Coupled fixed point theorems under new coupled implicit relation in Hilbert spaces

Kyung Soo Kim

doi:10.1515/dema-2022-0010

Article Open Access

Coupled fixed point theorems under new coupled implicit relation in Hilbert spaces

Published/Copyright: May 2, 2022

Published by

Become an author with De Gruyter Brill

Author Information Explore this Subject

From the journal Demonstratio Mathematica Volume 55 Issue 1

Abstract

The aim of this paper is to study existence and uniqueness of coupled fixed point for a family of self-mappings satisfying a new coupled implicit relation in a Hilbert space. We also prove well-posedness of a coupled fixed point problem.

Keywords: coupled implicit relation; coupled fixed point; coupled asymptotically F-regular; well-posedness; Hilbert space

MSC 2010: 47H10; 46C15; 49K40

1 Introduction

The fixed point theorem lies in the core of the famous and traditional theories in mathematics and has a broad set of applications.

Most of the substantial advances in fixed point theory started after the celebrated fixed point result of Banach, known as Banach’s contraction mapping principle in 1922. This principle can be stated as follows: any contraction in a complete metric space has a unique fixed point. When compared to Browder’s fixed point theorem, the power of Banach’s contraction principle comes from the fact that it guarantees the uniqueness of a fixed point and gives a method to determine the fixed point, i.e., Banach’s contraction principle gives an answer to the existence and uniqueness of a solution of an operator equation T x = x . These two strengths of Banach’s contraction principle have useful techniques for many prominent mathematicians who aim to broaden the applications of nonlinear functional analysis via fixed point theory in various quantitative sciences.

In the light of these developments, Guo and Lakshmikantham [1] defined the notion of a coupled fixed point in 1987. Later, Karapınar [2] improved the idea of a coupled fixed point in the category of cone metric space and partially ordered metric space. Interested readers may refer to [3,4, 5,6,7, 8,9]. Different generalizations of the Banach’s contraction mapping were studied by many authors in metric space [10,11,12, 13,14,15]. Also, these improvements and generalizations were made either by using the contractive conditions or by imposing some additional conditions on various spaces [16,17,18, 19,20,21].

Veerapandi and Kumar [22], and Pitchaimani and Kumar [23] investigated the properties of fixed points of sequence of mappings under contraction condition in Hilbert and metric spaces [24,25].

On the other hand, in [26,27,28], the authors studied implicit relation in Banach spaces. The study of well-posedness of a fixed point problem has been an active area of research and several mathematicians investigated this interesting concept, see [29,30, 31,32].

In spite of the above work, study of contraction condition through implicit relation in Hilbert spaces needs more investigation. In this paper, we study coupled fixed point theorems for a family of self-mappings in Hilbert space by applying a new coupled implicit relation of three dimensions. As an application of our results, we discuss the well-posedness problem for a coupled fixed point.

2 Preliminaries

Definition 2.1

Let X be a nonempty set and F : X × X → X be a mapping. Then F is said to be continuous at ( x , y ) ∈ X × X if for any sequence { x n } , { y n } in X with x n → x , y n → y implies that

F ( x n , y n ) → F ( x , y ) and F ( y n , x n ) → F ( y , x )

as n → ∞ .

Definition 2.2

(Coupled implicit relation) Let R + be the set of all nonnegative real numbers, Φ be the class of real valued continuous functions ϕ : R + 3 → R + non-decreasing in the third argument and satisfying the following condition: for x , y , a , b > 0 ,

x ≤ ϕ a + b 2 , x + a 2 , y + b and y ≤ ϕ a + b 2 , y + b 2 , x + a ,
or
x ≤ ϕ a + b 2 , 0 , b and y ≤ ϕ a + b 2 , 0 , a ,

there exists a real number 0 < h < 1 such that x + y ≤ h ( a + b ) .

The idea of coupled fixed point was given by Guo and Lakshmikantham [1] in 1987.

Definition 2.3

For any nonempty set X and a mapping F : X × X → X , ( x , y ) ∈ X × X is said to be a coupled fixed point of F , if

F ( x , y ) = x and F ( y , x ) = y .

