Fixed point theorems of enriched multivalued mappings via sequentially equivalent Hausdorff metric

Mujahid Abbas; Rizwan Anjum; Muhammad Haris Tahir

doi:10.1515/taa-2022-0136

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Fixed point theorems of enriched multivalued mappings via sequentially equivalent Hausdorff metric

Mujahid Abbas , Rizwan Anjum and Muhammad Haris Tahir

Published/Copyright: August 27, 2023

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From the journal Topological Algebra and its Applications Volume 11 Issue 1

Abstract

Recently, Abbas et al. [Enriched multivalued contractions with applications to differential inclusions and dynamic programming, Symmetry 13(8) (2021), 1350] obtained an interesting generalization of the Nadler fixed point theorem by introducing the concept of enriched multivalued contraction in the framework of Banach spaces. In this article, we define a new class of metrics on the family of closed and bounded subsets of a given metric space. Furthermore, fixed point theorems were established for enriched multi-valued contractions by substituting the Hausdorff metric with metrics from a specific class that are either metrically or sequentially equivalent to the Hausdorff metric. Some examples are provided to illustrate the concepts and results presented herein. These results improve, unify, and generalize several comparable results in the literature.

Keywords: fixed point; Hausdorff metric; enriched multivalued contraction; sequentially equivalent metric

MSC 2010: 47H10; 47H09

1 Introduction

Let ( X , d ) be a metric space and T : X → X . An element x ∈ X is called a fixed point of T if it remains invariant under the action of T , i.e., T x = x . A mapping T on a metric space ( X , d ) is said to be a Banach contraction if there exists 0 ≤ k < 1 such that d ( T x , T y ) ≤ k d ( x , y ) holds for all x , y ∈ X .

The Banach contraction principle [5] states that a Banach contraction mapping defined on a complete metric space has a unique fixed point. Several authors have extended and generalized the Banach contraction principle in different directions. One of the results known as Nadler’s fixed point theorem extended the Banach contraction principle from single-valued maps to multivalued maps.

Let P ( X ) ( C B ( X ) ) be the family of all nonempty subsets (the family of all nonempty closed and bounded subsets) of X . Let T : X → P ( X ) . An element x ∈ T x is called a fixed point of T . The set { x ∈ X : x ∈ T ( x ) } of all fixed points of a multivalued T is denoted by Fix ( T ) .

The symmetric functional H : C B ( X ) × C B ( X ) → [ 0 , ∞ ) defined by

H ( A , B ) = max { D ( A , B ) , D ( B , A ) } ,

where D ( A , B ) = sup a ∈ A { η ( a , B ) } and η ( a , B ) = inf { d ( a , b ) , b ∈ B } , for all A , B ∈ C B ( X ) is a metric on C B ( X ) called the Pompeiu-Hausdorff metric (see [10]).

A mapping T : X → C B ( X ) is called a multivalued contraction mapping if

(1) H ( T x , T y ) ≤ k d ( x , y )

holds for all x , y ∈ X , where k is a constant such that k ∈ ( 0 , 1 ) .

The study of fixed point theorems for multivalued mappings was initiated by Markin [13] and Nadler [14]. The following result is due to Nadler [14].

Theorem 1.1

[14] Let ( X , d ) be a complete metric space. Then, a multivalued contraction mapping T : X → C B ( X ) has a fixed point.

Later, several interesting fixed point theorems for multivalued mappings were obtained (see [2,7,11] and especially the monographs of Rus [16–18]).

One of the interesting generalizations was proved by Abbas et al. [1] by extending the concept of multivalued contraction to enriched multivalued contraction.

Following [1], let ( X , ∥ ⋅ ∥ ) be a normed space. A multivalued mapping T : X → C B ( X ) is called ( b , θ ) -enriched multivalued contraction if there exist b ∈ [ 0 , ∞ ) and θ ∈ [ 0 , b + 1 ) such that

(2) H ( b x + T x , b y + T y ) ≤ θ ‖ x − y ‖ ∀ x , y ∈ X .

Note that a ( 0 , θ ) -enriched multivalued contraction is a multivalued contraction (1). It was proved that any ( b , θ ) -enriched multivalued contraction defined on a Banach space has at least one fixed point. For more details on the study of enriched contraction mappings, we refer to [3,4,6,8,9] and references therein.

