

([25]). Let (X, d) be a metric space. A binary relation ℜ defined on X is called d-self-closed if whenever {x_n} is an ℜ-preserving sequence and

then there exists a subsequence {x_nk} of {x_n} with [ x n k , x ] ∈ ℜ for all k ∈ ℕ.

Definition 2.8

([26]). Let X be a nonempty set and ℜ a binary relation on X. A subset E of X is called ℜ-directed if for each x, y ∈ E, there exists z ∈ X such that (x, z) ∈ ℜ and (y, z)∈ ℜ.

In this paper, we use the following notations for a binary relation ℜ on a nonempty set X:

where Τ : X → X is a given mapping.

Definition 3.1

A function ψ : [0, ∞) → [0, ∞) is said to be a weak altering distance function if it satisfies the following conditions:

1. ψ is lower semicontinuous and nondecreasing;

2. ψ(t) = 0 if and only if t = 0.

Every continuous function is lower semicontinuous and so the class of weak altering distance functions is larger than the class of altering distance functions. In general, a weak altering distance function need not necessarily be an altering distance function. Next, we give some examples of the weak altering distance functions which show that the weak altering distance functions are real generalization of altering distance functions.

Example 3.2

Define ψ₁, ψ₂, ψ₃: [0, ∞) → [0, ∞) by

We see that ψ₁, ψ₂ and ψ₃ are weak altering distance functions because ψ₁, ψ₂ and ψ₃ are lower semicontinuous and nondecreasing. Moreover, ψ_i(t) = 0 if and only if t = 0 for all i = 1,2,3. (The graphs of functions ψ₁, ψ₂ and ψ₃ show in Figure 2).

Fig. 2

Graphs of ψ₁, ψ₂, ψ₃in Example 3.2.

Now we give an useful proposition concerning a contractive condition given for comparable elements with respect to a binary relation.

Proposition 3.3

If (X, d) is a metric space, ℜ is a binary relation on Χ, Τ is a self-mapping on Χ, ψ is a weak altering distance function and φ : [0, ∞) → [0, ∞) is a right upper semicontinuous function, then the following contractivity conditions are equivalent:

1. ψ(d(Tx, Ty)) ≤ φ(d(x, y)),   ∀x, y ∈ X with (x, y)∈ ℜ,

2. ψ(d(Tx, Ty)) ≤ φ(d(x, y)),   ∀x, y ∈ X with [x, y] ∈ ℜ.

Proof

First, we will show that the implication (i)⇒(ii) holds.

Assume that (i) holds. Take x, y ∈ X with [x, y]∈ℜ. If (x, y)∈ℜ, then (ii) directly follows from (i). Now, suppose that (y, x)∈ℜ, then using the symmetry of d and (i), we get

This shows that (i)⇒(ii).

Conversely, the implication (ii)⇒(i) is trivial. This completes the proof.

Theorem 3.4

Let (X, d) be a complete metric space and ℜ be a transitive relation on X. Suppose that Τ : X → X is a continuous mappings and ℜ is Τ -closed such that

where ψ is a weak altering distance function and φ : [0, ∞) → [0, ∞) is a right upper semicontinuous function such that ψ(t) > φ(t) for all t > 0. If X(T; ℜ) is a nonempty set, then Τ has a fixed point.

Proof

Let x₀ be an arbitrary point in X(T;ℜ). Put x_n = Tx_n-1 = Tⁿ_x0for all n ∈ℕ. If x_n* = χ_{n* + 1for some} n* ∈ℕ₀, then x_n*is a fixed point of Τ. Thus we will assume that x_n ≠ x_n+1for all n ∈ ℕ0. Since (x₀, Tx₀)∈ℜ, using the T-closedness of ℜ, we get

and so (x_n, x_n+1) ∈ ℜ for all n ∈ ℕ₀. Thus the sequence {x_n}is ℜ-preserving. From contractive condition (1), we have

for all n ∈ℕ. Since ψ is a nondecreasing function, we have

for all n ∈ ℕ. Thus, the sequence {d{x_n, x_n+1)} is decreasing and bounded below. Consequently, there exists s ≥ 0 such that

From (3), letting n → ∞ and using the property of ψ and φ we get

Since ψ(t) > φ(t) for all t > 0, we have 5=0 and so {d{x_n, x_n+1)} converges to 0. Now, we will show that {x_n} is a Cauchy sequence.

