Fixed-point results for convex orbital operators

Ovidiu Popescu

doi:10.1515/dema-2022-0184

Article Open Access

Fixed-point results for convex orbital operators

Ovidiu Popescu

Published/Copyright: January 25, 2023

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From the journal Demonstratio Mathematica Volume 56 Issue 1

Abstract

The aim of this article is to introduce a new type of operator similar to those of A. Petruşel and G. Petruşel type (Fixed point results for decreasing convex orbital operators, J. Fixed Point Theory Appl. 23 (2021), no. 35) and prove some fixed-point theorems which generalize and complement several results in the theory of nonlinear operators.

Keywords: Hilbert space; graphic contraction; convex orbital Lipschitz operator; weakly Picard operator; fixed-point

MSC 2010: 47H10; 54H25

1 Introduction and preliminaries

We recall some important concepts for the fixed-point theory.

Definition 1.1

Let ( X , d ) be a metric space. Then, T : X → X is called a Picard operator if:

F T = { x ∗ } , where F T = { x ∈ X : x = T ( x ) } is the fixed-point set of T ;
the sequence of iterates ( T n ( x ) ) n ∈ N → x ∗ as n → ∞ , for all x ∈ X .

Definition 1.2

Let ( X , d ) be a metric space. Then, T : X → X is called a weakly Picard operator if, for any x ∈ X , the sequence of iterates ( T n ( x ) ) n ∈ N converges to a fixed-point of T .

In this case, the mapping T ∞ : X → F T , given by T ∞ ≔ lim n → ∞ T n ( x ) is a set retraction on F T .

Definition 1.3

[1] Let ( X , d ) be a metric space and T : X → X be an operator such that F T is nonempty. Let r : X → F T be a set retraction. We say that T satisfies a retraction-displacement condition if there exists c > 0 such that for every x ∈ X

d ( x , r ( x ) ) ≤ c d ( x , T ( x ) ) .

Definition 1.4

Let ( X , d ) be a metric space, T : X → X be an operator such that F T is nonempty, and r : X → F T be a set retraction. Then:

the fixed-point equation x = T ( x ) is called well posed in the sense of Reich and Zaslavski (see [2,3]) if for each x ∗ ∈ F T and any sequence ( u n ) n ∈ N in r − 1 ( x ∗ ) for which d ( u n , T ( u n ) ) → 0 as n → ∞ , we have that u n → x ∗ as n → ∞ .
the operator T has the Ostrowski property (see [4,5]) if for each x ∗ ∈ F T and any sequence ( u n ) n ∈ N in r − 1 ( x ∗ ) for which d ( u n + 1 , T ( u n ) ) → 0 as n → ∞ , we have that u n → x ∗ as n → ∞ .
the fixed-point equation x = T ( x ) is Ulam-Hyers stable (see [6,7]) if there exists c > 0 such that for every ε > 0 and every ε -fixed-point y ∗ ∈ X of T (i.e., d ( y ∗ , T ( y ∗ ) ) ≤ ε ), there exists a fixed-point x ∗ ∈ X of T such that d ( x ∗ , y ∗ ) ≤ c ε .

Recently, Petruşel and Rus [8] proved the following important theorem :

Theorem 1.1

(Graphic contraction principle) Let ( X , d ) be a complete metric space and f : X → X be a graphic k -contraction, i.e., there exists k ∈ ( 0 , 1 ) such that

d ( f ( x ) , f 2 ( x ) ) ≤ k d ( x , f ( x ) ) ,

for every x ∈ X .

If f has a closed graph, then:

the sequence of iterates ( f n ( x 0 ) ) n ∈ N converges in ( X , d ) to a fixed-point x ∗ ( x 0 ) of f ;
F f = F f n ≠ ∅ for all n ∈ N ∗ ;
f is a weakly Picard operator;
d ( x , f ∞ ( x ) ) ≤ 1 1 − k d ( x , f ( x ) ) , for every x ∈ X , i.e., f is a 1 1 − k -weakly Picard operator;
the fixed-point equation x = f ( x ) is well posed in the sense of Reich and Zaslavski;
the fixed-point equation x = f ( x ) is Ulam-Hyers stable:
if k < 1 / 3 , then d ( f ( x ) , f ∞ ( x ) ) ≤ k 1 − 2 k d ( x , f ∞ ( x ) ) , for every x ∈ X , i.e., f is a k 1 − 2 k -quasicontraction;
if k < 1 / 3 , then f has the Ostrowski stability property.