Definition 2.4

A sequence { x n } , { y n } in a Hilbert space X , ( { x n } , { y n } ) is said to be coupled asymptotically F -regular if

lim n → ∞ ‖ x n − F ( x n , y n ) ‖ = 0 and lim n → ∞ ‖ y n − F ( y n , x n ) ‖ = 0 .

3 Main results

Lemma 3.1

Let H be a Hilbert space, for any positive integer n ,

( x 1 + x 2 + ⋯ + x n ) 2 ≤ n ( x 1 2 + x 2 2 + ⋯ + x n 2 ) ,

for all x i ∈ H , i = 1 , 2 , … , n .

Theorem 3.2

Let C be a closed subset of a Hilbert space H and F : C × C → C be continuous self-mapping such that

(1) ‖ F ( x , y ) − F ( p , q ) ‖ 2 ≤ ϕ ‖ x − p ‖ 2 + ‖ y − q ‖ 2 2 , ‖ x − F ( x , y ) ‖ 2 + ‖ p − F ( p , q ) ‖ 2 2 , ‖ q − F ( y , x ) ‖ 2 + ‖ y − F ( q , p ) ‖ 2 2

for all x , y , p , q ∈ C . Then F have a unique coupled fixed point in C × C .

Proof

Let x 0 , y 0 ∈ C , we obtain a sequence in C such that

x n + 1 = F ( x n , y n ) , y n + 1 = F ( y n , x n ) , ∀ n ≥ 0 .

By using (1) and parallelogram law, we have

‖ x n + 1 − x n ‖ 2 = ‖ F ( x n , y n ) − F ( x n − 1 , y n − 1 ) ‖ 2 ≤ ϕ ‖ x n − x n − 1 ‖ 2 + ‖ y n − y n − 1 ‖ 2 2 , ‖ x n − F ( x n , y n ) ‖ 2 + ‖ x n − 1 − F ( x n − 1 , y n − 1 ) ‖ 2 2 , ‖ y n − 1 − F ( y n , x n ) ‖ 2 + ‖ y n − F ( y n − 1 , x n − 1 ) ‖ 2 2 = ϕ ‖ x n − x n − 1 ‖ 2 + ‖ y n − y n − 1 ‖ 2 2 , ‖ x n − x n + 1 ‖ 2 + ‖ x n − 1 − x n ‖ 2 2 , ‖ y n − 1 − y n + 1 ‖ 2 + ‖ y n − y n ‖ 2 2 ≤ ϕ ‖ x n − x n − 1 ‖ 2 + ‖ y n − y n − 1 ‖ 2 2 , ‖ x n − x n + 1 ‖ 2 + ‖ x n − 1 − x n ‖ 2 2 , ‖ y n − 1 − y n ‖ 2 + ‖ y n − y n + 1 ‖ 2

and

‖ y n + 1 − y n ‖ 2 = ‖ F ( y n , x n ) − F ( y n − 1 , x n − 1 ) ‖ 2 ≤ ϕ ‖ y n − y n − 1 ‖ 2 + ‖ x n − x n − 1 ‖ 2 2 , ‖ y n − F ( y n , x n ) ‖ 2 + ‖ y n − 1 − F ( y n − 1 , x n − 1 ) ‖ 2 2 , ‖ x n − 1 − F ( x n , y n ) ‖ 2 + ‖ x n − F ( x n − 1 , y n − 1 ) ‖ 2 2 = ϕ ‖ x n − x n − 1 ‖ 2 + ‖ y n − y n − 1 ‖ 2 2 , ‖ y n − y n + 1 ‖ 2 + ‖ y n − 1 − y n ‖ 2 2 , ‖ x n − 1 − x n + 1 ‖ 2 2 ≤ ϕ ‖ x n − x n − 1 ‖ 2 + ‖ y n − y n − 1 ‖ 2 2 , ‖ y n − y n + 1 ‖ 2 + ‖ y n − 1 − y n ‖ 2 2 , ‖ x n − 1 − x n ‖ 2 + ‖ x n − x n + 1 ‖ 2 .