On the other hand, Kirk and Shahzad [12] extended Nadler’s fixed point theorem by replacing the Hausdorff metric with a sequentially equivalent metric and obtained fixed points of multivalued contraction mappings.

Before stating the main result of [12], let us recall the following definition.

Definition 1

[12] A metric U : CB ( X ) × CB ( X ) → [ 0 , ∞ ) is called sequentially equivalent to the Hausdorff metric H if for Q ∈ CB ( X ) and for the sequence { Q n } ⊆ CB ( X ) , we have

lim n → ∞ U ( Q n , Q ) = 0 if and only if lim n → ∞ H ( Q n , Q ) = 0 .

Example 1

[12] Define H + : CB ( X ) × CB ( X ) → [ 0 , + ∞ ) by

H + ( P , Q ) = 1 2 [ D ( P , Q ) + D ( Q , P ) ] for every P , Q ∈ CB ( X ) .

Note that H + is sequentially equivalent to the Hausdorff metric H . Indeed, H + is a metric on CB ( X ) (see [15]). In addition, for any P , Q ∈ CB ( X ) , we have

(3) 1 2 H ( P , Q ) ≤ H + ( P , Q ) ≤ H ( P , Q ) .

Thus, for any sequence { Q n } ⊆ C B ( X )

lim n → ∞ H + ( Q n , Q ) = 0 if and only if lim n → ∞ H ( Q n , Q ) = 0 for every Q ∈ CB ( X ) .

In the above Definition 1, if we replace ( CB ( X ) , H ) with the metric space ( X , d ) , then any metric p on X is called sequentially equivalent to the metric d , if for z ∈ X and for the sequence { z n } ⊆ X , we have

lim n → ∞ p ( z n , z ) = 0 if and only if lim n → ∞ d ( z n , z ) = 0 .

Now we state the main result in [12].

Theorem 1.2

[12] Let ( X , d ) be a complete metric space and U be a metric on CB ( X ) , which is sequentially equivalent to H. Assume that there exists a constant k ∈ [ 0 , 1 ) and a multivalued mapping T : X → CB ( X ) such that

for each y , z ∈ X
(4) U ( T y , T z ) ≤ k d ( y , z ) ;
and
for y ∈ X and z ∈ T y ,
(5) η ( z , T z ) ≤ U ( T z , T y ) .

Then, T has a fixed point in X.

Recall that two metrics defined on a metric space are called equivalent if they induce the same metric topology.

The aim of this article is to extend the fixed point theorem of enriched multivalued contraction (2) by replacing the Hausdorff metric with metrics on C B ( X ) , which are either metrically or sequentially equivalent to the Pompeiu-Hausdorff metric.

2 H r class of metrics

For any metric space ( X , d ) , we now introduce a new class of metrics on CB ( X ) , named as H r class of metrics.

Definition 2

For any positive real number r , define H r : CB ( X ) × CB ( X ) → [ 0 , ∞ ) by

(6) H r ( P , Q ) = r [ D ( P , Q ) + D ( Q , P ) ] , ∀ P , Q ∈ CB ( X ) .

Now we prove that for each fixed r , H r is a metric on CB ( X ) .

Proposition 2.1

Let ( X , d ) be any metric space. Then, for any real number r > 0 , H r is a metric on CB ( X ) . Moreover, H r is sequentially equivalent to the Hausdorff metric H.

Proof

First, we prove that, for any r > 0 , H r satisfies all the properties of a metric.