Very recently, Petruşel and Petruşel [9] gave the following notion of convex orbital β -Lipschitz operator.

Definition 1.5

Let ( X , ∥ ⋅ ∥ ) be a normed space and Y be a nonempty and convex subset of X . Let T : Y → Y be an operator and λ ∈ ( 0 , 1 ] . We say that T is a convex orbital β -Lipschitz operator if β > 0 and for any x ∈ Y

∥ T ( x ) − T ( ( 1 − λ ) x + λ T ( x ) ) ∥ ≤ β λ ∥ x − T ( x ) ∥ .

They proved that this class of operators includes the Banach contractions, Kannan contractions, Ćirić-Reich-Rus contractions, Berinde contractions, nonexpansive operators, enriched ( β , θ ) contractions, and Lipschitz operators [10,11, 12,13,14, 15,16]. From now on, we will use the symbol T x instead of T ( x ) . The main results in [9] are the following theorems:

Theorem 1.2

Let ( X , ⟨ ⋅ ⟩ ) be a Hilbert space, Y be a nonempty closed and convex subset of X, and T : Y → Y be an operator with closed graph. We suppose that:

T is a convex orbital β -Lipschitz;
T is decreasing, i.e., Re ( ⟨ T u − T v , u − v ⟩ ) ≤ 0 , for every u , v ∈ Y .

Then, there exists λ ∈ ( 0 , 1 ) such that, for every x 0 ∈ Y , the sequence ( x n ) n ∈ N ⊂ Y , defined by

x n + 1 = ( 1 − λ ) x n + λ T x n , n ∈ N ,

converges to the unique fixed-point x ∗ ∈ Y of T .

Theorem 1.3

Let ( X , ⟨ ⋅ ⟩ ) be a Hilbert space, Y be a nonempty closed and convex subset of X, and T : Y → Y be an operator satisfying all the assumptions in Theorem 1.2. If x ∗ ∈ Y is the unique fixed-point of T, then the following conclusions hold:

T satisfies the following retraction-displacement condition
∥ x − x ∗ ∥ ≤ ( 1 + k ) ∥ x − T x ∥ ,
for every x ∈ Y , where k ≔ β 1 + β 2 ;
the fixed-point equation x = T x is Ulam-Hyers stable;
the fixed-point equation x = T x is well posed.

Theorem 1.4

Let ( X , ⟨ ⋅ ⟩ ) be a Hilbert space, Y be a nonempty closed and convex subset of X , and T : Y → Y be an operator satisfying all the assumptions in Theorem 1.2. If β < 5 − 2 5 5 , then T is a quasicontraction.

In this article we will clarify and complement the notion of convex orbital β -Lipschitz, by introducing some new classes of convex orbital operators. Then similar results as Theorems 1.2, 1.3, and 1.4 for these new types of operators are proved.

2 Main results

Definition 2.1

[9] Let ( X , ∥ ⋅ ∥ ) be a normed space and Y be a nonempty and convex subset of X . Let T : Y → Y be an operator. We say that T is a convex orbital β -Lipschitz operator if there exists β > 0 such that for any λ ∈ ( 0 , 1 ] , and for any x ∈ Y ,

∥ T x − T T λ x ∥ ≤ β λ ∥ x − T x ∥ ,

where T λ x ≔ ( 1 − λ ) x + λ T x .

Definition 2.2

Let ( X , ∥ ⋅ ∥ ) be a normed space and Y be a nonempty and convex subset of X . Let T : Y → Y be an operator. We say that T is a weak convex orbital Lipschitz operator if for any λ ∈ ( 0 , 1 ] there exists β > 0 such that for any x ∈ Y ,

∥ T x − T T λ x ∥ ≤ β λ ∥ x − T x ∥ .

Definition 2.3

Let ( X , ∥ ⋅ ∥ ) be a normed space and Y be a nonempty and convex subset of X . Let T : Y → Y be an operator. We say that T is a convex orbital ( λ , β ) -Lipschitz operator if there exists λ ∈ ( 0 , 1 ] and β > 0 such that for any x ∈ Y

∥ T x − T T λ x ∥ ≤ β λ ∥ x − T x ∥ .

Obviously, every convex orbital β -Lipschitz operator is a weak convex orbital Lipschitz operator and every weak convex orbital Lipschitz operator is a convex orbital ( λ , β ) -Lipschitz operator.

Example 2.1

(See Example 2.7 of [9]) Let ( X , ∥ . ∥ ) be a normed space, Y be a nonempty and convex subset of X , and T : Y → Y be an L -Lipschitz operator, i.e., L > 0 and for each x , y ∈ Y

∥ T x − T y ∥ ≤ L ∥ x − y ∥ .

