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# Remarks on some recent fixed point theorems

- Hassen Aydi
^{1}, - Erdal Karapınar
^{2}and - Bessem Samet
^{3}Email author

**2012**:76

https://doi.org/10.1186/1687-1812-2012-76

© Aydi et al; licensee Springer. 2012

**Received:**1 February 2012**Accepted:**8 May 2012**Published:**8 May 2012

## Abstract

The purpose of this article is to show that some recent fixed point theorems are particular results of previous existing theorems in the literature.

**Mathematics Subject Classification 2000:** 54H25; 47H10.

## Keywords

- fixed point
- coupled fixed point
- ordered set
- metric space
- cone metric space

## 1 Introduction

In this section, we recall some known results on fixed point theory.

We start with the well-known Banach contraction principle [1].

**Theorem 1**.

*Let*(

*X, d*)

*be a complete metric space and let F*:

*X → X be a mapping such that for each pair of points x, y*∈

*X*,

*where k is a constant in* [0, 1). *Then F has a unique fixed point*.

In 2008, Dutta and Choudhury [2] obtained the following generalization of the Banach contraction principle.

**Theorem 2**.

*Let*(

*X, d*)

*be a complete metric space and let F*:

*X → X be a mapping such that for each pair of points x, y*∈

*X*,

*where ψ, φ:* [0, *∞*) *→* [0, *∞*) *are continuous, non-decreasing and ψ* ^{-1}({0}) = *φ* ^{1}({0}) = {0}. *Then F has a unique fixed*.

**Remark 3**. *Note that Theorem* 2 *remains true if φ satisfies only the following assumptions: φ is lower semi-continuous and φ*^{-1}({0}) = {0} *(see, for example, Abbas and Dorić* [3]*and Dorić* [4]).

Using the above remark, we have also

**Theorem 4**.

*Let*(

*X, d*)

*be a complete metric space and let F*:

*X → X be a mapping such that for each pair of points x, y*∈

*X*,

*where ψ:* [0, *∞*) *→* [0, *∞*) *is continuous, non-decreasing, ψ* ^{-1}({0}) = {0}, *and φ:* [0, *∞*) *→* [0, *∞*) *is lower semi-continuous and φ* ^{-1}({0}) = {0}. *Then F has a unique fixed*.

We have also an ordered version of Theorem 4 (see [3–5]).

**Theorem 5**.

*Let*(

*X*,≼)

*be a partially ordered set and suppose that there is a metric d on X such that*(

*X, d*)

*is a complete metric space. Let F: X → X be a continuous non-decreasing mapping such that*

*for all x, y* ∈ *X with x* ≼ *y, where ψ:* [0, *∞*) *→* [0, *∞*) *is continuous, non-decreasing, ψ* ^{-1}({0}) = {0}, *and φ:* [0, *∞*) *→* [0, *∞*) *is lower semi-continuous and φ* ^{-1}({0}) = {0}. *If there exists x*_{0} ∈ *X such that x*_{0} ≼ *Fx*_{0}, *then F has a fixed point*.

The following result was obtained by Olaleru [6].

**Theorem 6**.

*Let*(

*X, d*)

*be a cone metric space with a cone P having non-empty interior. Let f, g: X → X be mappings such that*

*for all x, u* ∈ *X, where α*_{1}, *α*_{2}, *α*_{3}, *α*_{4}, *α*_{5} ∈ [0, 1) *and α*_{1} + *α*_{2} + *α*_{3} + *α*_{4} + *α*_{5} *<* 1. *Suppose that f* (*X*) ⊆ *g*(*X*) *and g*(*X*) *is a complete subspace of X. Then f and g have a coincidence point. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point*.

The purpose of this article is to show that some recent fixed point theorems are particular cases of the above mentioned results. This article can be considered as a continuation of the recent work of Haghi et al. [7].

## 2 Main results

Beiranvand et al. [8] introduced a new class of mappings *T*: *X → X* as follows.

**Definition 7**. *The mapping T*: *X → X is said to be sequentially convergent, if the sequence* {*y*_{
n
}} *in X is convergent whenever* {*T y*_{
n
}} *is convergent*.

In the same article, the authors established the following result.

**Theorem 8**.

*Let*(

*X, d*)

*be a complete metric space and T, f*:

*X → X be two mappings satisfying*

*for all x, y* ∈ *X, where k is a constant in* [0, 1) *and T is continuous, injective and sequentially convergent. Then f has a unique fixed point*.