Hence from Definition 2.2(i), there exists 0 < h < 1 such that

‖ x n + 1 − x n ‖ 2 + ‖ y n + 1 − y n ‖ 2 ≤ h ( ‖ x n − x n − 1 ‖ 2 + ‖ y n − y n − 1 ‖ 2 ) .

Continuing this processing, we have

‖ x n + 1 − x n ‖ 2 + ‖ y n + 1 − y n ‖ 2 ≤ h n ( ‖ x 1 − x 0 ‖ 2 + ‖ y 1 − y 0 ‖ 2 ) , n ≥ 1 .

For any positive integer p , by Lemma 3.1, we obtain

‖ x n − x n + p ‖ 2 + ‖ y n − y n + p ‖ 2 ≤ ( ‖ x n − x n + 1 ‖ + ‖ x n + 1 − x n + 2 ‖ + ⋯ + ‖ x n + p − 1 − x n + p ‖ ) 2 + ( ‖ y n − y n + 1 ‖ + ‖ y n + 1 − y n + 2 ‖ + ⋯ + ‖ y n + p − 1 − y n + p ‖ ) 2 ≤ p ( ‖ x n + 1 − x n ‖ 2 + ‖ y n + 1 − y n ‖ 2 + ‖ x n + 2 − x n + 1 ‖ 2 + ‖ y n + 2 − y n + 1 ‖ 2 + ⋯ + ‖ x n + p − x n + p − 1 ‖ 2 + ‖ y n + p − y n + p − 1 ‖ 2 ) ≤ p ( h n + h n + 1 + ⋯ + h n + p − 1 ) ( ‖ x 1 − x 0 ‖ 2 + ‖ y 1 − y 0 ‖ 2 ) ≤ p h n 1 − h ( ‖ x 1 − x 0 ‖ 2 + ‖ y 1 − y 0 ‖ 2 ) .

Since 0 < h < 1 , we obtain

lim n → ∞ ( ‖ x n − x n + p ‖ 2 + ‖ y n − y n + p ‖ 2 ) = 0 .

Hence,

lim n → ∞ ‖ x n − x n + p ‖ = 0 and lim n → ∞ ‖ y n − y n + p ‖ = 0 .

Since C is closed, there exists an element x , y ∈ C such that

x n → x and y n → y

as n → ∞ . By the continuity of F ( ⋅ , ⋅ ) , it is clear that

x = lim n → ∞ x n + 1 = lim n → ∞ F ( x n , y n ) = F ( x , y ) , y = lim n → ∞ y n + 1 = lim n → ∞ F ( y n , x n ) = F ( y , x ) .

Therefore, ( x , y ) is a coupled fixed point of F .

For the uniqueness, let ( u , v ) ∈ C × C be another coupled fixed point of F , where ( x , y ) ≠ ( u , v ) . Then

‖ x − u ‖ 2 = ‖ F ( x , y ) − F ( u , v ) ‖ 2 ≤ ϕ ‖ x − u ‖ 2 + ‖ y − v ‖ 2 2 , ‖ x − F ( x , y ) ‖ 2 + ‖ u − F ( u , v ) ‖ 2 2 , ‖ v − F ( y , x ) ‖ 2 + ‖ y − F ( v , u ) ‖ 2 2 = ϕ ‖ x − u ‖ 2 + ‖ y − v ‖ 2 2 , 0 , ‖ v − y ‖ 2

and

‖ y − v ‖ 2 = ‖ F ( y , x ) − F ( v , u ) ‖ 2 ≤ ϕ ‖ y − v ‖ 2 + ‖ x − u ‖ 2 2 , ‖ y − F ( y , x ) ‖ 2 + ‖ v − F ( v , u ) ‖ 2 2 , ‖ u − F ( x , y ) ‖ 2 + ‖ x − F ( u , v ) ‖ 2 2 = ϕ ‖ y − v ‖ 2 + ‖ x − u ‖ 2 2 , 0 , ‖ u − x ‖ 2 .