Assume on the contrary, there is an ϵ > 0 and subsequences { x m k } a n d { x n k } of {x_n} with n_k > m_k ≥ k such that

Choosing n_k to be the smallest integer exceeding m_k for which (5) holds, we obtain that

Using (5) and (6), we get

Hence, d(x_mk, x_nk)→ ∊as k→∞. Furthermore, we have

and

Letting k→ ∞ in (7) and (8) and using the fact that lim n → ∞ d ( x n , x n + 1 ) = 0 a n d lim k → 8 d ( x m k , x n k ) = ϵ , we have

Since n_k > m_k and ℜ is a transitive relation, we get (x_mk−1, x_nk–1)∈ℜ. This implies that

From (9), letting k→∞ and using the property of ψ and ϕ we get

It yields that ∊ = 0, which is a contradiction. Therefore, {x_n} is a Cauchy sequence.

Since (X, d) is a complete metric space, there exists x* ∈ X such that x_n→ x* as n→ ∞. Thus, by the continuity of T, we get Tx* = x*.This completes the proof.

Remark 3.5

It is fascinating to point out that we use the result in Theorem 3.4 to derive a criterion for the existence of fixed points in some cases wherein several results contained in [17, 20-22] cannot guarantee the existence of fixed points. Indeed, the main results of Yan et al. in [17] (Theorem 1.3) are not applicable in the following cases:

1. (X,⪯) is not a partially ordered set;

2. ψ is not an altering distance function;

3. φ is not continuous, the main results of Rus in [20] are not applicable in the following cases:

4. (X,⪯) is not a partially ordered set (or quasi-ordered set);

5. Τ does not satisfy the contractive condition with the control function φ ∈ Φ, the main results of Soleimani Rad et al. in [21] are not applicable in the following cases:

6. Τ does not satisfy the contractive condition with several contractive constant, the main results of Su et al. in [22] are not applicable in the following cases:

7. Τ does not satisfy the contractive condition in metric space without the transitive relation;

8. Τ does not satisfy the contractive condition with two control functions ψ and φ (see the detail in [22]).

Now, we give an example to illustrate utility of Theorem 3.4.

Example 3.6

Let X = [0, ∞) with usual metric d. Thus (X, d) is a complete metric space. Define a binary relationℜ on X by

It is easy to prove that ℜ is a transitive relation on X. Also, we define two functions ϕ, ψ: [0, ∞) → [0, ∞) by

and ϕ ( t ) = 2 t 3 . Then ψ is a weak altering distance function and ϕ is a right upper semicontinuous function such that ψ(t) > ϕ(t) for all t > 0.

Let Τ: X→ X be defined by

for all x ∈ X. Then Τ is continuous.

Next, we will show that ℜ is Τ-closed. Assume that x, y ∈ X such that (x, y) ∈ ℜ and then ln(x² + 2x +1) = ln(y² + 2y +1). This means that Τx = Ty and so (Tx)² + TT χ = (Ty)² + 2 Ty. This implies that (Tx, Ty) ∈ ℜ. Therefore, ℜ is Τ-closed. Moreover, there exists0 e X such that (0, T0) ∈ ℜ. This shows that X(T; ℜ) is a nonempty set.

Finally, it is clear that for each x, y e X with (x, y) ∈ ℜ, we get

So Τ satisfies the contractive condition (1).

Now all of the conditions in Theorem 3.4 hold and hence Τ has at least one fixed point. For instance, the point x =0 is one of the fixed point of T.

Remark 3.7

We note that Yan et al.’s result in [17] (Theorem 1.3) is not applicable in the above example since ψ is not an altering distance function. This implies that the Banach contraction principle is not also applicable in the above example.

The following theorem guarantees the uniqueness of fixed point in Theorem 3.4.

Theorem 3.8

In addition to the hypothesis of Theorem 3.4, suppose that ϕ (0)= 0 and X is ℜ-directed. Then Τ has a unique fixed point.