It is obvious to note that, for any P , Q ∈ CB ( X ) , H r ( P , Q ) ≥ 0 .
Let H r ( P , Q ) = 0 . This implies that D ( P , Q ) = D ( Q , P ) = 0 . It is easy to show that P ⊆ Q ¯ = Q and Q ⊆ P ¯ = P , where Q ¯ and P ¯ stand for closure of Q and P , respectively. Then, we have P = Q . On the other hand, P = Q gives D ( P , Q ) = D ( Q , P ) = 0 . Therefore, H r ( P , Q ) = 0 .
H r ( P , Q ) = r [ D ( P , Q ) + D ( Q , P ) ] = r [ D ( Q , P ) + D ( P , Q ) ] = H r ( Q , P ) for any P , Q ∈ CB ( X ) .
If P , Q , R ∈ CB ( X ) . Then, for any p ∈ P , q ∈ Q and r ∈ R , we have
d ( p , r ) ≤ d ( p , q ) + d ( q , r ) ,
and
(7) inf r ∈ R d ( p , r ) ≤ d ( p , q ) + inf r ∈ R d ( q , r ) ≤ d ( p , q ) + D ( Q , R ) .
As equation (7) is true for each p ∈ P , we obtain that
inf r ∈ R d ( p , r ) ≤ d ( p , q ) + D ( Q , R ) ≤ D ( P , Q ) + D ( Q , R ) ,
and hence,
(8) D ( P , R ) ≤ D ( P , Q ) + D ( Q , R ) .
Interchanging the roles of P and R , we obtain
(9) D ( R , P ) ≤ D ( R , Q ) + D ( Q , P ) .
Taking sum of equations (8) and (9) and then multiplying it by r , we obtain
H r ( R , P ) ≤ H r ( R , Q ) + H r ( Q , P ) .

Furthermore, it is easy to check that H r is metrically equivalent to the Hausdorff metric; indeed, for P , Q ∈ CB ( X ) , we have

(10) r H ( P , Q ) ≤ H r ( P , Q ) ≤ 2 r H ( P , Q ) .

By (10), we have

lim n → ∞ H r ( Q n , Q ) = 0 ⇔ lim n → ∞ H ( Q n , Q ) = 0 ,

for each Q ∈ CB ( X ) and for any sequence { Q n } ⊆ CB ( X ) .□

Let ( X , ‖ ⋅ ‖ ) be a normed space. We denote by Ω 1 the class of metrics U on CB ( X ) that satisfy the following conditions:

U is sequentially equivalent to the Hausdorff metric H ;
for each β ∈ [ 0 , ∞ ) and for P , Q ∈ CB ( X ) , we have
U ( β P , β Q ) = β U ( P , Q ) .

Example 2

Let ( X , ‖ ⋅ ‖ ) be a normed space. As, for any r > 0 , the metric H r on CB ( X ) is sequentially equivalent to the Hausdorff metric H . Moreover, we can obtain that

H r ( β P , β Q ) = β H r ( P , Q ) ,

where β ∈ [ 0 , ∞ ) . Therefore, H r ∈ Ω 1 . In particular, H + ∈ Ω 1 .

Remark 1

For each positive real number r and for any P , Q ∈ CB ( X ) , we have

(11) H r ( P , Q ) = 2 r H + ( P , Q ) .

In particular,

H 1 2 ( P , Q ) = H + ( P , Q ) .

Proof

The proof is straightforward by the definitions of H r and H + . □

We now define another class Ω 2 of metrics U on CB ( X ) that satisfy the following conditions:

U ∈ Ω 1 ;
for any P , Q ∈ CB ( X ) ,
η ( q , P ) ≤ 2 U ( Q , P ) ∀ q ∈ Q .

Clearly, Ω 2 ⊆ Ω 1 .

Example 3

Let ( X , ‖ ⋅ ‖ ) be a normed space. We know that H + ∈ Ω 1 .

In addition, from equations (10) and (11), it is easy to see that for any P , Q ∈ CB ( X ) , we have

η ( q , P ) ≤ 2 H + ( Q , P ) ∀ q ∈ Q .

Therefore, H + ∈ Ω 2 .

Example 4

For a normed space ( X , ‖ ⋅ ‖ ) , define H 1 3 : CB ( X ) × CB ( X ) → [ 0 , ∞ ) by

H 1 3 ( P , Q ) = 1 3 [ D ( P , Q ) + D ( P , Q ) ]

We know that H 1 3 ∈ Ω 1 .

On the other hand, if we take X = R , P = [ − 5 , 2 ] , and Q = [ 1 , 3 ] , we have

D ( P , Q ) = 6 and D ( Q , P ) = 1 ,

which implies that

H 1 3 ( P , Q ) = 7 3 .

If we choose − 5 ∈ P = [ − 5 , 2 ] , then

η ( − 5 , Q ) = 6 > 14 3 = 2 H 1 3 ( P , Q ) .