By Definition 2.2(ii), there exists 0 < h < 1 such that

‖ x − u ‖ 2 + ‖ y − v ‖ 2 ≤ h ( ‖ x − u ‖ 2 + ‖ y − v ‖ 2 ) .

Thus,

‖ x − u ‖ 2 + ‖ y − v ‖ 2 = 0 ,

that is,

x = u and y = v .

Therefore,

( x , y ) = ( u , v ) ,

which shows that ( x , y ) is a unique coupled fixed point of F . This completes the proof.□

Theorem 3.3

Let C be a closed subset of a Hilbert space X and F : C × C → C be a mapping satisfying (1). Then F has a unique coupled fixed point in C × C and ( { z n } , { w n } ) be coupled asymptotically F -regular if and only if F is continuous at the coupled fixed point of F .

Proof

( ⇒ ) Let ( u , v ) ∈ C × C be a coupled fixed point of F and z n → u , w n → v as n → ∞ . Now

‖ F ( z n , w n ) − F ( u , v ) ‖ 2 ≤ ϕ ‖ z n − u ‖ 2 + ‖ w n − v ‖ 2 2 , ‖ z n − F ( z n , w n ) ‖ 2 + ‖ u − F ( u , v ) ‖ 2 2 , ‖ v − F ( w n , z n ) ‖ 2 + ‖ w n − F ( v , u ) ‖ 2 2

and

‖ F ( w n , z n ) − F ( v , u ) ‖ 2 ≤ ϕ ‖ w n − v ‖ 2 + ‖ z n − u ‖ 2 2 , ‖ w n − F ( w n , z n ) ‖ 2 + ‖ v − F ( v , u ) ‖ 2 2 , ‖ u − F ( z n , w n ) ‖ 2 + ‖ z n − F ( u , v ) ‖ 2 2 .

Since F ( u , v ) = u , F ( v , u ) = v , ( { z n } , { w n } ) is coupled asymptotically F -regular and z n → u , w n → v as n → ∞ , we obtain

lim n → ∞ ‖ z n − F ( z n , w n ) ‖ = 0 and lim n → ∞ ‖ w n − F ( w n , z n ) ‖ = 0 .

Thus,

lim n → ∞ ‖ F ( z n , w n ) − F ( u , v ) ‖ 2 ≤ lim n → ∞ ϕ ‖ F ( z n , w n ) − F ( u , v ) ‖ 2 + ‖ F ( w n , z n ) − F ( v , u ) ‖ 2 2 , 0 , ‖ F ( w n , z n ) − F ( v , u ) ‖ 2

and

lim n → ∞ ‖ F ( w n , z n ) − F ( v , u ) ‖ 2 ≤ lim n → ∞ ϕ ‖ F ( w n , z n ) − F ( v , u ) ‖ 2 + ‖ F ( z n , w n ) − F ( u , v ) ‖ 2 2 , 0 , ‖ F ( z n , w n ) − F ( u , v ) ‖ 2 .

By Definition 2.2(ii), there exists 0 < h < 1 such that

lim n → ∞ ( ‖ F ( z n , w n ) − F ( u , v ) ‖ 2 + ‖ F ( w n , z n ) − F ( v , u ) ‖ 2 ) ≤ h ⋅ lim n → ∞ ( ‖ F ( z n , w n ) − F ( u , v ) ‖ 2 + ‖ F ( w n , z n ) − F ( v , u ) ‖ 2 ) ,

which yields that

F ( z n , w n ) → F ( u , v ) and F ( w n , z n ) → F ( v , u )

as n → ∞ . Hence, F is continuous at ( u , v ) ∈ X × X .