Proof. Suppose that there exist x*, y* ∈ X which are fixed points. We distinguish two cases.

Case 1. If (x*, y*) ∈ ℜ, then (Tⁿ x*, Tⁿ y*) ∈ ℜ for all n ∈ ℕ₀. It yields that

for all n ∈ ℕ. From the fact that ψ(t) > ϕ (t) for all t > 0, we get d(x*, y*) = 0. Therefore, x* = y*.

Case 2. If (x*, y*) ∉ ℜ, then there exists z*∈ X such that (x*, z*) ∈ ℜ and (y*, z*) ∈ ℜ. Since ℜ is T-closed, we get (Tⁿx*, Tⁿ z*)∈ ℜ and (Tⁿ y*, Tⁿ z*)∈ ℜ for all n ∈ ℕ₀. Moreover, we have

for all n ∈ ℕ.Since ψ is a nondecreasing function, we have

for all n ∈ ℕ.Thus, the sequence {d(x*, Tⁿ z*)}is non-increasing. Thus, there exists ξ such that

From (10), letting n → ∞ and using the property of ψ and ϕ we get

From (12) and the condition: ψ(t) > ϕ(t) for all t > 0, it implies that ξ = 0. Similarly, we can show that

Therefore, Tⁿ z*→ x* and Tⁿ z*→ y* as n → ∞. This implies that x* = y *.This completes the proof.

Now we use the following notation for a binary relation ℜ on a nonempty set X for all Ν ∈ ℕ,

where Τ : X^N→ Χ is a given mapping.

Definition 3.9

([27]). Let X be a nonempty set. Given Ν ∈ ℕ and Τ: Χ N → Χ is a mapping. A binary relation ℜ defined on X is called ΤN -closed if for any (x₁, x₂,. . ., x_N), (y₁. y₂, . . ., y_N) ∈ X^N

Definition 3.10

Let X be a nonempty set and ℜ a binary relation on X. Given Ν ∈ ℕ. A subset E^N of X ^N is called ℜ^N -directed if for each(x₁, x₂, . . ., x_N) (y₁, x₂, . . ., y_N) ∈ E_N, there exists(z₁, z₂, . . ., z_N) ∈ X suchthat

and

Next, we illustrate how to prove multidimensional results from the unidimensional result by involving simple tools. Given Ν ∈ ℕ and a mapping Τ : X^N→ X, let us denote by G T N : X N → X N the mapping

The following lemmas will be useful later.

Lemma 3.11

([27]). Given Ν ∈ ℕ, Τ : Χ^Ν → Χ, a point (x₁, x₂, . . ., x_N) ∈ Χ^Ν is a fixed point of Ν-order of mapping Τ if and only if it is a fixed point of G T N .

Lemma 3.12

([27]). Given Ν ∈ ℕ, Τ : Χ^Ν→ Χ and G T N : X N → X N are two mappings. A binary relation ℜ defined on X is T_N -closed if and only if a binary relation ℜ^N defined on X ^N is G T N -closed.

Lemma 3.13

([27]). Given Ν ∈ Ν, Τ : Χ^Ν→ Χ and G T N : X N → x N are two mappings. A point(x₁, x₂, . . ., x_n) ∈ X^N (T; ℜ^N) if and only if a point (x₁, x₂,..., x_N) ∈ X^N ( G T N ; R N ) .

Lemma 3.14

([27]). Given Ν ∈ ℕ. Let(X, d) be a metric space and a mapping D^N : X^N× X^N → ℝ defined by

for all A =(a₁, a₂,..., a_N), Β = (b₁, b₂, ...b_N) ∈ X^N . Then the following properties hold.

1. (X^N , D^N) is also a metric space.

2. Let { A n = ( a n 1 , a n 2 , … a n N ) } be a sequence on X^N and let A = (a₁,a₂,..., a_N)∈ ΧN . Then { A n } → D N A if and only if { a n i } → d a i for all i ∈ {1, 2,..., Ν}.

3. If { A n = ( a n 1 , a n 2 , … a n N ) } is a sequence on X^N, then {A_n} is D^N -Cauchy if and only if { a n i } is Cauchy for all i ∈ {l,2,...,N}.

