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# Generalization of fixed point theorems in ordered metric spaces concerning generalized distance

- Elham Graily
^{1}, - Seiyed Mansour Vaezpour
^{2}, - Reza Saadati
^{1}Email author and - Yeol JE Cho
^{3}

**2011**:30

https://doi.org/10.1186/1687-1812-2011-30

© Graily et al; licensee Springer. 2011

**Received:**14 March 2011**Accepted:**11 August 2011**Published:**11 August 2011

## Abstract

In this article, we consider ordered metric spaces concerning generalized distance and prove some fixed point theorems in these spaces. Our results generalize, improve, and simplify the proof of the previous results given by some authors.

**Mathematics Subject Classification (2000)**

47H10, 54H25

## Keywords

- Ordered metric space
- Fixed point
- Generalized distance

## 1. Introduction and Preliminary

Recently, Nieto and Rodriguez-Lopez [1, 2], Ran and Reurins [3], Petrusel and Rus [4] presented some new results in partially ordered metric spaces. Their main idea was to combine the ideas of iterative technique in the contractive mapping with these in monotone technique.

Recently, Kada et al. [5, 6] in 1996 introduced the concept of *w*-distance in a metric space and prove some fixed point theorems. For the study of fixed point theorem concerning generalized distance followed in other articles, see [5, 7–15].

The aim of this article is to use the concept of *w*-distance to generalize the fixed point theorems in partially ordered metric spaces. Our results not only generalize some fixed point theorems, but also improve and simplify the previous results.

In the sequel, we state some definitions and a lemma which we will use in our main results.

**Definition 1.1**. ([5, 8, 10]) Let (

*X*,

*d*) be a metric space. Then, a function

*p*:

*X*×

*X*→ [0, ∞) is called a

*w*-distance on X if the following conditions are satisfied:

- (a)
*p*(*x*,*z*) ≤*p*(*x*,*y*) +*p*(*y*,*z*) for any*x*,*y*,*z*∈*X*; - (b)
for any

*x*∈*X*,*p*(*x*, .) :*X*→ [0, ∞) is lower semi-continuous; - (c)
for any

*ε >*0, there exists*δ >*0 such that*p*(*x*,*z*) ≤*δ*and*p*(*z*,*y*) ≤*δ*imply*d*(*x*,*y*) ≤*ε*.

We know that a real-valued function *f* defined in a metric space *X* is said to be lower semi-continuous at a point *x*_{0} ∈ *X* if either
or
, whenever *x*_{
n
} ∈ *X* for each *n* ∈ **N** and *x*_{
n
} → *x*_{0}.

**Lemma 1.2**. ([5, 7])

*Let*(

*X*,

*d*)

*be a metric space and p be a w-distance on X. Let*{

*x*

_{ n }}, {

*y*

_{ n }}

*be sequences in X*, {

*α*

_{ n }}, {

*β*

_{ n }}

*be sequences in*[0, ∞)

*converging to zero and let x*,

*y*,

*z*∈

*X. Then, the following conditions hold:*

- (1)
*If p*(*x*_{ n },*y*) ≤*α*_{ n }*and p*(*x*_{ n },*z*) ≤*β*_{ n }*for any n*∈**N**,*then y*=*z. In particular, if p*(*x.y*) = 0*and p*(*x*,*z*) = 0,*then y*=*z;* - (2)
*If p*(*x*_{ n },*y*_{ n }) ≤*α*_{ n }*and p*(*x*_{ n },*z*) ≤*β*_{ n }*for any n*∈**N**,*then d*(*y*_{ n },*z*) → 0; - (3)
*If p*(*x*_{ n },*x*_{ m }) ≤*α*_{ n }*for any n*,*m*∈**N***with m > n, then*{*x*_{ n }}*is a Cauchy sequence;* - (4)
*If p*(*y*,*x*_{ n }) ≤*α*_{ n }*for any n*∈**N**,*then*{*x*_{ n }}*is a Cauchy sequence*.