Then, T is a convex orbital L -Lipschitz operator. Indeed, if we choose y ≔ T λ x in the aforementioned inequality, we have that

∥ T x − T T λ x ∥ ≤ L ∥ x − T λ x ∥ = L λ ∥ x − T x ∥ ,

for any λ ∈ ( 0 , 1 ] and for every x ∈ Y .

Example 2.2

(See Example 2.1 of [9]) Let ( X , ∥ . ∥ ) be a normed space, Y be a nonempty and convex subset of X , and T : Y → Y be an α -contraction, i.e., α ∈ ( 0 , 1 ) and for each x , y ∈ Y

∥ T x − T y ∥ ≤ α ∥ x − y ∥ .

Since every α -contraction is a Lipschitz operator with L ≔ α , then T is a convex orbital α -Lipschitz operator.

Example 2.3

(See Example 2.5 of [9]) Let ( X , ∥ . ∥ ) be a normed space, Y be a nonempty and convex subset of X , and T : Y → Y be a nonexpansive operator, i.e.,

∥ T x − T y ∥ ≤ ∥ x − y ∥ .

Since every nonexpansive operator is a Lipschitz operator with L ≔ 1 , then T is a convex orbital 1-Lipschitz operator.

Example 2.4

(See Example 2.6 of [9]) Let ( X , ∥ . ∥ ) be a normed space, Y be a nonempty and convex subset of X , and T : Y → Y be an enriched ( b , θ ) -contraction, i.e., there exist b ≥ 0 , θ ∈ [ 0 , b + 1 ) such that for each x , y ∈ Y

∥ b ( x − y ) + T x − T y ∥ ≤ θ ∥ x − y ∥ .

Then, T is a convex orbital ( b + θ ) -Lipschitz operator. Indeed, if we choose y ≔ T λ x in the aforementioned relation, we obtain that

∥ b λ ( x − T x ) + T x − T T λ x ∥ ≤ θ λ ∥ x − T x ∥ ,

from which we obtain

∥ T x − T λ x ∥ − b λ ∥ x − T x ∥ ≤ θ λ ∥ x − T x ∥ ,

for any λ ∈ ( 0 , 1 ] and for every x ∈ Y . Hence, we obtain that

∥ T x − T T λ x ∥ ≤ ( b + θ ) λ ∥ x − T x ∥ .

Example 2.5

(See Example 2.2 of [9]) Let ( X , ∥ . ∥ ) be a normed space, Y be a nonempty and convex subset of X , and T : Y → Y be a Kannan γ -contraction, i.e., γ ∈ [ 0 , 1 / 2 ) , and for each x , y ∈ Y ,

∥ T x − T y ∥ ≤ γ [ ∥ x − T x ∥ + ∥ y − T y ∥ ] .

Then, T is a weak convex orbital Lipschitz operator. Indeed, if we insert in the aforementioned inequality y ≔ T λ x , then we obtain for λ ∈ ( 0 , 1 ] and x ∈ Y that

∥ T x − T T λ x ∥ ≤ γ [ ∥ x − T x ∥ + ∥ ( 1 − λ ) x + λ T x − T T λ x ∥ ] ≤ γ [ ∥ x − T x ∥ + ( 1 − λ ) ∥ x − T x ∥ + ∥ T x − T T λ x ∥ ] .

Hence, we obtain

∥ T x − T T λ x ∥ ≤ γ ( 2 − γ ) 1 − γ ∥ x − T x ∥ .

Therefore, T is a weak convex orbital Lipschitz operator β = γ ( 2 − γ ) λ ( 1 − γ ) . Obviously, lim λ → 0 β ( λ ) = ∞ , so T is not a convex orbital β -Lipschitz operator.

Example 2.6

(See Example 2.3 of [9]) Let ( X , ∥ . ∥ ) be a normed space, Y be a nonempty and convex subset of X , and T : Y → Y be a Ćirić-Reich-Rus ( α , γ ) -contraction, i.e., α , γ ∈ R + with α + 2 γ < 1 , and for each x , y ∈ Y ,

∥ T x − T y ∥ ≤ α ∥ x − y ∥ + γ [ ∥ x − T x ∥ + ∥ y − T y ∥ ] .