Therefore, H 1 3 ∉ Ω 2 .

3 Fixed point theorem for a new class of enriched multivalued contraction

In this section, we first define a new class of multivalued contractions in normed spaces by using the metrics of class Ω 1 . Afterward, we prove the existence theorem for the fixed point of such multivalued contractions in a Banach space.

We need the following remark in order to prove the main results of this article.

Remark 2

[1] Let M be a convex subset of a normed space X and T : M → C B ( M ) . Then, for any λ ∈ ( 0 , 1 ) , consider the mapping T λ : M → C B ( M ) given by

(12) T λ ( x ) = ( 1 − λ ) x + λ T x

(13) = { ( 1 − λ ) x + λ s : s ∈ T x } .

In other words, for each x in M , T λ ( x ) is the translation of the set λ T x by the vector ( 1 − λ ) x . Clearly,

Fix ( T λ ) = Fix ( T ) .

We now introduce the following class of multivalued mappings.

Definition 3

Let ( X , ‖ ⋅ ‖ ) be a normed space and T : X → CB ( X ) . Then, T is said to be a ( b , k , U ) -enriched multivalued contraction if there exist two constants b ∈ [ 0 , ∞ ) , k ∈ [ 0 , b + 1 ) and a metric U ∈ Ω 1 such that

for all y , z ∈ X ,
(14) U ( b y + T y , b z + T z ) ≤ k ‖ y − z ‖ ,
for y ∈ X and z ∈ T λ y ,
(15) η ( z , T z ) ≤ U ( b z + T z , b y + T y ) ,
where λ = 1 b + 1 .

We start with the following result.

Theorem 3.1

Let ( X , ‖ ⋅ ‖ ) be a Banach space and T : X → CB ( X ) a ( b , k , U ) -enriched multivalued contraction. Then, Fix ( T ) ≠ ∅ .

Proof

Take λ = 1 b + 1 . Clearly, 0 < λ < 1 . In this case, equation (14) becomes

U 1 λ − 1 y + T y , 1 λ − 1 z + T z ≤ k ‖ y − z ‖

and hence,

U ( ( 1 − λ ) y + λ T y , ( 1 − λ ) z + λ T z ) ≤ λ k ‖ y − z ‖ .

Equivalently, it can be written as follows:

(16) U ( T λ y , T λ z ) ≤ θ ‖ y − z ‖ , for each y , z ∈ X ,

where θ = λ k ∈ [ 0 , 1 ) . Similarly, equation (15) becomes

η ( z , T z ) ≤ U 1 λ − 1 z + T z , 1 λ − 1 y + T y = 1 λ U ( ( 1 − λ ) z + λ T z , ( 1 − λ ) y + λ T y ) ,

hence

(17) λ η ( z , T z ) ≤ U ( ( 1 − λ ) z + λ T z , ( 1 − λ ) y + λ T y ) .

Note that

λ η ( z , T z ) = λ inf { ∥ z − g ∥ : g ∈ T z } = inf { ∥ λ z − λ g ∥ : g ∈ T z } = inf { ∥ λ z + z − z − λ g ∥ : g ∈ T z } = inf { ∥ z − z ( 1 − λ ) − λ g ∥ : g ∈ T z } = inf { ∥ z − ( z ( 1 − λ ) + λ g ) ∥ : g ∈ T z } = inf { ∥ z − w ∥ : z ( 1 − λ ) + λ g = w ∈ T λ z } = η ( z , T λ z ) .

Then, for y ∈ X and z ∈ T λ y , we have

(18) η ( z , T λ z ) ≤ U ( T λ z , T λ y ) .

Now, let z 0 ∈ X and z 1 ∈ T λ z 0 . By equations (16) and (18), there exists z 2 ∈ T λ z 1 such that

‖ z 2 − z 1 ‖ ≤ U ( T λ z 0 , T λ z 1 ) + θ ≤ θ ‖ z 1 − z 0 ‖ + θ .

Similarly, there exists z 3 ∈ T λ z 2 such that

‖ z 3 − z 2 ‖ ≤ U ( T λ z 1 , T λ z 2 ) + θ 2 ≤ θ ‖ z 2 − z 1 ‖ + θ 2 ≤ θ [ θ ‖ z 1 − z 0 ‖ + θ ] + θ 2 = θ 2 ‖ z 1 − z 0 ‖ + 2 θ 2 .