4. (X, d) is complete if and only if (X^N,D^N) is complete.

Here, we show how to use Theorem 3.4 in order to deduce multidimensional fixed point results.

Theorem 3.15

Let (X, d) be a complete metric space and ℜ be a transitive relation on X. Given Ν ∈ ℕ. Suppose that Τ: X^N → X is a continuous and ℜ is T_N-closed such that for each (x₁, x₂,..., x_N), (y₁, y₂, …, y_N) ∈ X^N with((x₁, x₂,..., x_N), (y₁, y₂,..., y_N) ∈ ℜ^N,

where ψ is a weak altering distance function and ϕ: [0, ∞) → [0, ∞) is a right upper semicontinuous function such that ψ(t) > ϕ (t) for all t > 0. If ΧN (Τ; ℜ) is a nonempty set, then Τ has a fixed point of Ν-order.

Proof.Using items 1 and 4 of Lemma 3.14, we obtain that (X^N, D^N) is a complete metric space. By Lemma 3.12, a binary relation ℜ^N defined on X_N is G T N -closed. Assume that ( x 0 1 , x 0 2 , … , x 0 N ) ∈ X N ( T ; R N ) , by Lemma 3.13 we get ( x 0 1 , x 0 2 , … , x 0 N ) ∈ X N ( G T N ; R N ) . .

Now, let A = (a₁,a₂,...,a_N),B = (b₁, b₂ ,...,b_N) ∈ X^N such that (A, B) ∈ ℜ^Ν. Then

Applying Theorem 3.4, there exists X̂ = (x̂₁,x̂₂, . . ., .....x̂) ∈ X^N such that G T N X ^ = X ^ . That is, (x̂₁, x̂₂, . . ., x̂_N) is a fixed point of G T N . Using Lemma 3.11, (x̂₁, x̂₂, _{· · ·}, x̂_N) is a fixed point of N-order of mapping Τ.This completes the proof.

By using Theorem 3.8, we get the following uniqueness result of fixed point of N-order.

Theorem 3.16

In addition to the hypothesis of Theorem 3.15, suppose that ϕ(0) = 0 and X^N is ℜ^N -directed. Then Τ has a unique fixed point of Ν -order.

Lemma 4.1

([28]). If A, B ∈ H⁺(n), then

Lemma 4.2

([32]). If A ∈ Η (n) and A≺ I, then ||A||<1.

Theorem 4.3

Consider the matrix equation (13). Assume that there is a positive number M such that:

(i) for every X, Y ∈ H(n) such that (X, Y) ∈ ⪯, we have

where h: [0, ∞) → [0, →) is a right upper semicontinuous function with h(t) < t² for all t > 0; ∑ i = 1 m A i A i ∗ ≺ M I n .

If ∑ i = 1 m A i ∗ G ( Q ) A i ≻ 0 then the matrix equation (13) has a solution. Moreover, the iteration

where X₀∈ H(n) such that X 0 ≺ _ Q + ∑ i = 1 n A i ∗ G ( X 0 ) A i , converges in the sense of trace norm|| ·||_tr to the solution of matrix equation (13).

Proof. Throughout this proof, we define the mapping 𝓕 : H(n) → H(n) by

Then 𝓕 is well defined and ⪯on H(n) is 𝓕-closed and a fixed point of 𝓕 is a solution of equation (13). ¹ 𝓖 is order preserving if A, B ∈ Η (n) with A ⪯ Β implies that 𝓖(A) ⪯ (B).

Next, we will show that the contractive condition (1) holds with 𝓕.

Let X, Y ∈ Η (n) such that (X, Y) ∈⪯. This mean that Χ ⪯ Y and then 𝓖(Χ)⪯𝓖(Y). Therefore,

This yields that,

Putting ψ(t)= t² and ϕ(t) = h(t), obviously ψ is a weak altering distance function and φ is a right upper semicontinuous function such that ψ(t) > ϕ(t) for all t > 0.