Theorem 8 has attracted the attention of many authors, see, for example, [9–15], where extensions and generalizations of Theorem 8 were considered.

We shall prove the following result.

**Theorem 9**. *Theorem* 1 *and Theorem* 8 *are equivalent*.

*Proof*. Clearly, if

*T*is the identity mapping, Theorem 8 reduces to Theorem 1. Now, we shall prove that Theorem 8 can be deduced for the Banach contraction principle. Define the mapping

*δ*:

*X × X →*[0,

*∞*) by

*x, y, z*∈

*X*, we have

*δ*(

*x, y*) =

*δ*(

*y, x*),

*δ*(

*x, y*)

*≤ δ*(

*x, y*) +

*δ*(

*y, z*) and

*δ*is a metric on

*X*. Moreover, (

*X, δ*) is a complete metric space. Indeed, let {

*x*

_{ n }} be a Cauchy sequence in (

*X, δ*). From the definition of

*δ*, this implies that {

*T x*

_{ n }} is a Cauchy sequence in (

*X, d*). Since (

*X, d*) is complete, there exists

*y*∈

*X*such that

*d*(

*T x*

_{ n }

*, y*)

*→*0 as

*n → ∞*. But

*T*is sequentially convergent, then there exists

*x*∈

*X*, such that

*d*(

*x*

_{ n }

*, x*)

*→*0 as

*n → ∞*. Since

*T*is continuous, this implies that

*d*(

*T x*

_{ n }

*, Tx*)

*→*0 as

*n → ∞*, that is,

*δ*(

*x*

_{ n }

*, x*)

*→*0 as

*n → ∞*. This proves that (

*X, δ*) is complete. Now, condition (4) reduces to

for all *x, y* ∈ *X*. Thus Theorem 8 follows immediately from the Banach contraction principle (Theorem 1). □

Recently, Eslamian and Abkar [16] etablished the following result.

**Theorem 10**.

*Let*(

*X, d*)

*be a complete metric space and f*:

*X*→

*X be such that*

*for all x, y*∈

*X, where ψ, α, β:*[0,

*∞*) → [0,

*∞*)

*are such that ψ is continuous and non-decreasing, α is continuous, β is lower semi-continuous*,

*Then f has a unique fixed point*.

We shall prove the following result.

**Theorem 11**. *Theorem* 10 *and Theorem* 4 *are equivalent*.

*Proof*. Taking

*α*=

*ψ*in Theorem 10, we obtain immediately Theorem 4. Now, we shall prove that Theorem 10 can be deduced from Theorem 4. Indeed, let

*f*:

*X → X*be a mapping satisfying (5) with

*ψ, α, β*: [0,

*∞*)

*→*[0,

*∞*) satisfy conditions (6) and (7). From (5), for all

*x, y*∈

*X*, we have

*θ*: [0,

*∞*)

*→*[0,

*∞*) by

for all *x, y* ∈ *X*. Clearly, from (6) and (7), *θ* is lower semi-continuous and *θ* ^{1}({0}) = {0}. Now, Theorem 10 follows immediately from Theorem 4. □

Binayak et al. [17] extended Theorem 10 to the ordered case.

**Theorem 12**.

*Let*(

*X*, ≼)

*be a partially ordered set and suppose that there is a metric d on X such that*(

*X, d*)

*is a complete metric space. Let f: X → X be a continuous non-decreasing mapping such that*

*for all x, y*∈

*X with x*≼

*y, where ψ, α, β:*[0,

*∞*)

*→*[0,

*∞*)

*are such that ψ is continuous and non-decreasing, α is continuous, β is lower semi-continuous*,

*If there exists x*_{0} ∈ *X such that x*_{0} ≼ *fx*_{0}, *then f has a fixed point*.

Following similar arguments as in the proof of Theorem 11, we obtain

**Theorem 13**. *Theorem* 5 *and Theorem* 12 *are equivalent*.

Abbas et al. [18] introduced the concept of *w*-compatibility for a pair of mappings *F*: *X × X* → *X* and *g: X* → *X*.

**Definition 14**. *The mappings F*: *X × X* → *X and g*: *X* → *X are called w-compatible if g*(*F* (*x, y*)) = *F*(*gx, gy*) *whenever gx* = *F*(*x, y*) *and gy* = *F*(*y, x*).

