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# Common fixed points of mappings satisfying implicit contractive conditions

- Vasile Berinde
^{1}Email author and - Francesca Vetro
^{1}

**2012**:105

https://doi.org/10.1186/1687-1812-2012-105

© Berinde; licensee Springer. 2012

**Received:**18 April 2012**Accepted:**24 June 2012**Published:**24 June 2012

## Abstract

In this article we obtain, in the setting of metric spaces or ordered metric spaces, coincidence point, and common fixed point theorems for self-mappings in a general class of contractions defined by an implicit relation. Our results unify, extend, generalize many related common fixed point theorems from the literature.

**Mathematics Subject Classification (2000):** 47H10, 54H25.

## Keywords

- implicit relation
- contraction
- coincidence point
- fixed point
- common fixed point

## Introduction and preliminaries

It is well known that the contraction mapping principle, formulated and proved in the Ph.D. dissertation of Banach in 1920, which was published in 1922 [1], is one of the most important theorems in classical functional analysis. The study of fixed and common fixed points of mappings satisfying a certain metrical contractive condition attracted many researchers, see for example [2, 3] and for existence results for fixed points of contractive non-self-mappings, see [4–6]. Among these (common) fixed point theorems, only a few give a constructive method for finding the fixed points or the common fixed points of the mappings involved. Berinde in [7–15] obtained (common) fixed point theorems, which were called constructive (common) fixed point theorems, see [12]. These results have been obtained by considering self-mappings that satisfy an explicit contractive-type condition. On the other hand, several classical fixed point theorems and common fixed point theorems have been recently unified by considering general contractive conditions expressed by an implicit relation, see Popa [16, 17] and Ali and Imdad [18]. Following Popa's approach, many results on fixed point, common fixed point and coincidence point has been obtained, in various ambient spaces, see [16–25] and references therein.

In [21], Berinde obtained some constructive fixed point theorems for almost contractions satisfying an implicit relation. These results unify, extend, generalize related results (see [2, 3, 7–16, 21, 25–38]).

In this article we obtain, in the setting of metric spaces or ordered metric spaces, coincidence point, and common fixed point results for self-mappings in a general class of contractions defined by an implicit relation. Our results unify, extend, generalize many of related common fixed point theorems from literature.

Let *X* be a non-empty set and *f*, *T*: *X → X*. A point *x* ∈ *X* is called a coincidence point of *f* and *T* if *Tx* = *fx*. The mappings *f* and *T* are said to be weakly compatible if they commute at their coincidence point (i.e., *Tfx* = *fTx* whenever *Tx* = *fx*). Suppose *TX* ⊂ *fX*. For every *x*_{0} ∈ *X* we consider the sequence {*x*_{
n
}} ⊂ *X* defined by *fx*_{
n
} = *Tx*_{n-1}for all *n* ∈ ℕ, we say that {*Tx*_{
n
}} is a *T* -*f* -sequence with initial point *x*_{0}.

Let *X* be a non-empty set. If (*X*, *d*) is a metric space and (*X*, ≼) is partially ordered, then (*X*, *d*, ≼) is called an ordered metric space. Then, *x*, *y* ∈ *X* are called comparable if *x* ≼ *y* or *y* ≼ *x* holds. Let *f*, *T*: *X → X* be two mappings, *T* is said to be *f* -non-decreasing if *fx* ≼ *fy* implies *Tx* ≼ *Ty* for all *x*, *y* ∈ *X*. If *f* is the identity mapping on *X*, then *T* is non-decreasing.

Throughout this article the letters ℝ_{+} and ℕ will denote the set of all non-negative real numbers and the set of all positive integer numbers.

## Fixed point theorems for mappings satisfying an implicit relation

A simple and natural way to unify and prove in a simple manner several metrical fixed point theorems is to consider an implicit contraction type condition instead of the usual explicit contractive conditions. Popa [16, 17] initiated this direction of research which produced so far a consistent literature (that cannot be completely cited here) on fixed point, common fixed point, and coincidence point theorems, for both single-valued and multi-valued mappings, in various ambient spaces; see the recent nice paper [21] of Berinde, for a partial list of references.