*f*:

*X*→

*X*be an operator:

- (1)
*I*(*f*) is the set of all nonempty invariant subsets of*f*, i.e.,*I*(*f*) = {*Y*⊂*X*:*f*(*Y*) ⊂*Y*} and*F*_{ f }= {*x*∈*X*:*x*=*f*(*x*)}. - (2)
The operator

*f*is called Picard operator (briefly, PO) if there exists*x** ∈*X*such that*F*_{ f }= {*x**} and, for all*x*∈*X*, {*f*^{ n }(*x*)} converges to*x**. - (3)
The operator

*f*is called orbitally*U*-continuous for any*U*⊂*X*×*X*if the following condition holds:

*x*,

*y*]

_{≤}= {

*z*∈

*X*:

*x*≤

*z*≤

*y*}, where

*x*,

*y*∈

*X*and

*x*≤

*y*.

- (5)
If

*g*:*Y*→*Y*is an operator, then the Cartesian product of*f*and*g*is the mapping*f*×*g*:*X*×*Y*→*X*×*Y*defined by (*f*×*g*)(*x*,*y*) = (*f*(*x*),*g*(*y*)) for all (*x*,*y*) ∈*X*×*Y*. - (6)
*φ*:*R*_{+}→*R*_{+}is said to be a comparison function if it is increasing and*φ*^{ n }(*t*) → 0 as*n*→ ∞. As a consequence, we also have*φ*(*t*)*< t*for any*t >*0,*φ*(0) = 0, and*φ*is right continuous at 0.

## 2. Main Results

Now, we give the main results of this article.

**Theorem 2.1**.

*Let*(

*X*,

*d*, ≤)

*be an ordered metric space and f*:

*X*→

*X be an operator. Let p be a w-distance on*(

*X*,

*d*)

*and suppose that*

- (a)
*X*_{≤}∈*I*(*f*×*f*)*;* - (b)
*there exists x*_{0}∈*X such that*(*x*_{0},*f*(*x*_{0})) ∈*X*_{≤}*;* - (c)
(

*c*_{1})*f is orbitally continuous or*

*c*

_{2})

*f is orbitally X*

_{≤}

*-continuous and there exists a subsequence*

*of*{

*f*

^{ n }(

*x*

_{0})}

*such that*

*for any k*∈

**N**

*;*

- (d)

*Then F*_{
f
} ≠ ∅.

**Proof**. If *f*(*x*_{0}) = *x*_{0}, then the proof is completed. Let *x*_{0} ∈ *X* be such that (*x*_{0}, *f* (*x*_{0})) ∈ *X*_{≤}. By (a), since (*f* × *f* )(*X*_{≤}) ⊂ *X*_{≤}, we have (*f* × *f* )(*x*_{0}, *f* (*x*_{
o
} )) ∈ *X*_{≤} and so (*f*(*x*_{0}), *f*^{2}(*x*_{
o
} )) ∈ *X*_{≤}.

for any *n* ∈ **N**.

*n*∈

**N**. Let

*p*

_{0}=

*p*(

*x*

_{0},

*f*(

*x*

_{0})) and

*p*

_{ n }=

*p*(

*f*

^{ n }(

*x*

_{0}),

*f*

^{n+1}(

*x*

_{0})) for any

*n*∈

**N**. Then we have

*n*∈

**N**. If max{

*p*

_{n-1},

*p*

_{ n }} =

*p*

_{n-1}, then (3.1) follows. Otherwise, max{

*p*

_{n-1},

*p*

_{ n }} =

*p*

_{ n }Then, by (3.2), we have

*p*

_{ n }≤

*φ*(

*p*

_{ n }) ≤

*p*

_{ n }and so

*p*

_{ n }= 0 and (3.1) follows. By induction, we obtain

as *n* → ∞.

as *n* → ∞ for any *k >* 0. Therefore, {*f*^{
n
} (*x*_{0})} is a Cauchy sequence in *X*. Since *X* is complete, there exists *x** ∈ *X* such that *f*^{
n
} (*x*_{0}) → *x** as *n* → ∞.