( ⇐ ) Assume that F is continuous at ( u , v ) ∈ X × X . Note that, by Theorem 3.2, F has a unique coupled fixed point ( u , v ) ∈ C × C . Let { z n } , { w n } in X with z n → u , w n → v as n → ∞ . Since F is continuous at ( u , v ) ∈ C × C , we obtain

F ( z n , w n ) → F ( u , v ) and F ( w n , z n ) → F ( v , u )

as n → ∞ . From ( u , v ) is the coupled fixed point, we obtain

‖ z n − F ( z n , w n ) ‖ → ‖ u − F ( u , v ) ‖ = 0 and ‖ w n − F ( w n , z n ) ‖ → ‖ v − F ( v , u ) ‖ = 0

as n → ∞ . This means

lim n → ∞ ‖ z n − F ( z n , w n ) ‖ = 0 and lim n → ∞ ‖ w n − F ( w n , z n ) ‖ = 0 .

This completes the proof.□

We define as T x = F ( x , x ) . Then, we have the following corollaries.

Corollary 3.4

Let C be a closed subset of a Hilbert space X and T : C → C be a continuous self-mapping such that

(2) ‖ T x − T p ‖ 2 ≤ ϕ ‖ x − p ‖ 2 , ‖ x − T x ‖ 2 + ‖ p − T p ‖ 2 2 , ‖ x − T p ‖ 2 + ‖ p − T x ‖ 2 2 ,

for all x , p ∈ C . Then T has a unique fixed point in C .

Proof

Taking x = y and p = q in Theorem 3.2, then (1) is equal to (2). Thus, we have the conclusion of the Corollary from Theorem 3.2.□

Remark 3.5

If x = y and a = b in Definition 2.2, the coupled implicit relation conditions restricted follows implicit relation conditions:

Let R + be the set of all nonnegative real numbers, Φ be the class of real valued continuous functions ϕ : R + 3 → R + non-decreasing in the third argument and satisfying the following condition: for x , a > 0 ,

x ≤ ϕ a , x + a 2 , x + a
or
x ≤ ϕ ( a , 0 , a ) ,

there exists a real number 0 < h < 1 such that x ≤ h a .

Corollary 3.6

[23, Theorem 3.3] Let C be a closed subset of a Hilbert space X and T : C → C be a self-mapping satisfying (2). Then T has a unique fixed point in C and { z n } is asymptotically T -regular if and only if T is continuous at the fixed point of T .

Proof

Taking x = y , p = q in (1) and z n = w n , u = v in Theorem 3.3, then we have the conclusion of the Corollary from Theorem 3.3.□

Remark 3.7

If z n = w n in Definition 2.4, then the coupled asymptotically F -regular is equal to asymptotically T -regular:

A sequence { x n } in a Hilbert space X is said to be asymptotically T -regular if lim n → ∞ ‖ x n − T x n ‖ = 0 .

4 Well-posedness theorem

In this section, we prove well-posedness of coupled fixed point problem of mapping in Theorem 3.2.

Definition 4.1

[23] Let X be a Hilbert space and f : X → X be a self-mapping. The fixed point problem of f is said to be well-posed if

f has a unique fixed point x 0 ∈ X ,
for any sequence { x n } ⊂ X and lim n → ∞ ‖ x n − f x n ‖ = 0 , we have lim n → ∞ ‖ x n − x 0 ‖ = 0 .

Let C o ℱ P ( F , X × X ) denote a coupled fixed point problem of mapping F on X × X and C o ℱ ( F ) denote the set of all common fixed points of F .

Definition 4.2

Let X be a Hilbert space and F : X × X → X be a mapping. C o ℱ P ( F , X × X ) is called well-posed if C o ℱ ( F ) is unique and for any sequences { x n } , { y n } in X with

( x ¯ , y ¯ ) ∈ C o ℱ ( F ) and lim n → ∞ ‖ F ( x n , y n ) − x n ‖ = 0 = lim n → ∞ ‖ F ( y n , x n ) − y n ‖

implies

x ¯ = lim n → ∞ x n , y ¯ = lim n → ∞ y n .

Theorem 4.3

Let C be a closed subset of a Hilbert space X , F be continuous mapping on C × C as in Theorem 3.2. For any sequences { x n } , { y n } in X and ( u , v ) ∈ C o ℱ ( F ) , if

lim n → ∞ ‖ u − F ( x n , y n ) ‖ = 0 = lim n → ∞ ‖ v − F ( y n , x n ) ‖ ,

then the coupled fixed point problem of F is well-posed.