In general, for each i ∈ N , there exists z i + 1 ∈ T λ z i such that

‖ z i + 1 − z i ‖ ≤ U ( T λ z i − 1 , T λ z i ) + θ i ≤ θ ‖ z i − z i − 1 ‖ + θ i ≤ θ [ U ( T λ z i − 2 , T λ z i − 1 ) + θ i − 1 ] + θ i ≤ θ 2 ‖ z i − 1 − z i − 2 ‖ + 2 θ i ≤ … ≤ θ i ‖ z 1 − z 0 ‖ + i θ i .

Therefore,

∑ i = 0 ∞ ‖ z i + 1 − z i ‖ ≤ ‖ z 1 − z 0 ‖ ∑ i = 0 ∞ θ i + ∑ i = 0 ∞ i θ i < ∞ .

Hence, { z n } is a Cauchy sequence in X , there exists z ∗ ∈ X such that lim n → ∞ z n = z ∗ . It follows from equation (16) that lim n → ∞ U ( T λ z n , T λ z ∗ ) = 0 . We know that, U and H are sequentially equivalent so lim n → ∞ H ( T λ z n , T λ z ∗ ) = 0 .

It follows from the definition of the Hausdorff metric, we get

η ( z n + 1 , T λ z ∗ ) ≤ H ( T λ z n , T λ z ∗ ) ,

On taking the limit as n → ∞ , we obtain that

lim n → ∞ η ( z n + 1 , T λ z ∗ ) = η ( z ∗ , T λ z ∗ ) = 0 .

Since T λ z ∗ is closed, this gives z ∗ ∈ T λ z ∗ . Hence, z ∗ ∈ T z ∗ .□

If we take U = H in Theorem 3.1, we obtain the Corollary 1 of [1] in the setting of Banach spaces.

Corollary 3.1.1

[1] Let ( X , ‖ ⋅ ‖ ) be a Banach space and T : X → CB ( X ) be an enriched multivalued contraction. Then, Fix ( T ) ≠ ∅ .

By using the class Ω 2 , we define enriched weak multivalued contraction and prove a fixed point theorem for such mappings.

Definition 4

Let ( X , ‖ ⋅ ‖ ) be a normed space and T : X → CB ( X ) . Then, T is said to be a ( b , L , k , U ) -enriched weak multivalued contraction if there exist three constants L ∈ 0 , 1 2 , b ∈ [ 0 , ∞ ) , k ∈ [ 0 , b + 1 ) , and a metric U ∈ Ω 2 such that

for all y , z ∈ X ,
(19) U ( b y + T y , b z + T z ) ≤ k ‖ y − z ‖ + L . η ( ( b + 1 ) z , b y + T y ) ,
or equivalently,
(20) U ( b y + T y , b z + T z ) ≤ k ‖ y − z ‖ + L . η ( ( b + 1 ) y , b z + T z ) ,
for y ∈ X and z ∈ T λ y ,
(21) η ( z , T z ) ≤ U ( b z + T z , b y + T y ) ,
where λ = 1 b + 1 .

Now we present following result.

Theorem 3.2

Let ( X , ‖ ⋅ ‖ ) be a Banach space. Assume that T : X → CB ( X ) is a ( b , L , k , U ) -enriched weak multivalued contraction. Then, Fix ( T ) ≠ ∅ .

Proof

Take λ = 1 b + 1 . Clearly, 0 < λ < 1 . In this case, equation (20) becomes

U 1 λ − 1 y + T y , 1 λ − 1 z + T z ≤ k ‖ y − z ‖ + L . η 1 λ − 1 + 1 y , 1 λ − 1 z + T z ,

and hence, we have

U ( ( 1 − λ ) y + λ T y , ( 1 − λ ) z + λ T z ) ≤ λ k ‖ y − z ‖ + L . η ( y , ( 1 − λ ) z + λ ( T z ) ) .