In [21], Berinde considered the family of all continuous real functions $F:{\mathbb{R}}_{+}^{6}\to {\mathbb{R}}_{+}$ and the following conditions:

(F_{1a}) *F* is non-increasing in the fifth variable and *F* (*u*, *v*, *v*, *u*, *u* + *v*, 0) *≤* 0 for *u*, *v ≥* 0 implies that there exists *h* ∈ [0, 1) such that *u ≤ hv*;

(F_{1b}) *F* is non-increasing in the fourth variable and *F* (*u, v*, 0, *u* + *v*, *u*, *v*) *≤* 0 for *u*, *v ≥* 0 implies that there exists *h* ∈ [0, 1) such that *u ≤ hv*;

(F_{1c}) *F* is non-increasing in the third variable and *F* (*u*, *v*, *u*+*v*, 0, *v*, *u*) *≤* 0 for *u*, *v ≥* 0 implies that there exists *h* ∈ [0, 1) such that *u ≤ hv*;

(F_{2}) *F* (*u*, *u*, 0, 0, *u*, *u*) *>* 0, for all *u >* 0.

He gave many examples of functions corresponding to well-known fixed point theorems and satisfying most of the conditions (*F*_{1a})-(*F*_{2}) above, see Examples 1-11 of [21].

*Example*

**1**. The following functions $F\in \mathcal{F}$ satisfy properties

*F*

_{2}and

*F*

_{1a}-

*F*

_{1c}(see Examples 1-6, 9, and 11 of [21]).

- (i)
*F*(*t*_{1},*t*_{2},*t*_{3},*t*_{4},*t*_{5},*t*_{6}) =*t*_{1}*− at*_{2}, where*a*∈ [0, 1); - (ii)
*F*(*t*_{1},*t*_{2},*t*_{3},*t*_{4},*t*_{5},*t*_{6}) =*t*_{1}*− b*(*t*_{3}+*t*_{4}), where*b*∈ [0, 1*/*2); - (iii)
*F*(*t*_{1},*t*_{2},*t*_{3},*t*_{4},*t*_{5}*, t*_{6}) =*t*_{1}*− c*(*t*_{5}+*t*_{6}), where*c*∈ [0, 1*/*2); - (iv)
$F\left({t}_{1},{t}_{2},{t}_{3},{t}_{4},{t}_{5},{t}_{6}\right)={t}_{1}-a\mathsf{\text{max}}\left\{{t}_{2},\frac{{t}_{3}+{t}_{4}}{2},\frac{{t}_{5}+{t}_{6}}{2}\right\}$, where

*a*∈ [0, 1); - (v)
*F*(*t*_{1},*t*_{2},*t*_{3},*t*_{4},*t*_{5},*t*_{6}) =*t*_{1}*− at*_{2}*− b*(*t*_{3}+*t*_{4})*− c*(*t*_{5}+*t*_{6}), where*a*,*b*,*c*∈ [0, 1) and*a*+ 2*b*+ 2*c <*1; - (vi)
$F\left({t}_{1},{t}_{2},{t}_{3},{t}_{4},{t}_{5},{t}_{6}\right)={t}_{1}-a\mathsf{\text{max}}\left\{{t}_{2},\frac{{t}_{3}+{t}_{4}}{2},{t}_{5},{t}_{6}\right\}$, where

*a*∈ [0, 1); - (vii)
*F*(*t*_{1},*t*_{2},*t*_{3},*t*_{4},*t*_{5},*t*_{6}) =*t*_{1}*− at*_{2}*− L*min{*t*_{3},*t*_{4},*t*_{5},*t*_{6}}, where*a*∈ [0, 1); - (viii)
$F\left({t}_{1},{t}_{2},{t}_{3},{t}_{4},{t}_{5},{t}_{6}\right)={t}_{1}-a\mathsf{\text{max}}\left\{{t}_{2},{t}_{3},{t}_{4},\frac{{t}_{5}+{t}_{6}}{2}\phantom{\rule{2.77695pt}{0ex}}\right\}-L\mathsf{\text{min}}\left\{{t}_{3},{t}_{4},{t}_{5},{t}_{6}\right\}$, where

*a*∈ [0, 1) and*L ≥*0.