*x** is a fixed point. If (

*c*

_{1}) holds, then

*f*

^{n+1}(

*x*

_{0}) →

*f*(

*x**) and, by lower semi-continuity of

*p*(

*f*

^{ n }(

*x*

_{0}), ·), we have

and *α*_{
n
} , *β*_{
n
} → 0 as *n* → ∞. Thus, by (3.3) and Lemma 1.2, we conclude that *f* (*x**) = *x**.

Now, suppose that (*c*_{2}) holds. Since
converges to *x** and *f* is *X*_{≤}-orbitally continuous, it follows that
converges to *f* (*x**). Similarly, by lower semi-continuity of *p*(*f*^{
n
} (*x*_{0}), ·), we conclude that *f* (*x**) = *x**. This completes the proof. □

**Corollary 2.2**. *Let* (*X*, *d*, ≤) *be an ordered metric space and f* : *X* → *X be an operator*.

*Let p be a w-distance on*(

*X*,

*d*)

*and suppose that*

- (a)
*X*_{≤}∈*I*(*f*×*f*)*;* - (b)
*there exists x*_{0}∈*X such that*(*x*_{0},*f*(*x*_{0})) ∈*X*_{≤}*;* - (c)
(

*c*_{1}))*f is orbitally continuous or*

*c*

_{2})

*f is orbitally X*

_{≤}

*-continuous and there exists a subsequence*

*of*{

*f*

^{ n }(

*x*

_{0})}

*such that*

*for any k*∈

**N**

*;*

- (d)

*for any*(

*x*,

*y*) ∈

*X*

_{≤},

*where*

- (e)
*the metric d is complete;* - (f)
*if*(*x*,*y*) ∈*X*_{≤}*and*(*y*,*z*) ∈*X*_{≤}*.vskip 1 mm*

*Then, F*_{
f
} ≠ ∅.

**Theorem 2.3**. *Let* (*X*, *d*, ≤) *be an ordered metric space and f* : *X* → *X be an operator*.

*Let p be a w-distance on*(

*X*,

*d*)

*and suppose that*

- (a)
*X*_{≤}∈*I*(*f*×*f*)*;* - (b)
*There exists x*_{0}∈*X such that*(*x*_{0},*f*(*x*_{0})) ∈*X*_{≤}*;* - (c)
(

*c*_{1})*f is orbitally continuous or*

*c*

_{2})

*f is orbitally X*

_{≤}

*-continuous and there exists a subsequence*

*of*{

*f*

^{ n }(

*x*

_{0})}

*such that*

*for any k*∈

**N**

*;*

- (d)

*for any*(

*x*,

*y*) ∈

*X*

_{≤},

*where*

- (e)
*the metric d is complete;* - (f)
*if x*,*y*∈*X with*(*x*,*y*) ∉*X*_{≤},*then there exists c*(*x*,*y*) ∈*X such that*(*x*,*c*(*x*,*y*)) ∈*X*_{≤}*and*(*y*,*c*(*x*,*y*)) ∈*X*_{≤}^{ . }.

*Then, f is PO*.

**Proof**. According to Theorem 2.1, there exists *x** ∈ *X* such that *f*(*x**) = *x**. Take *x* ∈ *X*.

for any *n* ∈ **N**. Thus, by Lemma 1.2, *f*^{
n
} (*x*) → *x** as *n* → ∞.

*x*,

*x*

_{0}) ∉

*X*

_{≤}, then there exists

*z*∈

*X*such that (

*x*,

*z*) ∈

*X*

_{≤}and (

*x*

_{0},

*z*) ∈

*X*

_{≤}and so

for any *n* ∈ *N*. Thus, by Lemma 1.2, we have *f*^{
n
} (*z*) → *x** as *n* → ∞. Also, since (*x*, *z*) ∈ *X*_{≤}, we have *f* ^{
n
} (*z*) → *x** as *n* → ∞. Consequently, *f* ^{
n
} (*x*) → *x** as *n* → ∞.

and so, by Lemma 2.1, *y* = *x**, i.e., *F*_{
f
} = {*x**}. This completes the proof. □

**Corollary 2.4**. *Let* (*X*, *d*, ≤) *be an ordered metric space and f* : *X* → *X be an operator*.

*Let p be a w-distance on*(

*X*,

*d*)