Then, T is a weak convex orbital Lipschitz operator. Indeed, if we insert in the aforementioned inequality y ≔ T λ x , we obtain that

∥ T x − T T λ x ∥ ≤ α ∥ x − T λ x ∥ + γ [ ∥ x − T x ∥ + ∥ T λ x − T T λ x ∥ ] .

This implies

∥ T x − T T λ x ∥ ≤ α λ ∥ x − T x ∥ + γ [ ∥ x − T x ∥ + ( 1 − λ ) ∥ x − T x ∥ + ∥ T x − T T λ x ∥ ] .

Hence,

∥ T x − T T λ x ∥ ≤ α λ + γ ( 2 − λ ) 1 − γ ∥ x − T x ∥ .

Therefore, T is a weak convex orbital Lipschitz operator β = α λ + γ ( 2 − λ ) λ ( 1 − γ ) . Obviously, lim λ → 0 β ( λ ) = ∞ , so T is not a convex orbital β -Lipschitz operator.

Example 2.7

(See Example 2.4 of [9]) Let ( X , ∥ . ∥ ) be a normed space, Y be a nonempty and convex subset of X , and T : Y → Y be a Berinde ( α , L ) -contraction, i.e., α , L ∈ R + with α < 1 , and for each x , y ∈ Y ,

∥ T x − T y ∥ ≤ α ∥ x − y ∥ + L ∥ y − T x ∥ .

Then, T is a weak convex orbital Lipschitz operator. Indeed, if we insert in the aforementioned inequality y ≔ T λ x , we obtain that

∥ T x − T T λ x ∥ ≤ α ∥ x − T x ∥ + L ∥ T λ x − T x ∥ ≤ α ∥ x − T x ∥ + L ( 1 − λ ) ∥ x − T x ∥ ,

where we obtain for every λ ∈ ( 0 , 1 ] and each x ∈ Y

∥ T x − T T λ x ∥ ≤ ( α λ + L ( 1 − λ ) ) ∥ x − T x ∥ .

Therefore, T is a weak convex orbital Lipschitz operator β = α λ + L ( 1 − λ ) λ . Obviously, lim λ → 0 β ( λ ) = ∞ , so T is not a convex orbital β -Lipschitz operator.

The following examples show that there exist convex orbital ( λ , β ) -Lipschitz operators, which are not weak convex orbital Lipschitz operators.

Example 2.8

Let X = Y = R , T : R → R be a mapping defined by T x ≔ − x if x ≠ 0 and T x ≔ 1 if x = 0 . We have T 1 x = T x , T T 1 x = T 2 x = x if x ≠ 0 and T T 1 x = T 2 x = − 1 if x = 0 . Then, we obtain that T x − T T 1 x = − 2 x if x ≠ 0 , T x − T T 1 x = 2 if x = 0 , x − T x = 2 x if x ≠ 0 , and x − T x = − 1 if x = 0 . It is easy to see that the inequality ∥ T x − T T 1 x ∥ ≤ 2 ∥ x − T x ∥ holds for every x , so T is a convex orbital ( 1 , 2 ) -Lipschitz operator. Moreover, T 1 / 2 x = ( x + T x ) / 2 = 0 if x ≠ 0 and T 1 / 2 x = 1 / 2 if x = 0 . Thus, T T 1 / 2 x = 1 if x ≠ 0 , T T 1 / 2 x = − 1 / 2 if x = 0 , T x − T T 1 / 2 x = − x − 1 if x ≠ 0 , and T x − T T 1 / 2 x = 3 / 2 if x = 0 . For x ≠ 0 , the inequality ∥ T x − T T 1 / 2 x ∥ ≤ β / 2 ∥ x − T x ∥ is equivalent to ∣ x + 1 ∣ ≤ β ∣ x ∣ . For x → 0 , we obtain a contradiction. Then, T is not a weak convex orbital Lipschitz operator.