*Example*

**2**. The function $F\in \mathcal{F}$, given by

where *a* ∈ [0, 1*/* 2) satisfies properties *F*_{2} and *F*_{1a}-*F*_{1c}with $h=\frac{a}{1-a}<1$.

Motivated by [21], the following theorem is one of the main results in this article.

**Theorem 1**.

*Let*(

*X*,

*d*)

*be a metric space and T*,

*f*:

*X → X be self-mappings such that TX*⊆

*fX. Assume that there exists*$F\in \mathcal{F}$,

*satisfying*(

*F*

_{1a})

*, such that for all x*,

*y*∈

*X*

*If fX is a complete subspace of X, then T and f have a coincidence point. Moreover, if T and f are weakly compatible and F satisfies also F*_{2}*, then T and f have a unique common fixed point. Further, for any x*_{0} ∈ *X, the T-f-sequence* {*Tx*_{
n
}} *with initial point x*_{0} *converges to the common fixed point*.

*Proof*. Let

*x*

_{0}∈

*X*be an arbitrary point. As

*TX*⊆

*fX*, one can choose a

*T*-

*f*-sequence {

*Tx*

_{ n }} with initial point

*x*

_{0}. If we take

*x*=

*x*

_{ n }and

*y*=

*x*

_{n+1}in (1) and denote with

*u*=

*d*(

*Tx*

_{ n },

*Tx*

_{n+1}) and

*v*=

*d*(

*Tx*

_{n-1},

*Tx*

_{ n }) we get that

*d*(

*Tx*

_{n-1},

*Tx*

_{n+1})

*≤ d*(

*Tx*

_{n-1},

*Tx*

_{ n }) +

*d*(

*Tx*

_{ n },

*Tx*

_{n+1}) =

*u*+

*v*and, since

*F*is non-increasing in the fifth variable, we have

*F*

_{1a}), there exists

*h*∈ [0, 1) such that

*u ≤ hv*, i.e.,

*Tx*

_{ n }} is a Cauchy sequence. Since

*fX*is complete, there exist

*z*,

*w*∈

*X*such that

*z*=

*fw*and

*x*=

*x*

_{ n }and

*y*=

*w*in (1), we obtain that

*F*is continuous, using (3) and letting

*n →*+

*∞*in (4), we get

which, by assumption (*F*_{1a}), yields *d*(*fw*, *Tw*) *≤* 0, i.e., *fw* = *Tw* = *z*. Thus, we have proved that *T* and *f* have a coincidence point.

Now, we assume that *T* and *f* are weakly compatible, then *fz* = *fTw* = *Tfw* = *Tz*.

We show that *Tz* = *z* = *Tw*.

*d*(

*Tz*,

*Tw*)

*>*0, by taking

*x*=

*z*and

*y*=

*w*in (1), we get

which is a contradiction by assumption (*F*_{2}). This implies that *d*(*Tz*, *Tw*) = 0 and hence *fz* = *Tz* = *Tw* = *z*. So *T* and *f* have a common fixed point.

The uniqueness of the common fixed point is a consequence of assumption (*F*_{2}). Clearly, for any *x*_{0} ∈ *X*, the *T*-*f*-sequence {*Tx*_{
n
}} with initial point *x*_{0} converges to the unique common fixed point. □

*Remark*1. From (2) we deduce the unifying error estimate

which are extremely important in applications, especially when approximating the solutions of nonlinear equations.

If *f* = *I*_{
X
} from Theorem 1, we deduce the following result of fixed point for one self-mapping, see [21].

**Corollary 1**.

*Let*(

*X*,

*d*)

*be a complete metric space and T*:

*X → X. Assume that there exists*, $F\in \mathcal{F}$

*satisfying*(

*F*

_{1a})

*, such that for all x*,

*y*∈

*X*

*Then T has a fixed point. Moreover, if F satisfies also F*_{2}*, then T has a unique fixed point. Further, for any x*_{0} ∈ *X, the Picard sequence* {*T*^{
n
}*x*_{0}} *with initial point x*_{0} *converges to the fixed point*.