Proof

From Theorem 3.2, the mapping F has a unique coupled fixed point, say ( x 0 , y 0 ) ∈ X × X . Let { x n } , { y n } be sequences in X and

lim n → ∞ ‖ F ( x n , y n ) − x n ‖ = 0 = lim n → ∞ ‖ F ( y n , x n ) − y n ‖ .

Without loss of generality, we assume that ( x 0 , y 0 ) ≠ ( x n , y n ) for any non-negative integer n . Using F ( x 0 , y 0 ) = x 0 and F ( y 0 , x 0 ) = y 0 , we obtain

‖ x 0 − x n ‖ 2 = ‖ F ( x 0 , y 0 ) − x n ‖ 2 = ‖ F ( x 0 , y 0 ) − F ( x n , y n ) ‖ 2 + ‖ F ( x n , y n ) − x n ‖ 2 + 2 ⟨ F ( x 0 , y 0 ) − F ( x n , y n ) , F ( x n , y n ) − x n ⟩ ≤ ϕ ‖ x 0 − x n ‖ 2 + ‖ y 0 − y n ‖ 2 2 , ‖ x 0 − F ( x 0 , y 0 ) ‖ 2 + ‖ x n − F ( x n , y n ) ‖ 2 2 , ‖ y n − F ( y 0 , x 0 ) ‖ 2 + ‖ y 0 − F ( y n , x n ) ‖ 2 2 + ‖ F ( x n , y n ) − x n ‖ 2 + 2 ⟨ F ( x 0 , y 0 ) − F ( x n , y n ) , F ( x n , y n ) − x n ⟩

and

‖ y 0 − y n ‖ 2 = ‖ F ( y 0 , x 0 ) − y n ‖ 2 = ‖ F ( y 0 , x 0 ) − F ( y n , x n ) ‖ 2 + ‖ F ( y n , x n ) − y n ‖ 2 + 2 ⟨ F ( y 0 , x 0 ) − F ( y n , x n ) , F ( y n , x n ) − y n ⟩ ≤ ϕ ‖ y 0 − y n ‖ 2 + ‖ x 0 − x n ‖ 2 2 , ‖ y 0 − F ( y 0 , x 0 ) ‖ 2 + ‖ y n − F ( y n , x n ) ‖ 2 2 , ‖ x n − F ( x 0 , y 0 ) ‖ 2 + ‖ x 0 − F ( x n , y n ) ‖ 2 2 + ‖ F ( y n , x n ) − y n ‖ 2 + 2 ⟨ F ( y 0 , x 0 ) − F ( y n , x n ) , F ( y n , x n ) − y n ⟩ .

Since

lim n → ∞ ‖ x 0 − F ( x n , y n ) ‖ = 0 = lim n → ∞ ‖ y 0 − F ( y n , x n ) ‖ ,

for ( x 0 , y 0 ) ∈ C o ℱ ( F ) , we obtain

lim n → ∞ ‖ x 0 − x n ‖ 2 ≤ lim n → ∞ ϕ ‖ x 0 − x n ‖ 2 + ‖ y 0 − y n ‖ 2 2 , 0 , ‖ y n − y 0 ‖ 2

and

lim n → ∞ ‖ y 0 − y n ‖ 2 ≤ lim n → ∞ ϕ ‖ y 0 − y n ‖ 2 + ‖ x 0 − x n ‖ 2 2 , 0 , ‖ x n − x 0 ‖ 2 .

By the definition of implicit relation, there exists 0 < h < 1 such that

lim n → ∞ ‖ x 0 − x n ‖ 2 + lim n → ∞ ‖ y 0 − y n ‖ 2 ≤ h ( lim n → ∞ ‖ x 0 − x n ‖ 2 + lim n → ∞ ‖ y 0 − y n ‖ 2 ) .

Thus,

lim n → ∞ x n = x 0 and lim n → ∞ y n = y 0 .

This completes the proof.□

Acknowledgements

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

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: Author states no conflict of interest.