Equivalently, it can be written as follows:

(22) U ( T λ y , T λ z ) ≤ θ ‖ y − z ‖ + L . η ( y , T λ z ) for each y , z ∈ X ,

where θ = λ k ∈ [ 0 , 1 ) . Similarly, equation (21) becomes

η ( z , T z ) ≤ U 1 λ − 1 z + T z , 1 λ − 1 y + T y ,

and hence,

η ( z , T λ z ) ≤ U ( ( 1 − λ ) z + λ ( T z ) , ( 1 − λ ) y + λ ( T y ) ) .

Then, for y ∈ X and z ∈ T λ y , we have

(23) η ( z , T λ z ) ≤ U ( T λ z , T λ y ) .

Now choose z 0 ∈ X and z 1 ∈ T λ z 0 . By equations (22) and (23), there exists z 2 ∈ T λ z 1 such that

‖ z 2 − z 1 ‖ ≤ U ( T λ z 0 , T λ z 1 ) + θ ≤ θ ‖ z 1 − z 0 ‖ + L . η ( z 1 , T λ z 0 ) + θ = θ ‖ z 1 − z 0 ‖ + θ .

Similarly, there exists z 3 ∈ T λ z 2 such that

‖ z 3 − z 2 ‖ ≤ U ( T λ z 1 , T λ z 2 ) + θ 2 ≤ θ ‖ z 2 − z 1 ‖ + L . η ( z 2 , T λ z 1 ) + θ 2 = θ ‖ z 2 − z 1 ‖ + θ 2 ≤ θ [ θ ‖ z 1 − z 0 ‖ + θ ] + θ 2 = θ 2 ‖ z 1 − z 0 ‖ + 2 θ 2 .

In general, for each i ∈ N , there exists z i + 1 ∈ T λ z i such that

‖ z i + 1 − z i ‖ ≤ U ( T λ z i − 1 , T λ z i ) + θ i ≤ θ ‖ z i − z i − 1 ‖ + L . η ( z i , T λ z i − 1 ) + θ i = θ ‖ z i − z i − 1 ‖ + θ i ≤ θ [ U ( T λ z i − 2 , T λ z i − 1 ) + θ i − 1 ] + θ i ≤ θ [ θ ‖ z i − 1 − z i − 2 ‖ + L . η ( z i − 1 , T λ z i − 2 ) + θ i − 1 ] + θ i = θ 2 ‖ z i − 1 − z i − 2 ‖ + 2 θ i ≤ ⋯ ≤ θ i ‖ z 1 − z 0 ‖ + i θ i .

Therefore,

∑ i = 0 ∞ ‖ z i + 1 − z i ‖ ≤ ‖ z 1 − z 0 ‖ ∑ i = 0 ∞ θ i + ∑ i = 0 ∞ i θ i < ∞ .

Hence, { z n } is a Cauchy sequence in X , and there exists z ∗ ∈ X such that lim n → ∞ z n = z ∗ . Now by using (22), we have

U ( T λ z n , T λ z ∗ ) ≤ θ ‖ z n − z ∗ ‖ + L . η ( z n , T λ z ∗ ) ≤ θ ‖ z n − z ∗ ‖ + 2 L . U ( T z n − 1 , T λ z ∗ ) ≤ θ ‖ z n − z ∗ ‖ + 2 L . [ θ ‖ z n − 1 − z ∗ ‖ + L . η ( z n − 1 , T λ z ∗ ) ] = θ ‖ z n − z ∗ ‖ + 2 L . θ ‖ z n − 1 − z ∗ ‖ + 2 L 2 . η ( z n − 1 , T λ z ∗ ) .

Continuing this way, we obtain

U ( T λ z n , T λ z ∗ ) ≤ θ ‖ z n − z ∗ ‖ + 2 L . θ ‖ z n − 1 − z ∗ ‖ + … + 2 n − 1 L n − 1 . θ ‖ z 0 − z ∗ ‖ + 2 n − 1 L n . η ( z 0 , T λ z ∗ ) .

As L ∈ 0 , 1 2 , it follows that lim n → ∞ U ( T λ z n , T λ z ∗ ) = 0 . We know that U and H are sequentially equivalent, therefore, H ( T λ z n , T λ z ∗ ) = 0 . Also z n + 1 ∈ T λ z n gives that

lim n → ∞ η ( z n , T λ z ∗ ) = 0 .