*and suppose that*

- (a)
*if x*,*y*∈*X with*(*x*,*y*)*X*_{≤}*there exists c*(*x*,*y*) ∈*X such that*(*x*,*c*(*x*,*y*)) ∈*X*_{≤}*and*(*y*,*c*(*x*,*y*)) ∈*X*_{≤}*;* - (b)
*X*_{≤}∈*I*(*f*×*f*)*;* - (c)
*There exists x*_{0}∈*X such that*(*x*_{0},*f*(*x*_{0})) ∈*X*_{≤}*;* - (d)
(

*d*_{1})*f is orbitally continuous or*

*d*

_{2})

*f is orbitally X*

_{≤}

*-continuous and there exists a subsequence*

*of*{

*f*

^{ n }(

*x*

_{0})}

*such that*

*for any k*∈

**N**

*;*

- (e)

*Then, f is PO*.

**Corollary 2.5**. *Let* (*X*, *d*, ≤) *be an ordered metric space and f* : *X* → *X be an operator*.

*Let p be a w-distance on*(

*X*,

*d*)

*and suppose that*

- (a)
*if x*,*y ∈**X with*(*x*,*y*)*X*_{≤},*then there exists c*(*x*,*y*) ∈*X such that*(*x*,*c*(*x*,*y*)) ∈*X*_{≤}*and*(*y*,*c*(*x*,*y*)) ∈*X*_{≤}*;* - (b)
*if*(*x*,*y*) ∈*X*_{≤}*and*(*y*,*z*) ∈*X*_{≤},*then*(*x*,*z*) ∈*X*_{≤}*;* - (c)

*Then, f is PO*.

**Corollary 2.6**. *Let* (*X*, *d*, ≤) *be an ordered metric space and f* : *X* → *X be an operator*.

*Let p be a w-distance on*(

*X*,

*d*)

*and suppose that*

- (a)
*if x*,*y*∈*X with*(*x*,*y*)*X*_{≤},*then there exists c*(*x*,*y*) ∈*X such that*(*x*,*c*(*x*,*y*)) ∈*X*_{≤}*and*(*y*,*c*(*x*,*y*)) ∈*X*_{≤}*;* - (b)
*X*_{≤}∈*I*(*f*×*f*)*;* - (c)
*there exists x*_{0}∈*X such that*(*x*_{0},*f*(*x*_{0})) ∈*X*_{≤}*;* - (d)
*if*(*x*,*y*) ∈*X*_{≤}*and*(*y*,*z*) ∈*X*_{≤},*then*(*x*,*z*) ∈*X*_{≤}*;* - (e)
(

*e*_{1})*f is orbitally continuous or*

*e*

_{2})

*f is orbitally X*

_{≤}

*-continuous and there exists a subsequence*

*of*{

*f*

^{ n }(

*x*

_{0})}

*such that*

*for any k*∈

**N**

*;*

- (f)

*Then, f is PO*.

**Corollary 2.7**. *Let* (*X*, *d*, ≤) *be an ordered metric space and f* : *X* → *X be an operator*.

*Let p be a w-distance on*(

*X*,

*d*)

*and suppose that*

- (a)
*if x*,*y*∈*X with*(*x*,*y*)*X*_{≤},*then there exists c*(*x*,*y*) ∈*X such that*(*x*,*c*(*x*,*y*)) ∈*X*_{≤}*and*(*y*,*c*(*x*,*y*)) ∈*X*_{≤}*;* - (b)
*f is increasing or decreasing;* - (c)
*there exists x*_{0}∈*X such that*(*x*_{0},*f*(*x*_{0})) ∈*X*_{≤}*;* - (d)
(

*d*_{1})*f is orbitally continuous or*

*d*

_{2})

*f is orbitally X*

_{≤}

*-continuous and there exists a subsequence*

*of*{

*f*

^{ n }(

*x*

_{0})}

*such that*

*for any k*∈

**N**

*;*

- (e)

*Then, f is PO*.

## Declarations

### Acknowledgements

The authors would like to thank the referees and area editor Professor Simeon Reich for giving useful suggestions and comments for the improvement of this article. Y. J. Cho was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).

## Authors’ Affiliations

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