Example 2.9

Let X = Y = R , T : R → R be a mapping defined by T x ≔ 1 + 1 x − 1 if x > 1 , T x ≔ 2 if x ∈ [ − 1 , 1 ] , and T x ≔ 1 + 1 − x − 1 if x < − 1 . For x > 1 , we have T 1 x = T x , T T 1 x = T 2 x = x , T x − T T 1 x = 1 + 1 x − 1 − x , and x − T x = x − 1 − 1 x − 1 . Then, ∥ T x − T T 1 x ∥ ≤ ∥ x − T x ∥ . If x ∈ [ − 1 , 1 ] , we have T T 1 x = T 2 x = 2 , T x − T T 1 x = 2 − x and x − T x = x − 2 , by where we obtain ∥ T x − T T 1 x ∥ ≤ ∥ x − T x ∥ . For x < − 1 , we have T T 1 x = T 2 x = − x , T x − T T 1 x = 1 − 1 1 + x + x = x 2 + 2 x 1 + x and x − T x = x − 1 + 1 1 + x = x 2 1 + x . Since 1 + 2 / x ∈ ( − 1 , 1 ) , we obtain ∣ x 2 + 2 x ∣ ≤ ∣ x 2 ∣ , by where ∥ T x − T T 1 x ∥ ≤ ∥ x − T x ∥ . Therefore, T is a convex orbital ( 1 , 1 ) -Lipschitz operator. Since T 1 / 2 1 n = 1 + 1 2 n , T T 1 / 2 1 n = 1 + 2 n , T 1 n − T T 1 / 2 1 n = 1 − 2 n , and 1 n − T 1 n = 1 n − 1 , the inequality T 1 n − T T 1 / 2 1 n ≤ ( 1 / 2 ) β 1 n − T 1 n is not satisfied for n sufficiently large. Hence, T is not a weak convex orbital Lipschitz operator. Also, it is easy to see that T has a closed graph.

In the following example, we present a weak convex orbital Lipschitz operator with a closed graph, which is not a convex orbital β - Lipschitz operator.

Example 2.10

Let X = Y = R , T : R → R be a mapping defined by T x ≔ 1 / x if x ≠ 0 and T x ≔ 1 if x = 0 . Obviously, T has a closed graph. For x ≠ 0 , we have T λ x = ( 1 − λ ) x + λ x > 0 , T T λ x = x ( 1 − λ ) x 2 + λ , T x − T T λ x = λ ( 1 − x 2 ) x [ ( 1 − λ ) x 2 + λ ] , and x − T x = x 2 − 1 x . Taking β ≔ 1 / λ , we have 1 ( 1 − λ ) x 2 + λ ≤ β , so λ ∣ 1 − x 2 ∣ x [ ( 1 − λ ) x 2 + λ ] ≤ λ β ∣ x 2 − 1 ∣ ∣ x ∣ . Then, we obtain ∥ T x − T T λ x ∥ ≤ λ β ∥ x − T x ∥ . If x = 0 , we have T λ x = λ , T T λ x = 1 λ , T x − T T λ x = 1 − 1 λ , and x − T x = − 1 . Taking β = 1 − λ λ 2 , we obtain 1 − 1 λ ≤ λ β , so ∥ T x − T T λ x ∥ ≤ λ β ∥ x − T x ∥ . Therefore, for any λ ∈ ( 0 , 1 ] , there exists β ≔ max 1 λ , 1 − λ λ 2 such that ∥ T x − T T λ x ∥ ≤ λ β ∥ x − T x ∥ , i.e., T is a weak convex orbital Lipschitz operator. Now, if we take x = λ = 1 / n with n ≥ 2 , we have T x − T T λ x = n 3 − n n 2 + n − 1 , x − T x = 1 − n 2 n . Then, the inequality ∥ T x − T T λ x ∥ ≤ λ β ∥ x − T x ∥ is equivalent to the inequality n 3 n 2 + n − 1 ≤ β , which is not satisfied for n sufficiently large. Hence, T is not a convex orbital β - Lipschitz operator.

Now, we give similar results of Theorems 1.2, 1.3, and 1.4 (which hold for convex orbital β -Lipschitz operators) for convex orbital ( λ , β ) -Lipschitz operators.

Theorem 2.1

Let ( X , ∥ ⋅ ∥ ) be a Banach space and Y be a nonempty closed and convex subset of X. Let T : Y → Y be a convex orbital ( λ , β ) -Lipschitz operator with closed graph, where β < 1 . Then, for every x 0 ∈ Y , the sequence ( x n ) n ∈ N ⊂ Y , defined by

x n + 1 = ( 1 − λ ) x n + λ T x n , n ∈ N ,

converges to a fixed-point x ∗ ( x 0 ) of T .

Proof

Let the operator T λ : Y → Y defined by

T λ ≔ ( 1 − λ ) x + λ T x , x ∈ Y .

It is easy to see that F T = F T λ and T λ has a closed graph. For every x , y ∈ Y , we have

∥ T λ x − T λ y ∥ = ∥ ( 1 − λ ) ( x − y ) + λ ( T x − T y ) ∥ ≤ ( 1 − λ ) ∥ x − y ∥ + λ ∥ T x − T y ∥ .