## Common fixed point in ordered metric spaces

The existence of fixed points in ordered metric spaces was investigated by Turinici [39], Ran and Reurings [40], Nieto and Rodríguez-López [41]. See, also [42–45], and references therein. A common fixed point result in ordered metric spaces for mappings satisfying implicit contractive conditions is given by the next theorem.

**Theorem 2**.

*Let*(

*X*,

*d*, ≼)

*be a complete ordered metric space and T*,

*f*:

*X → X be self-mappings such that TX*⊆

*fX. Assume that there exists*$F\in \mathcal{F}$

*, satisfying*(

*F*

_{1a})

*, such that for all x*,

*y*∈

*X with fx*≼

*fy*

*If the following conditions hold:*

- (i)
*there exists x*_{0}∈*X such that fx*_{0}≼*Tx*_{0}*;* - (ii)
*T is f-non-decreasing;* - (iii)
*for a non-decreasing sequence*{*fx*_{ n }} ⊆*X converging to fw*∈*X, we have fx*_{ n }≼*fw for all n*∈ ℕ*and fw*≼*f fw;*

*then T and f have a coincidence point in X. Moreover, if*

- (iv)
*T and f are weakly compatible;* - (v)
*F satisfies also F*_{2},

*then T and f have a common fixed point. Further, for any x*_{0} ∈ *X, the T-f-sequence* {*Tx*_{
n
}} *with initial point x*_{0} *converges to a common fixed point*.

*Proof*. Let

*x*

_{0}∈

*X*such that

*fx*

_{0}≼

*Tx*

_{0}and let {

*Tx*

_{ n }} be a

*T*-

*f*-sequence with initial point

*x*

_{0}. Since

*fx*

_{0}≼

*Tx*

_{0}and

*Tx*

_{0}=

*fx*

_{1}, we have

*fx*

_{0}≼

*fx*

_{1}. As

*T*is

*f*-non-decreasing we get that

*Tx*

_{0}≼

*Tx*

_{1}. Continuing this process we obtain

*d*(

*Tx*

_{ n },

*Tx*

_{n+1})

*>*0 for all

*n*∈ ℕ, since if

*Tx*

_{ n }=

*Tx*

_{n+1}for some

*n*, then

*fx*

_{n+1}=

*Tx*

_{ n }=

*Tx*

_{n+1}. This implies that

*x*

_{n+1}is a coincidence point for

*T*and

*f*and the result is proved. As

*fx*

_{ n }≼

*fx*

_{n+1}for all

*n*∈ ℕ, if we take

*x*=

*x*

_{ n }and

*y*=

*x*

_{n+1}in (5) and denote

*u*=

*d*(

*Tx*

_{ n },

*Tx*

_{n+1}) and

*v*=

*d*(

*Tx*

_{n- 1},

*Tx*

_{ n }) we get that

*d*(

*Tx*

_{n-1},

*Tx*

_{n+1})

*≤ d*(

*Tx*

_{n-1},

*Tx*

_{ n }) +

*d*(

*Tx*

_{ n },

*Tx*

_{n+1}) =

*u*+

*v*and, since

*F*is non-increasing in the fifth variable, we have

*F*

_{1a}), there exists

*h*∈ [0, 1) such that

*u ≤ hv*, i.e.,

*Tx*

_{ n }} is a Cauchy sequence. Since (

*X*,

*d*) is complete, there exist

*z*,

*w*∈

*X*such that

*z*=

*fw*and

*iii*),

*fx*

_{ n }≼

*fw*for all

*n*∈ ℕ, if we take

*x*=

*x*

_{ n }and

*y*=

*w*in (5) we get

*F*is continuous, using (7) and letting

*n →*+

*∞*we obtain

which, by assumption (*F*_{1a}), yields *d*(*fw*, *Tw*) *≤* 0, i.e., *fw* = *Tw*. Thus we have proved that *T* and *f* have a coincidence point.