From the inequality (16), we have

Thus, the contractive condition (1) in Theorem 3.4 is satisfied for all X, Y ∈ H(n) such that (X, Y) ∈⪯. From ∑ i = 1 m A i ∗ G ( Q ) A i ≻ 0 , we have Q ⪯ 𝓕(Q)· This means that Q ∈ H(n)(𝓕; ⪯). Now from Theorem 3.4, there exists X̂ ∈ 𝓕(X̂) such that 𝓕(X̂) = X̂, that is, the matrix equation (13) has a solution.

Theorem 4.4

Consider the matrix equation (13). Assume that there is a positive number M such that:

1. for every X, Y ∈ Η (n) such that (X, Y) ∈ ⪯, we have

   | t r ( G ( Y ) − G ( X ) ) | ≤ 1 M | ln ⁡ ( 1 + h ( t r ( Y − X ) ) ) |

   where h : [0, ∞) → [0, ∞) is a right upper semicontinuous function with h(t) < e^t — 1, for all t > 0;

2. ∑ i = 1 m A i A i ∗ ≺ M I n . If ∑ i = 1 m A i ∗ G ( Q ) A i ≻ 0 , , then the matrix equation (13) has a solution. Moreover, the iteration

   (17) X n = Q + ∑ i = 1 n A i ∗ G ( X n − 1 ) A i

   where X₀ ∈ Η (n) such that X 0 ≺ _ Q + ∑ i = 1 n A i ∗ G ( X 0 ) A i , converges in the sense of trace norm || · ||_{t r} to the solution of matrix equation (13).

Proof

Throughout this proof, we define the mapping 𝓕 : Η (n) → H(n) by

Then 𝓕 is well defined and ⪯on H(n) is 𝓕-closed and a fixed point of 𝓕 is a solution of equation (13).

Next, we will show that the contractive condition (1) holds with 𝓕.

Let X, Y ∈ Η (n) such that (X, Y)∈⪯. This mean that Χ ⪯ Y and then 𝓖(X)⪯ 𝓖(Y). Using the same technique to Theorem 4.3, we get

Putting ψ(t) = t, ϕ(t) = ln(l + h(t)), obviously ψ is a weak altering distance function and ϕ is a right upper semicontinuous function such that ψ(t) > ϕ(t) for all t > 0. From the inequality (19), we have

Thus, the contractive condition (1) in Theorem 3.4 is satisfied for all X, Y ∈ H(n) such that (X, Y)∈⪯. From ∑ i = 1 m A i ∗ G ( Q ) A i ≻ 0 , we have Q ⪯ 𝓕This means that Q ∈ H(n)(𝓕; ⪯). Now from Theorem 3.4, there exists X̂ ∈(n) such that 𝓕X̂ = X̂, that is, the matrix equation (13) has a solution.

Theorem 4.5

In addition to the hypothesis of Theorem 4.3 (resp. Theorem 4.4), suppose that h(0) = 0. Then the equation (13) has a unique solution X̂ ∈ H(n).

Proof

It follows from h(0) = 0 that ϕ (0) = 0. Since for every X, Y ∈ Η (n) there is the greatest lower bound and the least upper bound, we obtain that H(n) is ⪯ directed. Thus, we deduce from Theorem 3.8 that 𝓕 has a unique fixed point in H(n). This implies that Equation (13) has a unique solution in H(n).

Example 5.1

Let

Define h : [0, ∞) → [0, ∞) by M = 1 2 . We consider Equation (13) with ∞(X) = X that is

All the hypotheses of Theorem 4.5 are satisfied with M = 1 6 . We will consider the iteration

where X₀ = Q, and the error E_n := ||X_n – X_n—1||_tr · After 8 iterations, we can approximate a solution X̂ of Equation (20) by

with E₈ = 2.3565e – 05.

Fig. 3

The error of iteration process 21 for the Equation (20) given in Example 5.1.

Example 5.2

Let

Define h : [0, ∞) ∞ [0, ∞) by h ( t ) = t 2 9 . We consider Equation (13) with 𝓖(X)= 2X that is

All the hypotheses of Theorem 4.5 are satisfied with M = 1 6 . We will consider the iteration

where X₀= Q, and the error E_n := ||X_n – X_n–1||_tr. After 7 iterations, we can approximate a solution X̂ of Equation (20) by

with Ε₇ = 2.3990e – 004.