Taking y ≔ T λ x , we obtain

∥ T λ x − T λ 2 x ∥ ≤ ( 1 − λ ) ∥ x − T λ x ∥ + λ ∥ T x − T T λ x ∥ ≤ ( 1 − λ ) ∥ x − T λ x ∥ + β λ 2 ∥ x − T x ∥ = ( 1 − λ ) ∥ x − T λ x ∥ + β λ ∥ x − T λ x ∥ = ( 1 − λ + β λ ) ∥ x − T λ x ∥ .

Since β < 1 , if we denote k ≔ 1 − λ + β λ , then k < 1 and

∥ T λ x − T λ 2 x ∥ ≤ k ∥ x − T λ x ∥ ,

for every x ∈ Y . This shows that T λ : Y → Y is a graphic k -contraction. Hence, by the graphic contraction principle, T λ is a weakly Picard operator. Since F T = F T λ , we have F T ≠ ∅ and the sequence ( T λ n x 0 ) n ∈ N converges to T λ ∞ x 0 ≔ x ∗ ( x 0 ) ∈ F T , for every x 0 ∈ Y .□

The following theorem is our main result.

Theorem 2.2

Let ( X , ⟨ ⋅ ⟩ ) be a Hilbert space, Y be a nonempty closed and convex subset of X, and T : Y → Y be an operator with a closed graph. We suppose that:

T is a convex orbital ( λ , β ) -Lipschitz operator with β ≥ 1 ;
Re ( ⟨ T u − T v , u − v ⟩ ) ≤ μ ∥ u − v ∥ 2 , for every u , v ∈ Y , where μ < 2 − λ ( 1 + β 2 ) 2 ( 1 − λ ) .

Then, for every x 0 ∈ Y , the sequence ( x n ) n ∈ N ⊂ Y , defined by

x n + 1 = ( 1 − λ ) x n + λ T x n , n ∈ N ,

converges to the unique fixed-point x ∗ ∈ Y of T .

Proof

Consider the operator T λ : Y → Y defined by

T λ ≔ ( 1 − λ ) x + λ T x , x ∈ Y .

Obviously, F T = F T λ and T λ has a closed graph. By using (ii), for every x , u ∈ Y , we have:

∥ T λ x − T λ u ∥ 2 = ∥ ( 1 − λ ) ( x − u ) + λ ( T x − T u ) ∥ 2 ≤ ( 1 − λ ) 2 ∥ x − u ∥ 2 + λ 2 ∥ T x − T u ∥ 2 + 2 λ ( 1 − λ ) Re ( ⟨ T x − T u , x − u ⟩ ) ≤ ( 1 − λ ) 2 ∥ x − u ∥ 2 + λ 2 ∥ T x − T u ∥ 2 + 2 λ ( 1 − λ ) μ ∥ x − u ∥ 2 .

Taking u ≔ T λ x in the aforementioned inequality, we obtain

∥ T λ x − T λ 2 x ∥ 2 ≤ [ ( 1 − λ ) 2 + 2 λ ( 1 − λ ) μ ] ∥ x − T λ x ∥ 2 + λ 2 ∥ T x − T T λ x ∥ 2 = [ ( 1 − λ ) 2 + 2 λ ( 1 − λ ) μ ] ∥ x − T λ x ∥ 2 + λ 4 β 2 ∥ x − T x ∥ 2 = [ ( 1 − λ ) 2 + 2 λ ( 1 − λ ) μ ] ∥ x − T λ x ∥ 2 + λ 2 β 2 ∥ x − T λ x ∥ 2 = [ ( 1 − λ ) 2 + 2 λ ( 1 − λ ) μ + λ 2 β 2 ] ∥ x − T λ x ∥ 2 .

If we denote by k ≔ ( 1 − λ ) 2 + 2 λ ( 1 − λ ) μ + λ 2 β 2 , we have by (ii) that k < 1 and

∥ T λ x − T λ 2 x ∥ ≤ k ∥ x − T λ x ∥ ,

for every x ∈ Y . Thus, by graphic contraction principle, T λ is a weakly Picard operator and the sequence ( T λ n x 0 ) n ∈ N converges to T λ ∞ x 0 ≔ x ∗ ( x 0 ) ∈ F T , for every x 0 ∈ Y .

Now, let us suppose that there exist x ∗ , y ∗ ∈ F T with x ∗ ≠ y ∗ . Then, we have x ∗ = T x ∗ = T λ x ∗ and y ∗ = T y ∗ = T λ y ∗ . Taking u ≔ x ∗ and v ≔ y ∗ in (ii), we obtain

Re ( ⟨ T x ∗ − T y ∗ , x ∗ − y ∗ ⟩ ) ≤ μ ∥ x ∗ − y ∗ ∥ 2 .