If *T* and *f* are weakly compatible we show that *z* is a common fixed point for *T* and *f* . As *fz* = *fTw* = *Tfw* = *Tz*, by condition (*iii*), we have that *fw* ≼ *f fw* = *fz*.

*x*=

*w*and

*y*=

*z*in (5) we get

Assumption (*F*_{2}) implies *d*(*Tz*, *Tw*) = 0 and hence *fz* = *Tz* = *Tw* = *z*. So *T* and *f* have a common fixed point. From the proof it follows that, for any *x*_{0} ∈ *X*, the *T* -*f* -sequence {*Tx*_{
n
}} with initial point *x*_{0} converges to a common fixed point. □

We shall give a sufficient condition for the uniqueness of the common fixed point in Theorem 2.

**Theorem 3**.

*Let all the conditions of Theorem 2 be satisfied. If the following conditions hold*

- (vi)
*for all x*,*y*∈*fX there exists v*_{0}∈*X such that fv*_{0}≼*x, fv*_{0}≼*y;* - (vii)
*F satisfies F*_{1c},

*then T and f have a unique common fixed point*.

*Proof*. Let

*z*,

*w*be two common fixed points of

*T*and

*f*with

*z*≠

*w*. If

*z*and

*w*are comparable, say

*z*≼

*y*. Then taking

*x*=

*z*and

*y*=

*w*in (5), we obtain

which is a contradiction by assumption (*F*_{2}) and so *z* = *w*.

If *z* and *w* are not comparable, then there exists *v*_{0} ∈ *X* such that *fv*_{0} ≼ *fz* = *z* and *fv*_{0} ≼ *fw* = *w*.

*T*is

*f*-non-decreasing from

*fv*

_{0}≼

*fz*we get that

*x*=

*v*

_{ n }and

*y*=

*z*in (5) we obtain

*u*=

*d*(

*Tv*

_{ n },

*Tz*) and

*v*=

*d*(

*Tv*

_{n-1},

*Tz*). As

*F*is non-increasing in the third variable, we get

*F*

_{1c}, there exists

*h*∈ [0, 1) such that

*u ≤ hv*, i.e.,

This implies that *d*(*Tv*_{
n
}, *Tz*) = *d*(*Tv*_{
n
}, *z*) *→* 0 as *n →* +*∞*.

*d*(

*Tv*

_{ n },

*w*)

*→*0 as

*n →*+

*∞*. Hence

as *n →* +*∞*, which is a contradiction. Thus *T* and *f* have a unique common fixed point. □

If *f* = *I*_{
X
} from Theorems 2 and 3, we deduce the following results of fixed point for one self-mapping.

**Corollary 2**.

*Let*(

*X*,

*d*, ≼)

*be a complete ordered metric space and T*:

*X → X. Assume that there exists*$F\in \mathcal{F}$

*, satisfying*(

*F*

_{1a})

*, such that for all x*,

*y*∈

*X with x*≼

*y*

*If the following conditions hold:*

- (i)
*there exists x*_{0}∈*X such that x*_{0}≼*Tx*_{0}*;* - (ii)
*T is non-decreasing;* - (iii)
*for a non-decreasing sequence*{*x*_{ n }} ⊆*X converging to w*∈*X, we have x*_{ n }≼*w for all n*∈ ℕ,

*then T has a fixed point in X. Further, for any x*_{0} ∈ *X, the Picard sequence* {*T*^{
n
}*x*_{0}} *with initial point x*_{0} *converges to a fixed point*.

**Corollary 3**.

*Let all the conditions of Corollary 2 be satisfied. If the following conditions hold*

- (v)
*F satisfies F*_{2}*;* - (vi)
*for all x, y*∈*X there exists v*_{0}∈*X such that v*_{0}≼*x, v*_{0}≼*y;* - (vii)
*F satisfies F*_{1c},

*then T has a unique fixed point*.

If *F* is the function in Example 2, then by Theorem 3 we obtain a fixed point theorem that extends the result of Theorem 3 of [44].

## Declarations

### Acknowledgements

The first author research's was supported by the Grant PN-II-RU-TE-2011-3-239 of the Romanian Ministry of Education and and Research.

## Authors’ Affiliations

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