References

[1] D. Guo and V. Lakshmikantham, Coupled fixed points of nonlinear operators with applications, Nonlinear Anal. 11 (1987), 623–632. 10.1016/0362-546X(87)90077-0Search in Google Scholar

[2] E. Karapınar, Couple fixed point theorems for nonlinear contractions in cone metric spaces, Comput. Math. Appl. 59 (2010), no. 12, 3656–3668, https://doi.org/10.1016/j.camwa.2010.03.062. Search in Google Scholar

[3] H. Afshari, S. Kalantari, and E. Karapınar, Solution of fractional differential equations via coupled fixed point, Electron. J. Differ. Equ. 2015 (2015), 286. Search in Google Scholar

[4] I. M. Erhan, E. Karapınar, A.-F. Roldán-López-de-Hierro, and N. Shahzad, Remarks on ‘Coupled coincidence point results for a generalized compatible pair with applications’, Fixed Point Theory Appl. 2014 (2014), 207, DOI: https://doi.org/10.1186/1687-1812-2014-207. 10.1186/1687-1812-2014-207Search in Google Scholar

[5] S. Gülyaz and E. Karapınar, Coupled fixed point result in partially ordered partial metric spaces through implicit function, Hacet. J. Math. Stat. 42 (2013), no. 4, 347–357. Search in Google Scholar

[6] B. Samet, E. Karapınar, H. Aydi, and V. C. Rajić, Discussion on some coupled fixed point theorems, Fixed Point Theory Appl. 2013 (2013), 50, https://doi.org/10.1186/1687-1812-2013-50. Search in Google Scholar

[7] S. Gülyaz, E. Karapınar, and I. S. Yüce, A coupled coincidence point theorem in partially ordered metric spaces with an implicit relation, Fixed Point Theory Appl. 2013 (2013), 38, https://doi.org/10.1186/1687-1812-2013-38. Search in Google Scholar

[8] I. Beg and A. R. Butt, Fixed point for set-valued mappings satisfying an implicit relation in partially ordered metric spaces, Nonlinear Anal. 71 (2009), no. 9, 3699–3704. 10.1016/j.na.2009.02.027Search in Google Scholar

[9] I. Beg and A. R. Butt, Fixed points for weakly compatible mappings satisfying an implicit relation in partially ordered metric spaces, Carpathian J. Math. 25 (2009), no. 1, 1–12. Search in Google Scholar

[10] M. Nazam and Ö. Acar, Fixed points of (α,ψ)-contractions in Hausdorff partial metric spaces, Math. Methods Appl. Sci. 42 (2019), no. 16, 5159–5173, https://doi.org/10.1002/mma.5251. Search in Google Scholar

[11] Ö. Acar, Generalization of (α−Fd)-contraction on quasi-metric space, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat. 68 (2019), no. 1, 35–42, https://doi.org/10.31801/cfsuasmas.443587. Search in Google Scholar

[12] A. S. Babu, T. Došenović, Md. M. Ali, S. Radenović, and K. P. R. Rao, Some Preśić type results in b-dislocated metric spaces, Constr. Math. Anal. 2 (2019), no. 1, 40–48, https://doi.org/10.33205/cma.499171. Search in Google Scholar

[13] C. Vetro, A fixed-point problem with mixed-type contractive condition, Constr. Math. Anal. 3 (2020), no. 1, 45–52, https://doi.org/10.33205/cma.684638. Search in Google Scholar

[14] L. B. Ćirić, A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc. 45 (1974), 267–273. 10.1090/S0002-9939-1974-0356011-2Search in Google Scholar

[15] Lj. B. Ćirić, Generalized contractions and fixed point theorems, Publ. Inst. Math. (Beograd) (N.S.) 12 (1971), no. 26, 19–26. Search in Google Scholar

[16] F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl. 20 (1967), 197–228. 10.1016/0022-247X(67)90085-6Search in Google Scholar

[17] Y. Cui and J. Sun, Fixed point theorems for a class of nonlinear operators in Hilbert spaces and applications, Positivity 15 (2011), 455–464. 10.1007/s11117-010-0095-3Search in Google Scholar