In the same article, the authors established the following result.

**Theorem 15**.

*Let*(

*X, d*)

*be a cone metric space with a cone P having non-empty interior, F: X × X*→

*X and g: X*→

*X be mappings satisfying*

*for all x, y, u, v* ∈ *X, where a*_{
i
}*, i* = 1, 2,..., 6 *are nonnegative real numbers such that* ${\sum}_{i=1}^{6}{a}_{i}\phantom{\rule{0.3em}{0ex}}<\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}1$. *If F* (*X* × *X*) ⊆ *g*(*X*) *and g*(*X*) *is complete subset of X, then F and g have a coupled coincidence point in X, that is, there exists* (*x, y*) ∈ *X* × *X such that gx* = *F*(*x, y*) *and gy* = *F*(*y, x*). *Moreover, if F and g are w-compatible, then there exists a unique u* ∈ *X such that u* = *gu* = *F*(*u, u*).

We shall prove the following result.

**Theorem 16**. *Theorem* 6 *and Theorem* 15 *are equivalent*.

*Proof*. (i) Theorem 15 ⇒ Theorem 6. Let

*f, g: X*→

*X*be mappings satisfying the hypotheses of Theorem 6. Define the mapping

*F*:

*X × X*→

*X*by

*x, y, u, v*∈

*X*. Then condition (8) is satisfied with (

*a*

_{1},

*a*

_{2},

*a*

_{3},

*a*

_{4},

*a*

_{5},

*a*

_{6}) = (

*α*

_{5},

*α*

_{1}, 0,

*α*

_{2},

*α*

_{4},

*α*

_{3}). On the other hand, from the definition of

*F*, we have

*F*(

*X*×

*X*) =

*f*(

*X*) ⊆

*g*(

*X*). Also,

*g*(

*X*) is a complete subspace of (

*X, d*). Now, applying Theorem 15, we obtain that

*F*and

*g*have a coupled coincidence point in

*X*, that is, there exists (

*x, y*)

*X × X*such that

*gx*=

*F*(

*x, y*) and

*gy*=

*F*(

*y, x*). From the definition of

*F*, this implies that

*gx*=

*fx*, that is,

*x*is a coincidence point of

*f*and

*g*. Suppose now that

*f*and

*g*are weakly compatible. Let

*x, y*∈

*X*such that

*gx*=

*F*(

*x, y*). This implies that

*gx*=

*fx*. Since

*f*and

*g*are weakly compatible, we get

*fgx*=

*gfx*, that is,

*F*(

*gx, gy*) =

*g*(

*F*(

*x, y*)). This implies that

*F*and

*g*are

*w*-compatible. From Theorem 15, there exists a unique

*u*∈

*X*such that

*u*=

*gu*=

*F*(

*u, u*), that is, there exists a unique

*u ∈ X*such that

*u*=

*gu*=

*fu*. Then

*f*and

*g*have a unique common fixed point. Thus we proved Theorem 6.

- (ii)
Theorem 6 ⇒ Theorem 15.

*F*:

*X × X*→

*X*and

*g*:

*X*→

*X*be mappings satisfying the hypotheses of Theorem 15. Define the mapping

*f*:

*X*→

*X*by

for all *x, u* ∈ *X*, where (*α*_{1}, *α*_{2}, *α*_{3}, *α*_{4}, *α*_{5}) = (*a*_{2}, *a*_{4}, *a*_{6}, *a*_{5}, *a*_{1} + *a*_{3}). Then condition (3) of Theorem 6 is satisfied. On the other hand, we have *f*(*X*) ⊆ *F*(*X × X*) ⊆ *g*(*X*) and *g*(*X*) is a complete subspace of (*X, d*). Applying Theorem 6, we obtain that there exists *x* ∈ *X* (a coincidence point) such that *fx* = *gx*, that is, *F*(*x, x*) = *gx*. Moreover, if *F* and *g* are *w*-compatible, then *f* and *g* are weakly compatible. Applying again Theorem 6, we obtain that *f* and *g* have a unique common fixed point, that is, there exists a unique *u* ∈ *X* such that *u* = *gu* = *fu* = *F* (*u, u*). Thus we proved Theorem 15. □

## Declarations

### Acknowledgements

The authors thank the referees for their helpful remarks and suggestions which improved the final presentation of this article.

## Authors’ Affiliations

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