As T λ z ∗ is closed, z ∗ ∈ T λ z ∗ . Hence, z ∗ ∈ T z ∗ .□

If we take b = 0 in our main Theorem 3.2, we obtain the following result.

Corollary 3.2.1

Let ( X , ‖ ⋅ ‖ ) be a Banach space. Assume that T : X → CB ( X ) is a ( 0 , L , k , U ) -enriched weak multivalued contraction, i.e., there exist two constants L ∈ 0 , 1 2 and k ∈ [ 0 , 1 ) such that,

for every y , z ∈ X ,
(24) U ( T y , T z ) ≤ k ‖ y − z ‖ + L η ( y , T z ) ;
and
for y ∈ X and z ∈ T y ,
(25) η ( z , T z ) ≤ U ( T z , T y ) ,
where U ∈ Ω 2 . Then, T has a fixed point in X.

If we take U = H in Theorem 3.2, we obtain the following result.

Corollary 3.2.2

Let ( X , ‖ ⋅ ‖ ) be a Banach space. Assume that T : X → CB ( X ) is a multivalued mapping such that there exist three constants L ∈ [ 0 , 1 2 ) , b ∈ [ 0 , + ∞ ) , and k ∈ [ 0 , b + 1 ) such that

H ( b x + T x , b y + T y ) ≤ k ‖ x − y ‖ + L . η ( ( b + 1 ) x , b y + T y ) for every x , y ∈ X .

Then, T has a fixed point in X.

If we take U = H in the Corollary 3.2.1, we obtain Theorem 3 of [7] in the setting of Banach space with restricted values of constant L .

Corollary 3.2.3

[7] Let ( X , ‖ ⋅ ‖ ) be a Banach space. Assume that T : X → CB ( X ) is a multivalued mapping such that there exist two constants L ∈ [ 0 , 1 2 ) and k ∈ [ 0 , 1 ) such that we have the following inequality:

H ( T x , T y ) ≤ k ‖ x − y ‖ + L . η ( x , T y ) for e v e r y x , y ∈ X .

Then, T has a fixed point in X.

Example 5

Let ( Y , μ ) be a finite measure of space. The classical Lebesgue space X = L 2 ( Y , μ ) is defined as the collection of all Borel measurable functions f : Y → R such that ∫ Y ∣ f ( y ) ∣ 2 d μ ( y ) < ∞ . We know that the space X equipped with the norm

‖ f ‖ X = ∫ Y ∣ f ∣ 2 d μ 1 2

is a Banach space. Define the mapping T : L 2 ( Y , μ ) → C B ( L 2 ( Y , μ ) ) by

T g = { − g , h − g } ,

where h ( y ) = 1 ∀ y ∈ Y . Clearly, h ∈ X as μ ( Y ) < ∞ . If we take b = 1 , L = 0 , and U = H 1 3 . Then, we obtain

g + T g = { 0 , h } ,

where 0 is the zero measurable function on Y . Clearly, we have

η ( 0 , f + T f ) = inf { ‖ 0 ‖ , ‖ 0 − g ‖ } = 0

and

η ( h , f + T f ) = 0 ,

which gives that

D ( g + T g , f + T f ) = 0 .

Similarly, we have D ( f + T f , g + T g ) = 0 . Thus,

H 1 3 ( g + T g , f + T f ) = 1 2 { D ( g + T g , f + T f ) + D ( f + T f , g + T g ) } = 0 ≤ d ( g , f ) ∀ g , f ∈ X .

Now, if we take h ∗ ∈ X . Then, T 1 2 h ∗ = { 0 , 1 2 h } , and for w ∈ T 1 2 h ∗ , we have

η ( w , T 1 2 w ) = inf { ‖ w − v ‖ : v ∈ T 1 2 w } .

Case 1. If w = 0 . Then, T 1 2 0 = 0 , 1 2 h . Hence,

η 0 , T 1 2 0 = inf ‖ 0 − 0 ‖ , 0 − 1 2 h = 0 .

Case 2. If w = 1 2 h . Then, T 1 2 1 2 h = 0 , 1 2 h . So,

η 1 2 h , T 1 2 1 2 h = inf 1 2 h − 0 , 1 2 h − 1 2 h = 0 .