[18] K. S. Kim, Best proximity point of contraction type mapping in metric space, J. Comput. Anal. Appl. 27 (2019), no. 6, 1023–1033. Search in Google Scholar

[19] K. S. Kim, A constructive scheme for a common coupled fixed point problems in Hilbert space, Mathematics 8 (2020), no. 10, 1717, https://doi.org/10.3390/math8101717. Search in Google Scholar

[20] K. S. Kim, Existence theorem of a fixed point for asymptotically nonexpansive nonself-mapping in CAT(0) spaces, Nonlinear Func. Anal. Appl. 25 (2020), no. 2, 355–362, https://doi.org/10.22771/nfaa.2020.25.02.11. Search in Google Scholar

[21] K. S. Kim, Convergence theorem for a generalized varphi-weakly contractive nonself-mapping in metrically convex metric spaces, Nonlinear Func. Anal. Appl. 26 (2021), no. 3, 601–610, https://doi.org/10.22771/nfaa.2021.26.03.10. Search in Google Scholar

[22] T. Veerapandi and S. A. Kumar, Common fixed point theorems of a sequence of mappings on Hilbert space, Bull. Calcutta Math. Soc. 91 (1999), no. 4, 299–308. Search in Google Scholar

[23] M. Pitchaimani and D. R. Kumar, On construction of fixed point theory under implicit relation in Hilbert spaces, Nonlinear Func. Anal. Appl. 21 (2016), no. 3, 513–522. Search in Google Scholar

[24] S. Chandok, E. Karapınar, and M. S. Khan, Existence and uniqueness of common coupled fixed point results via auxiliary functions, spaces, Bull. Iranian Math. Soc. 40 (2014), no. 1, 199–215. Search in Google Scholar

[25] H. H. Alsulami, E. Karapınar, M. A. Kutbi, and A.-F. Roldán-López-de-Hierro, An illusion: “A Suzuki type coupled fixed point theorem”, Abstr. Appl. Anal. 2014 (2014), 235731, https://doi.org/10.1155/2014/235731. Search in Google Scholar

[26] M. Pitchaimani and D. R. Kumar, Some common fixed point theorems using implicit relation in 2-Banach spaces, Surv. Math. Appl. 10 (2015), 159–168. Search in Google Scholar

[27] M. Pitchaimani and D. R. Kumar, Common and coincidence fixed point theorems for asymptotically regular mappings in 2-Banach Space, Nonlinear Func. Anal. Appl. 21 (2016), no. 1, 131–144. Search in Google Scholar

[28] G. S. Saluja, On common fixed point theorems under implicit relation in 2-Banach spaces, Nonlinear Func. Anal. Appl. 20 (2015), no. 2, 215–221. Search in Google Scholar

[29] M. Akkouchi, Well posedness of the fixed point problem for certain asymptotically regular mappings, Ann. Math. Sil. 23 (2009), 43–52. Search in Google Scholar

[30] M. Akkouchi and V. Popa, Well-posedness of fixed point problem for mappings satisfying an implicit relation, Demonstr. Math. 43 (2010), no. 4, 923–929. 10.1515/dema-2010-0418Search in Google Scholar

[31] B. K. Lahiri and P. Das, Well-posedness and porosity of a certain classes of operators, Demonstr. Math. 38 (2005), 169–176. 10.1515/dema-2005-0119Search in Google Scholar

[32] V. Popa, Well-posedness of fixed problem in compact metric space, Bull. Univ. Petrol-Gaze, Ploicsti, sec. Mat Inform. Fiz. 60 (2008), no. 1, 1–4. Search in Google Scholar

Received: 2022-01-20

Revised: 2022-03-16

Accepted: 2022-04-04

Published Online: 2022-05-02

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

Articles in the same Issue

https://doi.org/10.1515/dema-2022-0010

Keywords for this article

coupled implicit relation; coupled fixed point; coupled asymptotically F-regular; well-posedness; Hilbert space

Creative Commons

BY 4.0