Hence,

∥ x ∗ − y ∗ ∥ 2 ≤ μ ∥ x ∗ − y ∗ ∥ 2 .

Since β ≥ 1 , we have 2 − λ ( 1 + β ) 2 ≤ 2 ( 1 − λ ) , and hence, μ < 1 . Therefore, ∥ x ∗ − y ∗ ∥ = 0 , which is a contradiction. Thus, F T = F T λ = { x ∗ } and T λ is a Picard operator.□

We will illustrate the aforementioned theorem by the following example:

Example 2.11

Let T : R 2 → R 2 be a mapping defined by

T ( x , y ) ≔ 3 4 ( x − y , x + y ) .

Then:

T is a convex orbital ( 1 / 2 , 3 2 / 2 ) -Lipschitz operator;
T satisfies (ii) of Theorem 2.2 with μ = 3 / 4 ;
T is continuous on R 2 ;
T is not decreasing on R 2 ;
F T = { ( 0 , 0 ) } and ∥ T 1 / 2 n ( x , y ) ∥ = ( 58 / 8 ) n ∥ ( x , y ) ∥ → 0 as n → ∞ .
1. For ( x , y ) ∈ R 2 , we have:
  T 1 / 2 ( x , y ) = ( 1 / 2 ) ( x , y ) + ( 1 / 2 ) T ( x , y ) = ( 1 / 8 ) ( 7 x − 3 y , 3 x + 7 y ) ,
  
  T T 1 / 2 ( x , y ) = ( 3 / 16 ) ( 2 x − 5 y , 5 x + 2 y ) .
  Hence, we obtain:
  T ( x , y ) − T T 1 / 2 ( x , y ) = ( 3 / 16 ) ( 2 x + y , − x + 2 y ) .
  Since ( x , y ) − T ( x , y ) = ( 1 / 4 ) ( x + 3 y , x − 3 y ) , we obtain:
  ∥ T ( x , y ) − T T 1 / 2 ( x , y ) ∥ = ( 3 5 / 16 ) x 2 + y 2
  and
  ∥ ( x , y ) − T ( x , y ) ∥ = ( 10 / 4 ) x 2 + y 2 .
  Therefore, we have
  ∥ T ( x , y ) − T T 1 / 2 ( x , y ) ∥ = ( 1 / 2 ) ( 3 2 / 4 ) ∥ ( x , y ) − T ( x , y ) ∥ .
  Thus, T is a convex orbital ( 1 / 2 , 3 2 / 4 ) -Lipschitz operator.
2. If ( x 1 , y 1 ) , ( x 2 , y 2 ) ∈ R 2 , then we have:
  T ( x 1 , y 1 ) − T ( x 2 , y 2 ) = ( 3 / 4 ) ( x 1 − x 2 − ( y 1 − y 2 ) , x 1 − x 2 + y 1 − y 2 ) .
  Hence,
  Re ( ⟨ T ( x 1 , y 1 ) − T ( x 2 , y 2 ) , ( x 1 , y 1 ) − ( x 2 , y 2 ) ⟩ ) = ( 3 / 4 ) [ ( x 1 − x 2 ) − ( y 1 − y 2 ) ] ( x 1 − x 2 ) + ( 3 / 4 ) [ x 1 − x 2 + y 1 − y 2 ] ( y 1 − y 2 ) = ( 3 / 4 ) [ ( x 1 , − x 2 ) 2 + ( y 1 − y 2 ) 2 ] = ( 3 / 4 ) ∥ ( x 1 , y 1 ) − ( x 2 , y 2 ) ∥ 2 .
  Therefore, T satisfies (ii) of Theorem 2.2 with μ = 3 / 4 . We note that μ < 2 − λ ( 1 + β 2 ) 2 ( 1 − λ ) = 15 / 16 .
3. It is obvious.
4. For ( x 1 , y 1 ) = ( 2 , 0 ) and ( x 2 , y 2 ) = ( 1 , 0 ) , we have
  Re ( ⟨ T ( x 1 , y 1 ) − T ( x 2 , y 2 ) , ( x 1 , y 1 ) − ( x 2 , y 2 ) ⟩ ) = 3 / 4 > 0 ,
  hence T is not decreasing.
5. It is easy to see that F T = { ( 0 , 0 ) } and ∥ T 1 / 2 ( x , y ) ∥ = ( 58 / 8 ) ∥ ( x , y ) ∥ . This implies that ∥ T 1 / 2 n ( x , y ) ∥ = ( 58 / 8 ) n ∥ ( x , y ) ∥ → 0 as n → ∞ .