Therefore,

η w , T 1 2 w = 0 ≤ H 1 3 T 1 2 w , T 1 2 x for x ∈ X and w ∈ T 1 2 x .

In addition,

Fix ( T ) = 0 , g 2 .

Hence, T satisfies all the conditions of Theorem 3.2.

We now define a new class of single-valued enriched mappings in the following.

Definition 5

Let ( X , ‖ ⋅ ‖ ) be a normed space. Suppose that u is any metric on X induced by some other norm on X , which is sequentially equivalent to the metric d induced by the norm ‖ ⋅ ‖ . A mapping T : X → X is called ( b , k , u ) -enriched contraction if there exist b ∈ [ 0 , + ∞ ) and k ∈ [ 0 , b + 1 ) such that the following conditions are satisfied:

for each y , z ∈ X ,
(26) u ( b y + T y , b z + T z ) ≤ k d ( y , z ) ;
and
for y ∈ X ,
(27) d ( b y + T y , b ( T y ) + T 2 y ) ≤ u ( b y + T y , b ( T y ) + T 2 y ) .

We now obtain the following results.

Corollary 3.2.4

Let ( X , ‖ ⋅ ‖ ) be a Banach space. Suppose that u is any metric on X induced by some other norm on X, which is sequentially equivalent to the metric d induced by the norm ‖ ⋅ ‖ . Then, a ( b , k , u ) -enriched contraction mapping T : X → X has a fixed point.

Proof

Take λ = 1 b + 1 . Clearly, 0 ≤ λ < 1 , and equation (26) becomes

u 1 λ − 1 y + T y , 1 λ − 1 z + T z ≤ k d ( y , z ) .

Hence,

u ( ( 1 − λ ) y + λ ( T y ) , ( 1 − λ ) z + λ ( T z ) ) ≤ λ k d ( y , z ) .

Equivalently, we obtain

(28) u ( T λ y , T λ z ) ≤ θ d ( y , z ) for y , z ∈ X ,

where θ = λ k ∈ [ 0 , 1 ) . Similarly, equation (27) becomes

d 1 λ − 1 T y + T 2 y , 1 λ − 1 y + T y ≤ u 1 λ − 1 y + T y , 1 λ − 1 T y + T 2 y .

So,

d ( ( 1 − λ ) T y + λ ( T 2 y ) , ( 1 − λ ) y + λ ( T y ) ) ≤ u ( ( 1 − λ ) y + λ ( T y ) , ( 1 − λ ) T y + λ ( T 2 y ) ) .

Equivalently, we obtain

(29) d ( T λ ( T y ) , T λ y ) ≤ u ( T λ y , T λ ( T y ) ) for y ∈ X .

Combining equations (28) and (29), we have

(30) d ( T λ y , T λ ( T y ) ) ≤ θ d ( T y , y ) .

This implies that { T λ n y } is a Cauchy sequence in X , so lim n → ∞ T λ n y = y 0 exists. Equation (28) implies that T λ is continuous, and hence, T λ y 0 = y 0 .□

If we take b = 0 in the Corollary 3.2.4, we obtain the following result as in [12], in the setting of Banach spaces.

Corollary 3.2.5

[12] Let ( X , ‖ ⋅ ‖ ) be a Banach space. Suppose u is any metric on X, which is sequentially equivalent to the metric d induced by the norm ‖ ⋅ ‖ . Assume that T satisfies the following conditions:

there exists k ∈ [ 0 , 1 ) such that
u ( T y , T z ) ≤ k d ( y , z ) for e a c h y , z ∈ X ,
and
for y ∈ X ,
d ( T 2 y , T y ) ≤ U ( T y , T 2 y ) .

Then, T has a fixed point.

Acknowledgment

The authors are thankful to the reviewers for their useful comments and remarks, which helped us improve the presentation of the article.

Funding information: No funding source.
Conflict of interest: The authors declare that they have no conflict of interest.

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Received: 2022-08-03

Revised: 2023-01-28

Accepted: 2023-05-12

Published Online: 2023-08-27

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

Articles in the same Issue

https://doi.org/10.1515/taa-2022-0136

Keywords for this article

fixed point; Hausdorff metric; enriched multivalued contraction; sequentially equivalent metric

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