By the previous theorems, we obtain some additional properties of the fixed-point equation x = T x .

Theorem 2.3

Let ( X , ∥ ⋅ ∥ ) be a Banach space and Y be a nonempty closed and convex subset of X . Let T : Y → Y be a convex orbital ( λ , β ) -Lipschitz operator with a closed graph, where β < 1 . Then, the following conclusions hold:

T satisfies the following retraction-displacement condition
∥ x − x ∗ ( x ) ∥ ≤ 1 1 − β ∥ x − T x ∥ ,
for every x ∈ Y ;
the fixed-point equation x = T x is Ulam-Hyers stable;
if β < 1 / 3 and λ > 2 3 ( 1 − β ) , then T has the Ostrowski stability property.

Proof

(a) By the proof of Theorem 2.1, the operator T λ : Y → Y , given by T λ x ≔ ( 1 − λ ) x + λ T x is weakly Picard. By graphic contraction principle, we obtain

∥ x − x ∗ ( x ) ∥ ≤ 1 1 − k ∥ x − T λ x ∥ ,

for every x ∈ Y , where ( T λ n x ) n ∈ N converges to x ∗ ( x ) and k = 1 − λ + λ β . Since ∥ x − T λ x ∥ = λ ∥ x − T x ∥ , we obtain that

∥ x − x ∗ ( x ) ∥ ≤ λ 1 − k ∥ x − T x ∥ = 1 1 − β ∥ x − T x ∥ ,

for every x ∈ Y . This proves that T satisfies the ( c , r ) -retraction-displacement condition, where c ≔ 1 1 − β and r : Y → F T is given by r ( x ) ≔ x ∗ ( x ) , x ∈ Y .

(b) Let ε > 0 and y ∈ Y such that ∥ y − T y ∥ ≤ ε . Then, we have

∥ y − x ∗ ( y ) ∥ ≤ 1 1 − β ∥ y − T y ∥ ≤ ε 1 − β .

(c) By the graphic contraction principle, we know that T has the Ostrowski stability property if k < 1 / 3 . This means that 1 − λ + λ β < 1 / 3 , i.e., λ > 2 3 ( 1 − β ) . Since β < 1 / 3 , we have 2 3 ( 1 − β ) < 1 , by where there exists λ ≤ 1 such that λ > 2 3 ( 1 − β ) . Also, in this case, T is a k 1 − 2 k -quasicontraction.□

Theorem 2.4

Let ( X , ⟨ ⋅ ⟩ ) be a Hilbert space, Y be a nonempty closed and convex subset of X , and T : Y → Y be an operator satisfying all the conditions in Theorem 2.2. If x ∗ is the unique fixed-point of T , then the following conclusions hold:

T satisfies the retraction-displacement condition
∥ x − x ∗ ∥ ≤ λ 1 − k ∥ x − T x ∥ ,
for every x ∈ Y , where k ≔ ( 1 − λ ) 2 + 2 λ ( 1 − λ ) μ + λ 2 β 2 ;
the fixed-point equation x = T x is Ulam-Hyers stable;
the fixed-point equation x = T x is well posed.

Proof

By graphic contraction principle and the proof of Theorem 2.2, we have
∥ x − x ∗ ∥ ≤ 1 1 − k ∥ x − T λ x ∥ = λ 1 − k ∥ x − T x ∥ ,
for every x ∈ Y .
Similarly with (b) from Theorem 2.3.
Let ( u n ) n ∈ N be a sequence in Y such that lim n → ∞ ∥ u n − T u n ∥ = 0 . Then, by (a), we have that
∥ u n − x ∗ ∥ ≤ λ 1 − k ∥ u n − T u n ∥ → 0
as n → ∞ . Hence, u n → x ∗ as n → ∞ .□

Acknowledgements

The author would like to thank the referees for their valuable comments and suggestions that improved the presentation of this article.

Funding information: None declared.
Author contributions: The author has accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The author states no conflict of interest.

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Received: 2022-07-10

Revised: 2022-11-03

Accepted: 2022-11-24

Published Online: 2023-01-25

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

Articles in the same Issue

https://doi.org/10.1515/dema-2022-0184

Keywords for this article

Hilbert space; graphic contraction; convex orbital Lipschitz operator; weakly Picard operator; fixed-point

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